A continuous rating method for preferential voting

A CONTINUOUS RA TING METHOD F OR PREFERENTIAL V OTING R osa Camps, Xa vier Mora and Laia Saumell Departamen t de Matem` atiques, Univ ersitat Aut` onoma de Barcelona, Catalonia, Spain xmora @ mat.uab.cat 29th July 2008, revised 26th No v em b er 2008 Abstract A metho d is giv en for quan titatively rating the social acceptance of diﬀeren t options whic h are the matter of a preferential vote. The prop osed method is pro ved to satisfy certain desirable conditions, among whic h there is a ma jority principle, a prop ert y of clone con- sistency , and the con tinuit y of the rates with respect to the data. One can view this metho d as a quantitativ e complemen t for a qualita- tiv e method in tro duced in 1997 by Markus Sc h ulze. It is also related to certain methods of one-dimensional scaling or cluster analysis. Keyw ords: pr efer ential voting, Condor c et, p air e d c omp arisons, ma- jority principle, clone c onsistency, appr oval voting, c ontinuous r ating, one-dimensional sc aling, ultr ametrics, R obinson c ondition, Gr e enb er g c ondition. AMS sub ject classiﬁcations: 05C20, 91B12, 91B14, 91C15, 91C20. The outcome of a v ote is often exp ected to en tail a quantitativ e rating of the candidate options according to their so cial acceptance. Some v oting metho ds are directly based up on such a rating. This is the case when each v oter is asked to choose one option and each option is rated by the fraction of the v ote in its fav our. The resulting rates can b e used for ﬁlling a single seat (ﬁrst past the p ost) or for distributing a n umber of them (prop ortional represen tation). A more elaborate v oting metho d based up on quantitativ e rates was in tro duced in 1433 by Nik olaus von Kues [ 23 : § 1.4.3, § 4 ] and again in 1770–1784 b y Jean-Charles de Borda [ 23 : § 1.5.2, § 5 ]. Here, eac h voter is ask ed to rank the diﬀerent options in order of preference and each option is rated b y the av erage of its ranks, i. e. the ordinal num b ers that give its p osi- tion in these diﬀerent rankings (this formulation diﬀers from the traditional one b y a linear function). F or future reference in this pap er, these t wo rating 2 R. Camps, X. Mora, L. Sa umell metho ds will b e called resp ectiv ely the metho d of ﬁrst-c hoice fractions and the metho d of av erage ranks . Ho w ever, b oth of these metho ds ha v e imp ortant drawbac ks, whic h leads to the p oint of view of paired comparisons , where eac h option is con- fron ted with ev ery other by coun ting how man y voters prefer the former to the latter and vice versa. F rom this p oint of view it is quite natural to abide b y the so-called Condorcet principle: an option should b e deemed the win- ner whenev er it defeats every other one in this sort of tournamen t. This approac h w as introduced as early as in the thirteenth cen tury by Ramon Llull [ 23 : § 1.4.2, § 3 ], and later on it was prop ounded again by the marquis of Condorcet in 1785–1794 [ 23 : § 1.5.4, § 7 ], and by Charles Do dgson, alias Lewis Carroll, in 1873–1876 [ 2 ; 23 : § 12 ]. Its developmen t gives rise to a v a- riet y of metho ds, some of them with remark ably go o d prop erties. This is particularly the case of the metho d of rank ed pairs , prop osed in 1986/87 b y Thomas M. Za vist and T. Nicolaus Tideman [ 37, 40 ], and the metho d in tro duced in 1997 by Markus Sc hulze [ 34, 35 ], which we will refer to as the metho d of paths . In spite of the fact that generally sp eaking they can pro- duce diﬀeren t results, b oth of them comply with the Condorcet principle and they share the remark able prop erty of clone consistency [ 38, 34 ]. Nev ertheless, these metho ds do not immediately giv e a quan titative rating of the candidate options. Instead, they are deﬁned only as algorithms for determining a winner or at most a purely ordinal ranking. On the other hand, they are still based up on the quan titative information provided b y the table of paired-comparison scores, which raises the question of whether their qualitativ e results can b e consistently con v erted in to quan titative ratings. In [ 26 : § 10 ] a quan titative rating algorithm w as devised with the aim of complementing the metho d of rank ed pairs. Although a strong evidence w as given for its fulﬁlling certain desirable conditions —lik e the ones stated b elo w—, it w as also pointed out that it fails a most natural one, namely that the output, i. e. the rating, b e a con tinuous function of the input, i. e. the frequency of each p ossible con tent of an individual vote. In fact, suc h a lack of contin uity seems una v oidable when the metho d of ranked pairs is considered and those other conditions are imp osed. In con trast, in this pap er w e will see that the method of paths do es admit suc h a con tinuous rating pro cedure. Our method can b e view ed as a pro jection of the matrix of paired- comparison scores onto a special set of suc h matrices. This pro jection is com bined with a subsequent application of t w o standard rating metho ds, one of which the metho d of av erage ranks. The o v erall idea has some p oints in common with [ 32 ]. Continuous ra ting for preferential voting , § 1 3 W e will refer to the metho d describ ed in this pap er as the CLC rating metho d , where the capital letters stand for “Con tinuous Llull Condorcet”. The pap er is organized as follo ws: In section 1 w e state the problem whic h is to b e solved and w e mak e some general remarks. Section 2 presen ts an heuristic outline of the prop osed metho d. Section 3 gives a summary of the procedure, after whic h certain v ariants are in tro duced. Section 4 presents some illustrativ e examples. Finally , sections 5–18 give detailed mathematical pro ofs of the claimed prop erties for the main v ariant. The reader in terested to try the CLC metho d can make use of the to ol whic h is av ailable at http://mat.uab.cat/ ~ xmora/CLC calculator/ . 1 Statemen t of the problem and general remarks 1.1 W e consider a set of N options which are the matter of a vote. Al- though more general cases will b e included later on ( § 3.3), for the momen t w e assume that each v oter expresses his preferences in the form of a ranking; b y it w e mean an ordering of the options in question b y decreasing degree of preference, with the p ossibility of ties and/or truncation (i. e. expressing a top segment only). W e wan t to aggregate these individual preferences into a so cial r ating , where eac h option is assigned a rate that quan tiﬁes its so cial acceptance. In some places we will restrict our atten tion to the case of complete v otes. F or ranking v otes, w e are in suc h a situation whenev er we are dealing with non-truncated rankings. As w e will see, the incomplete case will give us m uch more work than the complete one. W e will consider t w o kinds of ratings, whic h will be referred to resp ectiv ely as rank-lik e ratings and fraction-like ones. As it is suggested by these names, a rank-lik e rating will b e reminiscent of a ranking, whereas a fraction-lik e one will ev oke the notion of prop ortional representation. Our metho d will pro duce b oth a rank-like rating and a fraction-like one. They will agree with eac h other in the ordering of the candidate options, except that the ordering implied by the fraction-like rating ma y b e restricted to a top segmen t of the other one. Quan titativ ely sp eaking, the tw o ratings hav e diﬀerent meanings. In particular, the fraction-like rates can be viewed as an estimate of the ﬁrst- c hoice fractions based not only on the ﬁrst c hoices of the voters, but also on the whole set of preferences expressed b y them. In contrast, the rank-lik e rates are not fo cused on choosing, but they aim simply at p ositioning all the candidate options on a certain scale. 4 R. Camps, X. Mora, L. Sa umell More sp eciﬁcally , the tw o ratings are asked to satisfy the follo wing con- ditions: A Sc ale invarianc e . The rates dep end only on the relative frequency of each p ossible conten t of an individual v ote. In other words, if ev ery individual vote is replaced b y a ﬁxed num b er of copies of it, the rates remain exactly the same. B Permutation e quivarianc e . Applying a certain permutation of the options to all of the individual votes has no other eﬀect than getting the same p ermutation in the so cial rating. C Continuity . The rates dep end con tinuously on the relativ e frequency of each p ossible con tent of an individual vote. The next conditions apply to the rank-lik e rating: D R ank-like r ange . Each rank-lik e rate is a num b er, in teger or frac- tional, b et w een 1 and N . The best p ossible v alue is 1 and the w orst p ossible one is N . E R ank-like de c omp osition . Let us restrict the attention to the com- plete case. Consider a splitting of the options into a ‘top class’ X plus a ‘lo w class’ Y . Assume that all of the voters ha ve put eac h mem b er of X ab ov e ev ery member of Y . In that case, and only in that case, the rank-like rates can b e obtained separately for eac h of these tw o classes according to the corresponding restriction of the ranking v otes (with the proviso that the unassembled low-class rates diﬀer from the assem bled ones b y the n umber of top-class mem b ers). In its turn, the fraction-lik e rating is required to satisfy the following condi- tions: F F r action-like char acter . Each fraction-like rate is a num b er greater than or equal to 0 . Their sum is equal to a ﬁxed v alue. More sp eciﬁcally , we will tak e this v alue to b e the participation fraction, i. e. the fraction of non-empt y v otes. G F r action-like de c omp osition . Consider the same situation as in E with the additional assumption that there is no prop er subset of X with the same splitting prop ert y as X (namely , that all voters hav e put eac h option from that set ab o v e every one outside it). In that case, and only in that case, the top-class fraction-like rates are all of them positive and they can b e obtained according to the cor- resp onding restriction of the ranking votes, whereas the low-class fraction-lik e rates are all of them equal to 0 . Continuous ra ting for preferential voting , § 1 5 H Case of plumping votes . Assume that eac h v oter plumps for a sin- gle option. In that case, the fraction-like rates coincide with the fractions of the vote obtained b y eac h option. F urthermore, we ask for some prop erties that concern only the concomitant so cial ranking, i. e. the purely ordinal information con tained in the social rating: I Majority principle . Consider a splitting of the options in to a ‘top class’ X plus a ‘low class’ Y . Assume that for each mem b er of X and ev ery mem b er of Y there are more than half of the individual v otes where the former is preferred to the latter. In that case, the so cial ranking also prefers eac h member of X to ev ery mem b er of Y . J Clone c onsistency . A set C of options is said to b e a cluster (of clones ) for a given ranking when each element from outside C compares with all elemen ts of C in the same wa y (i. e. either it lies ab ov e all of them, or it lies b elo w all of them, or it ties with all of them). In this connection, it is required that if a set of options is a cluster for eac h of the individual v otes, then: (a) it is a cluster for the so cial ranking; and (b) con tracting it to a single option in all of the in- dividual votes has no other eﬀect in the so cial ranking than getting the same contraction. 1.2 Let us emphasize that the individual v otes that w e are dealing with do not hav e a quan titative character (at least for the moment): eac h v oter is allo w ed to express a preference for x rather than y , or vice versa, or maybe a tie b etw een them, but he is not allow ed to quantify suc h a preference. This contrasts with ‘range voting’ metho ds, where each individual vote is already a quan titative rating [ 36, 1 ]. Suc h metho ds are free from man y of the diﬃculties that lurk b ehind the present setting. How ever, they make sense only as long as all voters mean the same b y eac h p ossible v alue of the rating v ariable. This h yp othesis ma y b e reasonable in some cases, but quite often it is hardly applicable (a typical symptom of its not b eing appropriate is a concen tration of the rates in a small set indep enden tly of whic h particular options are under consideration). In suc h cases, it is quite natural that the individual v otes express only qualitativ e comparisons b et w een pairs of op- tions. If the issue is not to o complicated, one can exp ect these comparisons to form a ranking. In the own w ords of [ 1 a ], “When there is no common lan- guage, a judge’s only meaningful input is the order of his grades”. Certainly , the judges will agree up on the qualitative comparison b et w een t w o options m uc h more often than they will agree up on their resp ective rates in a certain 6 R. Camps, X. Mora, L. Sa umell scale. Suc h a lac k of quan titativ e agreement may b e due to truly diﬀeren t opinions; but quite often it is rather meaningless. Of course, the rates will coincide more easily if a discrete scale of few grades is used. But then it ma y happ en that the judges rate equally t w o options ab out whic h they all share a deﬁnite preference for one o v er the other, in which case these discrete rates are thro wing a w ay genuine information. An yw a y , voting is often used in connection with moral, psyc hological or aesthetic qualities, whose appre- ciation may b e as little quantiﬁable, but also as m uc h “comparable”, as, for instance, the feelings of pleasure or pain. So, in our case the quantitativ e c haracter of the output is not presen t in the individual v otes (unless w e adopt the general setting considered at the end of § 3.3), but it deriv es from the fact of having a n umber of them. The larger this num b er, the more meaningful is the quantitativ e c haracter of the so cial rating. This is esp ecially applicable to the contin uity prop erty C, according to which a small v ariation in the prop ortion of votes with a giv en con ten t pro duces only small v ariations in the rates. In fact, if all individual v otes ha v e the same weigh t, a few v otes will be a small proportion only in the measure that the total num b er of v otes is large enough. In this connection, it should b e noticed that prop erty C diﬀers from the contin uity prop ert y adopted in [ 1 ] (axiom 6), whic h do es not refer to small v ariations in the prop ortion of votes with a given conten t, but to small v ariations in the quantitativ e conten t of eac h individual vote. In the general setting considered at the end of § 3.3, the CLC metho d satisﬁes not only the con tin uity prop erty C, but also the axiom 6 of [ 1 ]; in con trast, the “ma jority- grade” metho d considered in [ 1 ] satisﬁes the latter but not the former. 1.3 One can easily see that the method of a v erage ranks satisﬁes conditions A–E. In principle that metho d assumes that all of the individual votes are complete rankings; how ever, one can extend it to the general case of rankings with ties and/or truncation while keeping those conditions (it suﬃces to use form ula (6) of § 2.5). In their turn, the ﬁrst-c hoice fractions are easily seen to satisfy conditions A–C and F–H. Ho w ever, neither of these tw o metho ds satisﬁes conditions I and J. In fact, these conditions w ere introduced precisely as particularly desirable prop erties that are not satisﬁed by those metho ds [ 23, 2, 38 ]. Of course, one can go for a particular ranking metho d that satisﬁes con- ditions I and J and then lo ok for an appropriate algorithm to con vert the ranking result in to the desired rating according to the quan titativ e informa- tion coming from the v ote. But this should b e done in such a wa y that the ﬁnal rating b e alwa ys in agreement with the ranking metho d as w ell as in Continuous ra ting for preferential voting , § 1 7 compliance with conditions A–H, which is not so easy to achiev e. F rom this p oin t of view, our prop osal can b e viewed as pro viding suc h a complement for one of the v ariants of the metho d of paths [ 34, 35 ]. 1.4 When the set X consists of a single option, the ma jorit y principle I tak es the following form: I1 Majority principle, winner form . If an option x has the property that for ev ery y 6 = x there are more than half of the individual votes where x is preferred to y , then x is the so cial winner. In the complete case the preceding condition is equiv alen t to the following one: I1 0 Condor c et principle . If an option x has the prop ert y that for ev ery y 6 = x there are more individual votes where x is pre- ferred to y than vice versa, then x is the so cial winner. Ho w ever, we wan t to admit the p ossibility of individual votes where no in- formation is given ab out certain pairs of options. F or instance, in the case of a truncated ranking it makes sense to interpret that there is no information ab out tw o particular options which are not present in the list. In that case condition I1 is w eaker than I1 0 , and the CLC metho d will satisfy only the w eak er v ersion. This lac k of compliance with the Condorcet principle and its b eing re- placed by a w eaker condition may b e c onsidered undesirable. How ever, other authors hav e already remark ed that suc h a weak ening of the Condorcet principle is necessary in order to b e able to keep other properties [ 39 ] (see also [ 18 ]). In our case, Condorcet principle seems to conﬂict with the con- tin uit y prop ert y C (see § 3.3). On the other hand, the Condorcet principle w as originally prop osed in connection with the complete case [ 23 ], its gener- alization in the form I1 0 instead of I1 b eing due to later authors. Ev en so, no w adays it is a common practice to refer to I1 0 b y the name of “Condorcet principle’. 1.5 As w e mentioned in the preceding subsection, we w an t to admit the p ossibilit y of individual v otes where no information is giv en ab out certain pairs of options. In this connection, the CLC metho d will carefully dis- tinguish a deﬁnite indiﬀer enc e ab out t w o or more options from a lack of information ab out them (see [ 13 ]). F or instance, if all of the individual votes are complete rankings but they balance in to an exact so cial indiﬀerence —in particular if each individual v ote expresses such a complete indiﬀerence—, the resulting rank-lik e rates will b e all of them equal to ( N + 1) / 2 and the cor- resp onding fraction-like rates will b e equal to 1 / N . In con trast, in the case of 8 R. Camps, X. Mora, L. Sa umell a full absten tion, i. e. where no v oter has expressed an y opinion, the rank-lik e rates will b e all of them equal to N and the corresp onding fraction-like rates will b e equal to 0 . Although the decomp osition conditions E and G hav e b een stated only for the complete case, some partial results of that sort will hold under more general conditions. In particular, the following condition will b e satisﬁed for general, p ossibly incomplete, ranking v otes: The winner will b e rated exactly 1 (in b oth the rank-lik e rating and the fraction-like one) if and only if all of the voters ha ve put that option into ﬁrst place. Conditions E and G, as well as the preceding prop ert y , refer to cases where “all of the voters” pro ceed in a certain w a y . Of course, it should b e clear whether we mean all of the “actual” voters or maybe all of the “p oten tial” ones (i. e. actual v oters plus abstainers). W e assume that one has made a c hoice in that connection, thus deﬁning a total n um b er of voters V . Considering all p oten tial voters instead of only the actual ones has no other eﬀect than contracting the ﬁnal rating tow ards the p oint where all rates take the minimal v alue (namely , N for rank-lik e rates and 0 for fraction-like ones). 1.6 It is in teresting to lo ok at the results of the CLC rating metho d when it is applied to the approv al v oting situation, i. e. the case where eac h voter giv es only a list of approv ed options, without an y expression of preference b et w een them. In suc h a situation it is quite natural to rate each option b y the num b er of receiv ed approv als; the resulting metho d has prett y go o d prop erties, not the least of whic h is its eminent simplicity [ 6 ]. No w, an individual v ote of approv al type can b e view ed as a truncated ranking which ties up all of the options that app ear in it. So it mak es sense to apply the CLC rating metho d. Quite remark ably , one of its v ariants turns out to order the options in exactly the same wa y as the num b er of receiv ed appro v als (see § 17). More sp eciﬁcally , the v arian t in question corresp onds to interpreting that the non-appro ved options of an individual v ote are tied to each other. Ho w ever, the main v arian t, which ac knowledges a lack of comparison b etw een non-approv ed options, can lead to diﬀerent results. 2 Heuristic outline This section presents our prop osal as the result of a quest for the desired prop erties. Hop efully , this will comm unicate the main ideas that lie b ehind the formulas. Continuous ra ting for preferential voting , § 2 9 2.1 The aim of complying with conditions I and J calls for the p oint of view of paired comparisons . In accordance with it, our pro cedure will b e based up on considering every pair of options and coun ting how many voters prefer one to the other or vice versa. T o that eﬀect, we must adopt some rules for translating the ranking votes (p ossibly truncated or with ties) in to binary preferences. In principle, these rules will b e the follo wing: (a) When x and y are b oth in the list and x is rank ed ab ov e y (without a tie), we certainly in terpret that x is preferred to y . (b) When x and y are b oth in the list and x is ranked as go o d as y , w e interpret it as b eing equiv alent to half a v ote preferring x to y plus another half a vote preferring y to x . (c) When x is in the list and y is not in it, we in terpret that x is preferred to y . (d) When neither x nor y are in the list, w e in terpret nothing about the preference of the voter b et w een x and y . Later on ( § 3.2, 3.3) we will consider certain alternatives to rules (d) and (c). The preceding rules allo w us to coun t how man y voters support a giv en bi- nary preference, i. e. a particular statemen t of the form “ x is preferable to y ”. By doing so for each p ossible pair of options x and y , the whole v ote gets summarized in to a set of N ( N − 1) n um b ers (since x and y must be diﬀeren t from eac h other). W e will denote these num b ers b y V xy and w e will call them the binary scores of the v ote. The collection of these n um b ers will b e called the Llull matrix of the vote. Since we lo ok for scale in v ariance, it makes sense to divide all of these num b ers by the total n um b er of v otes V , which normalizes them to range from 0 to 1 . In the follo wing we will work mostly with these normalized scores, whic h will b e denoted by v xy . In practice, ho w ever, the absolute scores V xy ha v e the adv antage that they are in teger n um b ers, so we will use them in the examples. In general, the n umbers V xy are bound to satisfy V xy + V y x ≤ V , or equiv- alen tly v xy + v y x ≤ 1 . The sp ecial case where the ranking votes are all of them complete, i. e. without truncation, is characterized by the condition that V xy + V y x = V , or equiv alently v xy + v y x = 1 . F rom now on we will refer to such a situation as the case of complete v otes . Besides the scores v xy , in the sequel we will often deal with the margins m xy and the turnov ers t xy , which are deﬁned resp ectiv ely b y m xy = v xy − v y x , t xy = v xy + v y x . (1) Ob viously , their dep endence on the pair xy is resp ectiv ely an tisymmetric 10 R. Camps, X. Mora, L. Sa umell and symmetric, that is m y x = − m xy , t y x = t xy . (2) It is clear also that the scores v xy and v y x can b e recov ered from m xy and t xy b y means of the formulas v xy = ( t xy + m xy ) / 2 , v y x = ( t xy − m xy ) / 2 . (3) 2.2 A natural candidate for deﬁning the so cial preference is the follo wing: x is so cially preferred to y whenev er v xy > v y x . Of course, it can happ en that v xy = v y x , in whic h case one w ould consider that x is so cially equiv alen t to y . The binary relation that includes all pairs xy for which v xy > v y x will b e denoted by µ ( v ) and will b e called the comparison relation ; together with it, w e will consider also the adjoin t comparison relation ˆ µ ( v ) which is deﬁned by the condition v xy ≥ v y x . As it is w ell-known, the main problem with paired comparisons is that the comparison relations µ ( v ) and ˆ µ ( v ) may lac k transitivit y even if the individual preferences are all of them transitiv e [ 23, 2 ]. More sp eciﬁcally , µ ( v ) can contain a ‘Condorcet cycle’, i. e. a sequence x 0 x 1 . . . x n suc h that x n = x 0 and x i x i +1 ∈ µ ( v ) for all i . A most natural reaction to it is going for the transitiv e closure of ˆ µ ( v ) , whic h we will denote b y ˆ µ ∗ ( v ) . By deﬁnition, ˆ µ ∗ ( v ) includes all (ordered) pairs xy for which there is a path x 0 x 1 . . . x n from x 0 = x to x n = y whose links x i x i +1 are all of them in ˆ µ ( v ) . In other w ords, we can say that ˆ µ ∗ ( v ) includes all pairs that are “indirectly related” through ˆ µ ( v ) . Ho wev er, this op eration replaces eac h cycle of intransivit y b y an equiv alence b etw een its members. Instead of that, w e would rather break these equiv alences according to the quantitativ e information provided by the scores v xy . This is what is done in such metho ds as rank ed pairs or paths. How ev er, these metho ds use that quantitativ e information to reach only qualitative results. In contrast, our results will k eep a quantitativ e character un til the end. 2.3 The next dev elopmen ts rely up on an op eration ( v xy ) → ( v ∗ xy ) that transforms the original binary scores into a new one. This op eration is deﬁned in the following wa y: for every pair xy , one considers all p ossible paths x 0 x 1 . . . x n going from x 0 = x to x n = y ; ev ery suc h path is asso ciated with the score of its w eakest link, i. e. the smallest v alue of v x i x i +1 ; ﬁnally , v ∗ xy is deﬁned as the maximum v alue of this asso ciated score o v er all paths from x Continuous ra ting for preferential voting , § 2 11 to y . In other w ords, v ∗ xy = max x 0 = x x n = y min i ≥ 0 i < n v x i x i +1 , (4) where the max op erator considers all p ossible paths from x to y , and the min op erator considers all the links of a particular path. The scores v ∗ xy will b e called the indirect scores asso ciated with the (direct) scores v xy . If ( v xy ) is the table of 0’s and 1’s asso ciated with a binary relation ρ (b y putting v xy = 1 if and only if xy ∈ ρ ), then ( v ∗ xy ) is exactly the table asso ciated with ρ ∗ , the transitive closure of ρ . So, the operation ( v xy ) 7→ ( v ∗ xy ) can b e view ed as a quantitativ e analog of the notion of transitiv e closure (see [ 8 : Ch. 25 ]). The main p oin t, remark ed in 1998 by Markus Sch ulze [ 34 b ], is that the comparison relation asso ciated with a table of indirect scores is alw a ys tran- sitiv e (Theorem 6.3). So, µ ( v ∗ ) is alw a ys transitiv e, no matter what the case is for µ ( v ) . This is true in spite of the fact that µ ( v ∗ ) can easily diﬀer from µ ∗ ( v ) . In the follo wing we will refer to µ ( v ∗ ) as the indirect com- parison relation . R emark Somewhat surprisingly , in the case of incomplete v otes the transitiv e re- lation µ ( v ∗ ) may diﬀer from µ ( v ) even when the latter is already transitive. An example is given b y the following proﬁle, where each indicated preference is preceded by the num b er of p eople who voted in that wa y: 17 a , 24 c , 16 a  b  c , 16 b  a  c , 8 b  c  a , 8 c  b  a ; in this case the direct comparison relation µ ( v ) is the ranking a  b  c , whereas the indirect comparison relation µ ( v ∗ ) is the ranking b  a  c . More sp eciﬁcally , we ha v e V ab = 33 > 32 = V ba but V ∗ ab = 33 < 40 = V ∗ ba . The agreement with µ ( v ) can be forced by suitably redeﬁning the indirect scores; more sp eciﬁcally , formula (4) can b e replaced b y an analogous one where the max op erator is not concerned with all p ossible paths from x to y but only those contained in µ ( v ) . This idea is put forward in [ 35 ]. Generally sp eaking, how ever, suc h a metho d cannot b e made into a contin uous rating pro cedure since one do es quite diﬀerent things dep ending on whether v xy > v y x or v xy < v y x . On the other hand, w e will see that in the complete case the indirect comparison relation do es not change when the paths are restricted to b e contained in µ ( v ) ( § 7). 2.4 In the following w e put ν = µ ( v ∗ ) , ˆ ν = ˆ µ ( v ∗ ) , m ν xy = v ∗ xy − v ∗ y x . (5) 12 R. Camps, X. Mora, L. Sa umell So, xy ∈ ν if and only if v ∗ xy > v ∗ y x , i. e. m ν xy > 0 , and xy ∈ ˆ ν if and only if v ∗ xy ≥ v ∗ y x , i. e. m ν xy ≥ 0 . F rom no w on w e will refer to m ν xy as the indirect margin asso ciated with the pair xy . As it has been stated ab ov e, the relation ν is transitiv e. Besides that, it is clearly antisymmetric (one cannot hav e b oth v ∗ xy > v ∗ y x and vice v ersa). On the other hand, it may b e not complete (one can hav e v ∗ xy = v ∗ y x ). When it diﬀers from ν , the complete relation ˆ ν is not an tisymmetric and —somewhat surprisingly— it may b e not transitiv e either. F or instance, consider the proﬁle given by 4 b  a  c , 3 a  c  b , 2 c  b  a , 1 c  a  b ; in this case the indirect comparison relation ν = µ ( v ∗ ) contains only the pair ac ; as a consequence, ˆ ν con tains cb and ba but not ca . Ho w ev er, one can alwa ys ﬁnd a total order ξ whic h satisﬁes ν ⊆ ξ ⊆ ˆ ν (Theorem 8.2). F rom now on, an y total order ξ that satisﬁes this condition will b e called an admissible order . The rating that w e are looking for will be based on suc h an order ξ . More sp eciﬁcally , it will b e compatible with ξ in the sense that the rates r x will satisfy the inequalit y r x ≤ r y whenev er xy ∈ ξ . If ν is already a total order, so that ξ = ν , the preceding inequality will be satisﬁed in the strict form r x < r y , and this will happ en if and only if xy ∈ ν (Theorem 10.2). If there is more than one admissible order then some options will ha ve equal rates. In fact, we will ha ve r x = r y whenev er xy ∈ ξ 1 and y x ∈ ξ 2 , where ξ 1 , ξ 2 are t wo admissible orders. This will b e so b ecause w e wan t the rating to b e indep enden t of the choice of ξ . This indep endence with resp ect to ξ seems essen tial for achieving the con tinuit y prop erty C; in fact, eac h p ossible choice of ξ for a giv en proﬁle of v ote frequencies may easily b ecome the only one for a slight p erturbation of that proﬁle (but not necessarily , as it is illustrated by example 10 of [ 35 : § 4.6 ]). The following steps assume that one has ﬁxed an admissible order ξ . F rom now on the situation xy ∈ ξ will be expressed also b y x  ξ y . According to the deﬁnitions, the inclusions ν ⊆ ξ ⊆ ˆ ν are equiv alen t to saying that v ∗ xy > v ∗ y x implies x  ξ y and that the latter implies v ∗ xy ≥ v ∗ y x . In other w ords, if the diﬀeren t options are ordered according to  ξ , the matrix v ∗ xy has then the prop ert y that each elemen t ab ov e the diagonal is larger than or equal to its symmetric ov er the diagonal. 2.5 Rating the diﬀerent options means p ositioning them on a line. Besides complying with the qualitative restriction of b eing compatible with ξ in the sense ab o v e, we wan t that the distances b et w een items reﬂect the quan titativ e information pro vided b y the binary scores. Ho wev er, a rating is expressed b y N n umbers, whereas the binary scores are N ( N − 1) n um b ers. So w e Continuous ra ting for preferential voting , § 2 13 are b ound to do some sort of pro jection. Problems of this kind hav e a cer- tain tradition in com binatorial data analysis and cluster analysis [ 15, 24, 14 ]. In fact, some of the op erations that will b e used b elow can b e viewed from that p oint of view. Let us assume for a while that we are dealing with complete ranking v otes, so that it makes sense to talk ab out the a verage ranks. It is w ell- kno wn [ 2 : Ch. 9 ] that their v alues, whic h we will denote by ¯ r x , can be ob- tained from the Llull matrix b y means of the following formula: ¯ r x = N − X y 6 = x v xy . (6) Equiv alen tly , we can write ¯ r x = ( N + 1 − X y 6 = x m xy ) / 2 , (7) where the m xy are the margins of the original scores v xy , i. e. m xy = v xy − v y x . In fact, the hypothesis of complete v otes means that v xy + v y x = 1 , so that m xy = 2 v xy − 1 , which giv es the equiv alence b etw een (6) and (7). Let us look at the meaning of the margins m xy in connection with the idea of pro jecting the Llull matrix into a rating: If there are no other items than x and y , w e can certainly view the sign and magnitude of m xy as giving resp ectiv ely the qualitative and quan titative asp ects of the relative p ositions of x and y on the rating line, that is, the order and the distance b etw een them. When there are more than tw o items, how ever, we ha ve sev eral pieces of information of this kind, one for ev ery pair, and these diﬀeren t pieces ma y b e incompatible with each other, quan titativ ely or ev en qualitativ ely , whic h motiv ates indeed the problem that w e are dealing with. In particular, the av erage ranks often violate the desired compatibilit y with the relation ξ . In order to construct a rating compatible with ξ , we will use a form ula analogous to (6) where the scores v xy are replaced by certain pro jected scores v π xy to b e deﬁned in the following paragraphs. T ogether with them, w e will make use of the corresp onding pro jected margins m π xy = v π xy − v π y x and the corresp onding pro jected turno v ers t π xy = v π xy + v π y x . So, the rank- lik e rates that we are lo oking for will b e obtained in the following wa y: r x = N − X y 6 = x v π xy . (8) This formula will b e used not only in the case of complete ranking votes, but also in the general case where the votes are allow ed to b e incomplete and/or in transitiv e binary relations. 14 R. Camps, X. Mora, L. Sa umell 2.6 Let us b egin b y the case of complete votes, i. e. t xy = v xy + v y x = 1 . In this case, w e put also t π xy = 1 . Analogously to (6) and (7), formula (8) is then equiv alen t to the following one: r x = ( N + 1 − X y 6 = x m π xy ) / 2 . (9) W e wan t to deﬁne the pro jected margins m π xy so that the rating deﬁned b y (9) b e compatible with the ranking ξ , i. e. xy ∈ ξ implies r x ≤ r y . No w, this ranking deriv es from the relation ν , which is concerned with the sign of m ν xy = v ∗ xy − v ∗ y x . This clearly p oin ts tow ards taking m π xy = m ν xy . Ho w ever, this is still not enough for ensuring the compatibilit y with ξ . In order to ensure this prop erty , it suﬃces that the pro jected margins, whic h w e assume antisymmetric, b ehav e in the follo wing w a y: x  ξ y = ⇒ m π xy ≥ 0 and m π xz ≥ m π y z for an y z 6∈ { x, y } . (10) On the other hand, we also w an t the rates to b e indep endent of ξ when there are several p ossibilities for it. T o this eﬀect, w e will require the pro- jected margins to hav e already such an indep endence. The next op eration will transform the indirect margins so as to satisfy these conditions. It is deﬁned in the follo wing wa y , where we assume x  ξ y and x 0 denotes the item that immediately follo ws x in the total order ξ : m ν xy = v ∗ xy − v ∗ y x , (11) m σ xy = min { m ν pq | p  − ξ x, y  − ξ q } , (12) m π xy = max { m σ pp 0 | x  − ξ p  ξ y } , (13) m π y x = − m π xy . (14) One can easily see that the m σ xy obtained in (12) already satisfy a condition analogous to (10). How ev er, the indep endence of ξ is not b e ensured until steps (13–14). This prop erty is a consequence of the fact that the pro jected margins giv en b y the preceding form ulas satisfy not only condition (10) but also the following one: m π xy = 0 = ⇒ m π xz = m π y z for any z 6∈ { x, y } . (15) In particular, this will happ en whenev er m ν xy = 0 (since this implies m σ pp 0 = 0 for all p suc h that x  − ξ p  ξ y ). More particularly , in the even t of having t w o admissible orders that in terc hange tw o consecutive elemen ts p and p 0 w e will ha v e m π pp 0 = m σ pp 0 = m ν pp 0 = 0 and consequen tly m π pz = m π p 0 z for an y z 6∈ { p, p 0 } , as it is required by the desired indep endence of ξ . Continuous ra ting for preferential voting , § 2 15 An yw ay , the pro jected margins are ﬁnally introduced in (9), whic h de- termines the rank-lik e rates r x . The corresp onding fraction-like rates will b e in tro duced in § 2.9. R emarks 1. Condition (10) gives the pattern of growth of the pro jected mar- gins m π pq when p and q v ary according to an admissible order ξ . This pattern is illustrated in ﬁgure 1 b elo w, where the square represents the matrix ( m π pq ) with p and q ordered according to ξ , from b etter to w orse. As usual, the ﬁrst index lab els the rows, and the second one lab els the columns. The diagonal corresp onds to the case p = q , whic h w e systematically leav e out of con- sideration. Having said that, here it would b e appropriate to put m π pp = 0 . An yw ay , the pro jected margins are greater than or equal to zero ab o v e the diagonal and smaller than or equal to zero b elow it, and they increase or re- main the same as one mov es along the indicated arrows. The righ t-hand side of the ﬁgure follo ws from the left-hand one b ecause the pro jected margins are antisymmetric. x y z 1 z 2 z 3 x y z 1 z 2 z 3 Figure 1. Directions of gro wth of the pro jected margins. Of course, the absolute v alues d xy = | m π xy | keep this pattern in the up- p er triangle but they b eha v e in the reverse wa y in the lo w er one. Such a b eha viour is often considered in combinatorial data analysis, where it is asso ciated with the name of W. S. Robinson, a statistician who in 1951 in- tro duced a condition of this kind as the cornerstone of a metho d for se- riating arc haeological dep osits (i. e. placing them in chronological order) [ 31 ; 14 : § 4.1.1, 4.1.2, 4.1.4 ; 33 ]. Condition (15), more precisely its expression in terms of the d xy , is also considered in cluster analysis, where it is referred to by sa ying that the ‘dissimilarities’ d xy are ‘ev en’ [ 15 : § 9.1 ] (‘semideﬁnite’ according to other authors). 16 R. Camps, X. Mora, L. Sa umell In the presen t case of complete votes, the pro jected margins m π xy deﬁned b y (11–14) satisfy not only (10) and (15), but also the stronger condition m π xz = max ( m π xy , m π y z ) , whenev er x  ξ y  ξ z . (16) Besides (10) and (15), this prop erty implies also that the d xy satisfy the follo wing inequalit y , which mak es no reference to the relation ξ : d xz ≤ max ( d xy , d y z ) , for any x, y , z . (17) This condition, called the ultrametric inequalit y , is also w ell kno wn in clus- ter analysis, where it app ears as a necessary and suﬃcient condition for the dissimilarities d xy to deﬁne a hy erarchical classiﬁcation of the set under con- sideration [ 14 : § 3.2.1 ; 29 ]. Our problem diﬀers from the standard one of combinatorial data analysis in that our dissimilarities, namely the margins, are an tisymmetric, whereas the standard problem considers symmetric dissimilarities. In other words, our dissimilarities hav e b oth magnitude and direction, whereas the standard ones ha v e magnitude only . This mak es an imp ortan t diﬀerence in connection with the seriation problem, i. e. positioning the items on a line. Let us remark that the case of directed dissimilarities is considered in [ 14 : § 4.1.2 ]. 2. The op eration ( m ν xy ) → ( m π xy ) deﬁned b y (12–13) is akin to the single-link metho d of cluster analysis, which can b e viewed as a contin uous metho d for pro jecting a matrix of dissimilarities onto the set of ultrametric distances; suc h a contin uous pro jection is achiev ed by taking the maximal ultrametric distance whic h is b ounded by the giv en matrix of dissimilarities [ 15 : § 7.3, 7.4, 8.3, 9.3 ]. The op eration ( m ν xy ) → ( m π xy ) do es the same kind of job under the constraint that the clusters —in the sense of cluster analysis— b e interv als of the total order ξ . 2.7 In order to get more insight in to the case of incomplete votes, it is in teresting to lo ok at the case of plumping votes, i. e. the case where eac h v ote plumps for a single option. In this case, and assuming in terpretation (d), the binary scores of the vote ha ve the form v xy = f x for every y 6 = x , where f x is the fraction of voters who choose x . In the spirit of condition H, in this case we exp ect the pro jected scores v π xy to coincide with the original ones v xy = f x . So, b oth the pro jected margins m π xy and the pro jected turno v ers t π xy should also coincide with the original ones, namely f x − f y and f x + f y . In this connection, one easily sees that the indirect scores v ∗ xy coincide with v xy (see Prop osition 9.4). As a consequence, ξ is any total order for which the f x are non-increasing. Continuous ra ting for preferential voting , § 2 17 If we apply formulas (11–14), w e ﬁrst get m ν xy = v ∗ xy − v ∗ y x = v xy − v y x = f x − f y , and then m σ xx 0 = f x − f x 0 , but the pro jected margins resulting from (13) cease to coincide with the original ones. Most interestingly , suc h a coincidence would hold if the max op erator of formula (13) was replaced b y a sum. No w, these t w o apparen tly diﬀeren t op erations —maxim um and addition— can b e viewed as particular cases of a general procedure whic h in volv es taking the union of certain interv als, namely γ xx 0 = [ ( t xx 0 − m σ xx 0 ) / 2 , ( t xx 0 + m σ xx 0 ) / 2 ] . In fact, in the case of complete v otes, all the turno vers are equal to 1 , so these in terv als are all of them centred at 1 / 2 and the union op eration is equiv alent to lo oking for the maxim um of the widths. In the case of plumping v otes, w e know that t xx 0 = f x + f x 0 and we hav e just seen that m σ xx 0 = f x − f x 0 , whic h implies that γ xx 0 = [ f x 0 , f x ] ; so, the interv als γ xx 0 and γ x 0 x 00 are then adjacen t to eac h other (the right end of the latter coincides with the left end of the former) and their union in v olves adding up the widths. This remark strongly suggests that the general metho d should rely on suc h in terv als. In the follo wing we will refer to them as score in terv als . A score interv al can b e viewed as giving a pair of scores ab out t w o options, the tw o scores b eing resp ectiv ely in fa vour and against a sp eciﬁed preference relation ab out the t w o options. Alternativ ely , it can b e viewed as giving a certain margin together with a certain turno v er. More sp eciﬁcally , one is immediately tempted to replace the minim um and maximum op erations of (12–13) by the intersection and union of score in terv als. The starting p oint would b e the score in terv als that com bine the original turno vers t xy with the indirect margins m ν xy . Suc h a pro cedure w orks as desired b oth in the case of complete votes and that of plumping ones. Unfortunately , ho w ever, it breaks down in other cases of incomplete v otes whic h pro duce empt y intersections or disjoint unions. So, a more elab orate metho d is required. 2.8 In this subsection w e will ﬁnally describ e a rank-like rating pro cedure whic h is able to cop e with the general case. This pro cedure will use score in terv als. How ever, these in terv als will not b e based directly on the original turno v ers, but on certain transformed ones. This prior transformation of the turno v ers will hav e the virtue of av oiding the problems p oin ted out at the end of the preceding paragraph. So, w e are giv en as input from one side the indirect margins m ν xy , and from the other side the original turnov ers t xy . The output to b e pro duced is a set of pro jected scores v π xy . They should hav e the virtue that the associated rank-lik e rating given b y (8) has the follo wing properties: (a) it is the exactly 18 R. Camps, X. Mora, L. Sa umell the same for all admissible orders ξ ; and (b) it is compatible with any such order ξ , i. e. xy ∈ ξ implies r x ≤ r y . As w e did in the complete case, we will require the pro jected scores v π xy to satisfy the condition of indep endence with resp ect to ξ . On the other hand, in order to ensure the compatibility condition (b), it suﬃces that the pro jected scores b ehav e in the follo wing w ay: x  ξ y = ⇒ v π xy ≥ v π y x and v π xz ≥ v π y z for any z 6∈ { x, y } . (18) If w e think in terms of the asso ciated margins m π xy and turno vers t π xy —whic h add up to 2 v π xy — it suﬃces that b oth of them satisfy conditions analogous to (18). More, sp eciﬁcally , it suﬃces that the pro jected margins b e antisym- metric and satisfy condition (10) of § 2.6 and that the pro jected turnov ers b e symmetric and satisfy x  ξ y = ⇒ t π xz ≥ t π y z for any z 6∈ { x, y } . (19) So, we w ant the pro jected scores to b e indep endent of ξ , and their asso- ciated margins and turno v ers to satisfy conditions (10) and (19). These re- quiremen ts are fulﬁlled b y the pro cedure form ulated in (20–26) b elow. These form ulas use the following notations: Ψ is an op erator to b e describ ed in a while; [ a, b ] means the closed interv al { x ∈ R | a ≤ x ≤ b } ; | γ | means the length of such an in terv al γ = [ a, b ] , i. e. the num b er b − a ; and • γ means its barycen tre, or centroid, i. e. the n um b er ( a + b ) / 2 . As in (11–14), the follo w- ing form ulas assume that x  ξ y , and x 0 denotes the option that immediately follo ws x in the total order ξ . m ν xy = v ∗ xy − v ∗ y x , t xy = v xy + v y x , (20) m σ xy = min { m ν pq | p  − ξ x, y  − ξ q } , t σ xy = Ψ [ ( t pq ) , ( m σ pp 0 ) ] xy , (21) γ xx 0 = [ ( t σ xx 0 − m σ xx 0 ) / 2 , ( t σ xx 0 + m σ xx 0 ) / 2 ] , (22) γ xy = S { γ pp 0 | x  − ξ p  ξ y } , (23) m π xy = | γ xy | , t π xy = 2 • γ xy , (24) m π y x = − m π xy , t π y x = t π xy , (25) v π xy = max γ xy = ( t π xy + m π xy ) / 2 , v π y x = min γ xy = ( t π xy − m π xy ) / 2 . (26) Lik e (11–14), the preceding pro cedure can b e viewed as a tw o-step trans- formation. The ﬁrst step is given by (21) and it transforms the input mar- gins and turnov ers ( m ν xy , t xy ) into certain in termediate pro jections ( m σ xy , t σ xy ) Continuous ra ting for preferential voting , § 2 19 whic h already satisfy conditions analogous to (10) and (19) but are not in- dep enden t of ξ . The condition of indep endence requires a second step which is describ ed by (22–26). As in § 2.6, the sup erdiagonal ﬁnal pro jections co- incide with the intermediate ones, i. e. m π xx 0 = m σ xx 0 and t π xx 0 = t σ xx 0 . Notice also that the intermediate margins m σ xy are constructed exactly as in (12). The main diﬃcult y lies in constructing the in termediate turnov ers t σ xy so that they do not depend on ξ . The reason is that this condition in v olv es the admissible orders, whic h dep end on the relation ν asso ciated with the indirect margins m ν xy . So, that construction must tak e into accoun t not only the original turno v ers but also the indirect margins. This connection with the m ν xy will b e controlled indirectly through the m σ xx 0 . In fact, we will lo ok for the t σ xy so as to satisfy the follo wing conditions: m σ xx 0 ≤ t σ xx 0 ≤ 1 , (27) 0 ≤ t σ py − t σ p 0 y ≤ m σ pp 0 , (28) 0 ≤ t σ xq − t σ xq 0 ≤ m σ q q 0 . (29) Notice that (28) ensures that t σ py and t σ p 0 y will coincide with each other when- ev er m σ pp 0 = 0 . Since m σ pp 0 = m ν pp 0 , we are in the case of ha ving tw o admis- sible orders that interc hange p with p 0 . The fact that this implies t σ py = t σ p 0 y ev en tually ensures the indep endence of ξ (Theorem 9.2; w e say ‘ev en tually’ b ecause the full pro of is quite long). In the case of complete votes we will hav e t σ xy = 1 , so that condition (27) will b e satisﬁed with an equalit y sign in the righ t-hand inequalit y , whereas (28) and (29) will b e satisﬁed with an equality sign in the left-hand inequality . In the case of plumping v otes, where we kno w that m σ xx 0 = m xx 0 = f x − f x 0 ( § 2.7), w e will hav e t σ xy = t xy = f x + f y , so that (28) and (29) will be satisﬁed with an equality sign in the right-hand inequalities (equation (27) is satisﬁed to o, but in this case the inequalities can b e strict). Notice also that conditions (28–29) imply the following one: 0 ≤ t σ xx 0 − t σ x 0 x 00 ≤ m σ xx 0 + m σ x 0 x 00 . (30) In geometrical terms, the inequalities in (27) mean that (a) the inter- v al γ xx 0 is con tained in [0 , 1] . On the other hand, the inequalities in (30) mean that the interv als γ xx 0 and γ x 0 x 00 are related to each other in the fol- lo wing wa y: (b) the barycentre of the ﬁrst one lies to the righ t of that of the second one; (c) the t wo interv als ov erlap each other. Conditions (27–29) can b e easily ac hiev ed b y taking simply t σ xy = 1 . Ho w ever, this choice go es against our aim of distinguishing b et w een deﬁnite 20 R. Camps, X. Mora, L. Sa umell indiﬀerence and lac k of information; in particular, condition H requires that in the case of plumping votes the pro jected turno v ers should coincide with the original ones (whic h are then less than 1 ). Now, conditions (27–29) are con v ex with resp ect to the t σ xy (the whole set of them), i. e. if they are satisﬁed b y t w o diﬀeren t c hoices of these n um b ers, they are satisﬁed also b y any con vex combination of them. This implies that for an y given set of original turnov ers t xy there is a unique set of v alues t σ xy whic h minimizes the euclidean distance to the given one while satisfying those conditions. So, the op erator Ψ can deﬁned in the following w ay: t σ xy is the set of turno v ers whic h is determined by conditions (27–29) together with that of minimizing the following measure of deviation with resp ect to the t xy : Φ = X x X y ( t σ xy − t xy ) 2 . (31) The actual computation of the t σ xy can b e carried out in a ﬁnite num b er of steps by means of a quadratic programming algorithm [ 22 : § 14.1 (2nd ed.) ]. An yw ay , the preceding op erations hav e the virtue of ensuring the desired prop erties. R emarks 1. Condition (19) is illustrated in ﬁgure 2, where the arrows indicate the directions of growth of the pro jected turno v ers. The right-hand side of the ﬁgure follows from the left-hand one b ecause the pro jected turnov ers are symmetric. x y z 1 z 2 z 3 x y z 1 z 2 z 3 Figure 2. Directions of gro wth of the pro jected turnov ers. This condition can b e asso ciated with the name of Marshall G. Greenberg, a mathematical psyc hologist who in 1965 considered a condition of this form —at the suggestion of Clyde H. Co om bs— in connection with the problem of pro ducing a rating after paired-comparison data, sp ecially in the incomplete Continuous ra ting for preferential voting , § 2 21 case [ 12 ; 14 : § 4.1.2 ]. Ha ving said that, we strongly diﬀer from that author in that he applies a prop ert y lik e (19) to the scores, whereas we consider more appropriate to apply it to the turno v ers. In fact, under the general assumption that each v ote is a ranking, p os- sibly incomplete, and that eac h ranking is translated into a set of binary preferences according to rules (a–d) of § 2.1, it is fairly reasonable to exp ect that the turnov er for a pair xy , i. e. the num b er of v oters who expressed an opinion ab out x in comparison with y , should increase as x and/or y are higher in the so cial ranking. In practice, the original turnov ers can deviate to a certain extent from this ideal b eha viour. In con trast, our pro jected turno v ers are alwa ys in agreement with it (with respect to the total order ξ ). 2. The pro jected scores turn out to satisfy not only (18), but also the follo wing stronger prop erty: if x  ξ y then v π xy ≥ v π y x and v π xz ≥ v π y z , v π z x ≤ v π z y for any z 6∈ { x, y } . (32) So, the pro jected scores increase or remain constant in the directions sho wn in ﬁgure 1. F urthermore, we will see that the quotien ts m π xy /t π xy ha v e also the same prop erty . 3. In contrast to the case of complete v otes, in this case the pro jected margins do not satisfy (16) but only m π xz ≤ m π xy + m π y z , whenev er x  ξ y  ξ z . (33) As a consequence, the absolute v alues d xy = | m π xy | satisfy the triangular inequalit y: d xz ≤ d xy + d y z , for any x, y , z . (34) 4. Under the assumption of ranking votes (but not necessarily in a more general setting) one can see that the original turnov ers already satisfy (27). In this case, the preceding deﬁnition of Ψ turns out to b e equiv alen t to an analogous one where conditions (27–29) are replaced simply b y (28–29). F rom this it follows that the intermediate turnov ers ha v e then the same sum as the original ones: X x X y t σ xy = X x X y t xy . (35) 2.9 Finally , let us see ho w shall we deﬁne the fraction-lik e rates ϕ x . As in the case of the rank-lik e rates, w e will use a classical metho d which would usually b e applied to the original scores, but here w e will apply it to the pro jected scores. This metho d was in tro duced in 1929 by Ernst Zermelo [ 41 ] 22 R. Camps, X. Mora, L. Sa umell and it w as redisco v ered by other authors in the 1950s [ 4, 11 ] (see also [ 16 ]). Zermelo’s w ork was motiv ated b y chess tournaments, whereas the other au- thors were considering comparative judgmen ts. An ywa y , all of them were esp ecially in terested in the incomplete case, i. e. the case where turnov ers ma y dep end on the pair xy . More sp eciﬁcally , the fraction-lik e rates ϕ x will b e determined by the follo wing system of equations (together with the condition that ϕ x ≥ 0 for ev ery x ): X y 6 = x t π xy ϕ x / ( ϕ x + ϕ y ) = X y 6 = x v π xy (= N − r x ) , (36) X x ϕ x = f , (37) where (36) con tains one equation for every x , and f stands for the fraction of non-empt y v otes (i. e. f = F /V where F is the n um b er of non-empt y votes and V is the total n umber of v otes). In spite of having N + 1 equations, the N equations con tained in (36) are not indep enden t, since their sum results in the identit y ( P x P y 6 = x t π xy ) / 2 = P x P y 6 = x v π xy . On the other hand, it is clear that (36) is insensitive to all of the ϕ x b eing multiplied b y a constant factor. This indeterminacy disapp ears once (36) is supplemented with equation (37). In the case of plumping votes, where we know that v π xy = f x and t π xy = f x + f y , the solution of (36–37) is easily seen to b e ϕ x = f x , as required by condition H. The problem of solving the system (36–37) is well p osed when the pro- jected Llull matrix ( v π xy ) is irreducible. This means that there is no splitting of the options in to a ‘top class’ X plus a ‘low class’ Y so that v π y x = 0 for an y x ∈ X and y ∈ Y . When suc h a splitting exists, one is forced to put ϕ y = 0 for all y ∈ Y . F or more details, the reader is referred to section 11. Zermelo (and the other authors) dealed also with the problem of nu- merically solving a non-linear system of the form (36). In this connection, he show ed that in the irreducible case its solution (up to a multiplicativ e constan t) can b e appro ximated to an arbitrary degree of accuracy by means of an iterative scheme of the form ϕ n +1 x =  X y 6 = x v π xy  X y 6 = x t π xy / ( ϕ n x + ϕ n y )  , (38) starting from an arbitrary set of v alues ϕ 0 x > 0 . Continuous ra ting for preferential voting , § 3 23 The fraction-like rates ϕ x determined by (36–37) can b e view ed as an estimate of the ﬁrst-c hoice fractions using not only the ﬁrst choices but the whole rankings. Prop erly sp eaking, Zermelo’s metho d (with the original scores and turnov ers) corresp onds to a maximum lik eliho o d estimate of the parameters of a certain probabilistic mo del for the outcomes of a tournament b et w een seve ral play ers, or, more in the lines of our applications, for the outcomes of comparativ e judgments. This mo del will b e brieﬂy describ ed in section 11. Although w e are far from its h yp otheses, we will see that Zermelo’s metho d is quite suitable for translating our rank-like rates into fraction-lik e ones. 3 Summary of the metho d. V arian ts. General forms of v ote 3.1 Let us summarize the whole pro cedure. In the gener al c ase , where the v otes are not necessarily complete, it consists of the following steps: 1. F orm the Llull matrix ( v xy ) ( § 2.1). W ork out the turnov ers t xy = v xy + v y x . 2. Compute the indirect scores v ∗ xy deﬁned b y (4). An eﬃcient wa y to do it is the Floyd-W arshall algorithm [ 8 : § 25.2 ]. W ork out the indirect margins m ν xy = v ∗ xy − v ∗ y x and the asso ciated indirect comparison relation ν = { xy | m ν xy > 0 } . 3. Find an admissible order ξ ( § 2.4) and arrange the options according to it. F or instance, it suﬃces to arrange the options by non-decreasing v al- ues of the ‘tie-splitting’ Cop eland scores κ x = 1 + |{ y | y 6 = x, m ν y x > 0 }| + 1 2 |{ y | y 6 = x, m ν y x = 0 }| (Prop osition 8.5). 4. Starting from the indirect margins m ν xy , w ork out the sup erdiagonal in termediate pro jected margins m σ xx 0 as deﬁned in (21.1). 5. Starting from the original turno v ers t xy , and taking in to accoun t the sup erdiagonal intermediate pro jected margins m σ xx 0 , determine the in termediate pro jected turnov ers t σ xy so as to minimize (31) under the constraints (27–29). This can b e carried out in a ﬁ- nite num b er of steps by means of a quadratic programming algorithm [ 22 : § 14.1 (2nd ed.) ]. 6. F orm the interv als γ xx 0 deﬁned by (22), derive their unions γ xy as deﬁ- ned by (23), and read oﬀ the pro jected scores v π xy (26). 7. Compute the rank-like ranks r x according to (8). 8. Determine the fraction-lik e rates ϕ x b y solving the system (36–37). This can b e done n umerically b y means of the iterative scheme (38). 24 R. Camps, X. Mora, L. Sa umell In the c omplete c ase , the scores v xy and the margins m xy are related to eac h other b y the monotone increasing transformation v xy = (1 + m xy ) / 2 . Because of this fact, the preceding pro cedure can then b e simpliﬁed in the follo wing w a y: • Step 2 computes m ∗ xy instead of v ∗ xy and takes m ν xy = ( m ∗ xy − m ∗ y x ) / 2 . • Step 5 is not needed. • Step 6 reduces to (13–14). • Step 7 mak es use of formula (9). 3.2 The preceding procedure admits of certain v arian ts which might b e appropriate to some sp ecial situations. Next w e will distinguish four of them, namely 1. Main 2. Dual 3. Balanced 4. Margin-based The ab o ve-described procedure is included in this list as the main v arian t. The four v ariants are exactly equiv alen t to each other in the complete case, but in the incomplete case they can produce diﬀerent results. In spite of this, they all share the main prop erties. The dual v ariant is analogous to the main one except that the max-min indirect scores v ∗ xy are replaced by the follo wing min-max ones: ∗ v xy = min x 0 = x x n = y max i ≥ 0 i < n v x i x i +1 . (39) Equiv alen tly , ∗ v xy = 1 − ˆ v ∗ y x where ˆ v xy = 1 − v y x . In the complete case one has ∗ v xy = 1 − v ∗ y x , so that ∗ v xy − ∗ v y x = v ∗ xy − v ∗ y x and µ ( ∗ v ) = µ ( v ∗ ) ; as a consequence, the dual v ariant is then equiv alen t to the main one. The balanced v ariant takes ν = µ ( v ∗ ) ∩ µ ( ∗ v ) together with m ν xy =      min ( v ∗ xy − v ∗ y x , ∗ v xy − ∗ v y x ) , if xy ∈ µ ( v ∗ ) ∩ µ ( ∗ v ), − m ν y x , if y x ∈ µ ( v ∗ ) ∩ µ ( ∗ v ), 0 , otherwise. (40) The remarks made in connection with the dual v arian t sho w that in the complete case the balanced v ariant is also equiv alent to the preceding ones. Continuous ra ting for preferential voting , § 3 25 The margin-based v ariant follo ws the simpliﬁed pro cedure of the end of § 3.1 even if one is not originally in the complete case. Equiv alen tly , it corresp onds to replacing the original scores v xy b y the follo wing ones: v 0 xy = (1 + m xy ) / 2 . This amounts to replacing an y lac k of information ab out a pair of options b y a deﬁnite indiﬀerence b etw een them, whic h brings the problem into the complete case. So, the sp eciﬁc c haracter of this v ariant lies only in its interpretation of incomplete votes. Although this interpretation go es against the general principle stated in § 1.5, it ma y b e suitable to cer- tain situations where the voters are w ell acquain ted with all of the options. In the case of ranking votes, it amoun ts to replace rule (d) of § 2.1 by the follo wing one: (d 0 ) When neither x nor y are in the list, w e interpret that they are considered equally go o d (or equally bad), so we pro ceed as in (b). In other words, eac h truncated v ote is completed by app ending to it all the missing options tied to each other. R emark Other v ariants —in the incomplete case— arise when equation (8) is replaced by the following one: r x = 1 + X y 6 = x v π y x . (41) 3.3 Most of our results will hold if the “v otes” are not required to b e rank- ings, but they are allow ed to b e general binary relations. In particular, this allo ws to deal with certain situations where it makes sense to replace rule (c) of § 2.1 by the following one: (c 0 ) When x is in the list and y is not in it, w e interpret nothing ab out the preference of the voter b et w een x and y . One could ev en allo w the votes to b e non-transitiv e binary relations; such a lac k of transitivity in the individual preferences may arise when individuals are aggregating a v ariety of criteria [ 13 ]. A vote in the form of a binary relation ρ contributes to the binary scores with the following amounts: v xy =      1 , if xy ∈ ρ and y x / ∈ ρ 1 / 2 , if xy ∈ ρ and y x ∈ ρ 0 , if xy / ∈ ρ. (42) Ev en more generally , a vote could b e any set of normalized binary scores, i. e. an element of the set Ω = { v ∈ [0 , 1] Π | v xy + v y x ≤ 1 } , where Π denotes the set of pairs xy ∈ A × A with x 6 = y . 26 R. Camps, X. Mora, L. Sa umell An yw ay , the collectiv e Llull matrix is simply the center of gra vity of a distribution of individual votes: v xy = X k α k v k xy , (43) where α k are the relative frequencies or w eigh ts of the individual votes v k . 4 Examples 4.1 As a ﬁrst example of a vote whic h inv olv ed truncated rankings, we lo ok at an election which to ok place the 16th of F ebruary of 1652 in the Spanish roy al household. This election is quoted in [ 30 ], but w e use the sligh tly diﬀerent data which are giv en in [ 28 : vol. 2, p. 263–264 ]. The oﬃce under election w as that of “ap osentador ma yor de palacio”, and the king was assessed by six noblemen, who expressed the following preferences: Marqu ´ es de Ari¸ ca . . . . . . . . . . . . . . . . . . . . . . . . b  e  d  a Conde de Bara jas . . . . . . . . . . . . . . . . . . . . . . . . b  a  f Conde de Montalb´ an . . . . . . . . . . . . . . . . . . . . . a  f  b  d Marqu ´ es de Po v ar . . . . . . . . . . . . . . . . . . . . . . . . e  b  f  c Conde de Pu ˜ nonrostro . . . . . . . . . . . . . . . . . . . e  a  b  f Conde de Ysingui´ en . . . . . . . . . . . . . . . . . . . . . . b  d  a  f The candidates a – f nominated in these preferences were: a Alonso Carb onel (architect, 1583–1660) b Gaspar de F uensalida (died 1664) c Joseph Nieto d Sim´ on Ro dr ´ ıguez e F rancisco de Ro jas (1583–1659) f Diego V el´ azquez (painter, 1599–1660) The CLC computations are as follo ws: x a b c d e f V xy a b c d e f ∗ 2 5 3 3 5 4 ∗ 6 6 4 5 1 0 ∗ 1 0 0 2 0 3 ∗ 2 2 3 2 3 3 ∗ 3 1 1 5 4 3 ∗ x a b c d e f V ∗ xy a b c d e f ∗ 2 5 4 3 5 4 ∗ 6 6 4 5 1 1 ∗ 1 1 1 2 2 3 ∗ 2 2 3 2 3 3 ∗ 3 3 2 5 4 3 ∗ κ 2 1 2 1 6 5 3 3 1 2 Continuous ra ting for preferential voting , § 4 27 x b a e f d c M ν xy b a e f d c ∗ 2 2 3 4 5 ∗ ∗ 0 2 2 4 ∗ ∗ ∗ 0 1 2 ∗ ∗ ∗ ∗ 2 4 ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c T xy b a e f d c ∗ 6 6 6 6 6 ∗ ∗ 6 6 5 6 ∗ ∗ ∗ 6 5 3 ∗ ∗ ∗ ∗ 6 5 ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c M σ xy b a e f d c ∗ 2 2 3 4 5 ∗ ∗ 0 2 2 4 ∗ ∗ ∗ 0 1 2 ∗ ∗ ∗ ∗ 1 2 ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c T σ xy b a e f d c ∗ 6 6 6 6 6 ∗ ∗ 6 6 5 1 3 4 2 3 ∗ ∗ ∗ 6 5 1 3 4 2 3 ∗ ∗ ∗ ∗ 5 1 3 4 2 3 ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c M π xy b a e f d c ∗ 2 2 2 2 3 ∗ ∗ 0 0 1 2 1 6 ∗ ∗ ∗ 0 1 2 1 6 ∗ ∗ ∗ ∗ 1 2 1 6 ∗ ∗ ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c T π xy b a e f d c ∗ 6 6 6 6 5 ∗ ∗ 6 6 5 1 3 4 1 6 ∗ ∗ ∗ 6 5 1 3 4 1 6 ∗ ∗ ∗ ∗ 5 1 3 4 1 6 ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ ∗ x b a e f d c V π xy b a e f d c ∗ 4 4 4 4 4 2 ∗ 3 3 3 1 6 3 1 6 2 3 ∗ 3 3 1 6 3 1 6 2 3 3 ∗ 3 1 6 3 1 6 2 2 1 6 2 1 6 2 1 6 ∗ 3 1 1 1 1 1 ∗ x b a e f d c r x ϕ x 2.6667 0.3049 3.6111 0.1703 3.6111 0.1703 3.6111 0.1703 4.0833 0.1293 5.1667 0.0549 28 R. Camps, X. Mora, L. Sa umell According to these results, the oﬃce should ha v e b een given to candi- date b , who is also the winner by most other metho ds. In the CLC metho d, this candidate is follo w ed b y three runners-up tied to each other, namely candidates a , e and f . In spite of the clear adv an tage of candidate b , the king app oin ted candidate f , namely , the celebrated painter Diego V el´ azquez. 4.2 As an example where the v otes are complete strict rankings, we will con- sider the ﬁnal round of a dancesp ort comp etition. Sp eciﬁcally , we will take the ﬁnal round of the Professional Latin Rising Star section of the 2007 Black- p o ol Dance F estiv al (Blac kp o ol, England, 25th Ma y 2007). The data were tak en from http://www.scrutelle.info/results/estelle/2007/blackpool - 2007/ . As usual, the ﬁnal w as con tested b y six couples, whose n um b ers w ere 3 , 4 , 31 , 122 , 264 , 238 . Eleven adjudicators ranked their simultaneous p er- formances in four equiv alent dances. The all-round oﬃcial result was 3  122  264  4  31  238 . This result comes from the so-called “Sk ating System”, whose name reﬂects a prior use in ﬁgure-sk ating. The Sk ating System has a ﬁrst part which pro- duces a separate result for eac h dance. This is done mainly on the basis of the median rank obtained by each couple, a criterion which Condorcet prop osed as a “practical” metho d in 1792/93 [ 23 : ch. 8 ]. How ever, the ﬁne prop erties of this criterion are lost in the second part of the Sk ating System, where the all-round result is obtained b y adding the up the ﬁnal ranks obtained in the diﬀerent dances. F rom the p oin t of view of paired comparisons, it mak es sense to base the all-round result on the Llull matrix which collects the 44 rankings pro- duced b y the 11 adjudicators ov er the 4 dances [ 26 : § 11 ]. As one can see b elo w, in the presen t case this matrix exhibits sev eral Condorcet cycles, like for instance 3  4  264  3 and 3  122  264  3 , which means that the comp etition was closely con tested. In suc h close con tests, the Sk ating System often has to resort to certain tie-breaking rules whic h are virtually equiv alent to throwing the dice. In con trast, the all-round Llull matrix has the virtue of b eing a more accurate quan titativ e aggregate ov er the diﬀerent dances. On the basis of this more accurate aggregate, in this case the indirect scores rev eal an all-round ranking whic h is quite diﬀeren t from the one pro duced b y the Sk ating System (but it coincides with the one pro duced by other paired- comparison metho ds, lik e rank ed pairs). In consonance with all this, the CLC rates obtained b elow are quite close to each other, particularly for the couples 3 , 4 , 122 and 264 . Since w e are dealing with complete votes, in this case the CLC compu- Continuous ra ting for preferential voting , § 4 29 tations can b e carried out entirely in terms of the margins. In the following w e hav e c hosen to pass to margins after computing the indirect scores, but w e could hav e done it b efore that step. x 3 4 31 122 238 264 V xy 3 4 31 122 238 264 ∗ 23 28 23 28 20 21 ∗ 23 20 30 24 16 21 ∗ 15 25 18 21 24 29 ∗ 28 23 16 14 19 16 ∗ 19 24 20 26 21 25 ∗ x 3 4 31 122 238 264 V ∗ xy 3 4 31 122 238 264 ∗ 23 28 23 28 23 24 ∗ 24 23 30 24 21 21 ∗ 21 25 21 24 24 29 ∗ 28 24 19 19 19 19 ∗ 19 24 23 26 23 25 ∗ κ 4 2 5 1 6 3 x 122 4 264 3 31 238 M ν xy 122 4 264 3 31 238 ∗ 1 1 1 8 9 ∗ ∗ 1 1 3 11 ∗ ∗ ∗ 1 5 6 ∗ ∗ ∗ ∗ 7 9 ∗ ∗ ∗ ∗ ∗ 6 ∗ ∗ ∗ ∗ ∗ ∗ x 122 4 264 3 31 238 M π xy 122 4 264 3 31 238 ∗ 1 1 1 3 6 ∗ ∗ 1 1 3 6 ∗ ∗ ∗ 1 3 6 ∗ ∗ ∗ ∗ 3 6 ∗ ∗ ∗ ∗ ∗ 6 ∗ ∗ ∗ ∗ ∗ ∗ x 122 4 264 3 31 238 r x ϕ x 3.3636 0.1815 3.3864 0.1788 3.4091 0.1761 3.4318 0.1734 3.5682 0.1583 3.8409 0.1318 30 R. Camps, X. Mora, L. Sa umell 4.3 As a second example of an election inv olving truncated rankings we tak e the Debian Pro ject leader election, whic h is using the metho d of paths since 2003. So far, the winners of these elections ha v e been clear enough. Ho wev er, a quan titativ e measure of this clearness was lacking. In the follo wing w e consider the 2006 election, whic h had a participation of V = 421 actual v oters out of a total p opulation of 972 members. The individual votes are a v ailable in http://www.debian.org/vote/2006/vote - 002 . That election resulted in the follo wing Llull matrix: x 1 2 3 4 5 6 7 8 V xy 1 2 3 4 5 6 7 8 ∗ 321 144 159 1 2 193 1 2 347 1 2 246 320 51 ∗ 42 53 50 262 65 163 251 340 ∗ 198 1 2 253 362 300 345 245 1 2 341 204 1 2 ∗ 256 371 1 2 291 1 2 339 1 2 193 1 2 325 144 149 ∗ 357 254 321 1 2 26 1 2 77 24 22 1 2 21 ∗ 30 74 1 2 137 292 90 109 1 2 131 330 ∗ 296 76 207 54 71 1 2 75 1 2 302 1 2 89 ∗ Notice that candidate 4 is the winner according to the Condorcet principle (but not according to the ma jorit y principle, since V 43 do es not reach V / 2 ). Notice also that there is no Condorcet cycle. How ever, candidates 1 and 5 are in a tie for third place: b oth of them defeat all other candidates except 4 and 3 , and V 15 coincides exactly with V 51 . The ensuing CLC computations are as follo ws: x 1 2 3 4 5 6 7 8 V ∗ xy 1 2 3 4 5 6 7 8 ∗ 321 159 1 2 159 1 2 193 1 2 347 1 2 246 320 89 ∗ 89 89 89 262 89 163 251 340 ∗ 198 1 2 253 362 300 345 245 1 2 341 204 1 2 ∗ 256 371 1 2 291 1 2 339 1 2 193 1 2 325 159 1 2 159 1 2 ∗ 357 254 321 1 2 77 77 77 77 77 ∗ 77 77 137 292 137 137 137 330 ∗ 296 89 207 89 89 89 302 1 2 89 ∗ κ 3 1 2 7 2 1 3 1 2 8 5 6 Continuous ra ting for preferential voting , § 4 31 x 4 3 1 5 7 8 2 6 M ν xy 4 3 1 5 7 8 2 6 ∗ 6 86 96 1 2 154 1 2 250 1 2 252 294 1 2 ∗ ∗ 91 1 2 93 1 2 163 256 251 285 ∗ ∗ ∗ 0 109 231 232 270 1 2 ∗ ∗ ∗ ∗ 117 232 1 2 236 280 ∗ ∗ ∗ ∗ ∗ 207 203 253 ∗ ∗ ∗ ∗ ∗ ∗ 44 225 1 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 185 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x 4 3 1 5 7 8 2 6 T xy 4 3 1 5 7 8 2 6 ∗ 403 405 405 401 411 394 394 ∗ ∗ 395 397 390 399 382 386 ∗ ∗ ∗ 387 383 396 372 374 ∗ ∗ ∗ ∗ 385 397 375 378 ∗ ∗ ∗ ∗ ∗ 385 357 360 ∗ ∗ ∗ ∗ ∗ ∗ 370 377 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 339 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x 4 3 1 5 7 8 2 6 M σ xy 4 3 1 5 7 8 2 6 ∗ 6 86 96 1 2 154 1 2 250 1 2 252 294 1 2 ∗ ∗ 86 93 1 2 154 1 2 250 1 2 251 285 ∗ ∗ ∗ 0 109 231 232 270 1 2 ∗ ∗ ∗ ∗ 109 231 232 270 1 2 ∗ ∗ ∗ ∗ ∗ 203 203 253 ∗ ∗ ∗ ∗ ∗ ∗ 44 225 1 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 185 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 32 R. Camps, X. Mora, L. Sa umell x 4 3 1 5 7 8 2 6 T σ xy 4 3 1 5 7 8 2 6 ∗ 403.4 403.4 403.4 403.25 403.25 392 392 ∗ ∗ 397.4 397.4 397.25 397.25 386 386 ∗ ∗ ∗ 389.6 389.6 389.6 374.75 374.75 ∗ ∗ ∗ ∗ 389.6 389.6 374.75 374.75 ∗ ∗ ∗ ∗ ∗ 385 366 366 ∗ ∗ ∗ ∗ ∗ ∗ 366 366 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 339 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x 4 3 1 5 7 8 2 6 M π xy 4 3 1 5 7 8 2 6 ∗ 6 86 86 109 203 203 217 ∗ ∗ 86 86 109 203 203 217 ∗ ∗ ∗ 0 109 203 203 217 ∗ ∗ ∗ ∗ 109 203 203 217 ∗ ∗ ∗ ∗ ∗ 203 203 217 ∗ ∗ ∗ ∗ ∗ ∗ 44 185 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 185 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ x 4 3 1 5 7 8 2 6 T π xy 4 3 1 5 7 8 2 6 ∗ 403.4 397.4 397.4 389.6 385 385 371 ∗ ∗ 397.4 397.4 389.6 385 385 371 ∗ ∗ ∗ 389.6 389.6 385 385 371 ∗ ∗ ∗ ∗ 389.6 385 385 371 ∗ ∗ ∗ ∗ ∗ 385 385 371 ∗ ∗ ∗ ∗ ∗ ∗ 366 339 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 339 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Continuous ra ting for preferential voting , § 4 33 x 4 3 1 5 7 8 2 6 V π xy 4 3 1 5 7 8 2 6 ∗ 204.7 241.7 241.7 249.3 294 294 294 198.7 ∗ 241.7 241.7 249.3 294 294 294 155.7 155.7 ∗ 194.8 249.3 294 294 294 155.7 155.7 194.8 ∗ 249.3 294 294 294 140.3 140.3 140.3 140.3 ∗ 294 294 294 91 91 91 91 91 ∗ 205 262 91 91 91 91 91 161 ∗ 262 77 77 77 77 77 77 77 ∗ r x ϕ x 3.6784 0.2067 3.6926 0.2048 4.1105 0.1596 4.1105 0.1596 4.5720 0.1218 5.8100 0.0599 5.9145 0.0559 6.7197 0.0317 As one can see, the CLC results are in full agreement with the Cop eland scores of the original Llull matrix. In particular, they still give an exact tie b et w een candidates 1 and 5 . Ev en so, the CLC rates yield a quan titativ e information which is not presen t in the Cop eland scores. In particular, they sho w that the victory of candidate 4 o v er candidate 3 w as relativ ely narro w. F or the computation of the rates w e ha ve taken V = 421 (the actual n um b er of votes) instead of V = 972 (the n umber of p eople with the righ t to v ote); in particular, the fraction-like rates ϕ x ha v e b een computed so that they add up to f = 1 instead of the true participation ratio f = 421 / 972 . This is esp ecially justiﬁed in Debian elections since they systematically include “none of the ab o v e” as one of the alternativ es, so it is reasonable to in terpret that abstention do es not ha v e a critical character. In the presen t case, “none of the ab o v e” was alternative 8 , which obtained a b etter result than tw o of the real candidates. 4.4 Finally , we lo ok at an example of appro v al voting. Sp eciﬁcally , we consider the 2006 Public Choice Society election [ 5 ]. Besides an appro v al v ote, here the voters were also asked for a preferential v ote “in the spirit of research on public c hoice”. How ever, here w e will limit ourselv es to the appro v al v ote, whic h w as the oﬃcial one. The v ote had a participation of V = 37 voters, most of which approv ed more than one candidate. The actual votes are listed in the following table, 1 where we giv e not only the appro v al voting data but also the asso ciated preferential v otes. The appro v ed candidates are the ones which lie at the left of the slash. 1 W e are grateful to Prof. Stev en J. Brams, who w as the presiden t of the Public Choice So ciet y when that election took place, for his kind p ermission to repro duce these data. 34 R. Camps, X. Mora, L. Sa umell A  B / A  C  B / D /  A  B  E  C B  A /  D  C  E D  A  B  C /  E C  B  A / E /  D C  A  B  E / D  E /  C  A  B E / B  C / D  C /  B  E  A B / A / A / D /  A ∼ B ∼ C ∼ E A ∼ C / / B  E  A  D  C A ∼ B ∼ E / A ∼ B ∼ C ∼ D ∼ E / D  A  B / B  D  A /  C  E A /  B  E  C  D D / A ∼ C  B /  D  E A /  D  B  C  E C /  B  D  A  E C / D ∼ E /  A  B ∼ C B /  C  A  D  E D  C  E / C /  A  B ∼ D ∼ E C / B  D /  E  C  A B  C /  A  E  D D  A  C  B / D  E /  A  B The appro v al v oting scores are the following: A : 17, B : 16, C : 17, D : 14, D : 9. So according to approv al v oting there w as a tie b et w een candidates A and C , whic h w ere follo w ed at a minim um distance by candidate B . The CLC computations are as follo ws: x A B C D E V xy A B C D E ∗ 12 1 2 11 14 15 1 2 11 1 2 ∗ 12 13 1 2 14 1 2 11 13 ∗ 14 1 2 15 1 2 11 11 1 2 11 1 2 ∗ 11 1 2 7 1 2 7 1 2 7 1 2 6 1 2 ∗ x A B C D E V ∗ xy A B C D E ∗ 12 1 2 12 14 15 1 2 11 1 2 ∗ 12 13 1 2 14 1 2 11 1 2 13 ∗ 14 1 2 15 1 2 11 1 2 11 1 2 11 1 2 ∗ 11 1 2 7 1 2 7 1 2 7 1 2 7 1 2 ∗ κ 1 3 2 4 5 x A C B D E M ν xy A C B D E ∗ 1 2 1 2 1 2 8 ∗ ∗ 1 3 8 ∗ ∗ ∗ 2 7 ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ x A C B D E T xy A C B D E ∗ 22 24 25 23 ∗ ∗ 25 26 23 ∗ ∗ ∗ 25 22 ∗ ∗ ∗ ∗ 18 ∗ ∗ ∗ ∗ ∗ Continuous ra ting for preferential voting , § 5 35 x A C B D E M σ xy A C B D E ∗ 1 2 1 2 1 2 8 ∗ ∗ 1 2 1 2 8 ∗ ∗ ∗ 2 7 ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ x A C B D E T σ xy A C B D E ∗ 24 1 2 24 1 2 24 1 2 22 7 8 ∗ ∗ 24 1 2 24 1 2 22 3 8 ∗ ∗ ∗ 24 1 2 21 3 8 ∗ ∗ ∗ ∗ 19 3 8 ∗ ∗ ∗ ∗ ∗ x A C B D E M π xy A C B D E ∗ 1 2 1 2 5.56 ∗ ∗ 1 2 5.56 ∗ ∗ ∗ 2 5.56 ∗ ∗ ∗ ∗ 4 ∗ ∗ ∗ ∗ ∗ x A C B D E T π xy A C B D E ∗ 24 1 2 24 1 2 24 1 2 20.94 ∗ ∗ 24 1 2 24 1 2 20.94 ∗ ∗ ∗ 24 1 2 20.94 ∗ ∗ ∗ ∗ 19 3 8 ∗ ∗ ∗ ∗ ∗ x A C B D E V π xy A C B D E ∗ 12 1 2 12 3 4 13 1 4 13 1 4 12 ∗ 12 3 4 13 1 4 13 1 4 11 3 4 11 3 4 ∗ 13 1 4 13 1 4 11 1 4 11 1 4 11 1 4 ∗ 11.69 7.69 7.69 7.69 7.69 ∗ x A C B D E r x ϕ x 3.6014 0.2315 3.6149 0.2276 3.6486 0.2181 3.7720 0.1928 4.1689 0.1299 So, the winner b y the CLC metho d is candidate A . How ever, this is true only for the main v ariant. F or the other three v ariants (dual, balanced and margin-based) the result is a tie b et ween A and C , in full agreemen t with the appro v al v oting scores. In § 17 w e will see that the margin-based v ariant alw a ys giv es suc h a full agreement. R emark . In all of the preceding examples, the matrix of the indirect scores has a constant ro w which corresp onds to the loser. How ever, it is not alwa ys so. 5 Some terminology and notation W e consider a ﬁnite set A . Its elements represent the options whic h are the matter of a vote. The n um b er of elemen ts of A will b e denoted b y N . W e will b e particularly concerned with (binary) relations on A . Stating 36 R. Camps, X. Mora, L. Sa umell that t wo elements a and b are in a certain relation ρ is equiv alent to saying that the (ordered) pair formed by these t w o elements is a member of a certain set ρ . The pair formed b y a and b , in this order, will b e denoted simply as ab . The pairs that consist of t w o copies of the same elemen t, i. e. those of the form aa , are not relev an t for our purp oses. So, w e will systematically exclude them from our relations. This can b e view ed as a sort of normal- ization. The set of all prop er pairs, i. e. the pairs ab with a 6 = b , will b e denoted as Π , or if necessary as Π ( A ) . So, we will restrict our attention to relations contained in Π (suc h relations are sometimes called “strict”, or “irreﬂexiv e”). In particular, the relation that includes the whole of Π will b e called complete tie . A relation ρ ⊆ Π will b e called : • total , or complete, when at least one of ab ∈ ρ and ba ∈ ρ holds for ev ery pair ab . • an tisymmetric when ab ∈ ρ and ba ∈ ρ cannot occur sim ultane- ously . • transitiv e when the simultaneous o ccurrence of ab ∈ ρ and bc ∈ ρ implies ac ∈ ρ . • a partial order , when it is at the same time transitiv e and an tisym- metric. • a total order , or strict ranking , when it is at the same time tran- sitiv e, an tisymmetric and total. • a complete ranking when it is at the same time transitive and total. • a truncated ranking when it consists of a complete ranking on a subset X of A together with all pairs ab with a ∈ X and b 6∈ X . F or ev ery relation ρ ⊆ Π , we will denote b y ρ 0 the relation that consists of all pairs of the form ab where ba ∈ ρ ; ρ 0 will b e called the con v erse of ρ . On the other hand, we will denote by ¯ ρ the relation that consists of all pairs ab for which ab / ∈ ρ ; ¯ ρ will b e called the complement of ρ . F or certain purposes, it will be useful to consider also the relation ˆ ρ giv en b y the complement of the con verse of ρ , or equiv alen tly b y the con verse of its complemen t. So, ab ∈ ˆ ρ if and only if ba 6∈ ρ . This relation will b e called the adjoin t of ρ . This op eration will b e used mainly in § 8, in connection with the indirect comparison relation ν = µ ( v ∗ ) . The following prop osition collects several prop erties which are immediate consequences of the deﬁnitions: Continuous ra ting for preferential voting , § 5 37 Lemma 5.1. (a) ˆ ˆ ρ = ρ . (b) ρ ⊂ σ ⇐ ⇒ ˆ σ ⊂ ˆ ρ . (c) ρ is antisymmetric ⇐ ⇒ ρ ⊆ ˆ ρ ⇐ ⇒ ˆ ρ is total . (d) ρ is total ⇐ ⇒ ˆ ρ ⊆ ρ ⇐ ⇒ ˆ ρ is antisymmetric . Besides pairs, w e will be concerned also with longer sequences a 0 a 1 . . . a n . They will b e referred to as paths , and in the case a n = a 0 they are called cycles . When a i a i +1 ∈ ρ for every i , we will say that the path a 0 a 1 . . . a n is con tained in ρ , and also that a 0 and a n are indirectly related through ρ . When ρ is transitive, the condition “ a is indirectly related to b through ρ ” implies ab ∈ ρ . In general, how ever, it deﬁnes a new relation, which is called the transitive closure of ρ , and will be denoted b y ρ ∗ ; this is the minim um transitiv e relation that con tains ρ . The transitive-closure op era- tor is easily seen to ha ve the following prop erties: ρ ∗ ⊆ σ ∗ whenev er ρ ⊆ σ ; ( ρ ∩ σ ) ∗ ⊆ ( ρ ∗ ) ∩ ( σ ∗ ) ; ( ρ ∗ ) ∪ ( σ ∗ ) ⊆ ( ρ ∪ σ ) ∗ ; ( ρ ∗ ) ∗ = ρ ∗ . On the other hand, one can easily chec k that Lemma 5.2. The tr ansitive closur e ρ ∗ is antisymmetric if and only if ρ c on- tains no cycle. Mor e sp e ciﬁc al ly, ab, ba ∈ ρ ∗ if and only if ρ c ontains a cycle that includes b oth a and b . A subset C ⊆ A will b e said to b e a cluster for a relation ρ when, for an y x 6∈ C , ha ving ax ∈ ρ for some a ∈ C implies bx ∈ ρ for any b ∈ C , and similarly , having xa ∈ ρ for some a ∈ C implies xb ∈ ρ for an y b ∈ C . On the other hand, C ⊆ A will b e said to b e an in terv al for a relation ρ when the simultaneous occurrence of ax ∈ ρ and xb ∈ ρ with a, b ∈ C implies x ∈ C . The following facts are easy consequences of the deﬁnitions: If ρ is antisymmetric and C is a cluster for ρ then C is also an interv al for ρ . If ρ is total and C is an in terv al for ρ then C is also a cluster for ρ . As a corollary , if ρ is total and an tisymmetric, then C is a cluster for ρ if and only if it is an interv al for that relation. Later on we will make use of the following fact, which is also an easy consequence of the deﬁnitions: Lemma 5.3. The fol lowing c onditions ar e e quivalent to e ach other: (a) C is a cluster for ρ . (b) C is a cluster for ˆ ρ . (c) The simultane ous o c curr enc e of ax ∈ ρ and xb ∈ ˆ ρ with a, b ∈ C implies x ∈ C , and similarly, the simultane ous o c curr enc e of ax ∈ ˆ ρ and xb ∈ ρ with a, b ∈ C implies also x ∈ C . 38 R. Camps, X. Mora, L. Sa umell When C is a cluster for ρ , it will b e useful to consider a new set e A and a new relation e ρ deﬁned in the following wa y: e A is obtained from A b y replacing the set C b y a single element e c , i. e. e A = ( A \ C ) ∪ { e c } ; for x, y ∈ A \ C , x e c ∈ e ρ if and only if there exists c ∈ C such that xc ∈ ρ , e c y ∈ e ρ if and only if there exists c ∈ C suc h that cy ∈ ρ , and ﬁnally , xy ∈ e ρ if and only if xy ∈ ρ . W e will refer to this operation as the con- traction of ρ by the cluster C . If ρ is a strict ranking (resp. a complete ranking) on A , then e ρ is a strict ranking (resp. a complete ranking) on e A . Giv en a relation ρ , w e will asso ciate ev ery element x with the follo wing sets: • the set of predecessors , P x , i. e. the set of y ∈ A such that y x ∈ ρ . • the set of successors , S x , i. e. the set of y ∈ A suc h that xy ∈ ρ . • the set of collaterals , C x , i. e. the set of y ∈ A \ { x } whic h are neither predecessors nor successors of x in ρ . The sets P x , S x and C x are esp ecially meaningful when the relation ρ is a partial order. In that case, and it is quite natural to rank the elements of A by their num b er of predecessors, or by the num b er of elemen ts which are not their successors. More precisely , it makes sense to deﬁne the rank of x in ρ as κ x = 1 + | P x | + ϑ | C x | = 1 + (1 − ϑ ) | P x | + ϑ ( N − 1 − | S x | ) , (44) where ϑ is a ﬁxed n um b er in the interv al 0 ≤ ϑ ≤ 1 . If we do not sa y otherwise, w e will tak e ϑ = 1 / 2 . The following facts are easy consequences of the deﬁnitions: Lemma 5.4. Assume that ρ is a p artial or der. In that c ase, having xy ∈ ρ implies the fol lowing facts: P x ⊂ P y , S x ⊃ S y (b oth inclusions ar e strict), and κ x < κ y (for any ϑ in the interval 0 ≤ ϑ ≤ 1 ). F or ϑ = 1 / 2 , the aver age of the numb ers κ x is e qual to ( N + 1) / 2 . If ρ is a total or der, then κ x do es not dep end on ϑ ; furthermor e, having xy ∈ ρ is then e quivalent to κ x < κ y . As in § 2.2, given a set of binary scores s xy , we denote b y µ ( s ) the corre- sp onding comparison relation : xy ∈ µ ( s ) ≡ s xy > s y x . (45) F or such a relation, the adjoint ˆ µ ( s ) corresponds to replacing the strict in- equalit y b y the non-strict one. Continuous ra ting for preferential voting , § 6 39 6 The indirect scores and its comparison relation Let us recall that the indirect scores v ∗ xy are deﬁned in the following w a y: v ∗ xy = max { v α | α is a path x 0 x 1 . . . x n from x 0 = x to x n = y } , where the score v α of a path α = x 0 x 1 . . . x n is deﬁned as v α = min { v x i x i +1 | 0 ≤ i < n } . In the following statemen ts, and the similar ones whic h app ear elsewhere, “an y x, y , z ” should b e understoo d as meaning “an y x, y , z which are pairwise diﬀeren t from each other”. R emark . The matrix of indirect scores v ∗ can be viewed as a p ow er of v (supplemen ted with v xx = 1 ) for a matrix pro duct deﬁned in the following w a y: ( v w ) xz = max y min( v xy , w y z ) . More precisely , v ∗ coincides with suc h a p o w er for any exp onent greater than or equal to N − 1 . Lemma 6.1. The indir e ct sc or es satisfy the fol lowing ine qualities: v ∗ xz ≥ min ( v ∗ xy , v ∗ y z ) for any x, y , z . (46) Pr o of. Let α b e a path from x to y suc h that v ∗ xy = v α ; let β be a path from y to z such that v ∗ y z = v β . Consider no w their concatenation α β . Since αβ go es from x to z , one has v ∗ xz ≥ v αβ . On the other hand, the deﬁnition of the score of a path ensures that v αβ = min ( v α , v β ) . Putting these things together gives the desired result. The following lemma is someho w a con v erse of the preceding one: Lemma 6.2. Assume that the original sc or es satisfy the fol lowing ine qual- ities: v xz ≥ min ( v xy , v y z ) for any x, y , z . (47) In that c ase, the indir e ct sc or es c oincide with the original ones. Pr o of. The inequality v ∗ xz ≥ v xz is an immediate consequence of the deﬁni- tion of v ∗ xz . The conv erse inequality can b e obtained in the following wa y: Let γ = x 0 x 1 x 2 . . . x n b e a path from x to z such that v ∗ xz = v γ . By virtue of (47), we hav e min  v x 0 x 1 , v x 1 x 2 , v x 2 x 3 , . . . , v x n − 1 x n  ≤ min  v x 0 x 2 , v x 2 x 3 , . . . , v x n − 1 x n  . So, v ∗ xz ≤ v γ 0 where γ 0 = x 0 x 2 . . . x n . By iteration, one even tually gets v ∗ xz ≤ v xz . 40 R. Camps, X. Mora, L. Sa umell Theorem 6.3 (Sc h ulze, 1998 [ 34 b ]) . µ ( v ∗ ) is a tr ansitive r elation. Pr o of. W e will argue b y con tradiction. Let us assume that xy ∈ µ ( v ∗ ) and y z ∈ µ ( v ∗ ) , but xz / ∈ µ ( v ∗ ) . This means resp ectively that (a) v ∗ xy > v ∗ y x and (b) v ∗ y z > v ∗ z y , but (c) v ∗ z x ≥ v ∗ xz . On the other hand, Lemma 6.1 ensures also that (d) v ∗ xz ≥ min ( v ∗ xy , v ∗ y z ) . W e will distinguish tw o cases dep ending on whic h of the t wo last quantities is smaller: (i) v ∗ y z ≥ v ∗ xy ; (ii) v ∗ xy ≥ v ∗ y z . Case (i) : v ∗ y z ≥ v ∗ xy . W e will see that in this case (c) and (d) entail a con tradiction with (a). In fact, we hav e the following c hain of inequalities: v ∗ y x ≥ min ( v ∗ y z , v ∗ z x ) ≥ min ( v ∗ y z , v ∗ xz ) ≥ min ( v ∗ y z , v ∗ xy ) = v ∗ xy , where w e are using successively: Lemma 6.1, (c), (d) and (i). Case (ii) : v ∗ xy ≥ v ∗ y z . An entirely analogous argument shows that in this case (c) and (d) entail a con tradiction with (b). In fact, we ha v e v ∗ z y ≥ min ( v ∗ z x , v ∗ xy ) ≥ min ( v ∗ xz , v ∗ xy ) ≥ min ( v ∗ y z , v ∗ xy ) = v ∗ y z , where we are using successiv ely: Lemma 6.1, (c), (d) and (ii). 7 Restricted paths In this section we consider paths restricted to either µ ( v ) or ˆ µ ( v ) . Suc h restricted paths allow to ac hieve not only the ma jorit y principle I1, but also the Condorcet principle I1 0 . In exc hange, how ever, this idea can hardly b e made in to a contin uous rating metho d, since one is doing quite diﬀeren t things dep ending on whether v xy > v y x or v xy < v y x . Even so, we will see that in the complete case —where I1 is equiv alent to I1 0 — the indirect comparison relations which are obtained under such restrictions coincide with the one whic h is obtained when arbitrary paths are used. More sp eciﬁcally , w e will lo ok at the comparison relations asso ciated with u ∗ xy and w ∗ xy , where u xy and w xy are deﬁned as u xy = ( v xy , if v xy > v y x , 0 , otherwise; w xy = ( v xy , if v xy ≥ v y x , 0 , otherwise . (48) Prop osition 7.1. (a) µ ( u ∗ ) ⊆ µ ∗ ( v ) . (b) µ ( w ∗ ) ⊆ ˆ µ ∗ ( v ) . Pr o of. Part (a). Let us b egin b y recalling that µ ∗ ( v ) means the transitiv e closure of µ ( v ) . Let us assume that xy ∈ µ ( u ∗ ) , i. e. u ∗ xy > u ∗ y x . Since we Continuous ra ting for preferential voting , § 7 41 are dealing with non-negative n um b ers, this ensures that u ∗ xy > 0 . By the deﬁnition of u ∗ xy , this implies the existence of a path x 0 x 1 . . . x n from x 0 = x to x n = y such that u x i x i +1 > 0 for all i . According to (48.1), this ensures that v x i x i +1 > v x i +1 x i , i. e. x i x i +1 ∈ µ ( v ) , for all i . Therefore, xy ∈ µ ∗ ( v ) . An entirely analogous argument pro v es part (b). Lemma 7.2. (a) u ∗ xy ≤ w ∗ xy ≤ v ∗ xy . (b) v ∗ xy > 1 / 2 = ⇒ u ∗ xy = w ∗ xy = v ∗ xy . (c) v ∗ xy = 1 / 2 = ⇒ w ∗ xy = v ∗ xy . In the c omplete c ase one has: (d) v ∗ xy < 1 / 2 = ⇒ u ∗ xy = w ∗ xy = 0 . (e) v ∗ xy = 1 / 2 = ⇒ u ∗ xy = 0 . Pr o of. Part (a). It is simply a matter of noticing that u xy ≤ w xy ≤ v xy and c hecking that the inequality p xy ≤ q xy for all x, y implies p ∗ xy ≤ q ∗ xy for all x, y . As an intermediate result tow ards this implication, one can see that p γ ≤ q γ for all paths γ . In fact, if γ = x 0 x 1 . . . x n and i is suc h that q γ = q x i x i +1 , the deﬁnition of p γ and the inequalit y b et ween p xy and q xy giv e p γ ≤ p x i x i +1 ≤ q x i x i +1 = q γ . The second step of that implication uses an analogous argumen t: if γ is a path from x to y such that p ∗ xy = p γ , w e can write p ∗ xy = p γ ≤ q γ ≤ q ∗ xy , where we are using the intermediate result and the deﬁnition of q ∗ xy . P art (b). Let γ = x 0 x 1 . . . x n b e a path from x to y such that v ∗ xy = v γ . Since v ∗ xy > 1 / 2 , every link of that path satisﬁes v x i x i +1 > 1 / 2 , which implies that v x i x i +1 > v x i +1 x i (b ecause v x i x i +1 + v x i +1 x i ≤ 1 ). No w, that inequalit y en tails that u x i x i +1 = v x i x i +1 , from which it follo ws that u γ = v γ . Finally , it suﬃces to combine these facts with the inequality u γ ≤ u ∗ xy and the inequal- ities of part (a): v ∗ xy = v γ = u γ ≤ u ∗ xy ≤ w ∗ xy ≤ v ∗ xy . P art (c). The pro of is similar to that of part (b). Here we deal with the non-strict inequalit y v x i x i +1 ≥ 1 / 2 , whic h en tails v x i x i +1 ≥ v x i +1 x i and w x i x i +1 = v x i x i +1 . These facts allow to conclude that v ∗ xy = v γ = w γ ≤ w ∗ xy ≤ v ∗ xy . P art (d). The hypothesis that v ∗ xy < 1 / 2 means that for ev ery path γ = x 0 x 1 . . . x n from x to y there exists at least one i such that v x i x i +1 < 1 / 2 . By the assumption of completeness, this implies that v x i x i +1 < v x i +1 x i , so that 42 R. Camps, X. Mora, L. Sa umell u x i x i +1 = w x i x i +1 = 0 . This implies that u γ = w γ = 0 . Since γ is arbitrary , it follows that u ∗ xy = w ∗ xy = 0 . P art (e). The pro of is similar to that of part (d). Here we deal with the non-strict inequalit y v x i x i +1 ≤ 1 / 2 , which implies v x i x i +1 ≤ v x i +1 x i and u x i x i +1 = 0 . This holds for at least one link of every path γ from x to y . So, u ∗ xy = 0 . Theorem 7.3. In the c omplete c ase one has µ ( u ∗ ) = µ ( w ∗ ) = µ ( v ∗ ) . Pr o of. It suﬃces to pro v e the three following statements: v ∗ xy > v ∗ y x = ⇒ u ∗ xy > u ∗ y x and w ∗ xy > w ∗ y x (49) w ∗ xy > w ∗ y x = ⇒ v ∗ xy > v ∗ y x , (50) u ∗ xy > u ∗ y x = ⇒ v ∗ xy > v ∗ y x , (51) Pro of of (49). By combining the completeness assumption with the h y- p othesis of (49) w e can write 1 = v xy + v y x ≤ v ∗ xy + v ∗ y x < 2 v ∗ xy , so that v ∗ xy > 1 / 2 . According to part (b) of Lemma 7.2, this inequality implies that u ∗ xy = w ∗ xy = v ∗ xy . On the other hand, part (a) of the same lemma ensures that u ∗ y x ≤ w ∗ y x ≤ v ∗ y x . By combining these facts with the hypothesis of (49) w e obtain the right-hand side of it. Pro of of (50). Here we b egin by noticing that the left-hand side implies w ∗ xy > 0 , which by part (d) of Lemma 7.2 entails v ∗ xy ≥ 1 / 2 . If v ∗ y x < 1 / 2 , w e are ﬁnished. If, on the con trary , v ∗ y x ≥ 1 / 2 , then parts (a), (b) and (c) of Lemma 7.2 allow to conclude that v ∗ y x = w ∗ y x < w ∗ xy ≤ v ∗ xy . Pro of of (51). Similarly to ab o v e, the left-hand side implies u ∗ xy > 0 , whic h b y parts (d) and (e) of Lemma 7.2 entails v ∗ xy > 1 / 2 . If v ∗ y x ≤ 1 / 2 , w e are ﬁnished. If, on the contrary , v ∗ y x > 1 / 2 , then parts (a) and (b) of Lemma 7.2 allow to conclude that v ∗ y x = u ∗ y x < u ∗ xy ≤ v ∗ xy . 8 Admissible orders Let us recall that an admissible order is a total order ξ such that ν ⊆ ξ ⊆ ˆ ν . Here ν is the indirect comparison relation ν = µ ( v ∗ ) . So xy ∈ ν if and only if m ν xy = v ∗ xy − v ∗ y x > 0 , and xy ∈ ˆ ν if and only if m ν xy ≥ 0 . Lemma 8.1. Assume that ρ is an antisymmetric and tr ansitive r elation. If ρ c ontains neither xy nor y x , then ( ρ ∪ { xy } ) ∗ is also antisymmetric. Continuous ra ting for preferential voting , § 8 43 Pr o of. W e will proceed b y con tradiction. According to Lemma 5.2, if ( ρ ∪ { xy } ) ∗ w ere not antisymmetric, ρ ∪ { xy } would contain a cycle γ . On the other hand, the h yp otheses on ρ ensure, b y the same lemma, that ρ contains no cycles. Therefore, γ m ust in volv e the pair xy . By following this cycle from one o currence of the pair xy until the next o currence of x , one obtains a path from y to x which is contained in ρ . But, since ρ is transitiv e, this entails that y x ∈ ρ , which contradicts one of the hypotheses. Theorem 8.2. Given a tr ansitive antisymmetric r elation ρ on a ﬁnite set A , one c an always ﬁnd a total or der ξ such that ρ ⊆ ξ ⊆ ˆ ρ . If ρ c ontains neither xy nor y x , one c an c onstr ain ξ to include the p air xy . Pr o of. If ρ is total, it suﬃces to take ξ = ρ (notice that ˆ ρ = ρ b ecause of statemen ts (c) and (d) of Lemma 5.1). Otherwise, let us consider the relation ρ 1 = ( ρ ∪ { xy } ) ∗ , where xy is an y pair suc h that ρ contains neither xy nor y x . According to Lemma 8.1, ρ 1 is antisymmetri c. F urthermore, it is ob vious that ρ ⊂ ρ 1 . Therefore, the statemen ts (b) and (c) of Lemma 5.1 ensure that ρ ⊂ ρ 1 ⊆ ˆ ρ 1 ⊂ ˆ ρ . F rom here, one can rep eat the same pro cess with ρ 1 substituted for ρ : if ρ 1 is total w e take ξ = ρ 1 ; otherwise we consider ρ 2 = ( ρ 1 ∪ { x 1 y 1 } ) ∗ , where x 1 y 1 is an y pair suc h that ρ 1 con tains neither x 1 y 1 nor y 1 x 1 , and so on. This iteration will conclude in a ﬁnite num b er of steps since A is ﬁnite. Corollary 8.3. One c an always ﬁnd an admissible or der ξ . Pr o of. It follows from Theorem 8.2 b ecause ν = µ ( v ∗ ) is certainly antisym- metric and Theorem 6.3 ensures that it is transitive. Later on we will mak e use of the following fact: Theorem 8.4. Given a tr ansitive antisymmetric r elation ρ on a ﬁnite set A and a set C which is a cluster for ρ , one c an always ﬁnd a total or der ξ such that ρ ⊆ ξ ⊆ ˆ ρ and such that C is a cluster for ξ . Pr o of. As in the pro of of Theorem 8.2, we will progressively extend ρ until w e get a total order. Here, we will tak e care that b esides being transitive and an tisymmetric, the successiv e extensions ρ i k eep the property that C b e a cluster for ρ i . T o this eﬀect, the successiv e additions to ρ will follow a certain sp eciﬁc order, and w e will make an extensive use of the necessary and suﬃcient condition given b y Lemma 5.3. 44 R. Camps, X. Mora, L. Sa umell In a ﬁrst phase we will deal with pairs of the form cd with c, d ∈ C . Let us assume that neither cd nor dc is con tained in ρ , and let us consider ρ 1 = ( ρ ∪ { cd } ) ∗ . Besides the properties mentioned in the pro of of Theo- rem 8.2, we claim that this relation has the prop ert y that C is a cluster for ρ 1 . According to Lemma 5.3, it suﬃces to chec k that the simultaneous o ccurrence of ax ∈ ρ 1 and xb ∈ ˆ ρ 1 with a, b ∈ C implies x ∈ C , and similarly , that the simultaneous o ccurrence of ax ∈ ˆ ρ 1 and xb ∈ ρ 1 with a, b ∈ C implies also x ∈ C . So, let us assume ﬁrst that ax ∈ ρ 1 and xb ∈ ˆ ρ 1 with a, b ∈ C . Since ρ ⊂ ρ 1 , w e ha v e xb ∈ ˆ ρ (because ˆ ρ 1 ⊂ ˆ ρ ). If ax ∈ ρ , w e immediately get x ∈ C since C is kno wn to b e a cluster for ρ (Lemma 5.3). Otherwise, i. e. if ax ∈ ρ 1 \ ρ , we see that ρ 1 con tains a path of the form γ = a . . . cd . . . x . But this entails the existence of a path from d to x contained in ρ . So, by transitivit y , dx ∈ ρ . Again, this fact together with xb ∈ ˆ ρ ensures that x ∈ C since C is kno wn to b e a cluster for ρ . A similar argumen t tak es care of the case where ax ∈ ˆ ρ 1 and xb ∈ ρ 1 with a, b ∈ C . By rep eating the same pro cess we will even tually get an extension of ρ with the same prop erties plus the following one: it includes either cd or dc for any c, d ∈ C . In other words, its restriction to C is a total order. In the follo wing, this relation will b e denoted b y η . No w we will deal with pairs of the form cq or q c with c ∈ C and q 6∈ C . Let us assume that neither cq nor q c belong to η . In this case w e will pro ceed b y taking η 1 = ( η ∪ { q } ) ∗ , where  denotes the last element of C according to the total order determined by η (alternativ ely , one could take η 1 = ( η ∪ { q f } ) ∗ , where f denotes the ﬁrst elemen t of C b y η ). By so doing, w e make sure that η 1 con tains all pairs of the form z q with z ∈ C . As a consequence, C will keep the prop ert y of b eing a cluster for η 1 . In fact, let us assume, in the lines of Lemma 5.3, that ax ∈ η 1 and xb ∈ ˆ η 1 with a, b ∈ C . The h yp othesis that ax ∈ η 1 can b e divided in t wo cases, namely either ax ∈ η or ax ∈ η 1 \ η . Let us consider ﬁrst the case ax ∈ η 1 \ η . By the deﬁnition of η 1 , this means that η 1 con tains a path of the form γ = a . . . q . . . x , whose ﬁnal part shows that q x ∈ η 1 . On the other hand, w e kno w that η 1 con tains bq (since b ∈ C ). By transitivity , this entails bx ∈ η 1 and therefore xb 6∈ ˆ η 1 , in contradiction with the hypothesis that xb ∈ ˆ η 1 . So, the only p ossibilit y of having ax ∈ η 1 and xb ∈ ˆ η 1 is ax ∈ η . Besides, xb ∈ ˆ η 1 implies that xb ∈ ˆ η . So x ∈ C b ecause C is a cluster for η (Lemma 5.3). Let us assume now that ax ∈ ˆ η 1 and xb ∈ η 1 . Lik e before, the former implies ax ∈ ˆ η . Again, the h yp othesis that xb ∈ η 1 can b e divided in tw o cases, namely either xb ∈ η or xb ∈ η 1 \ η . In the ﬁrst case we hav e ax ∈ ˆ η and xb ∈ η . So x ∈ C b ecause C is a cluster for η . In the second Continuous ra ting for preferential voting , § 8 45 case we can still use the same argument since η 1 con tains a path of the form γ = x . . . q . . . b , which sho ws that x ∈ η . By rep eating the same pro cess w e will even tually get an extension of η with the same prop erties plus the following one: it includes either cq or q c for any c ∈ C and q 6∈ C . Finally , it rests to deal with any pairs of the form pq with p, q 6∈ C . Ho w ever, these pairs do not cause any problems since they do not app ear in the deﬁnition of C b eing a cluster. In practice, one can easily obtain admissible orders by suitably arranging the elemen ts of A according to their n umber of victories, ties and defeats against the others according to the indirect comparison relation ν . More precisely , it suﬃces to arrange the elements of A by non-decreasing v alues of their rank κ x in ν as deﬁned in (44). According to the the particular nature of ν and the deﬁnitions given in § 5, the sets P x , S x and C x whic h app ear in (44) are given b y P x = { y | y 6 = x, m ν xy < 0 } , (52) S x = { y | y 6 = x, m ν xy > 0 } , (53) C x = { y | y 6 = x, m ν xy = 0 } . (54) So, ranking b y κ x amoun ts to applying the Cop eland rule to the tourna- men t deﬁned b y the indirect comparison relation ν = µ ( v ∗ ) (see for instance [ 38 : p. 206–209 ]). Prop osition 8.5. Any total or dering of the elements of A by non-de cr e asing values of κ x ( ν ) is an admissible or der. This is true for any ﬁxe d value of ϑ in the interval 0 ≤ ϑ ≤ 1 . Pr o of. Let ξ b e a total order of A for whic h x 7→ κ x do es not decrease. This means that xy ∈ ξ = ⇒ κ x ≤ κ y , or equiv alen tly , κ y < κ x = ⇒ xy / ∈ ξ . F urthermore, the total c haracter of ξ allows to deriv e that κ y < κ x = ⇒ y x ∈ ξ . On the other h and, we kno w b y Theorem 6.3 that ν = µ ( v ∗ ) is transitive. As a consequence, by Lemma 5.4, xy ∈ ν implies κ x < κ y . By com bining this 46 R. Camps, X. Mora, L. Sa umell with the preceding implication (with x and y in terc hanged with eac h other), w e get that ν ⊆ ξ . In order to complete the pro of that ξ is admissible, we m ust c hec k that ξ ⊆ ˆ ν , or equiv alently , that xy 6∈ ˆ ν implies xy 6∈ ξ . This is true b ecause of the follo wing c hain of implications: xy 6∈ ˆ ν ⇐ ⇒ y x ∈ ν = ⇒ κ y < κ x = ⇒ xy 6∈ ξ , where w e used resp ectiv ely the deﬁnition of ˆ ν , Lemma 5.4, and the h yp othesis that κ x do es not decrease along ξ . In the following section w e will mak e use of the following fact: Lemma 8.6. Given two admissible or ders ξ and e ξ , one c an ﬁnd a se quenc e of admissible or ders ξ i ( i = 0 . . . n ) such that ξ 0 = ξ , ξ n = e ξ , and such that ξ i +1 diﬀers fr om ξ i only by the tr ansp osition of two c onse cutive elements. Pr o of. Given t w o total orders ρ and σ , we will denote as d ( ρ, σ ) the num b er of pairs ab such that ab ∈ ρ \ σ . Ob viously , ρ = σ if and only if d ( ρ, σ ) = 0 . F urthermore, w e will sa y that ab is a c onse cutive pair in ρ whenev er ab ∈ ρ and there is no x ∈ A such that ax, xb ∈ ρ . If all pairs ab whic h are consecutiv e in ξ b elong to e ξ , the transitivity of e ξ allo ws to derive that ξ ⊆ e ξ ; furthermore, the fact that all total orders on the ﬁnite set A ha ve the same n umber of pairs allo ws to conclude that ξ = e ξ . So, if e ξ 6 = ξ , there m ust b e some pair ab which is consecutiv e in ξ but it do es not b elong to e ξ . Since ab belongs to the admissible order ξ and ba b elongs to the admissible order e ξ , it follo ws that m ν ab = 0 . Let us take as ξ 1 the total order whic h diﬀers from ξ only b y the transp osition of the tw o consecutiv e elements a and b ; i. e. ξ 1 = ( ξ \ { ab } ) ∪ { ba } . This order is admissible since ξ is so and m ν ab = 0 . Ob viously , d ( ξ 1 , e ξ ) = d ( ξ , e ξ ) − 1 . F rom here, one can rep eat the same pro cess with ξ 1 substituted for ξ : if ξ 1 still diﬀers from e ξ w e take ξ 2 = ( ξ 1 \ { a 1 b 1 } ) ∪ { b 1 a 1 } , where a 1 b 1 is any pair whic h is consecutiv e in ξ 1 but it do es not b elong to e ξ , and so on. This iteration will conclude in a n um b er of steps equal to d ( ξ , e ξ ) , since d ( ξ i , e ξ ) decreases by one unit in eac h step. 9 The pro jection Let us recall that our rating metho d is based up on certain pro jected scores v π xy . These quan tities (or equiv alen tly , the pro jected margins m π xy = v π xy − v π y x and the pro jected turno v ers t π xy = v π xy + v π y x ) are w ork ed out by means of Continuous ra ting for preferential voting , § 9 47 the procedure (20–26) of page 18. Its starting point are the indirect mar- gins m ν xy = v ∗ xy − v ∗ xy and the original turno vers t xy = v xy + v y x . F rom these quan tities, equations (21.1) and (21.2), used in this order, determine what w e called the in termediate pro jected margins and turnov ers, m σ xy and t σ xy . After their construction, one b ecomes in terested only in their sup erdiagonal elemen ts m σ xx 0 and t σ xx 0 . In fact, these quantities are combined in to certain in terv als γ xx 0 whose unions give rise to the whole set of pro jected scores. Let us recall in more detail the meaning of the op erator Ψ which appears in step (21.2). This op erator pro duces the in termediate pro jected turno vers ( t σ xy ) as a function of the original turno v ers ( t xy ) and the sup erdiagonal in termediate pro jected margins ( m σ pp 0 ) . Here we are using paren theses to emphasize that we are dealing with the whole collection of turnov ers and the whole collection of sup erdiagonal in termediate pro jected margins. Specif- ically , ( t σ xy ) is found b y imp osing certain conditions, namely (27–29), and minimizing the function (31), which is nothing else than the euclidean dis- tance to ( t xy ) . Equiv alen tly , w e can think in the following wa y (where the pair xy is not restricted to b elong to ξ ): we consider a candidate ( τ xy ) which v aries o v er the set T which is determined b y the following conditions: τ y x = τ xy , (55) m σ xx 0 ≤ τ xx 0 ≤ 1; (56) 0 ≤ τ xy − τ xy 0 ≤ m σ y y 0 , (57) w e asso ciate each candidate ( τ xy ) with its euclidean distance from ( t xy ) ; ﬁnally , w e deﬁne ( t σ xy ) as the only v alue of ( τ xy ) which minimizes suc h a distance. The minimizer exists and it is unique as a consequence of the fact that T is a closed conv ex set [ 17 : c h. I, § 2 ]. In this connection, one can sa y that ( t σ xy ) is the orthogonal pro jection of ( t xy ) on to the conv ex set T . The pro cedure (20–26) pro duces the pro jected scores as the end p oin ts of the interv als (23) γ xy = [ { γ pp 0 | x  − ξ p  ξ y } , where (22) γ xx 0 = [ ( t σ xx 0 − m σ xx 0 ) / 2 , ( t σ xx 0 + m σ xx 0 ) / 2 ] . The desired properties of the pro jected scores and the asso ciated margins and turno v ers will b e based upon the follo wing prop erties of the interv als γ xy , where we recall that | γ | means the length of an in terv al, and • γ means its barycen tre, or centroid, i. e. the num b er ( a + b ) / 2 if γ = [ a, b ] . 48 R. Camps, X. Mora, L. Sa umell Lemma 9.1. The sets γ xy have the fol lowing pr op erties for x  ξ y  ξ z : (a) γ xy is a close d interval. (b) γ xy ⊆ [0 , 1] . (c) γ xz = γ xy ∪ γ y z . (d) γ xy ∩ γ y z 6 = ∅ . (e) | γ xz | ≥ max ( | γ xy | , | γ y z | ) . (f ) • γ xy ≥ • γ xz ≥ • γ y z . (g) | γ xz | / • γ xz ≥ max ( | γ xy | / • γ xy , | γ y z | / • γ y z ) . Pr o of. Let us start by recalling that the sup erdiagonal intermediate turnov ers and margins are ensured to satisfy the following inequalities: 0 ≤ m σ xx 0 ≤ t σ xx 0 ≤ 1 (58) (30) 0 ≤ t σ xx 0 − t σ x 0 x 00 ≤ m σ xx 0 + m σ x 0 x 00 . (59) F rom (58) it follows that 0 ≤ ( t σ xx 0 − m σ xx 0 ) / 2 ≤ ( t σ xx 0 + m σ xx 0 ) / 2 ≤ 1 . So, ev ery γ xx 0 is an in terv al (p ossibly reduced to one p oin t) and this interv al is contained in [0 , 1] . Also, the inequalities of (59) ensure on the one hand that • γ xx 0 ≥ • γ x 0 x 00 , and on the other hand that the in terv als γ xx 0 and γ x 0 x 00 o v erlap each other. In the follo wing w e will see that these facts ab out the elemen tary in terv als γ xx 0 en tail the stated prop erties of the sets γ xy deﬁned b y (59). P art (a). This is an obvious consequence of the fact that γ pp 0 and γ p 0 p 00 o v erlap eac h other. P art (b). This follo ws from the fact that γ pp 0 ⊆ [0 , 1] . P art (c). This is a consequence of the asso ciativ e property enj oy ed b y the set-union op eration. P art (d). This is again an obvious consequence of the fact that γ pp 0 and γ p 0 p 00 o v erlap eac h other (tak e p 0 = y ). P art (e). This follows from (c) b ecause γ ⊆ η implies | γ | ≤ | η | . P art (f ). This follo ws from the fact that • γ pp 0 ≥ • γ p 0 p 00 b ecause of the fol- lo wing general fact: If γ and η are tw o in terv als with • γ ≥ • η then • γ ≥ ( γ ∪ η ) • ≥ • η . This is clear if γ and η are disjoint and also if one of them is contained in the other. Otherwise, γ \ η and η \ γ are nonempty in terv als and the preceding disjoint case allo ws to pro ceed in the following w a y: • γ ≥ ( γ ∪ ( η \ γ )) • = ( γ ∪ η ) • = (( γ \ η ) ∪ η ) • ≥ • η . P art (g). This follows from (c) and (d) b ecause of the following general fact: If γ and η are tw o closed interv als with γ ⊆ η ⊂ [0 , + ∞ ) then | γ | / • γ ≤ Continuous ra ting for preferential voting , § 9 49 | η | / • η . In fact, let γ = [ a, b ] and η = [ c, d ] . The hypothesis that γ ⊆ η takes then the following form : c ≤ a and b ≤ d . On the other hand, the claim that | γ | / • γ ≤ | η | / • η takes the follo wing form: ( b − a ) / ( b + a ) ≤ ( d − c ) / ( d + c ) . An elemen tary computation sho ws that the latter is equiv alent to bc ≤ ad , whic h is a consequence of the preceding inequalities. The pro jection pro cedure mak es use of a particular admissible order ξ . In fact, this order o ccurs in equations (21–23), as w ell as in conditions (27– 29). In spite of this, the next theorem claims that the ﬁnal results are indep enden t of ξ . The pro of is not diﬃcult, but it is rather long. Theorem 9.2. The pr oje cte d sc or es do not dep end on the admissible or der ξ use d for their c alculation, i. e. the value of v π xy is indep endent of ξ for every xy ∈ Π . On the other hand, the matrix of the pr oje cte d sc or es in an admissible or der ξ is also inde p endent of ξ ; i. e. if x i denotes the element of r ank i in ξ , the value of v π x i x j is indep endent of ξ for every p air of indic es i, j . R emark . The t wo statemen ts say diﬀeren t things since the identit y of x i and x j ma y dep end on the admissible order ξ . Pr o of. F or the purp oses of this pro of it b ecomes necessary to c hange our set-up in a certain wa y . In fact, un til now the intermediate ob jects m σ xy , t σ xy and γ xy w ere considered only for x  ξ y , i. e. xy ∈ ξ . How ev er, since w e ha v e to deal with changing the admissible order ξ , here we will allo w their argumen t xy to b e an y pair (of diﬀerent elemen ts), no matter whether it b elongs to ξ or not. In this connection, we will certainly put m σ y x = − m σ xy and t σ y x = t σ xy . On the other hand, concerning γ xy and γ y x , w e will pro ceed in the follo wing w ay: if γ xy = [ a, b ] then γ y x = [ b, a ] . So, generally speaking the γ xy are here “orien ted in terv als”, i. e. ordered pairs of real num b ers. How ever, γ xy will alwa ys be “p ositiv ely orien ted” when xy b elongs to an admissible order (but it will b e reduced to a p oint whenev er there is another admissible order which includes y x ). In particular, the γ pp 0 whic h are combined in (23) are alwa ys p ositively orien ted interv als; so, the union op eration p erformed in that equation can alwa ys b e understo o d in the usual sense. In the following, γ ’ denotes the oriented interv al “reverse” to γ , i. e. γ ’ = [ b, a ] if γ = [ a, b ] . So, let us consider the eﬀect of replacing ξ b y another admissible order e ξ . In the following, the tilde is systematically used to distinguish b etw een hom- ologous ob jects whic h are asso ciated resp ectiv ely with ξ and e ξ ; in particular, suc h a notation will b e used in connection with the lab els of the equations whic h are formulated in terms of the assumed admissible order. 50 R. Camps, X. Mora, L. Sa umell With this terminology , we will prov e the tw o following equalities. First, γ xy = e γ xy , for any pair xy ( x 6 = y ) , (60) where γ xy are the interv als produced by (21–23) together with the op era- tion γ y x = γ ’ xy , and e γ xy are those pro duced by ( e 21 – e 23 ) together with the op eration e γ y x = e γ ’ xy . Secondly , w e will see also that γ x i x j = e γ ˜ x i ˜ x j , for any pair of indices ij ( i 6 = j ) , (61) where x i denotes the element of rank i in ξ , and analogously for ˜ x i in e ξ . These equalities contain the statemen ts of the theorem since the pro jected scores are nothing else than the end p oints of the γ in terv als. No w, b y Lemma 8.6, it suﬃces to deal with the case of t wo admissible orders ξ and e ξ which diﬀer from each other b y one in version only . So, w e will assume that there are t w o elemen ts a and b suc h that the only diﬀerence b et w een ξ and e ξ is that ξ contains ab whereas e ξ con tains ba . According to the deﬁnition of an admissible order, this implies that m ν ab = m ν ba = 0 . In order to control the eﬀect of the diﬀerences b et w een ξ and e ξ , w e will mak e use of the following notation: P and p will denote resp ectiv ely the set of predecessors of a in ξ and its low est element, i. e. the immediate predecessor of a in ξ ; in this connection, any statement ab out p will b e understo o d to imply the assumption that P is not empt y . Similarly , Q and q will denote resp ectiv ely the set of successors of b in ξ and its top element, i. e. the immediate successor of b in ξ ; here to o, any statemen t ab out q will b e understo o d to imply the assumption that Q is not empty . So, ξ and e ξ con tain resp ectiv ely the paths pabq and pbaq . Let us lo ok ﬁrst at the superdiagonal intermediate pro jected margins m σ hh 0 . According to (21.1), m σ hh 0 is the minim um of a certain set of v alues of m ν xy . In a table where x and y are ordered according to ξ , this set is an upp er- righ t rectangle with lo w er-left v ertex at hh 0 . Using e ξ instead of ξ amoun ts to in terchanging tw o consecutive columns and the corresp onding ro ws of that table, namely those lab eled by a and b . In spite of suc h a rearrangement, in all cases but one the underlying set from which the minim um is taken is exactly the same, so the mininum is the same. The only case where the underlying set is not the same o ccurs for h = a in the order ξ , or h = b in the order e ξ ; but then the minimum is still the same b ecause the underlying set includes m ν ab = m ν ba = 0 . So, m σ x i x i +1 = e m σ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 . (62) Continuous ra ting for preferential voting , § 9 51 In more sp eciﬁc terms, w e ha ve m σ xx 0 = e m σ xx 0 , whenev er x 6 = p, a, b, (63) m σ pa = e m σ pb , (64) m σ ab = e m σ ba = 0 , (65) m σ bq = e m σ aq . (66) In connection with equation (63) it should b e clear that for x 6 = p, a, b the imme diate suc c essor x 0 is the same in b oth or ders ξ and e ξ . Next we will see that the intermediate pro jected turnov ers t σ xy are inv ari- an t with resp ect to ξ : t σ xy = e t σ xy , for any pair xy ( x 6 = y ) , (67) where t σ xy are the num b ers pro duced by (21.2) together with the symmetry t σ y x = t σ xy , and e t σ xy are those pro duced b y ( e 21 .2) together with the symmetry e t σ y x = e t σ xy . W e will pro ve (67) by seeing that the set T determined b y conditions (55,56,57) coincides exactly with the set e T determined by (55, e 56 , e 57 ). In other w ords, conditions (56–57) are exactly equiv alen t to ( e 56 – e 57 ) under condition (55), which do es not dep end on ξ . In order to pro v e this equiv alence w e b egin b y noticing that condition (56) coincides exactly with ( e 56 ) when x 6 = p, a, b . This is true b ecause, on the one hand, x 0 is then the same in both orders ξ and e ξ , and, on the other hand, (63) ensures that the righ t-hand sides ha v e the same v alue. Similarly happ ens with conditions (57) and ( e 57 ) when y 6 = p, a, b . So, it remains to deal with conditions (56) and ( e 56 ) for x = p, a, b , and with conditions (57) and ( e 57 ) for y = p, a, b . No w, on account of the symmetry (55), one easily sees that condition (56) with x = a is equiv alen t to ( e 56 ) with x = b . In fact, b oth of them reduce to 0 ≤ τ ab ≤ 1 since m σ ab = e m σ ba = 0 , as it w as obtained in (65). This last equalit y ensures also the equiv alence b etw een condition (57) with y = a and condition ( e 57 ) with y = b . In this case b oth of them reduce to τ xa = τ xb . (68) This common equality plays a central role in the equiv alence b et w een the re- maining conditions. Th us, its combination with (66) ensures the equiv alence b et w een (56) with x = b and ( e 56 ) with x = a , as w ell as the equiv alence b et w een (57) with y = b and ( e 57 ) with y = a when x 6 = a, b . On the other hand, its com bination with (64) ensures the equiv alence b etw een (56) 52 R. Camps, X. Mora, L. Sa umell and ( e 56 ) when x = p , as well as the equiv alence b et w een (57) and ( e 57 ) when y = p and x 6 = a, b . Finally , we hav e the tw o following equiv alences: (57) with y = p and x = b is equiv alent to ( e 57 ) with y = p and x = a b ecause of the same equality (68) together with (64) and the symmetry (55); and similarly , (57) with y = b and x = a is equiv alent to ( e 57 ) with y = a and x = b b ecause of (68) together with (66) and (55). This completes the pro of of (67). Ha ving seen that condition (68) is included in b oth (57) and ( e 57 ), it follo ws that the intermediate pro jected turnov ers satisfy t σ xa = t σ xb , e t σ xa = e t σ xb . (69) By taking x = p, q and using also (67), it follows that t σ xx 0 = e t σ xx 0 , whenev er x 6 = p, a, b, (70) t σ pa = e t σ pb , (71) t σ ab = e t σ ba , (72) t σ bq = e t σ aq . (73) In other words, the sup erdiagonal intermediate turnov ers satisfy t σ x i x i +1 = e t σ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 . (74) On account of the deﬁnition of γ x i x i +1 and e γ ˜ x i ˜ x i +1 , the com bination of (62) and (74) results in γ x i x i +1 = e γ ˜ x i ˜ x i +1 , for any i = 1 , 2 , . . . N − 1 , (75) from which the union op eration (23) pro duces (61). Finally , let us see that (60) holds to o. T o this eﬀect, we begin by noticing that (65) together with (72) are saying not only that γ ab = e γ ba but also that this interv al reduces to a p oin t. As a consequence, w e ha ve γ ba = γ ab = e γ ba = e γ ab . (76) Let us consider no w the equation γ pa = e γ pb , which is contained in (75). Since γ ab reduces to a p oint, the ov erlapping prop erty γ pa ∩ γ ab 6 = ∅ (part (d) of Lemma 9.1) reduces to γ ab ⊆ γ pa . Therefore, γ pb = γ pa ∪ γ ab = γ pa (where w e used part (c) of Lemma 9.1). Analogously , e γ pa = e γ pb ∪ e γ ba = e γ pb . Altogether, this gives γ pb = γ pa = e γ pb = e γ pa . (77) Continuous ra ting for preferential voting , § 9 53 By means of an analogous argumen t, one obtains also that γ aq = γ bq = e γ aq = e γ bq . (78) On the other hand, (75) ensures that γ xx 0 = e γ xx 0 , whenev er x 6 = p, a, b . (79) Finally , part (c) of Lemma 9.1 allows to go from (76–79) to the desired general equalit y (60). Theorem 9.3. The pr oje cte d sc or es and their assso ciate d mar gins and turn- overs satisfy the fol lowing pr op erties with r esp e ct to any admissible or der ξ : (a) The fol lowing ine qualities hold whenever x  ξ y : v π xy ≥ v π y x m π xy ≥ 0 , (80) v π xz ≥ v π y z , v π z x ≤ v π z y , (81) m π xz ≥ m π y z , m π z x ≤ m π z y , (82) t π xz ≥ t π y z , t π z x ≥ t π z y , (83) m π xz /t π xz ≥ m π y z /t π y z , m π z x /t π z x ≤ m π z y /t π z y . (84) (b) If v π xy = v π y x , or e quivalently m π xy = 0 , then (81–84) ar e satisﬁe d al l of them with an e quality sign. (c) In the c omplete c ase, the pr oje cte d mar gins satisfy the fol lowing pr op erty: m π xz = max ( m π xy , m π y z ) , whenever x  ξ y  ξ z . (85) Pr o of. W e will see that these prop erties deriv e from those satisﬁed b y the γ interv als, whic h are collected in Lemma 9.1. F or the deriv ation one has to b ear in mind that v π xy and v π y x are resp ectively the right and left end p oin ts of the interv al γ xy , and that m π xy = − m π y x and t π xy = t π y x are resp ectively the width and twice the barycen tre of γ xy . P art (a). Let us b egin by noticing that (82) will b e an immediate con- sequence of (81), since m π xz = v π xz − v π z x and m π y z = v π y z − v π z y . On the other hand, (83.2) is equiv alen t to (83.1) and (84.2) is equiv alent to (84.1). These equiv alences hold b ecause the turnov ers and margins are resp ectively sym- metric and an tisymmetric. Now, (80) holds as so on as γ xy is an in terv al, as it is ensured b y part (a) of Lemma 9.1. So, it remains to prov e the inequali- ties (81), (83.1) and (84.1). In order to pro ve them we will distinguish three cases, namely: (i) x  ξ y  ξ z ; (ii) z  ξ x  ξ y ; (iii) x  ξ z  ξ y . 54 R. Camps, X. Mora, L. Sa umell Case (i) : By part (c) of Lemma 9.1, in this case w e hav e γ xz ⊇ γ y z . This immediately implies (81) b ecause [ a, b ] ⊇ [ c, d ] is equiv alent to saying that b ≥ d and a ≤ c . On the other hand, the inequalities (83.1) and (84.1) are con tained in parts (f ) and (g) of Lemma 9.1. Case (ii) is analogous to case (i). Case (iii) : In this case, (81) follows from part (d) of Lemma 9.1 since [ a, b ] ∩ [ c, d ] 6 = ∅ is equiv alent to sa ying that b ≥ c and a ≤ d . On the other hand, (83.1) is still con tained in part (f ) of Lemma 9.1 (b ecause of the symmetric c haracter of the turnov ers), and (84.1) holds since m π xz ≥ 0 ≥ m π y z . P art (b). The hypothesis that v π xy = v π y x is equiv alent to sa ying that γ xy reduces to a p oin t, i. e. γ xy = [ v , v ] for some v . The claimed equalities will b e obtained by showing that in these circumstances one has γ xz = γ y z . W e will distinguish the same three cases as in part (a). Case (i) : On account of the o v erlapping prop ert y γ xy ∩ γ y z 6 = ∅ (part (d) of Lemma 9.1), the one-p oint interv al γ xy = [ v , v ] m ust b e contained in γ y z . So, γ xz = γ xy ∪ γ y z = γ y z (where we used part (c) of Lemma 9.1). Case (ii) is again analogous to case (i). Case (iii) : By part (c) of Lemma 9.1 (with y and z in terchanged with eac h other), the fact that γ xy reduces to the one-p oin t in terv al [ v , v ] implies that b oth γ xz and γ z y reduce also to this one-p oin t in terv al P art (c). In the complete case the intermediate pro jected turnov ers are all of them equal to 1 , so the interv als γ pp 0 and γ xy are all of the cen tred at 1 / 2 . In these circumstances, (85) is exactly equiv alent to part (c) of Lemma 9.1. The follo wing prop ositions iden tify certain situations where the preceding pro jection reduces to the iden tit y . Prop osition 9.4. In the c ase of plumping votes the pr oje cte d sc or es c oin- cide with the original ones. Pr o of. Let us begin by recalling that in the case of plumping v otes the binary scores ha v e the form v xy = f x for every y 6 = x , where f x is the fraction of voters who c ho ose x . This implies that v ∗ xy = v xy = f x . In fact, any path γ from x to y starts with a link of the form xp , whose asso ciated score is v xp = f x . So v γ ≤ f x and therefore v ∗ xy ≤ f x . But on the other hand f x = v xy ≤ v ∗ xy . Consequen tly , w e get m ν xy = v ∗ xy − v ∗ y x = v xy − v y x = f x − f y , and the admissible orders are those for whic h the f x are non-increasing. Owing to this non-increasing c haracter, the in termediate pro jected margins are m σ xx 0 = m xx 0 = f x − f x 0 . On the other hand, the in termediate pro jected turno v ers are t σ xy = t xy = f x + f y . In fact these Continuous ra ting for preferential voting , § 9 55 n um b ers are easily seen to satisfy conditions (27–29) and they ob viously minimize (31). As a consequence, γ xx 0 = [ f x 0 , f x ] . In particular, the interv als γ xx 0 and γ x 0 x 00 are adjacen t to eac h other (the righ t end of the latter coincides with the left end of the former). This fact entails that γ xy = [ f y , f x ] whenever x  ξ y . Finally , the pro jected scores are the end p oints of these interv als, namely v π xy = f x = v xy and v π y x = f y = v y x . Prop osition 9.5. Assume that the votes ar e c omplete. Assume also that ther e exists a total or der ξ such that µ ( v ) ⊆ ξ ⊆ ˆ µ ( v ) and such that the original mar gins satisfy m xz = max ( m xy , m y z ) , whenever x  ξ y  ξ z in ξ . (86) In that c ase, the pr oje cte d sc or es c oincide with the original ones. Besides, c ondition (86) holds also for any other total or der e ξ which satisﬁes µ ( v ) ⊆ e ξ ⊆ ˆ µ ( v ) . R emark . The hypothesis that µ ( v ) ⊆ ξ ⊆ ˆ µ ( v ) is not the one which deﬁnes an admissible order, namely µ ( v ∗ ) ⊆ ξ ⊆ ˆ µ ( v ∗ ) . Ho wev er, in the course of the pro of w e will see that v ∗ xy = v xy . So, ξ will b e after all an admissible order. Pr o of. Since w e are in the complete case, the scores v xy and the margins m xy are related to each other b y the monotone increasing transformation v xy = (1 + m xy ) / 2 . Therefore, condition (86) on the margins is equiv alent to the following one on the scores: v xz = max ( v xy , v y z ) , whenev er x  ξ y  ξ z in ξ . (87) On the other hand, since v xy + v y x = 1 , the preceding condition is also equiv alen t to the following one: v z x = min ( v z y , v y x ) , whenev er x  ξ y  ξ z in ξ . (88) In fact, v z x = 1 − v xz = 1 − max ( v xy , v y z ) = min ( v z y , v y x ) . No w, w e claim that these prop erties imply the following one: v xz ≥ min( v xy , v y z ) , for any x, y , z . (89) In order to prov e (89) we will distinguish four cases dep ending on whether or not do xy and y z b elong to ξ : (a) If xy , y z ∈ ξ , then (89) is an immediate consequence of (87). (b) Similarly , if xy , y z 6∈ ξ , then (89) is an immediate 56 R. Camps, X. Mora, L. Sa umell consequence of (88) with x and z interc hanged with each other. (c) Consider no w the case where xy 6∈ ξ and y z ∈ ξ . In this case w e ha ve v xy ≤ 1 / 2 ≤ v y z , so min( v xy , v y z ) = v xy . No w w e must distinguish t wo sub cases: If xz ∈ ξ , then v xy ≤ 1 / 2 ≤ v xz , so we get (89). If, on the con trary , z x ∈ ξ , then (88) applied to y  ξ z  ξ x gives v xy = min ( v xz , v z y ) ≤ v xz as claimed. (d) Finally , the case where xy ∈ ξ and y z 6∈ ξ is analogous to the preceding one. No w we in vok e Lemma 6.2, according to whic h (89) implies that v ∗ xy = v xy . In particular, ξ is ensured to b e an admissible order. Let us consider an y pair xy con tained in ξ . By applying condition (86) we see that m σ xy = m ν xy = m xy . On the other hand, since the v otes are complete w e ha v e t σ xy = 1 . So, the in terv als γ pp 0 and their unions are all of them cen tred at 1 / 2 . In this case, the union op eration of (23) is equiv alent to a maximum op eration p erformed up on the margins. On accoun t of (86), this implies that m π xy = m xy . Since we also hav e t π xy = 1 = t xy , it follo ws that v π xy = v xy and v π y x = v y x . Ha ving prov ed that v xy = v ∗ xy = v π xy , and taking in to account that this en tails m xy = m π xy , one easily sees that condition (86) holds also for any other total order e ξ such that µ ( v ) ⊆ e ξ ⊆ ˆ µ ( v ) . In fact, suc h an order is an admissible one, since µ ( v ∗ ) = µ ( v ) , and that condition is guaran teed b y part (c) of Theorem 9.3. 10 The rank-lik e rates Let us recall that the rank-lik e rates r x are given by the form ula (8) r x = N − X y 6 = x v π xy . (90) where v π xy are the pro jected scores. In the sp ecial case of complete v otes, where v π xy + v π y x = 1 , the preceding form ula is equiv alen t to the following one: (9) r x = ( N + 1 − X y 6 = x m π xy ) / 2 . (91) Let us remark also that in this sp ecial case the rank-like rates ha v e the prop ert y that X x ∈ A r x = N ( N + 1) / 2 . (92) Continuous ra ting for preferential voting , § 10 57 In view of form ula (90), the prop erties of the pro jected scores obtained in Theorem 9.3 imply the follo wing facts: Lemma 10.1. (a) If x  ξ y in an admissible or der ξ , then r x ≤ r y . (b) r x = r y if and only if v π xy = v π y x , i. e. m π xy = 0 . (c) The ine qualities (80 – 84) ar e satisﬁe d whenever r x ≤ r y . In p articular, v π xy > v π y x implies r x < r y . Pr o of. Part (a). It is an immediate consequence of the preceding formula together with the inequalities (80) and (81.1) ensured by Theorem 9.3. P art (b). According to the form ula ab o v e, r y − r x = ( v π xy − v π y x ) + X z 6 = x z 6 = y ( v π xz − v π y z ) . (93) Let ξ b e an admissible order. By symmetry we can assume xy ∈ ξ . As a consequence, Theorem 9.3 ensures that the terms of (93) which appear in paren theses are all of them greater than or equal to zero. So the only p os- sibilit y for their sum to v anish is that each of them v anishes separately , i. e. v π xy = v π y x and v π xz − v π y z for any z 6∈ { x, y } . Finally , part (b) of The- orem 9.3 ensures that all of these equalities hold as so on as the ﬁrst one is satisﬁed. P art (c). It suﬃces to use the contrapositive of (a) in the case of a strict inequalit y and (b) together with Theorem 9.3.(b) in the case of an equality . Theorem 10.2. The r ank-like r ating given by (90) is r elate d to the indir e ct c omp arison r elation ν = µ ( v ∗ ) in the fol lowing way: (a) xy ∈ ˆ ν ⇒ r x ≤ r y . (b) r x < r y ⇒ xy ∈ ν . (c) If ν c ontains a set of the form X × Y with X ∪ Y = A , then r x < r y for any x ∈ X and y ∈ Y . (d) If ν is total, i. e. ˆ ν = ν , then xy ∈ ν ⇔ r x < r y . Pr o of. Part (a). Let us b egin by noticing that xy ∈ ν implies r x ≤ r y . This follo ws from part (a) of Lemma 10.1 since ν is included in an y admissible ordering ξ . Consider now the case xy ∈ ˆ ν \ ν . This is equiv alen t to saying that ν con tains neither xy nor y x . No w, in this case Theorem 8.2 ensures 58 R. Camps, X. Mora, L. Sa umell the existence of an admissible order which con tains suc h a pair xy . So, using again the preceding prop osition, w e are still ensured that r x ≤ r y . P art (b). It reduces to the the contrapositive of (a). P art (c). Let x ∈ X and y ∈ Y . Since X × Y ⊂ ν , part (a) ensures that r x ≤ r y . So, it suﬃces to exclude the p ossibilit y that r x = r y . This will b e done b y showing that this equalit y leads to a contradiction. By part (b) of Lemma 10.1, that equalit y implies v π xy = v π y x , or equiv alen tly , m π xy = 0 . But according to (22–24), this means that m σ hh 0 = 0 for all h suc h that x  − ξ h  ξ y . Here w e are making use of an admissible order ξ . In particular we hav e m σ `` 0 = 0 , where  denotes the lo w est elemen t of X according to ξ , and  0 is the top elemen t of Y . But this contradicts the fact that  0 ∈ X × Y ⊂ ν . P art (d). It suﬃces to show that r x < r x 0 , where x 0 denotes the item that immediately follows x in the total order ν . This follows from part (c) b y taking X = { p | p  − ξ x } and Y = { q | x 0  − ξ q } and using the transitivit y of ν . By construction, the rank-lik e rates are related to the pro jected scores in the same wa y as the av erage ranks are related to the original scores when the votes are complete rankings ( § 2.5). Therefore, if w e are in the case of complete ranking v otes and the pro jected scores coincide with the original ones, then the rank-like rates coincide with the av erage ranks: Prop osition 10.3. Assume that the votes ar e c omplete r ankings. Assume also that the Llul l matrix satisﬁes the hyp othesis of Pr op osition 9.5. In that c ase, the r ank-like r ates r x c oincide exactly with the aver age r anks ¯ r x . Pr o of. This is an immediate consequence of Prop osition 9.5. 11 Zermelo’s metho d The Llull matrix of a vote can b e view ed as corresp onding to a tournament b et w een the mem b ers of A where x and y ha ve play ed T xy matc hes (the n um b er of v oters who made a comparison b et w een x and y , even if this comparison resulted in a tie) and V xy of these matches w ere w on by x , whereas the other V y x w ere w on by y (one tied matc h will b e coun ted as half a matc h in fa v our of x plus half a matc h in fa v our of y ). F or such a scenario, Ernst Zermelo [ 41 ] devised in 1929 a rating metho d whic h turns out to b e quite suitable to con v ert our rank-lik e rates in to fraction-lik e ones. This metho d was redisco vered later on by other autors [ 4, 11 ]. Continuous ra ting for preferential voting , § 11 59 Zermelo’s method is based up on a probabilistic mo del for the outcome of a matc h b etw een tw o items x and y . This mo del assumes that suc h a matc h is won by x with probabilit y ϕ x / ( ϕ x + ϕ y ) whereas it is w on b y y with probabilit y ϕ y / ( ϕ x + ϕ y ) , where ϕ x is a non-negative parameter asso ciated with each play er x , usually referred to as its strength . If all matches are indep enden t ev ents, the probabilit y of obtaining a particular system of v alues for the scores ( V xy ) is given by P = Y { x,y }  T xy V xy   ϕ x ϕ x + ϕ y  V xy  ϕ y ϕ x + ϕ y  V yx , (94) where the pro duct runs through all unordered pairs { x, y } ⊆ A with x 6 = y . Notice that P depends only on the strength ratios; in other words, multi- plying all the strengths b y the same v alue has no eﬀect on the result. On accoun t of this, we will normalize the strengths b y requiring their sum to tak e a ﬁxed p ositive v alue f . In order to include certain extreme cases, one m ust allow for some of the strengths to v anish. Ho wev er, this may conﬂict with P being w ell deﬁned, since it could lead to indeterminacies of the t yp e 0 / 0 or 0 0 . So, one should b e careful in connection with v anishing strengths. With all this in mind, for the moment we will let the strengths v ary in the follo wing set: Q = { ϕ ∈ R A | ϕ x > 0 for all x ∈ A, X x ∈ A ϕ x = f } . (95) T ogether with this set, in the follo wing w e will consider also its closure Q , whic h includes v anishing strengths, and its b oundary ∂ Q = Q \ Q . In connection with our in terests, it is w orth noticing that Zermelo’s model can b e viewed as a sp ecial case of a more general one, prop osed in 1959 b y Rob ert Duncan Luce, which considers the outcome of making a choice out of multiple options [ 21 ]. According to Luce’s ‘c hoice axiom’, the proba- bilities of tw o diﬀerent c hoices x and y are in a ratio which do es not dep end on which other options are present. As a consequence, it follows that ev ery option x can b e asso ciated a n um b er ϕ x so that the probability of c ho os- ing x out of a set X 3 x is given b y ϕ x / ( P y ∈ X ϕ y ) . Ob viously , Zermelo’s mo del corresp onds to considering binary choices only . It is in teresting to notice that Luce’s mo del allo ws to associate every ranking with a certain probabilit y . In fact, a ranking can b e viewed as the result of ﬁrst choosing the winner out of the whole set A , then choosing the b est of the remainder, and so on. If these successive choices are assumed to b e indep enden t ev ents, then one can easily ﬁgure out the corresp onding probabilit y . Anyw ay , when 60 R. Camps, X. Mora, L. Sa umell the normalization condition P x ∈ A ϕ x = f ( ≤ 1 ) is adopted, Luce’s theory of choice allo ws to view ϕ x as the ﬁrst-c hoice probabilit y of x , and to view 1 − f as the probabilit y of abstaining from making a c hoice out of A . Let us men tion here also that the h yp othesis of indep endence which lies b ehind form ula (94) is certainly not satisﬁed by the binary comparisons which arise out of preferential voting. In order to satisfy that h yp othesis, the indi- vidual v otes should be based upon independent binary comparisons, in whic h case they could take the form of an arbitrary binary relation, as we consid- ered in § 3.3. How ever, even if the indep endence h yp othesis is not satisﬁed, w e will see that Zermelo’s metho d, which w e are ab out to discuss, has go o d prop erties for transforming our pro jected scores into fraction-lik e rates. Zermelo’s metho d corresp onds to a maxim um lik eliho o d estimate of the parameters ϕ x from a given set of actual v alues of V xy (and of T xy = V xy + V y x ). In other w ords, giv en the v alues of V xy , one lo oks for the v alues of ϕ x whic h maximize the probability P . Since V xy and T xy are now ﬁxed, this is equiv alen t to maximizing the following function of the ϕ x : F ( ϕ ) = Y { x,y } ϕ x v xy ϕ y v yx ( ϕ x + ϕ y ) t xy , (96) (recall that v xy = V xy /V and t xy = T xy /V where V is a p ositiv e constant greater than or equal to an y of the turno v ers T xy ; going from (94) to (96) in v olves taking the p o w er of exp onent 1 /V and disregarding a ﬁxed multi- plicativ e constant). The function F is certainly smo oth on Q . Besides, it is clearly bounded from ab o v e, since the probabilit y is alw ays less than or equal to 1 . How ever, generally sp eaking F needs not to achiev e a maximum in Q , b ecause this set is not compact. In the present situation, the only general fact that one can guarantee in this connection is the existence of maximizing sequences, i. e. sequences ϕ n in Q with the prop ert y that F ( ϕ n ) con v erges to the low est upp er b ound F = sup { F ( ψ ) | ψ ∈ Q } . In connection with maximizing the function F deﬁned by (96) it mak es a diﬀerence whether t w o particular items x and y satisfy or not the in- equalit y v xy > 0 , or more generally —as we will see— whether they satisfy v ∗ xy > 0 . By the deﬁnition of v ∗ xy , the last inequality deﬁnes a transitive relation —namely the transitiv e closure of the one deﬁned b y the former inequalit y—. In the follo wing we will denote this transitive relation by the sym b ol D . Thus, x D y ⇐ ⇒ v ∗ xy > 0 . (97) Asso ciated with it, it is in teresting to consider also the follo wing deriv ed rela- tions, which k eep the prop ert y of transitivity and are resp ectively symmetric Continuous ra ting for preferential voting , § 11 61 and antisymmetric: x ≡ y ⇐ ⇒ v ∗ xy > 0 and v ∗ y x > 0 , (98) x  y ⇐ ⇒ v ∗ xy > 0 and v ∗ y x = 0 . (99) Therefore, ≡ is an equiv alence relation and  is a partial order. In the follo wing, the situation where x  y will b e expressed b y sa ying that x dominates y . The equiv alence classes of A by ≡ are called the irreducible comp onen ts of A (for V ). If there is only one of them, namely A itself, then one says that the matrix V is irreducible. So, V is irreducible if and only if v ∗ xy > 0 for any x, y ∈ A . It is not diﬃcult to see that this prop ert y is equiv alen t to the following one formulated in terms of the direct scores only: there is no splitting of A in to t wo classes X and Y so that v y x = 0 for an y x ∈ X and y ∈ Y ; in other w ords, there is no ordering of A for whic h the matrix V takes the form  V X X V XY O V Y Y  , (100) where V X X and V Y Y are square matrices and O is a zero matrix. Besides, a subset X ⊆ A is an irreducible comp onen t if and only if X is maximal, in the sense of set inclusion, for the prop ert y of V X X b eing irreducible. On the other hand, it also happ ens that the relations D and  are compatible with the equiv alence relation ≡ , i. e. if x ≡ ¯ x and y ≡ ¯ y then x D y implies ¯ x D ¯ y , and analogously x  y implies ¯ x  ¯ y . As a consequence, the relations D and  can b e applied also to the irreducible comp onen ts of A for V . In the following we will b e interested in the case where V is irreducible, or more generally , when there is a top dominan t irreducible comp onent , i. e. an irreducible comp onen t which dominates any other. F rom no w on w e systematically use the notation V RS to mean the restriction of ( v xy ) to x ∈ R and y ∈ S , where R and S are arbitrary non-empty subsets of A . Similarly , ϕ R will denote the restriction of ( ϕ x ) to x ∈ R . The next theorems collect the basic results that we need ab out Zermelo’s metho d. Theorem 11.1 (Zermelo, 1929 [ 41 ]; see also [ 11, 16 ]) . If V is irr e ducible, then: (a) Ther e is a unique ϕ ∈ Q which maximizes F on Q . 62 R. Camps, X. Mora, L. Sa umell (b) ϕ is the solution of the fol lowing system of e quations: X y 6 = x t xy ϕ x ϕ x + ϕ y = X y 6 = x v xy , (101) X x ϕ x = f , (102) wher e (101) c ontains one e quation for every x . (c) ϕ is an inﬁnitely diﬀer entiable function of the sc or es v xy as long as they ke ep satisfying the hyp othesis of irr e ducibility. Pr o of. Let us b egin b y noticing that the h yp othesis of irreducibility entails that F can b e extended to a con tinuous function on Q by putting F ( ψ ) = 0 for ψ ∈ ∂ Q . In order to prov e this claim w e m ust sho w that F ( ψ n ) → 0 whenev er ψ n con v erges to a p oin t ψ ∈ ∂ Q . Let us consider the follo wing sets asso ciated with ψ : X = { x | ψ x > 0 } and Y = { y | ψ y = 0 } . The second one is not empt y since we are assuming ψ ∈ ∂ Q , whereas the ﬁrst one is not empt y b ecause the strengths add up to the p ositive v alue f . No w, for any x ∈ X and y ∈ Y , F ( ψ n ) con tains a factor of the form ( ψ n y ) v yx , which tends to zero as so on as v y x > 0 . So, the only wa y for F ( ψ n ) not to approach zero w ould b e V Y X = O , in con tradiction with the irreducibility of V . After suc h an extension, F is a con tinuous function on the compact set Q . So, there exists ϕ whic h maximizes F on Q . Ho wev er, since F ( ψ ) v anishes on ∂ Q whereas it is strictly p ositive for ψ ∈ Q , the maximizer ϕ must b elong to Q . This establishes the existence part of (a). Maximizing F is certainly equiv alen t to maximizing log F . According to Lagrange, any ϕ ∈ Q whic h maximizes log F under the condition of a ﬁxed sum is b ound to satisfy ∂ log F ( ϕ ) ∂ ϕ x = λ, (103) for some scalar λ and every x ∈ A . Now, a straigh tforw ard computation giv es ∂ log F ( ϕ ) ∂ ϕ x = X y 6 = x  v xy ϕ x − t xy ϕ x + ϕ y  . (104) On the other hand, using the fact that v xy + v y x = t xy , the preceding expres- sion is easily seen to imply that X x ∂ log F ( ϕ ) ∂ ϕ x ϕ x = 0 . (105) Continuous ra ting for preferential voting , § 11 63 In other w ords, the gradien t of log F at ϕ is orthogonal to ϕ , whic h w as foreseeable since F ( ϕ ) remains constant when ϕ is multiplied b y an arbitrary p ositiv e num b er. Notice that this is true for an y ϕ . In particular, (105) en tails that the ab o ve Lagrange m ultiplier λ is equal to zero; in fact, it suﬃces to plug (103) in (105) and to use the fact that P x ϕ x = f is p ositiv e. So, the conditions (103) reduce ﬁnally to ∂ log F ( ϕ ) ∂ ϕ x = 0 , (106) for ev ery x ∈ A , which is equiv alent to (101) on accoun t of (104) and the fact that ϕ x > 0 . So, any maximizer m ust satisfy the conditions stated in (b). Let us see no w that the maximizer is unique. Instead of following the in teresting pro of given b y Zermelo, here we will prefer to follow [ 16 ], which will ha ve the adv antage of preparing matters for part (c). More sp eciﬁcally , the uniqueness will b e obtained by seeing that an y critical p oin t of log F as a function on Q , i. e. any solution of (101–102), is a strict lo cal maxim um; this implies that there is only one critical p oint, b ecause otherwise one should ha v e other kinds of critical points [ 9 : § VI.6 ] (we are in voking the so-called moun tain pass theorem; here w e are using the fact that log F b ecomes −∞ at ∂ Q ). In order to study the c haracter of a critical p oint w e will look at the second deriv atives of log F with resp ect to ϕ . By diﬀerentiating (104), one obtains that ∂ 2 log F ( ϕ ) ∂ ϕ x 2 = − X y 6 = x  v xy ϕ 2 x − t xy ( ϕ x + ϕ y ) 2  , (107) ∂ 2 log F ( ϕ ) ∂ ϕ x ∂ ϕ y = t xy ( ϕ x + ϕ y ) 2 , for x 6 = y . (108) On the other hand, when ϕ is a critical p oin t, equation (101) transforms (107) into the following expression: ∂ 2 log F ( ϕ ) ∂ ϕ x 2 = − X y 6 = x t xy ( ϕ x + ϕ y ) 2 ϕ y ϕ x . (109) So, the Hessian bilinear form is as follows: X x,y  ∂ 2 log F ( ϕ ) ∂ ϕ x ∂ ϕ y  ψ x ψ y = − X x,y 6 = x t xy ( ϕ x + ϕ y ) 2  ϕ y ϕ x ψ 2 x − ψ x ψ y  = − X x,y 6 = x t xy ( ϕ x + ϕ y ) 2 ϕ x ϕ y  ϕ 2 y ψ 2 x − ϕ x ϕ y ψ x ψ y  = − X { x,y } t xy ( ϕ x + ϕ y ) 2 ϕ x ϕ y ( ϕ y ψ x − ϕ x ψ y ) 2 , (110) 64 R. Camps, X. Mora, L. Sa umell where the last sum runs through all unordered pairs { x, y } ⊆ A with x 6 = y . The last expression is non-p ositiv e and it v anishes if and only if ψ x /ϕ x = ψ y /ϕ y for an y x, y ∈ A (the ‘only if ’ part is immediate when t xy > 0 ; for arbitrary x and y the h yp othesis of irreducibilit y allo ws to connect them through a path x 0 x 1 . . . x n ( x 0 = x , x n = y ) with the property that t x i x i +1 ≥ v x i x i +1 > 0 for an y i , so that one gets ψ x /ϕ x = ψ x 1 /ϕ x 1 = · · · = ψ y /ϕ y ). So, the v anishing of (110) happ ens if and only if ψ = λϕ for some scalar λ . Ho w ev er, when ψ is restricted to v ariations within Q , i. e. to vec- tors in T Q = { ψ ∈ R A | P x ψ x = 0 } , the case ψ = λϕ reduces to ψ = 0 (since P x ϕ x = f is p ositiv e). So, the Hessian is negative deﬁnite on T Q . This ensures that ϕ is a strict lo cal maxim um of log F as a function on Q . In fact, one easily arriv es at such a conclusion when T a ylor’s formula is used to analyse the b ehaviour of log F ( ϕ + ψ ) for small ψ in T Q . Finally , let us consider the dep endence of ϕ ∈ Q on the matrix V . T o b egin with, we notice that the set I of irreducible matrices is op en since it is a ﬁnite intersection of op en sets, namely one op en set for eac h splitting of A into tw o sets X and Y . The dep endence of ϕ ∈ Q on V is due to the presence of v xy and t xy = v xy + v y x in the equations (101–102) whic h determine ϕ . How ever, w e are not in the standard setting of the implicit function theorem since w e are dealing with a system of N + 1 equations whilst ϕ v aries in a space of dimension N − 1 . In order to place oneself in a standard setting, it is conv enien t here to replace the condition of nor- malization P x ϕ x = f b y the alternative one ϕ a = 1 , where a is a ﬁxed elemen t of A . This c hange of normalization corresp onds to mapping Q to U = { ϕ ∈ R A | ϕ x > 0 for all x ∈ A, ϕ a = 1 } by means of the diﬀeo- morphism g : ϕ 7→ ϕ/ϕ a , whic h has the prop erty that F ( g ( ϕ )) = F ( ϕ ) . By taking as co ordinates the ϕ x with x ∈ A \ { a } =: A 0 , one easily c hec ks that the function F restricted to U —i. e. restricted to ϕ a = 1 — has the prop ert y that the matrix ( ∂ 2 log F ( ϕ ) /∂ ϕ x ∂ ϕ y | x, y ∈ A 0 ) is negativ e def- inite and therefore inv ertible, whic h en tails that the system of equations ( ∂ log F ( ϕ, V ) /∂ ϕ x = 0 | x ∈ A 0 ) determines ϕ ∈ U as a smo oth function of V ∈ I . Let us recall that a maximizing sequence means a sequence ϕ n ∈ Q suc h that F ( ϕ n ) approac hes the low est upp er b ound of F on Q . Theorem 11.2 (Statemen ts (a) and (b) are pro v ed in [ 41 ]; results related to (c) are contained in [ 7 ]) . Assume that ther e exists a top dominant irr e- ducible c omp onent X . In this c ase: (a) Ther e is a unique ϕ ∈ Q such that any maximizing se quenc e c on- ver ges to ϕ . Continuous ra ting for preferential voting , § 11 65 (b) ϕ X is the solution of a system analo gous to (101–102) wher e x and y vary only within X . ϕ A \ X = 0 . (c) ϕ is a c ontinuous function of the sc or es v xy as long as they ke ep satisfying the hyp otheses of the pr esent the or em. Pr o of. The deﬁnition of the low est upp er b ound immediately implies the existence of maximizing sequences. On the other hand, the compactness of Q guaran tees that an y maximizing sequence has a subsequence whic h con v erges in Q . Let ϕ n and ϕ denote resp ectiv ely one of such conv ergent maximizing sequences and its limit. In the follo wing w e will see that ϕ m ust be the unique point sp eciﬁed in statemen t (b). This en tails that any maximizing sequence conv erges itself to ϕ (without extracting a subsequence). So, our aim is now statemen t (b). F rom no w on we will use the follow- ing notations: a general element of Q will b e denoted by ψ ; w e will write Y = A \ X . F or con v enience, in this part of the pro of w e will replace the condition P x ψ x = f b y P x ψ x ≤ f (and similarly for ϕ n and ϕ ); since F ( λψ ) = F ( ψ ) for any λ > 0 , the properties that we will obtain will b e easily translated from b Q = { ψ ∈ R A | ψ x > 0 for all x ∈ A, P x ∈ A ψ x ≤ f } to Q . On the other hand, it will also b e conv enient to consider ﬁrst the case where Y is also an irreducible comp onent. In such a case, it is interesting to rewrite F ( ψ ) as a pro duct of three factors: F ( ψ ) = F X X ( ψ X ) F Y Y ( ψ Y ) F XY ( ψ X , ψ Y ) , (111) namely: F X X ( ψ X ) = Y { x, ¯ x }⊂ X ψ x v x ¯ x ψ ¯ x v ¯ xx ( ψ x + ψ ¯ x ) t x ¯ x , (112) F Y Y ( ψ Y ) = Y { y , ¯ y }⊂ Y ψ y v y ¯ y ψ ¯ y v ¯ yy ( ψ y + ψ ¯ y ) t y ¯ y , (113) F XY ( ψ X , ψ Y ) = Y x ∈ X y ∈ Y  ψ x ψ x + ψ y  v xy , (114) where we used that v y x = 0 and t xy = v xy . No w, let us lo ok at the eﬀect of replacing ψ Y b y λψ Y without v arying ψ X . The v alues of F X X and F Y Y remain unchanged, but that of F XY v aries in the following w ay: F XY ( ψ X , λψ Y ) F XY ( ψ X , ψ Y ) = Y x ∈ X y ∈ Y  ψ x + ψ y ψ x + λψ y  v xy . (115) 66 R. Camps, X. Mora, L. Sa umell In particular, for 0 < λ < 1 each of the factors of the righ t-hand side of (115) is greater than 1 . This remark leads to the following argumen t. First, we can see that ϕ n y /ϕ n x → 0 for any x ∈ X and y ∈ Y suc h that v xy > 0 (such pairs xy exist b ecause of the hypothesis that X dominates Y ). Otherwise, the pre- ceding remark en tails that the sequence e ϕ n = ( ϕ n X , λϕ n Y ) with 0 < λ < 1 w ould satisfy F ( e ϕ n ) > K F ( ϕ n ) for some K > 1 and inﬁnitely man y n , in contradiction with the h yp othesis that ϕ n w as a maximizing sequence. On the other hand, we see also that F XY ( ϕ n ) approaches its lo w est upp er b ound, namely 1 . Having ac hieved suc h a prop ert y , the problem of maxi- mizing F reduces to separately maximizing F X X and F Y Y , whic h is solv ed by Theorem 11.1. F or the momen t w e are dealing with relative strengths only , i. e. without an y normalizing condition like (102). So, we see that F Y Y gets optimized when each of the ratios ϕ n y /ϕ n ¯ y ( y , ¯ y ∈ Y ) approaches the homolo- gous one for the unique maximizer of F Y Y , and analogously with F X X . Since these ratios are ﬁnite p ositive quantities, the statemen t that ϕ n y /ϕ n x → 0 b ecomes extended to any x ∈ X and y ∈ Y whatso ev er (since one can write ϕ n y /ϕ n x = ( ϕ n y /ϕ n ¯ y ) × ( ϕ n ¯ y /ϕ n ¯ x ) × ( ϕ n ¯ x /ϕ n x ) with v ¯ x ¯ y > 0 ). Let us reco ver now the condition P x ∈ A ϕ n x = f . The preceding facts imply that ϕ n Y → 0 , whereas ϕ n X con v erges to the unique maximizer of F X X . This establishes (b) as well as the uniqueness part of (a). The general case where Y decomp oses into several irreducible comp o- nen ts, all of them dominated b y X , can b e taken care of b y induction o ver the diﬀeren t irreducible comp onents of A . A t eac h step, one deals with an irreducible comp onen t Z with the prop erty of b eing minimal, in the sense of the dominance relation  , among those which are still p ending. By means of an argumen t analogous to that of the preceding paragraph, one sees that: (i) ϕ n z /ϕ n x → 0 for an y z ∈ Z and x suc h that x  z with v xz > 0 ; (ii) the ratios ϕ n z /ϕ n ¯ z ( z , ¯ z ∈ Z ) approac h the homologous ones for the unique maximizer of F Z Z ; and (iii) ϕ n R is a maximizing sequence for F RR , where R denotes the union of the p ending components, Z excluded. Once this induction pro cess has b een completed, one can com bine its partial results to sho w that ϕ n z /ϕ n x → 0 for any x ∈ X and z 6∈ X (it suﬃces to consider a path x 0 x 1 . . . x n from x 0 ∈ X to x n = z with the prop ert y that v x i x i +1 > 0 for any i and to notice that each of the factors ϕ n x i +1 /ϕ n x i remains b ounded while at least one of them tends to zero). As ab o v e, one concludes that ϕ n A \ X → 0 , whereas ϕ n X con v erges to the unique maximizer of F X X . The t wo following remarks will b e useful in the proof of part (c): (1) According to the pro of ab ov e, ϕ X is determined (up to a m ultiplicativ e constan t) b y equations (101) with x and y v arying only within X : Continuous ra ting for preferential voting , § 11 67 F x ( ϕ X , V ) := X y ∈ X y 6 = x t xy ϕ x ϕ x + ϕ y − X y ∈ X y 6 = x v xy = 0 , ∀ x ∈ X . (116) Ho w ever, since y ∈ A \ X implies on the one hand ϕ y = 0 and on the other hand t xy = v xy , each of the preceding equations is equiv alen t to a similar one where y v aries o v er the whole of A \ { x } : F 0 x ( ϕ, V ) := X y ∈ A y 6 = x t xy ϕ x ϕ x + ϕ y − X y ∈ A y 6 = x v xy = 0 , ∀ x ∈ X . (117) (2) Also, it is in teresting to see the result of adding up the equations (117) for all x in some subset W of X . Using the fact that v xy + v y x = t xy , one sees that such an addition results in the following equalit y: X x ∈ W y 6∈ W t xy ϕ x ϕ x + ϕ y − X x ∈ W y 6∈ W v xy = 0 , ∀ W ⊆ X. (118) Let us pro ceed now with the pro of of (c). In the following, V and e V denote resp ectively a ﬁxed matrix satisfying the h yp otheses of the theo- rem and a slight p erturbation of it. As w e hav e done in similar o ccasions, w e systematically use a tilde to distinguish betw een homologous ob jects asso ciated resp ectiv ely with V and e V ; in particular, suc h a notation will b e used in connection with the lab els of certain equations. Our aim is to sho w that e ϕ approac hes ϕ as e V approac hes V . In this connection we will use the little-o and big-O notations made p opular by Edm und Landau (who b y the w a y is the author of a pap er on the rating of c hess pla y ers, namely [ 19 ], which inspired Zermelo’s w ork). This notation refers here to functions of e V and their b ehaviour as e V approaches V ; if f and g are tw o suc h functions, f = o ( g ) means that for every  > 0 there exists a δ > 0 suc h that k e V − V k ≤ δ implies k f ( e V ) k ≤  k g ( e V ) k ; on the other hand, f = O ( g ) means that there exist M and δ > 0 suc h that k e V − V k ≤ δ implies k f ( e V ) k ≤ M k g ( e V ) k . Ob viously , if e V is near enough to V then v xy > 0 implies e v xy > 0 . As a consequence, x D y implies x e D y . In particular, the irreducibilit y of V X X en tails that e V X X is also irreducible. Therefore, X is entirely contained in some irreducible comp onen t e X of A for e V . Besides, e X is a top dominant irreducible comp onen t for e V ; in fact, w e ha ve the following chain of impli- cations for x ∈ X ⊆ e X : y 6∈ e X ⇒ y 6∈ X ⇒ x  y ⇒ x e D y ⇒ x e  y , where we ha ve used successively: the inclusion X ⊆ e X , the hypothesis that X is top dominant for V , the fact that e V is near enough to V , and 68 R. Camps, X. Mora, L. Sa umell the hypothesis that y do es not b elong to the irreducible comp onen t e X . No w, according to part (b) and remark (1) from p. 66–67, ϕ X and e ϕ e X are determined resp ectiv ely b y the systems (116) and ( g 116 ), or equiv alently b y (117) and ( g 117 ), whereas ϕ A \ X and e ϕ A \ e X are b oth of them equal to zero. So w e must sho w that e ϕ y = o (1) for any y ∈ e X \ X , and that e ϕ x − ϕ x = o (1) for any x ∈ X . The pro of is organized in three main steps. Step (1). e ϕ y = O ( e ϕ x ) whenever v xy > 0 . F or the moment, we assume e V ﬁxed (near enough to V so that e v xy > 0 ) and x, y ∈ e X . Under these h yp otheses one can argue as follo ws: Since e ϕ e X maximizes e F e X e X , the corre- sp onding v alue of e F e X e X can b e b ounded from b elo w b y any particular v alue of the same function. On the other hand, w e can b ound it from ab ov e by the factor e ϕ x / ( e ϕ x + e ϕ y ) e v xy . So, we can write  1 2  N ( N − 1) ≤  1 2  P p,q ∈ e X e t pq = e F e X e X ( ψ ) ≤ e F e X e X ( e ϕ e X ) ≤  e ϕ x e ϕ x + e ϕ y  e v xy , (119) where ψ has b een taken so that ψ q has the same v alue for all q ∈ e X (and it v anishes for q 6∈ e X ). The preceding inequality en tails that e ϕ y ≤  2 N ( N − 1) / e v xy − 1  e ϕ x . (120) No w, this inequality holds not only for x, y ∈ e X , but it is also trivially true for y 6∈ e X , since then one has e ϕ y = 0 . On the other hand, the case y ∈ e X , x 6∈ e X is not p ossible at all, b ecause the hypothesis that e v xy > 0 w ould then con tradict the fact that e X is a top dominant irreducible comp onen t. Finally , w e let e V v ary to wards V . The desired result is a consequence of (120) since e v xy approac hes v xy > 0 . Step (2). e ϕ y = o ( e ϕ x ) for any x ∈ X and y 6∈ X . Again, w e will consider ﬁrst the sp ecial case where v xy > 0 . In this case the result is easily obtained as a consequence of the equalit y ( g 118 ) for W = X : X x ∈ X y 6∈ X e t xy e ϕ x e ϕ x + e ϕ y − X x ∈ X y 6∈ X e v xy = 0 . (121) In fact, this equality implies that X x ∈ X y 6∈ X e t xy  1 − e ϕ x e ϕ x + e ϕ y  = X x ∈ X y 6∈ X e v y x . (122) No w, it is clear that the right-hand side of this equation is o (1) and that each of the terms of the left-hand side is p ositive. Since e t xy − v xy = e t xy − t xy = Continuous ra ting for preferential voting , § 11 69 o (1) , the h yp othesis that v xy > 0 allo ws to conclude that e ϕ x / ( e ϕ x + e ϕ y ) approac hes 1 , or equiv alently , e ϕ y = o ( e ϕ x ) . Let us consider now the case of an y x ∈ X and y 6∈ X . Since X is top dominant, we know that there exists a path x 0 x 1 . . . x n from x 0 = x to x n = y suc h that v x i x i +1 > 0 for all i . According to step (1) w e ha v e e ϕ x i +1 = O ( e ϕ x i ) . On the other hand, there m ust b e some j suc h that x j ∈ X but x j +1 6∈ X , whic h has b een seen to imply that e ϕ x j +1 = o ( e ϕ x j ) . By com bining these facts one obtains the desired result. Step (3). e ϕ x − ϕ x = o (1) for any x ∈ X . Consider the equations ( g 117 ) for x ∈ X and split the sums in t wo parts dep ending on whether y ∈ X or y 6∈ X : X y ∈ X y 6 = x e t xy e ϕ x e ϕ x + e ϕ y − X y ∈ X y 6 = x e v xy = X y 6∈ X ( e v xy − e t xy e ϕ x e ϕ x + e ϕ y ) . (123) The last sum is o (1) since step (2) ensures that e ϕ y = o ( e ϕ x ) and w e also kno w that e t xy − e v xy = e v y x = o (1) (b ecause x ∈ X and y 6∈ X ). So e ϕ satisﬁes a system of the following form, where x and y v ary only within X and e w xy is a slight mo diﬁcation of e v xy whic h absorbs the right-hand side of (123): G x ( e ϕ X , e V , e W ) := X y ∈ X y 6 = x e t xy e ϕ x e ϕ x + e ϕ y − X y ∈ X y 6 = x e w xy = 0 , ∀ x ∈ X . (124) Here, the second argument of G refers to the dep endence on e V through e t xy . W e kno w that e t xy − t xy = o (1) and also that e w xy − v xy = ( e w xy − e v xy ) + ( e v xy − v xy ) = o (1) . So we are interested in the preceding equation near the p oin t ( ϕ X , V , V ) . Now in this p oin t w e hav e G ( ϕ X , V , V ) = F ( ϕ X , V ) = 0 , as w ell as ( ∂ G x /∂ e ϕ y )( ϕ X , V , V ) = ( ∂ F x /∂ ϕ y )( ϕ X , V ) . Therefore, the im- plicit function theorem can b e applied similarly as in Theorem 11.1, with the result that e ϕ X = H ( e V , e W ) , where H is a smo oth function whic h satisﬁes H ( V , V ) = ϕ X . In particular, the con tinuit y of H allo ws to conclude that e ϕ X approac hes ϕ X , since we know that b oth e V and e W approach V . Finally , b y com bining the results of steps (2) and (3) one obtains e ϕ y = o (1) for any y 6∈ X . R emarks 1. The conv ergence of ϕ n to ϕ is a necessary condition for ϕ n b eing a maximizing sequence but not a suﬃcien t one. The preceding proof sho ws that a necessary and suﬃcient condition is that the ratios ϕ n y /ϕ n z tend to 0 70 R. Camps, X. Mora, L. Sa umell whenev er y  z , whereas, if y ≡ z , i. e. if y and z b elong to the same irreducible comp onent Z , these ratios approac h the homologous ones for the unique maximizer of F Z Z . 2. If there is not a dominant comp onent then the maximizing sequences can ha v e multiple limit p oin ts. Ho w ev er, as we will see in the next section, the pro jected Llull matrices are alwa ys in the hypotheses of Theorem 11.2. 12 The fraction-lik e rates Let us recall from § 2.9 that the fraction-like rates ϕ x will be obtained b y applying Zermelo’s metho d to the pro jected Llull matrix ( v π xy ) . The next results sho w that this matrix has a very sp ecial structure in connection with irreducibility . Lemma 12.1. The pr oje cte d Llul l matrix ( v π xy ) has the fol lowing pr op erties for any admissible or der ξ ( p 0 denotes the imme diate suc c essor of p in ξ ): (a) If x  ξ y and v π y x = 0 , then v π p 0 p = 0 for some p such that x  − ξ p  ξ y . (b) If v π p 0 p = 0 for some p , then v π y x = 0 for al l x, y such that x  − ξ p  ξ y . (c) If x  ξ y and v π xy = 0 , then v π ab = 0 for al l a, b such that x  − ξ a . Pr o of. Part (a). Assume that x  ξ y . Then v π y x is the left end of the in terv al γ xy . No w, since γ xy = S { γ pp 0 | x  − ξ p  ξ y } , a v anishing left end for γ xy implies the same prop erty for some of the γ pp 0 , i. e. v π p 0 p = 0 . P art (b). According to Theorem 9.3.(a), x  − ξ p  ξ y implies the inequal- ities v π y x ≤ v π p 0 x ≤ v π p 0 p . Therefore, v π p 0 p = 0 implies v π y x = 0 . P art (c). F or x  ξ y , v π xy = 0 means that γ xy = [0 , 0] . This implies that γ pp 0 = [0 , 0] for all p such that x  − ξ p  ξ y . No w, according to Lemma 9.1, the barycentres of the interv als γ q q 0 decrease or sta y the same when q mov es to w ards the b ottom. So γ q q 0 = [0 , 0] for all q such that x  − ξ q . As a consequence, we immediately get v π ab = 0 for an y a, b such that x  − ξ a, b . F urthermore, for b  ξ x  − ξ a , part (a) of Theorem 9.3 gives the following inequalities: v π ab ≤ v π ax for a 6 = x , and v π ab ≤ v π ay for a = x , where the righ t-hand sides are already known to v anish. So v π ab v anishes also for such a and b . Prop osition 12.2. L et us assume that the pr oje cte d Llul l matrix ( v π xy ) is not the zer o matrix. L et us c onsider the set X = { x ∈ A | v π p 0 p > 0 for al l p such that p  ξ x } , (125) Continuous ra ting for preferential voting , § 12 71 wher e the right-hand side makes use of an admissible or der ξ . This set has the fol lowing pr op erties: (a) It do es not dep end on the admissible or der ξ . (b) v π xy > 0 for any x ∈ X and y ∈ A . (c) v π y x = 0 for any x ∈ X and y 6∈ X . (d) r x < r y for any x ∈ X and y 6∈ X . (e) X is the top dominant irr e ducible c omp onent of A for ( v π xy ) . Pr o of. Statement (a) will be prov ed at the end. The deﬁnition of X is equiv alen t to the following one: X = A if v π p 0 p > 0 for any p ; otherwise, X = { x ∈ A | x  − ξ h } , where h is the topmost (in ξ ) element of A whic h satisﬁes v π h 0 h = 0 . In particular, X reduces to the topmost element of A when v π p 0 p = 0 for an y p . Statemen t (b). In view of Lemma 12.1.(a), the deﬁnition of X implies that v π y x > 0 for any x, y ∈ X such that x  ξ y . This statement is empty when X reduces to a single element a , but then w e will mak e use of the fact that v π aa 0 > 0 , which is b ound to happ en b ecause otherwise Lemma 12.1.(c) w ould en tail that the whole matrix is zero, against our h yp othesis. These facts imply statement (b) b y virtue of Theorem 9.3.(a). Statemen t (c). If X = A there is nothing to prov e. Otherwise, if h is the ab o v e-mentioned topmost elemen t of A which satisﬁes v π h 0 h = 0 , then Lemma 12.1.(b) ensures that v π y x = 0 for any x, y such that x  − ξ h  ξ y , i. e. any x ∈ X and y 6∈ X . Statemen t (d). If X = A there is nothing to prov e. Otherwise, the result follo ws from parts (b) and (c) together with Lemma 10.1.(c). Statemen t (e). This is an immediate consequence of (b) and (c). Statemen t (a). A top dominan t irreducible comp onent is alwa ys unique b ecause the relation of dominance b etw een irreducible comp onents is an ti- symmetric. R emarks 1. In the complete case, the a v erage ranks ¯ r x deﬁned by equation (6) are easily seen to satisfy already a prop erty of the same kind as (d): if X and Y are tw o irreducible comp onents of ( v xy ) suc h that X dominates Y , then ¯ r x < ¯ r y for all x ∈ X and y ∈ Y [ 25 : Thm. 2.5 ]. 2. Ev en in the complete case, Zermelo’s rates asso ciated with the original Llull matrix ( v xy ) are not necessarily compatible with the av erage ranks ¯ r x . Ho w ever, as w e will see b elo w, the pro jected Llull matrices will alw ays enjoy suc h a compatibility . 72 R. Camps, X. Mora, L. Sa umell F rom now on, X denotes the top dominan t irreducible comp onent whose existence is established by the preceding prop osition. According to Theo- rem 11.2, the fraction-like rates ϕ x v anish if and only if x ∈ A \ X and their v alues for x ∈ X are determined by the restriction of ( v π xy ) to x, y ∈ X . More sp eciﬁcally , the latter are determined by the condition of maximizing the function F ( ϕ ) = Y { x,y } ϕ x v π xy ϕ y v π yx ( ϕ x + ϕ y ) t π xy , (126) under the restriction (37) X x ϕ x = f . (127) where w e will understand that x and y are restricted to X , and f denotes the fraction of non-empty v otes (i. e. f = F /V where F is the n umber of non - empt y votes and V is the total num b er of v otes). Moreo ver, we know that ( ϕ x | x ∈ X ) is the solution of the follo wing system of equations b esides (127): (36) X y 6 = x t π xy ϕ x ϕ x + ϕ y = X y 6 = x v π xy . (128) where the sums extend to all y 6 = x in X . The next result sho ws that the resulting fraction-lik e rates are fully compatible with the rank-lik e ones except for the v anishing of those outside the top dominant comp onen t. Theorem 12.3. (a) ϕ x > ϕ y = ⇒ r x < r y . (b) r x < r y = ⇒ either ϕ x > ϕ y or ϕ x = ϕ y = 0 . Pr o of. Let us b egin by noticing that b oth statements hold if ϕ y = 0 , i. e. if y 6∈ X . In the case of statement (a), this is true b ecause of Prop osi- tion 12.2.(d). So, w e can assume that ϕ y > 0 , i. e. y ∈ X . But in this case, each one of the hypotheses of the present theorem implies that ϕ x > 0 , i. e. x ∈ X . In the case of statement (b), this is true b ecause of Prop osi- tion 12.2.(d) (with x and y interc hanged with each other) and the fact that X is a top in terv al for an y admissible order. So, from now on w e can assume that x and y are b oth in X , or, on account of Theorem 11.2, that X = A . Statemen t (a): It will b e prov ed b y seeing that a sim ultaneous o ccurrence of the inequalities ϕ x > ϕ y and r x ≥ r y w ould entail a contradiction with the fact that ϕ is the unique maximizer of F ( ϕ ) . More sp eciﬁcally , w e will Continuous ra ting for preferential voting , § 12 73 see that one would ha v e F ( e ϕ ) ≥ F ( ϕ ) where e ϕ is obtained from ϕ b y in terc hanging the v alues of ϕ x and ϕ y , that is e ϕ z =      ϕ y , if z = x, ϕ x , if z = y , ϕ z , otherwise. (129) In fact, e ϕ diﬀers from ϕ only in the comp onents asso ciated with x and y , so that F ( e ϕ ) F ( ϕ ) =  e ϕ x ϕ x  v π xy Y z 6 = x,y  e ϕ x / ( e ϕ x + ϕ z ) ϕ x / ( ϕ x + ϕ z )  v π xz  ϕ x + ϕ z e ϕ x + ϕ z  v π zx ×  e ϕ y ϕ y  v π yx Y z 6 = x,y  e ϕ y / ( e ϕ y + ϕ z ) ϕ y / ( ϕ y + ϕ z )  v π yz  ϕ y + ϕ z e ϕ y + ϕ z  v π zy . (130) More particularly , in the case of (129) this expression b ecomes F ( e ϕ ) F ( ϕ ) =  ϕ y ϕ x  v π xy − v π yx Y z 6 = x,y  ϕ y / ( ϕ y + ϕ z ) ϕ x / ( ϕ x + ϕ z )  v π xz − v π yz  ϕ y + ϕ z ϕ x + ϕ z  v π zy − v π zx , (131) where all of the bases are strictly less than 1 , since ϕ x > ϕ y , and all of the the exp onen ts are non-p ositiv e, b ecause of Lemma 10.1.(c). Therefore, the pro duct is greater than or equal to 1 , as claimed. Statemen t (b): Since we are assuming x, y ∈ X , it is a matter of pro ving that r x < r y ⇒ ϕ x > ϕ y . On the other hand, by making use of the con tra- p ositiv e of (a), the problem reduces to proving that ϕ x = ϕ y ⇒ r x = r y . Similarly to ab o v e, this implication will b e prov ed by seeing that a si- m ultaneous o ccurrence of the equality ϕ x = ϕ y =: ω together with the inequalit y r x < r y (b y symmetry it suﬃces to consider this one) w ould en- tail a contradiction with the fact that ϕ is the unique maximizer of F ( ϕ ) . More sp eciﬁcally , here we will see that one w ould ha v e F ( e ϕ ) > F ( ϕ ) where e ϕ is obtained from ϕ by sligh tly increasing ϕ x while decreasing ϕ y , that is e ϕ z =      ω + , if z = x, ω − , if z = y , ϕ z , otherwise. (132) This claim will b e pro ved b y c hec king that d d  log F ( e ϕ ) F ( ϕ )      =0 > 0 . (133) 74 R. Camps, X. Mora, L. Sa umell In fact, (130) entails that log F ( e ϕ ) F ( ϕ ) = C + v π xy log e ϕ x + v π y x log e ϕ y + X z 6 = x,y  v π xz log e ϕ x e ϕ x + ϕ z + v π y z log e ϕ y e ϕ y + ϕ z  − X z 6 = x,y  v π z y log( e ϕ y + ϕ z ) + v π z x log( e ϕ x + ϕ z )  , (134) where C do es not dep end on  . Therefore, in view of (132) we get d d  log F ( e ϕ ) F ( ϕ )      =0 = ( v π xy − v π y x ) 1 ω + X z 6 = x,y ( v π xz − v π y z ) ϕ z ω ( ω + ϕ z ) + X z 6 = x,y ( v π z y − v π z x ) 1 ω + ϕ z . (135) No w, according to Lemma 10.1.(b, c), the assumption that r x < r y implies the inequalities v π xy > v π y x , v π xz ≥ v π y z and v π z y ≥ v π z x , whic h result indeed in (133). The next prop osition establishes prop ert y H: Prop osition 12.4. In the c ase of plumping votes the fr action-like r ates c oincide with the fr actions of the vote obtaine d by e ach option. Pr o of. Prop osition 9.4 ensures that the pro jected scores coincide with the original ones. So we ha v e v π xy = f x and t π xy = f x + f y . In these circumstances it is ob vious that equations (127–128) are satisﬁed if we tak e ϕ x = f x . So it suﬃces to inv oke the uniqueness of solution of this system. 13 Con tin uit y W e claim that b oth the rank-like rates r x and the fraction-lik e ones ϕ x are con tin uous functions of the binary scores v xy . The main diﬃculty in proving this statemen t lies in the admissible order ξ , which pla ys a cen tral role in the computations. Since ξ v aries in a discrete set, its dep endence on the data cannot b e con tinuous at all. Ev en so, w e claim that the ﬁnal result is still a contin uous function of the data. In this connection, one can consider as data the normalized Llull ma- trix ( v xy ) , its domain of v ariation b eing the set Ω in tro duced in § 3.3. Al- ternativ ely , one can consider as data the relative frequencies of the p ossible v otes, i. e. the co eﬃcien ts α k men tioned also in § 3.3. Continuous ra ting for preferential voting , § 13 75 Theorem 13.1. The fol lowing obje cts dep end c ontinuously on the Llul l ma- trix ( v xy ) : the pr oje cte d sc or es v π xy , the r ank-like r ates r x , and the fr action- like r ates ϕ x . Pr o of. Let us b egin by considering the dep endence of the rank-like rates and the fraction-like rates on the pro jected scores. In the case of the rank- lik e rates, this dep endence is giv en by formula (8), whic h is not only con- tin uous but even linear (non-homogeneous). In the case of the fraction-like rates, their dependence on the pro jected scores is more in volv ed, but is is still con tinuous. In fact, Theorem 11.2.(c) ensures such a con tinuit y under the hypothesis that there is a top irreducible comp onent, which hypothesis is satisﬁed by virtue of Prop osition 12.2.(e). So w e are left with the problem of showing that the pro jection P : ( v xy ) 7→ ( v π xy ) is contin uous. As it has been men tioned abov e, this is not so clear, since the pro jected scores are the result of certain operations whic h are based up on an admissible order ξ whic h is determined separately . How ever, w e will see that, on the one hand, P is contin uous as long as ξ remains unchanged, and on the other hand, the results of § 8, 9 allow to conclude that P is contin uous on the whole of Ω in spite of the fact that ξ can change. In the follo wing we will use the following notation: for every total order ξ , we denote by Ω ξ the subset of Ω whic h consists of the Llull matrices for which ξ is an admissible order, and we denote b y P ξ the restriction of P to Ω ξ . W e claim that the mapping P ξ is contin uous for ev ery total order ξ . In order to chec k the truth of this statemen t, one has to go ov er the dif- feren t mappings whose comp osition deﬁnes P ξ (see § 2.8), namely: ( v xy ) 7→ ( v ∗ xy ) 7→ ( m ν xy ) , ( v xy ) 7→ ( t xy ) , ( m ν xy ) 7→ ( m σ xy ) , Ψ : (( m σ xx 0 ) , ( t xy )) 7→ ( t σ xy ) , and ﬁnally (( m σ xx 0 ) , ( t σ xx 0 )) 7→ ( v π xy ) . Quite a few of these mappings inv olve the max and min op erations, which are certainly con tinuous. F or instance, the last mapping ab o v e can b e written as v π xy = max { ( t σ pp 0 + m σ pp 0 ) / 2 | x  − ξ p  ξ y } and v π y x = min { ( t σ pp 0 − m σ pp 0 ) / 2 | x  − ξ p  ξ y } for x  ξ y . Concerning the op erator Ψ , let us recall that its output is the orthogonal pro jection of ( t xy ) on to a certain conv ex set determined by ( m σ xx 0 ) ; a general result of con tin uity for such an op eration can b e found in [ 10 ]. Finally , the contin uit y of P (and the fact that it is well-deﬁned) is a con- sequence of the following facts (see for instance [ 27 : § 2-7 ]): (a) Ω = S ξ Ω ξ ; this is true b ecause of the existence of ξ (Corollary 8.3). (b) Ω ξ is a closed subset of Ω ; this is true because Ω ξ is describ ed by a set of non-strict inequal- ities which concern quan tities that are contin uous functions of ( v xy ) (namely the inequalities m ν xy ≥ 0 whenev er xy ∈ ξ ). (c) ξ v aries o v er a ﬁnite set. (d) P ξ coincides with P η at Ω ξ ∩ Ω η , as it is prov ed in Theorem 9.2. 76 R. Camps, X. Mora, L. Sa umell Corollary 13.2. The r ank-like r ates, as wel l as the fr action-like ones, dep end c ontinuously on the r elative fr e quency of e ach p ossible c ontent of an individual vote. Pr o of. It suﬃces to notice that the Llull matrix ( v xy ) is simply the cen ter of gra vit y of the distribution sp eciﬁed b y these relativ e frequencies (formula (43) of § 3.3). 14 Decomp osition Prop erties E and G are concerned with ha ving a partition of A in t w o sets X and Y such that the rates for x ∈ X can b e obtained by restricting the atten tion to V X X , i. e. the v x ¯ x with x, ¯ x ∈ X (and similarly for y ∈ Y in the case of prop erty E). More sp eciﬁcally , prop ert y E considers the case where the following equal- ities are satisﬁed: r x = e r x , for all x ∈ X , (136) r y = e r y + | X | , for all y ∈ Y , (137) where e r x and e r y denote the rank-lik e rates which are determined resp ectively b y the matrices V X X and V Y Y . Prop ert y E states that in the complete case these equalities are equiv alent to ha ving v xy = 1 (and therefore v y x = 0 ) whenever xy ∈ X × Y . (138) In the follo wing we will con tinue using a tilde to distinguish b etw een hom- ologous ob jects asso ciated resp ectiv ely with the whole matrix V and with its submatrices V X X and V Y Y . First of all we explore the implications of condition (138). Lemma 14.1. Given a p artition A = X ∪ Y in two disjoint nonempty sets, one has the fol lowing implic ations: v xy = 1 ∀ xy ∈ X × Y  = ⇒  m ν xy = 1 ∀ xy ∈ X × Y  ⇐ ⇒  v π xy = 1 ∀ xy ∈ X × Y (139) If the individual votes ar e c omplete, or alternatively, if they ar e tr ansitive r elations, then the c onverse of the ﬁrst implic ation holds to o. Continuous ra ting for preferential voting , § 14 77 Pr o of. Assume that v xy = 1 for all xy ∈ X × Y . Then v y x = 0 , for all suc h pairs, which implies that v γ v anishes for any path γ which goes from Y to X . This fact, together with the inequality v ∗ xy ≥ v xy , entails the follo wing equalities for all x ∈ X and y ∈ Y : v ∗ y x = 0 , v ∗ xy = 1 , and consequen tly m ν xy = 1 . Assume no w that m ν xy = 1 for all xy ∈ X × Y . Let ξ b e an admissible order. As an immediate consequence of the deﬁnition, it includes the set X × Y . Let  b e the last element of X according to ξ . F rom the presen t h yp othesis it is clear that m σ `` 0 = 1 , i. e. γ `` 0 = [0 , 1] , whic h entails that γ xy = [0 , 1] , i. e. v π xy = 1 , for ev ery xy ∈ X × Y . Assume now that v π xy = 1 for all xy ∈ X × Y . Let ξ b e an admissible order. Here too, w e are ensured that it includes the set X × Y ; this is so b y virtue of Theorem 9.3.(a). Let  be the last elemen t of X according to ξ . F rom the fact that m σ `` 0 = m π `` 0 = 1 , one infers that m ν xy = 1 for all xy ∈ X × Y . Finally , let us assume again that m ν xy = 1 for all xy ∈ X × Y . Since m ν xy = v ∗ xy − v ∗ y x and b oth terms of this diﬀerence b elong to [0 , 1] , the only p ossibilit y is v ∗ xy = 1 and v ∗ y x = 0 , which implies that v y x = 0 . In the complete case, this equality is equiv alent to v xy = 1 . In the case where the individual votes are transitiv e relations, one can reach the same conclusion in the follo wing w a y: The equality v ∗ xy = 1 implies the existence of a path x 0 x 1 . . . x n from x to y suc h that v x i x i +1 = 1 for all i . But this means that all of the votes include eac h of the pairs x i x i +1 of this path. So, if they are transitiv e relations, all of them include also the pair xy , i. e. v xy = 1 . Lemma 14.2. Condition (138) implies, for any admissible or der, the fol- lowing e qualities: m σ xx 0 = e m σ xx 0 , whenever x, x 0 ∈ X , (140) m σ y y 0 = e m σ y y 0 , whenever y , y 0 ∈ Y , (141) t π x ¯ x = 1 , for al l x, ¯ x ∈ X . (142) Pr o of. As we sa w in the pro of of Lemma 14.1, condition (138) implies the v anishing of v γ for any path γ whic h go es from Y to X . Besides the conclusions obtained in that lemma, this implies also the following equalities: v ∗ x ¯ x = e v ∗ x ¯ x , m ν x ¯ x = e m ν x ¯ x , for all x, ¯ x ∈ X , (143) v ∗ y ¯ y = e v ∗ y ¯ y , m ν y ¯ y = e m ν y ¯ y , for all y , ¯ y ∈ Y . (144) Let us ﬁx an admissible order ξ . The second equalit y of (139) ensures not only that ξ includes the set X × Y , but it can also b e com bined with (143) 78 R. Camps, X. Mora, L. Sa umell and (144) to obtain resp ectiv ely (140) and (141). On the other hand, the third equality of (139) implies that t π xy = 1 for all xy ∈ X × Y , from whic h the pattern of gro wth of the pro jected turnov ers —more sp eciﬁcally , equation (83.2)— allows to obtain (142). Theorem 14.3. In the c omplete c ase one has the fol lowing e quivalenc es: (136) ⇐ ⇒ (137) ⇐ ⇒ (138) . Pr o of. Since we are considering the complete case, w e can make use of the margin-based pro cedure ( § 2.6). The pro of is organized in tw o parts: P art (a): (138) = ⇒ (136) and (137). As a consequence of the equali- ties (140) and (141), the margin-based pro cedure —more sp eciﬁcally , steps (13) and (14)— results in the follo wing equalities: m π x ¯ x = e m π x ¯ x , for all x, ¯ x ∈ X , (145) m π y ¯ y = e m π y ¯ y , for all y , ¯ y ∈ Y . (146) On the other hand, the third equalit y of (139) is equiv alent to sa ying that m π xy = 1 , for all xy ∈ X × Y . (147) When the pro jected margins are in tro duced in (9) these equalities result in (136) and (137). P art (b): (136) ⇒ (138); (137) ⇒ (138). On accoun t of form ula (9), conditions (136) and (137) are easily seen to b e resp ectively equiv alen t to the following equalities: X y ∈ A y 6 = x m π xy = X ¯ x ∈ X ¯ x 6 = x e m π x ¯ x + | Y | , for all x ∈ X , (148) X x ∈ A x 6 = y m π y x = X ¯ y ∈ Y ¯ y 6 = y e m π y ¯ y − | X | , for all y ∈ Y . (149) Let us add up resp ectiv ely the equalities (148) ov er x ∈ X and the equalities (149) ov er y ∈ Y . Since m π pq + m π q p = e m π pq + e m π q p = 0 , we obtain X x ∈ X y ∈ Y m π xy = | X | | Y | , (150) X y ∈ Y x ∈ X m π y x = −| X | | Y | . (151) Continuous ra ting for preferential voting , § 14 79 Since the pro jected margins b elong to [ − 1 , 1] , the preceding equalities imply resp ectiv ely m π xy = 1 , for all x ∈ X and y ∈ Y , (152) m π y x = − 1 , for all x ∈ X and y ∈ Y , (153) (whic h are equiv alent to eac h other since m π xy + m π y x = 0 ). Finally , either of these equalities implies that v π xy = 1 for all xy ∈ X × Y , from which Lemma 14.1 allows to obtain (138). The follo wing prop ositions do not require the v otes to b e complete, but they require them to b e rankings, or, more generally , in the case of Prop osi- tion 14.6, to b e transitiv e relations. Lemma 14.4. In the c ase of r anking votes, c ondition (138) implies that t xy = 1 for any x ∈ X and y ∈ A . Pr o of. In fact, even if w e are dealing with truncated ranking votes, the rules that w e are using for translating them in to binary preferences —namely , rules (a–d) of § 2.1— en tail the following implications: (i) v xy = 1 for some y ∈ A implies that x is explicitly men tioned in all of the ranking v otes; and (ii) x b eing explicitly mentioned in all of the ranking v otes implies that t xy = 1 for an y y ∈ A . Prop osition 14.5. In the c ase of r anking votes, c ondition (138) implies (136) . Pr o of. Let us ﬁx an admissible order. According to Lemma 14.2, we hav e t π x ¯ x = 1 for all x, ¯ x ∈ X . On the other hand, Lemma 14.4 ensures that t x ¯ x = 1 for all x, ¯ x ∈ X , from which it follows that e t π x ¯ x = 1 for all x, ¯ x ∈ X (since e t π x ¯ x are the turno v ers obtained from the restriction to the matrix V X X , whic h b elongs to the complete case). In particular, we hav e t σ xx 0 = e t σ xx 0 = 1 whenev er x, x 0 ∈ X . On the other hand, Lemma 14.2 ensures also that m σ xx 0 = e m σ xx 0 whenev er x, x 0 ∈ X . These equalities entail that v π x ¯ x = e v π x ¯ x for all x, ¯ x ∈ X . By Lemma 14.1 w e know also that v π xy = 1 for all xy ∈ X × Y . Therefore, r x = N − X y 6 = x y ∈ A v π xy = | X | − X ¯ x 6 = x ¯ x ∈ X e v π x ¯ x = e r x , ∀ x ∈ X . 80 R. Camps, X. Mora, L. Sa umell Prop osition 14.6. Assume that the individual votes ar e tr ansitive r ela- tions. In this c ase, the e quality X x ∈ X r x = | X | ( | X | + 1) / 2 (154) implies (138) (with Y = A \ X ). Pr o of. Let us introduce form ula (8) for r x in to (154). By using the fact that v π x ¯ x + v π ¯ xx ≤ 1 , one obtains that X x ∈ X y ∈ Y v π xy ≥ | X | | Y | . (155) The only p ossible wa y to satisfy this inequality is having v π xy = 1 for all xy ∈ X × Y . Finally , (138) follo ws b y virtue of Lemma 14.1 since w e are assuming that the individual votes are transitive relations. Corollary 14.7. Assume that the votes ar e r ankings. Then r x = 1 if and only if al l voters have put x into ﬁrst plac e. Pr o of. It suﬃces to apply Prop ositions 14.5 and 14.6 with X = { x } . The next theorem establishes prop ert y G. Theorem 14.8. (a) In the c omplete c ase, or alternatively, under the hy- p othesis that the individual votes ar e r ankings, one has the fol lowing impli- c ation: Assume that X ⊂ A has the pr op erty that v xy = 1 whenever x ∈ X and y ∈ Y = A \ X , and that ther e is no pr op er subset with the same pr op- erty. In that c ase, the fr action-like r ates satisfy ϕ x = e ϕ x > 0 for al l x ∈ X and ϕ y = 0 for al l y ∈ Y . (b) In the c omplete c ase the c onverse implic ation holds to o. Pr o of. Statement (a). Let us ﬁx an admissible order ξ . By Lemma 14.1, the hypothesis that v xy = 1 for all xy ∈ X × Y implies the following facts for all xy ∈ X × Y : m ν xy = 1 , xy ∈ ξ , v π xy = 1 , v π y x = 0 . On the other hand, we can see that under the present hypothesis one has v π x ¯ x = e v π x ¯ x , for any x, ¯ x ∈ X . (156) In the complete case this follo ws from Lemma 14.2. Under the alternativ e h yp othesis that the individual votes are rankings, it can b e obtained as in Continuous ra ting for preferential voting , § 15 81 the pro of of Prop osition 14.5 as a consequence of Lemma 14.2 and the fact that in this case t π x ¯ x = e t π x ¯ x = 1 for an y x, ¯ x ∈ X . No w, according to Lemma 12.1, the matrix ( v π xy ) has a top dominan t irreducible comp onent b X . Since v π y x = 0 for all xy ∈ X × Y , it is clear that b X ⊆ X . How ever, a strict inclusion b X ⊂ X w ould imply v π x ˆ x = 0 and therefore v π ˆ xx = 1 for an y x ∈ X \ b X and ˆ x ∈ b X . Since we also hav e v π xy = 1 for x ∈ X and y 6∈ X , we w ould get v π ˆ x ˆ y = 1 for all ˆ x ∈ b X and ˆ y 6∈ b X , whic h w ould imply , b y Lemma 14.1, that v ˆ x ˆ y = 1 for all such pairs. This would con tradict the supp osed minimality of X . So, X itself is the top dominant irreducible comp onent of the matrix ( v π xy ) . By making use of Theorem 11.2, it follo ws that ϕ x = e ϕ x > 0 for all x ∈ X and ϕ y = 0 for all y ∈ Y . In principle, e ϕ x are here the fraction- lik e rates computed from the restriction of ( v π xy ) to the set X . Ho wev er, (156) allows to view them also as the fraction-lik e rates computed from the matrix ( e v π xy ) , which by deﬁnition has b een w ork ed out from the restriction of ( v xy ) to x, y ∈ X . Statemen t (b). Let us b egin b y noticing that the hypothesis that ϕ x > 0 for all x ∈ X and ϕ y = 0 for all y ∈ Y = A \ X implies that X is the top dominant irreducible comp onen t of the matrix ( v π xy ) . In fact, otherwise Theorem 11.2 w ould imply the existence of some x ∈ X with ϕ x = 0 or some y ∈ Y with ϕ y > 0 . In particular, w e ha v e v π y x = 0 for all xy ∈ X × Y . Because of the completeness assumption, this implies that v π xy = 1 and —by Lemma 14.1— v xy = 1 for all those pairs. Finally , let us see that X is minimal for this prop erty: If we had b X ⊂ X satisfying v ˆ x ˆ y = 1 for all ˆ x ˆ y ∈ b X × b Y with b Y = A \ X , then Lemma 14.1 would give v π ˆ x ˆ y = 1 and therefore v π ˆ y ˆ x = 0 for all such pairs, so X could not b e the top dominant irreducible comp onent of the matrix ( v π xy ) . 15 The ma jorit y principle Theorem 15.1. The r elation µ ( v ∗ ) c omplies with the majority principle: L et A b e p artitione d in two sets X and Y with the pr op erty that v xy > 1 / 2 whenever x ∈ X and y ∈ Y ; in that c ase, µ ( v ∗ ) includes any p air xy with x ∈ X and y ∈ Y . Pr o of. Assume that x ∈ X and y ∈ Y . Since v ∗ xy ≥ v xy , the hypothesis of the theorem entails that v ∗ xy > 1 / 2 . On the other hand, let γ b e a path from y to x such that v ∗ y x = v γ ; since it go es from Y to X , this path must con tain at least one link y i y i +1 with y i ∈ Y and y i +1 ∈ X ; no w, for this 82 R. Camps, X. Mora, L. Sa umell link we hav e v y i y i +1 ≤ 1 − v y i +1 y i < 1 / 2 , which en tails that v ∗ y x = v γ < 1 / 2 . Therefore, we get v ∗ y x < 1 / 2 < v ∗ xy , i. e. xy ∈ µ ( v ∗ ) . Corollary 15.2. The so cial r anking determine d by the r ank-like r ates c om- plies with the majority principle: L et A b e p artitione d in two sets X and Y with the pr op erty that v xy > 1 / 2 whenever x ∈ X and y ∈ Y ; in that c ase, the ine quality r x < r y holds for any x ∈ X and y ∈ Y . Pr o of. It follo ws from Theorem 15.1 by virtue of part (c) of Theorem 10.2. Corollary 15.3. In the c omplete c ase the so cial r anking determine d by the r ank-like r ates c omplies with the Condor c et principle: If x has the pr op erty that v xy > v y x for any y 6 = x , then r x < r y for any y 6 = x . Pr o of. In the complete case v xy > v y x implies v xy > 1 / 2 . So, it suﬃces to apply the preceding result with X = { x } and Y = A \ X . 16 Clone consistency The notion of a cluster (of clones) w as deﬁned in § 5 in connection with a binary relation: A subset C ⊆ A is said to b e a cluster for a relation ρ when, for an y x 6∈ C , having ax ∈ ρ for some a ∈ C implies bx ∈ ρ for any b ∈ C , and similarly , having xa ∈ ρ for some a ∈ C implies xb ∈ ρ for an y b ∈ C . Here w e will extend the notion of a cluster in the follo wing wa y: C ⊆ A is said to b e a cluster for a system of binary scores ( v xy ) when v ax = v bx , v xa = v xb , whenev er a, b ∈ C and x 6∈ C . (157) This deﬁnition can b e viewed as an extension of the preceding one b ecause of the following obvious fact: Lemma 16.1. C is a cluster for a r elation ρ if and only if C is a cluster for the c orr esp onding system of binary sc or es, which is deﬁne d in (42) . In particular, the extended notion allows the following results to include the case where the individual v otes b elong to the general class considered in § 3.3. In this section we will pro ve the clone consistency prop ert y J: If a set of options is a cluster for eac h of the individual votes, then: (a) it is a cluster Continuous ra ting for preferential voting , § 16 83 for the so cial ranking; and (b) con tracting it to a single option in all of the individual v otes has no other eﬀect in the social ranking than getting the same contraction. In the remainder of this section we assume the following standing h yp othesis : C is a cluster for al l of the individual votes. Since the collective binary scores are obtained by adding up the individual ones (equation (43)), the preceding h yp othesis immediately implies that C is a cluster for the c ol le ctive binary sc or es v xy . In the following w e will see that this prop ert y of b eing a cluster is maintained throughout the whole pro cedure whic h deﬁnes the so cial ranking. Lemma 16.2. Assume that either x or y , or b oth, lie outside C . In this c ase v ∗ xy = max { v γ | γ c ontains no mor e than one element of C } Pr o of. It suﬃces to see that an y path γ = x 0 . . . x n from x 0 = x to x n = y whic h con tains more than one element of C can b e replaced b y another one e γ which con tains only one suc h elemen t and satisﬁes v e γ ≥ v γ . Con- sider ﬁrst the case where x, y 6∈ C . In this case it will suﬃce to tak e e γ = x 0 . . . x j − 1 x k . . . x n , where j = min { i | x i ∈ C } and k = max { i | x i ∈ C } , whic h ob viously satisfy 0 < j < k < n . Since x j − 1 6∈ C and x j , x k ∈ C , w e ha v e v x j − 1 x j = v x j − 1 x k , so that v γ = min  v x 0 x 1 , . . . , v x n − 1 x n  ≤ min  v x 0 x 1 , . . . , v x j − 1 x j , v x k x k +1 , . . . , v x n − 1 x n  = min  v x 0 x 1 , . . . , v x j − 1 x k , v x k x k +1 , . . . , v x n − 1 x n  = v e γ . The case where x 6∈ C but y ∈ C can b e dealt with in a similar wa y b y taking e γ = x 0 . . . x j − 1 x n , and analogously , in the case where x ∈ C and y 6∈ C it suﬃces to take e γ = x 0 x k +1 . . . x n . Prop osition 16.3. C is a cluster for the indir e ct sc or es v ∗ xy . 84 R. Camps, X. Mora, L. Sa umell Pr o of. Consider a, b ∈ C and x 6∈ C . Let γ = x 0 x 1 x 2 . . . x n b e a path from a to x such that v ∗ ax = v γ . By Lemma 16.2, we can assume that a is the only elemen t of γ that b elongs to C . In particular, x 1 6∈ C , so that v ax 1 = v bx 1 , whic h allo ws to write v ∗ ax = v γ = min  v ax 1 , v x 1 x 2 , . . . , v x n − 1 x  = min  v bx 1 , v x 1 x 2 , . . . , v x n − 1 x  ≤ v ∗ bx . By in terchanging a and b , one gets the reverse inequality v ∗ bx ≤ v ∗ ax and there- fore the equality v ∗ ax = v ∗ bx . An analogous argument sho ws that v ∗ xa = v ∗ xb . Prop osition 16.4. C is a cluster for the indir e ct c omp arison r elation ν = µ ( v ∗ ) . Pr o of. This is an immediate consequence of the preceding prop osition. Prop osition 16.5. Ther e exists an admissible or der ξ such that C is a cluster for ξ . Pr o of. This result is given by Theorem 8.4 of p. 43. Theorem 16.6. C is a cluster for the r anking deﬁne d by the r ank-like r ates ( i. e. for the r elation σ = { xy ∈ Π | r x < r y } ) . Pr o of. W e must sho w that, for an y x 6∈ C and an y a, b ∈ C , r a < r x implies r b < r x and r x < r a implies r x < r b (from which it follo ws that r a = r x implies r b = r x ). Equiv alen tly , it suﬃces to sho w that: (a) r a < r x implies r b ≤ r x ; (b) r x < r a implies r x ≤ r b ; and (c) r a = r x implies r b = r x . The proof will mak e use of an admissible order ξ with the prop ert y that C is a cluster for ξ (whose existence is ensured by Prop osition 16.5). P arts (a) and (b) are then a straightforw ard consequence of part (a) of Lemma 10.1.(a). In fact, b y com bining this result, and its contrapositive, with the fact that C is a cluster for ξ , we ha ve the following implications: r a < r x ⇒ ax ∈ ξ ⇒ bx ∈ ξ ⇒ r b ≤ r x , and similarly , r x < r a ⇒ xa ∈ ξ ⇒ xb ∈ ξ ⇒ r x ≤ r b . P art (c): r a = r x implies r b = r x (for x 6∈ C and a, b ∈ C ). Since ξ is a total order, w e m ust hav e either ax ∈ ξ or xa ∈ ξ ; in the following we assume ax ∈ ξ (the other p ossibility admits of a similar treatmen t). In order to deal with this case w e will consider the last element of C according to ξ , Continuous ra ting for preferential voting , § 16 85 whic h we will denote as  , and its immediate successor  0 , whic h does not b elong to C . Since a  − ξ   ξ  0  − ξ x and r a = r x , w e m ust ha v e r ` = r ` 0 . No w, according to part (b) of Lemma 10.1, m π `` 0 = 0 ; in other w ords, m σ `` 0 = 0 . By the deﬁnition of m σ `` 0 , this means that there exist p and q with p  − ξ   ξ q suc h that m ν pq = v ∗ pq − v ∗ q p = 0 . Obviously , q 6∈ C , whereas p either b elongs to C or it precedes all elements of C . In the latter case, we immediately get m σ cc 0 = 0 for all c ∈ C (by the deﬁnition of m σ cc 0 ). If p ∈ C , w e arrive at the same conclusion thanks to Prop osition 16.3, whic h ensures that m ν cq = m ν pq . So, the interv als γ cc 0 with c ∈ C are all of them reduced to a p oint. Since C is a cluster for the total order ξ , this implies that γ ab is also reduced to the same p oin t (this holds for any a, b ∈ C ). According to part (b) of Lemma 10.1, this implies that r a = r b , as it was claimed. Finally , we consider the eﬀect of con tracting C to a single element. So we consider a new set e A = ( A \ C ) ∪ { e c } together with the scores e v xy ( x, y ∈ e A ) deﬁned b y the follo wing equalities, where p, q ∈ A \ C and c is an arbitrary elemen t of C : e v pq = v pq , e v p e c = v pc and e v e c q = v cq (the deﬁnition is not ambiguous since C is a cluster for the scores v xy ). In the following, a tilde is systematically used to distinguish b et ween homolo- gous ob jects asso ciated resp ectively with ( A, v ) and ( e A , e v ) . W e will also mak e use of the following notation: for every x ∈ A , e x denotes the element of e A deﬁned b y e x = e c if x ∈ C and by e x = x if x 6∈ C ; in terms of this mapping, the preceding equalities say simply that e v e x e y = v xy whenev er e x 6 = e y . Theorem 16.7. The r anking e σ = { xy ∈ e Π | e r x < e r y } c oincides with the c ontr action of σ = { xy ∈ Π | r x < r y } by the cluster C . Pr o of. W e b egin by noticing that the indirect scores e v ∗ xy ( x, y ∈ e A ) coincide with those obtained b y contraction of the v ∗ xy ( x, y ∈ A ) , i. e. e v ∗ e x e y = v ∗ xy whenev er e x 6 = e y . This follo ws from the analogous equality b etw een the direct scores b ecause of Lemma 16.2. As a consequence, e ν = µ ( e v ∗ ) coincides with the con traction of ν = µ ( v ∗ ) b y C . F rom this fact, parts (a) and (b) of Theorem 10.2, allow to deriv e that r x < r y implies e r e x ≤ e r e y whenev er e x 6 = e y , and that e r e x < e r e y implies r x ≤ r y . In order to complete the pro of, w e must c heck that r x = r y is equiv alent to e r e x = e r e y whenev er e x 6 = e y . According to part (b) of Lemma 10.1, it suﬃces to see that m π xy = 0 is equiv alent to e m π e x e y = 0 whenev er e x 6 = e y . In order to pro v e this equiv alence, we need to lo ok at the wa y that m π xy and e m π e x e y are ob- tained, whic h requires certain admissible orders ξ and e ξ ; in this connection, it will b e useful that ξ b e one of the admissible orders for which C is a clus- ter (Prop osition 16.5), and that e ξ be the corresp onding con traction, which is 86 R. Camps, X. Mora, L. Sa umell admissible as a consequence of Prop osition 16.3. No w, that prop osition en- tails not only that C is a cluster for the indirect margins m ν pq , but also that their contraction by C coincides with the margins of the con tracted indirect scores, i. e. e m ν e p e q = m ν pq whenev er e p 6 = e q . Moreo ver, by the deﬁnition of the in termediate pro jected margins, namely equation (21.1), it follo ws that C is also a cluster for the intermediate pro jected margins m σ pq and that their con traction by C coincides with the homologous quantities obtained from the con tracted indirect margins, i. e. e m σ e p e q = m σ pq whenev er e p 6 = e q . On the other hand, it is also clear from equation (21.1) that the in termediate pro jected margins b ehav e in the following wa y: m σ pq ≤ m σ ab whenev er a  − ξ p  ξ q  − ξ b. (158) After these remarks, we pro ceed with showing that m π xy = 0 is equiv alent to e m π e x e y = 0 whenev er e x 6 = e y . By symmetry , we can assume that xy ∈ ξ , which entails that e x e y ∈ e ξ . In view of (22–24), the equality m π xy = 0 is equiv alent to sa ying that m σ hh 0 = 0 for all h such that x  − ξ h  ξ y , and similarly , the equality e m π e x e y = 0 is equiv alent to e m σ η η 0 = 0 for all η suc h that e x  − ξ η  ξ e y . By considering a path x 0 x 1 . . . x n from x 0 = x to x n = y with x i x i +1 consecutiv e in ξ , it is clear that the problem reduces to proving the follo wing implications, where  denotes the last elemen t of C b y ξ , f denotes the ﬁrst one, and 0 h denotes the elemen t that immediately precedes h in ξ : (a) m σ `` 0 = 0 ⇒ e m σ e c ` 0 = 0 ; (b) e m σ e c ` 0 = 0 ⇒ m σ cc 0 = 0 for any c ∈ C ; (c) m σ 0 f f = 0 ⇒ e m σ 0 f e c = 0 ; and (d) e m σ 0 f e c = 0 ⇒ m σ 0 cc = 0 for an y c ∈ C . No w, (a) and (c) are immediate consequences of the fact that e m σ e p e q = m σ pq whenev er e p 6 = e q . On the other hand, (b) and (d) follo w from the same equalit y together with the inequalit y (158). In fact, these facts allo w us to write m σ cc 0 ≤ m σ c` 0 = e m σ e c ` 0 , whic h giv es (b), and similarly , m σ 0 cc ≤ m σ 0 f c = e m σ 0 f e c , whic h giv es (d). 17 Appro v al voting In appro v al voting, eac h voter is ask ed for a list of approv ed options, without an y expression of preference b et w een them, and each option x is then rated b y the num b er of appro v als for it [ 6 ]. In the following w e will refer to this n um b er as the appro v al score of x , and its v alue relative to V will b e denoted by α x . F rom the p oint of view of paired comparisons, an individual vote of ap- pro v al t yp e can b e view ed as a truncated ranking where all of the options that app ear in it are tied. In this section, we will see that the margin-based Continuous ra ting for preferential voting , § 17 87 v arian t orders the options exactly in the same w a y as the appro v al scores. In other w ords, the metho d of approv al voting agrees with ours under inter- pretation (d 0 ) of § 3.2, i. e. under the in terpretation that the non-appro v ed options of each individual v ote are tied. Ha ving said that, the preliminary results 17.1–17.3 will hold not only un- der in terpretation (d 0 ) but also under in terpretation (d), i. e. that there is no information ab out the preference of the voter b et ween tw o non-approv ed options, and also under the analogous in terpretation that there is no informa- tion ab out his preference b etw een tw o approv ed options. In terpretation (d 0 ) do es not play an essen tial role until Theorem 17.4, where we use the fact that it alwa ys brings the problem in to the complete case. In the following, λ ( α ) denotes the relation deﬁned b y xy ∈ λ ( α ) ≡ α x > α y . (159) Prop osition 17.1. In the appr oval voting situation, the fol lowing e quality holds: v xy − v y x = α x − α y . (160) In p articular, µ ( v ) = λ ( α ) . Pr o of. Obviously , the possible ballots are in one-to-one correspondence with the subsets X of A . In the follo wing, v X denotes the relative num b er of v otes that approv ed exactly the set X . With this notation it is ob vious that α x = X X 3 x v X = X X 3 x X 63 y v X + X X 3 x X 3 y v X . (161) On the other hand, one has v xy = X X 3 x X 63 y v X  + 1 2 X X 3 x X 3 y v X + 1 2 X X 63 x X 63 y v X  , (162) where the terms in brac kets are presen t or not dep ending on which in ter- pretation is used. Anyw a y , the preceding expressions, together with the analogous ones where x and y are in terc hanged with eac h other, result in the equality (160) indep enden tly of those alternative interpretations. Corollary 17.2. In the appr oval voting situation, a p ath x 0 x 1 . . . x n is c on- taine d in µ ( v ) (r esp. ˆ µ ( v ) ) if and only if the se quenc e α x i ( i = 0 , 1 , . . . n ) is de cr e asing (r esp. non-incr e asing). 88 R. Camps, X. Mora, L. Sa umell Prop osition 17.3. In the appr oval voting situation, one has µ ( w ∗ ) = λ ( α ) . Pr o of. Let us b egin by proving that α x > α y = ⇒ w ∗ xy > w ∗ y x . (163) W e will argue by contradiction. So, let us assume that w ∗ y x ≥ w ∗ xy . According to Prop osition 17.1, the h yp othesis that α x > α y is equiv alent to v xy > v y x , whic h en tails that w xy > 0 (b y the deﬁnition of w xy together with the strict inequalit y v xy > v y x ). No w, since w ∗ xy ≥ w xy and w e are assuming that w ∗ y x ≥ w ∗ xy , it follows that w ∗ y x > 0 . This implies the existence of a path from y to x which is contained in ˆ µ ( v ) (by the deﬁnitions of w ∗ y x and w pq ). Finally , Corollary 17.2 pro duces a contradiction with the presen t hypothesis that α x > α y . Let us see now that α x = α y = ⇒ w ∗ xy = w ∗ y x . (164) Again, w e will argue by contradiction. So, let us assume that w ∗ xy 6 = w ∗ y x . Ob viously , it suﬃces to consider the case w ∗ xy > w ∗ y x . No w, this inequal- it y implies that w ∗ xy > 0 , whic h tells us that w ∗ xy = w γ for a certain path γ : x 0 x 1 . . . x n whic h go es from x 0 = x to x n = y and is con tained in ˆ µ ( v ) . According to Corollary 17.2, we are ensured that the sequence α x i ( i = 0 , 1 , . . . n ) is non-increasing. How ever, the hypothesis that α x = α y lea v es no other p ossibilit y than α x i b eing constan t. So, the reverse path γ 0 : x n x n − 1 . . . x 1 x 0 is also contained in ˆ µ ( v ) . Besides, Prop osition 17.1 en- sures that v x i +1 x i = v x i x i +1 , so that w γ 0 = w γ . Since w ∗ y x ≥ w γ 0 , it follo ws that w ∗ y x ≥ w ∗ xy , which contradicts the h yp othesis that w ∗ xy > w ∗ y x . Finally , one easily c hec ks that the preceding implications entail that sgn ( α x − α y ) is alw a ys equal to sgn ( w ∗ xy − w ∗ y x ) . This is equiv alen t to the equalit y of the relations λ ( α ) and µ ( w ∗ ) . Theorem 17.4. In the appr oval voting situation, the mar gin-b ase d variant r esults in a ful l c omp atibility r elation b etwe en the r ank-like r ates r x and the appr oval sc or es α x : r x < r y ⇔ α x > α y . Pr o of. Recall that the margin-based v arian t amoun ts to using interpreta- tion (d 0 ), whic h alw a ys brings the problem into the complete case (when the terms in brack ets are included, equation (162) has indeed the prop ert y that v xy + v y x = 1 ). So we can in vok e Theorem 7.3. By combining it with Prop osition 17.3 we see that the inequalit y α x > α y is equiv alent to sa y- ing that xy ∈ ν . In the following we will keep this equiv alence in mind. Continuous ra ting for preferential voting , § 18 89 The implication r x < r y ⇒ α x > α y is then an immediate consequence of part (b) of Theorem 10.2. The con verse implication α x > α y ⇒ r x < r y can b e prov ed in the following wa y: Let ξ b e an admissible order. By def- inition, it contains ν . So, the inequalit y α x > α y implies xy ∈ ξ . On the other hand, that inequalit y implies also the existence of a consecutiv e pair hh 0 with x  − ξ h and h 0  − ξ y such that α h > α h 0 . As a consequence, one has α p > α q whenev er p  − ξ h and h 0  − ξ q . So, the sets X = { p | p  − ξ h } and Y = { q | h 0  − ξ q } are in the hypotheses of part (c) of Theorem 10.2, which ensures the desired inequality r x < r y . R emark So in this case we get a con verse of Theorem 10.2.(b). By following the same argumen ts as in the preceding pro of, one can see that suc h a conv erse holds whenever there exists a function s : A 3 x 7→ s x ∈ R , such that xy ∈ ν ⇔ s x < s y . Summing up, the standard approv al voting pro cedure is alw ays in full agreemen t with the margin-based v ariant of the CLC metho d. In the ap- pro v al v oting situation, this v arian t amoun ts to treat all of the candidates whic h are missing in an approv al ballot as equally ‘unpreferred’ (in the same w a y that all approv ed candidates are treated as equally preferred). This is quite reasonable if one can assume that the voters are well acquainted with all of the options. 18 Ab out monotonicit y In this section we consider the eﬀect of raising a particular option a to a more preferred status in the individual ballots without any change in the preferences ab out the other options. More generally , w e consider the case where the scores v xy are mo diﬁed into new v alues e v xy suc h that e v ay ≥ v ay , e v xa ≤ v xa , e v xy = v xy , ∀ x, y 6 = a. (165) In suc h a situation, one would exp ect the so cial rates to b ehav e in the fol- lo wing w a y , where y is an arbitrary element of A \ { a } : e r a < r a , (166) r a < r y = ⇒ e r a < e r y , r a ≤ r y = ⇒ e r a ≤ e r y , (167) where the tilde indicates the ob jects asso ciated with the mo diﬁed scores. Unfortunately , the rating metho d prop osed in this pap er does not satisfy 90 R. Camps, X. Mora, L. Sa umell these conditions, but generally sp eaking it satisﬁes only the following w eak er ones: r a < r y = ⇒ e r a ≤ e r y . (168) ( r a < r y , ∀ y 6 = a ) = ⇒ ( e r a < e r y , ∀ y 6 = a ) . (169) In particular, (169) is saying that if a w as the only winner for the scores v xy , then it is still the only winner for the scores e v xy . Let us remark that in the case of ranking votes, situation (165) includes the following ones: (a) the option a is raised to a b etter p osition in some of the ranking votes without any change in the preferences betw een the other op- tions; (b) the option a is app ended to some ballots which did not previously con tain it; (c) some ballots are added whic h plump for option a . Ho wev er, the third part of (165) lea ves out certain situations whic h are sometimes considered the matter of other “monotonicit y” conditions [ 39 ]. In the terminology of [ 3 ], prop erty (168) is saying that the metho d that w e are using is “v ery weakly monotonic” as a ranking pro cedure, whereas prop ert y (169) is related to what [ 3 ] calls “prop er monotonicity” of a choice pro cedure. In this connection, it is interesting to remark that the metho d of rank ed pairs enjo ys the c hoice - monotonicity prop ert y (169) [ 38 : p. 221–222 ], but it lac ks the ranking - monotonicit y prop erty (168). A proﬁle whic h ex- hibits such a failure of the ranking - monotonicit y for the metho d of ranked pairs is given in http://mat.uab.cat/ ~ xmora/CLC calculator/ (n um b er 9 of “Example inputs”). 18.1 This section is dev oted to giving a proof of prop erties (168) and (169). Theorem 18.1. Assume that ( v xy ) and ( e v xy ) ar e r elate d to e ach other in ac c or danc e with (165) . In this c ase, the fol lowing pr op erties ar e satisﬁe d for any x, y 6 = a : e v ∗ ay ≥ v ∗ ay , e v ∗ xa ≤ v ∗ xa , (170) P a ( e ν ) ⊆ P a ( ν ) , S a ( e ν ) ⊇ S a ( ν ) , (171) (168) r a < r y = ⇒ e r a ≤ e r y , (172) wher e ν = µ ( v ∗ ) and e ν = µ ( e v ∗ ) Pr o of. Let us b egin b y seeing that (172) will be a consequence of (171). In fact, w e ha ve the following c hain of implications: r a < r y ⇒ y ∈ S a ( ν ) ⇒ y ∈ S a ( e ν ) ⇒ e r a ≤ e r y , where the cen tral one is provided by (171.2) and the other tw o are guaranteed b y Theorem 10.2. Continuous ra ting for preferential voting , § 18 91 The pro of of (170–171) is organized in three steps. In the ﬁrst one, we lo ok at the sp ecial case where one increases the score of a single pair ab . After this, w e will consider the case where an increase in the score of ab is com bined with a decrease in the score of ba . Finally , the third step deals with the general situation (165). Sp ecial case 1. Assume that e v ab > v ab , e v xy = v xy , ∀ xy 6 = ab. (173) In this c ase, the fol lowing pr op erties ar e satisﬁe d: e v ∗ xy ≥ v ∗ xy , ∀ x, y (174) e v ∗ xa = v ∗ xa , ∀ x 6 = a (175) e v ∗ by = v ∗ by , ∀ y 6 = b (176) (171) P a ( e ν ) ⊆ P a ( ν ) , S a ( e ν ) ⊇ S a ( ν ) , (177) P b ( e ν ) ⊇ P b ( ν ) , S b ( e ν ) ⊆ S b ( ν ) . (178) In fact, under the hypothesis (173) it is obvious that e v γ ≥ v γ and that the strict inequalit y happ ens only when the path γ = x 0 . . . x n con tains the pair ab and the latter realizes the minimum of the scores v x i x i +1 . As a consequence, the indirect scores satisfy the inequality (174). F urthermore, a strict inequality in (174) implies that the maxim um whic h deﬁnes e v ∗ xy is realized b y a path γ whic h satisﬁes e v γ > v γ and therefore con tains the pair ab . No w, in order to obtain the indirect score for a pair of the form xa it is useless to consider paths inv olving ab , since suc h paths contains cycles whose omission results in paths not in v olving ab and ha ving a b etter or equal score. So, the maximum which deﬁnes e v ∗ xa is realized b y a path which do es not inv olve ab . According to the last statemen t of the preceding paragraph, this implies (175). An en tirely analogous argumen t establishes (176). Finally , (177) is obtained in the following w ay: x ∈ P a ( e ν ) means that e v ∗ xa > e v ∗ ax , from whic h (175) and (174) allow to derive that v ∗ xa = e v ∗ xa > e v ∗ ax ≥ v ∗ ax , i. e. x ∈ P a ( ν ) . Similarly , x ∈ S a ( ν ) implies x ∈ S a ( e ν ) b ecause one has e v ∗ ax ≥ v ∗ ax > v ∗ xa = e v ∗ xa . An analogous argument establishes (178). Sp ecial case 2. Pr op erties (170–171) ar e satisﬁe d in the fol lowing situation: e v ab ≥ v ab , e v ba ≤ v ba , e v xy = v xy , ∀ xy 6 = ab, ba. (179) 92 R. Camps, X. Mora, L. Sa umell This result will b e obtained from the preceding one b y going through an in termediate Llull matrix e e v deﬁned in the following wa y      e e v ab = v ab e e v ba = e v ba e e v xy = e v xy = v xy , ∀ xy 6 = ab, ba (180) If the h yp othesis e v ab ≥ v ab is satisﬁed with strict inequality , then e v and e e v are in the hypotheses of the sp ecial case 1 (they play resp ectiv ely the roles of e v and v ). In particular, w e get e v ∗ xy ≥ e e v ∗ xy , e v ∗ xa = e e v ∗ xa , P a ( e ν ) ⊆ P a ( e e ν ) , S a ( e ν ) ⊇ S a ( e e ν ) . (181) On the other hand, if e v ab = v ab then e e v = e v and the preceding relations hold as equalities. Similarly , if the hypothesis e v ba ≤ v ba is satisﬁed with strict inequality , then v and e e v are in the h yp otheses of the sp ecial case 1 with ab replaced b y ba (they play resp ectively the roles of e v and v ). In particular, we get v ∗ xy ≥ e e v ∗ xy , v ∗ ay = e e v ∗ ay , P a ( ν ) ⊇ P a ( e e ν ) , S a ( ν ) ⊆ S a ( e e ν ) . (182) As b efore, if e v ba = v ba then e e v = v and the preceding relations hold as equalities. Finally , (170–171) are obtained by combining (181) and (182): e v ∗ ay ≥ e e v ∗ ay = v ∗ ay , e v ∗ xa = e e v ∗ xa ≤ v ∗ xa , P a ( e ν ) ⊆ P a ( e e ν ) ⊆ P a ( ν ) , S a ( e ν ) ⊇ S a ( e e ν ) ⊇ S a ( ν ) . General case. In the general situation (165), prop erties (170–171) are a di- rect consequence of the successive application of the sp ecial case 2 to every pair ay . Corollary 18.2. Under the hyp othesis of The or em 18.1 one has also ϕ a > ϕ y ⇒ e ϕ a ≥ e ϕ y . (183) Pr o of. It suﬃces to com bine (172) with Theorem 12.3. Continuous ra ting for preferential voting , § 18 93 Corollary 18.3 ( 2 ) . Under the hyp othesis of The or em 18.1 one has also the pr op erty (169) . Pr o of. According to Theorem 10.2.(b), the left-hand side of (169) implies the strict inequalit y v ∗ ay > v ∗ y a for all y 6 = a . No w, this inequalit y can b e combined with (170) to derive that e v ∗ ay > e v ∗ y a for all y 6 = a . Finally , Theorem 10.2.(c) with X = { a } and Y = A \ { a } guaran tees that the righ t-hand side of (169) is satisﬁed. 18.2 The statements (166) and (167) can fail ev en in the complete case. Next we give an example of it, with 5 options (it seems to b e the minim um for the failure of (167) ) and 10 voters. The only change from left to right is one inv ersion in one of the votes; more sp eciﬁcally , the eighth ballot changes from the order d  b  c  a  e to the new one b  d  c  a  e . In spite of this c hange, fa vourable to b and disadv antageous to d , the rank-lik e rate of b is w orsened from 2 . 90 to 3 . 00 , whereas that of d is improv ed from 3 . 10 to 3 . 00 . This con tradicts (166) for a = b , as w ell as (167.1) for a = b and y = d , c , and also (167.2) for a = d and y = a , b (when one go es from righ t to left). How ever, it complies with (168). x a b c d e R anking votes 1 2 2 2 2 3 3 4 4 5 3 1 3 5 5 1 5 2 1 1 5 3 4 3 3 2 2 3 3 2 2 5 1 4 4 4 4 1 5 3 4 4 5 1 1 5 1 5 2 4 x a b c d e R anking votes 1 2 2 2 2 3 3 4 4 5 3 1 3 5 5 1 5 1 1 1 5 3 4 3 3 2 2 3 3 2 2 5 1 4 4 4 4 2 5 3 4 4 5 1 1 5 1 5 2 4 x a b c d e V xy a b c d e ∗ 5 5 7 5 5 ∗ 7 4 7 5 3 ∗ 7 5 3 6 3 ∗ 5 5 3 5 5 ∗ x a b c d e e V xy a b c d e ∗ 5 5 7 5 5 ∗ 7 5 7 5 3 ∗ 7 5 3 5 3 ∗ 5 5 3 5 5 ∗ 2 W e thank Markus Sch ulze for p ointing out this fact. 94 R. Camps, X. Mora, L. Sa umell x a b c d e V ∗ xy a b c d e ∗ 6 6 7 6 5 ∗ 7 7 7 5 6 ∗ 7 6 5 6 6 ∗ 6 5 5 5 5 ∗ x a b c d e e V ∗ xy a b c d e ∗ 5 5 7 5 5 ∗ 7 7 7 5 5 ∗ 7 5 5 5 5 ∗ 5 5 5 5 5 ∗ x a b c d e M ν xy a b c d e ∗ 1 1 2 1 ∗ ∗ 1 1 2 ∗ ∗ ∗ 1 1 ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ x a b c d e f M ν xy a b c d e ∗ 0 0 2 0 ∗ ∗ 2 2 2 ∗ ∗ ∗ 2 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ x a b c d e M π xy a b c d e ∗ 1 1 1 1 ∗ ∗ 1 1 1 ∗ ∗ ∗ 1 1 ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ ∗ ∗ r x 2.80 2.90 3.00 3.10 3.20 x a b c d e f M π xy a b c d e ∗ 0 0 0 0 ∗ ∗ 0 0 0 ∗ ∗ ∗ 0 0 ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ e r x 3.00 3.00 3.00 3.00 3.00 As one can see, the m ultiple zero es presen t in f M ν xy force a complete tie of the rank-lik e rates e r x in spite of the fact that e ν is not empty . Not only the latter contains the pair b d , but in fact f M ν bd = 2 > 1 = M ν bd . References [ 1 ] Michel Balinski, Rida Laraki, 2007. [ a ] A theory of measuring, electing, and ranking. Pr o c e e dings of the National A c ademy of Scienc es of the Unite d States of Americ a , 104 : 8720–8725. [ b ] One-V alue, One-V ote: Me asuring, Ele cting and R anking . (T o app ear). [ 2 ] Duncan Black, 1958 1 , 1986 2 , 1998 3 . The The ory of Committe es and Ele c- tions . Cam bridge Univ. Press 1 , 2 , Kluw er 3 . Continuous ra ting for preferential voting , § 18 95 [ 3 ] Denis Bouyssou, 2004. a Monotonicity of ‘ranking by c ho osing’ · A progress rep ort. So cial Choic e and Welfar e , 23 : 249–273. [ 4 ] Ralph Allan Bradley , Milton E. T erry , 1952. Rank analysis of incomplete blo c k designs: I. The metho d of paired comparisons. Biometrika , 39 : 324– 345. [ 5 ] Steven J. Brams, Michael W. Hanse n, Michael E. Orrison, 2006. Dead heat: the 2006 Public Choice So ciet y election. Public Choic e , 128 : 361–366. [ 6 ] Steven J. Brams, 2008. Mathematics and Demo cr acy · Designing Better V oting and F air-Division Pr o c e dur es . Princeton Univ. Press. [ 7 ] Gregory R. Conner, Christopher P . Grant, 2000. An extension of Zermelo’s mo del for ranking by paired comparisons. Eur op e an Journal of Applie d Mathematics , 11 : 225–247. [ 8 ] Thomas H. Cormen, Charles L. Leiserson, Ronald L. Rivest, Cliﬀord Stein, 1990 1 , 2001 2 . Intr o duction to Algorithms . MIT Press. [ 9 ] Richard Couran t, 1950 1 , 1977 2 . Dirichlet’s Principle, Conformal Mapping, and Minimal Surfac es . In terscience 1 , Springer 2 . [ 10 ] Stella Dafermos, 1988. Sensitivit y analysis in v ariational inequalities. Math- ematics of Op er ations R ese ar ch , 13 : 421–434. [ 11 ] Lester R. F ord, Jr., 1957. Solution of a ranking problem from binary com- parisons. The A meric an Mathematic al Monthly , 64, n. 8, part 2 : 28–33. [ 12 ] Marshall G. Green b erg, 1965. A metho d of successiv e cumulations for scaling of pair-comparison judgments. Psychometrika , 30 : 441–448. [ 13 ] Jobst Heitzig, 2002. Social c hoice under incomplete, cyclic preferences. http://arxiv.org/abs/math/0201285 . [ 14 ] Lawrence J. Hub ert, Phipps Arabie, Jacqueline Meulman, 2001. Combina- torial Data A nalysis: Optimization by Dynamic Pr o gr amming . So ciet y for Industrial and Applied Mathematics. [ 15 ] Nicholas Jardine, Robin Sibson, 1971. Mathematic al T axonomy . Wiley . [ 16 ] James P . Keener, 1993. The P erron-F robenius theorem and the ranking of fo otball teams. SIAM R eview , 35 : 80–93. [ 17 ] David Kinderlehrer, Guido Stampacc hia, 1980. A n Intr o duction to V aria- tional Ine qualities and their Applic ations . Academic Press. 96 R. Camps, X. Mora, L. Sa umell [ 18 ] Kathrin Konczak, J ´ erˆ ome Lang, 2005. V oting pro cedures with incom- plete preferences. Pr o c. Multidisciplinary IJCAI’05 Workshop on A d- vanc es in Pr efer enc e Hand ling (Edinburgh, Scotland, 31 July – 1 August 2005). http://www.irit.fr/recherches/RPDMP/persos/JeromeLang/papers/ konlan05a.pdf . [ 19 ] Edmund Landau, 1914. ¨ Ub er Preisverteilung b ei Spielturnieren. Zeitschrift f¨ ur Mathematik und Physik , 63 : 192–202. [ 20 ] Jean F ran¸ cois Laslier, 2004. L e vote et la r ` egle majoritair e · A nalyse math ´ ematique de la p olitique . CNRS ´ Editions. [ 21 ] Rob ert Duncan Luce, 1959. Individual Choic e Behavior · A The or etic al A nalysis . Wiley . [ 22 ] David G. Luenberger, 1973 1 , 1984 2 . Line ar and Nonline ar Pr o gr amming . Addison-W esley . [ 23 ] Iain McLean, Arnold B. Urk en (eds.), 1995. Classics of So cial Choic e . The Univ ersity of Michigan Press, Ann Arb or. [ 24 ] Boris Mirkin, 1996. Mathematic al Classiﬁc ation and Clustering . Kluw er. [ 25 ] John W. Mo on, Norman J. Pullman, 1970. On generalized tournament ma- trices. SIAM R eview , 12 : 384–399. [ 26 ] Xavier Mora, 2004. Impro ving the Sk ating system · I I : Metho ds and para- do xes from a broader p ersp eciv e. http://mat.uab.cat/ ~ xmora/articles/ iss2en.pdf . [ 27 ] James R. Munkres, 1975. T op olo gy . Prentice-Hall. [ 28 ] Jos´ e Manuel Pita Andrade, ´ Angel A terido F ern´ andez, Juan Manuel Mart ´ ın Garc ´ ıa (eds.), 2000. Corpus V elazque˜ no. Do cumentos y textos (2 vol.). Madrid, Ministerio de Educaci´ on, Cultura y Dep orte. [ 29 ] R. Rammal, G. T oulouse, M. A. Virasoro, 1986. Ultrametricit y for physi- cists. R eviews of Mo dern Physics , 58 : 765–788. [ 30 ] Mathias Risse, 2001. Arro w’s theorem, indeterminacy , and multiplicit y re- considered. Ethics , 111 : 706–734. [ 31 ] W. S. Robinson, 1951. A metho d for c hronologically ordering archaeological dep osits. A meric an Antiquity , 16 : 293–300. [ 32 ] Donald G. Saari, Vincen t R. Merlin, 2000. A geometric examination of Ke- men y’s rule. So cial Choic e and Welfar e , 17 : 403–438. Continuous ra ting for preferential voting , § 18 97 [ 33 ] A. Shuc hat, 1984. Matrix and netw ork mo dels in arc haeology. Mathematics Magazine , 57 (1) : 3–14. [ 34 ] Markus Sch ulze, 1997–2003. [ a ] P osted in the Ele ction Metho ds Mailing List . http://lists.electorama.com/ pipermail/election-methods-electorama.com/1997-October/001544.html (see also ibidem /1998-January/001576.html ). [ b ] Ibidem /1998-August/002044.html . [ c ] A new monotonic and clone-indep endent single-winner election metho d. V oting Matters , 17 (2003) : 9–19. http://www.mcdougall.org.uk/VM/ ISSUE17/I17P3.PDF . [ 35 ] Markus Sch ulze, 2003–2008. A new monotonic, clone- indep enden t, rev ersal symmetric, and Condorcet-consistent single-winner election method. W orking paper, a v ailable at http://home.versanet.de/ ~ chris1-schulze/schulze1.pdf . [ 36 ] W arren D. Smith, Jan Kok, since 2005. R angeV oting.or g · The Center for R ange V oting . http://rangevoting.org/ . [ 37 ] T. Nicolaus Tideman, 1987. Indep endence of clones as a criterion for v oting rules. So cial Choic e and Welfar e , 4 : 185–206. [ 38 ] T. Nicolaus Tideman, 2006. Col le ctive De cisions and V oting: The Potential for Public Choic e . Ashgate Publishing. [ 39 ] Douglas R. W o o dall, 1996. [ a ] Monotonicit y of single-seat preferen tial election rules. V oting Matters , 6 : 9–14. http://www.mcdougall.org.uk/VM/ISSUE6/P4.HTM . [ b ] Monotonicity of single-seat preferential election rules. Discr ete Applie d Mathematics , 77 (1997) : 81–98. [ 40 ] Thomas M. Zavist, T. Nicolaus Tideman, 1989. Complete indep endence of clones in the ranked pairs rule. So cial Choic e and Welfar e , 6 : 167–173. [ 41 ] Ernst Zermelo, 1929. Die Berechn ung der T urnier-Ergebnisse als ein Max- im umproblem der W ahrsc heinlic hkeitsrec hnung. Mathematische Zeitschrift , 29 : 436–460.

A continuous rating method for preferential voting

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