Voting in agreeable societies

V OTING IN A GREEABLE SOCI ETIES DEBORAH E. BER G, SERGUEI NORINE, FRANCIS EDW ARD SU, ROBIN THOMAS, P AUL W OLLAN My ide a of an agr e e able p erson... is a p erson who agr e es with me. —Benjamin Disraeli [8] 1. Introduction When is ag reemen t p ossible? An imp ortan t a sp ect of g roup deci sion-making is the questio n of ho w a group m ak es a c hoice when individual preferen ces ma y d iffer. Clearly , when making a single group c hoice, p eople cannot all h a ve their “ideal” pr eferences, i.e, th e options that they most desire, if th ose ideal p references are differen t. Ho wev er, for the sake of agreemen t, p eople ma y b e willing to accept as a group c hoice an option that is merely “close” to their ideal preferences. V oting is a situation in w h ic h p eople may b eha ve in this w a y . The usu al starting mo del is a one-dimensional p oliti cal sp ectrum , with conserv ative p ositions on the righ t and lib eral p ositions on the left, as in Figure 1. W e call eac h p osition on the sp ectrum a platform that a candidate or vote r may c ho ose to adopt. While a vot er ma y represent her id eal platform by some p oint x on this line, sh e migh t b e willing to vo te for a cand id ate who is p ositio ned at some p oint “close enough” to x , i.e., in an interv al ab ou t x . liberal conservative neutral x [ ] I Figure 1. A one-dimensional p olitic al sp ectrum, with a single vote r’s int erv al of app ro v ed platforms. In this article, we ask the follo win g: giv en such preferen ces on a p olitical sp ectrum, when can we guaran tee that some fraction (say , a ma jorit y) of the p opulation w ill agree on some candidate? By “agree”, w e mean in the sense of appr oval voting , in wh ic h vo ters declare wh ic h candidates they fin d acceptable. Appro v al v oting has not yet b een adopted for p olitical elections in the United States. Ho w- ev er, man y scient ific and m athematical so cieties, such as the Mathemati cal Asso ciation of America and the American Mathematica l So ciet y , use appro v al voting for their elections. Ad- ditionally , countries other than the United States ha v e u s ed approv al vo ting or an equiv alen t system; for details, see Brams and Fish bur n [2 ] wh o discus s the adv antag es of appro v al vo ting. Understanding whic h candidates can get v oter appr o v al can b e helpful when there are a large n umber of candidates. An extreme example is the 2003 California gub ernatorial recall election, whic h had 135 cand idates in the mix [6]. W e m igh t imagine these candidates p ositio ned at 135 p oin ts on the line in Figure 1, w h ic h w e think of as a subset of R . If eac h California v oter approv es of candidates “close enough” to her ideal platform, we ma y ask u nder what conditions there is a candidate that win s the appr o v al of a ma jorit y of th e v oters. 1 2 BERG , NORINE, SU, THOMAS, WOLLAN In this setting, we ma y assume that eac h v oter’s set of app ro v ed platforms (her appr oval set ) is a closed inte rv al in R , and that there is a set of candidates w ho tak e up p ositions at v arious p oints along this p oliti cal sp ectrum . W e shall call th is sp ectrum with a collection of candidates and v oters, together with vote rs’ approv al sets, a line ar so ciety (a more precise definition w ill b e give n so on). W e shall sa y th at the linear so ciet y is sup er-agr e e able if for eve ry pair of v oters there is some candidate that th ey wo uld b oth appr ov e, i.e., eac h pair of appro v al sets con tains a candidate in their intersectio n. F or linear so cieties this “lo cal” cond ition guarante es a strong “global” prop ert y , namely , that there is a candidate that eve ry vo ter appro v es! As w e sh all see in Theorem 5, this can b e view ed as a consequ ence of Helly’s theorem ab out inte rsections of con v ex sets. But p erhaps this is to o strong a conclusion. Is th ere a w eak er lo cal condition that wo uld guaran tee that only a ma jorit y (or some other fraction) of the voters would approv e a p articular candidate? F or instance, w e relax the condition ab ov e and call a linear so ciet y agr e e able if among ev ery three v oters, s ome pair of vot ers appro v e the same candidate. T hen it is n ot hard to s ho w: Theorem 1. In an agr e e able line ar so ciety, ther e is a c andidate who has the appr oval of at le ast half the voters. More generally , call a linear so ciet y ( k , m ) -agr e e able if it has at least m v oters, and among ev ery m vot ers, some su bset of k vo ters appr o ve the same candidate. Then our main theorem is a generalization of the previous result: Theorem 2 (Th e Agreeable Linear So cie t y Th eorem) . L et 2 ≤ k ≤ m . In a ( k , m ) -agr e e able line ar so ciety of n voters, ther e is a c andidate who has the appr oval of at le ast n ( k − 1 ) / ( m − 1) of the voters. W e pr ov e a sligh tly more general result in Theorem 8 and also b riefly study so cieties whose appro v al sets are con v ex su bsets of R d . As an example, consider a cit y with fourteen restaurants along its main b oulev ard : A B C D E F G H I J K L M N and su p p ose ev ery resident dines only at the fiv e restauran ts closest to his/her house (a set of consecutiv e r estauran ts, e.g., D E F GH ). A consequence of Theorem 1 is that there must b e a restaurant that is p atronized by at least half th e residents. Wh y? Th e pigeonhole pr inciple guaran tees that among ev ery 3 residents, eac h c ho osing 5 of 14 restaurants, there m ust b e a restauran t appr ov ed by at least 2 of them; hence this linear so ciet y is agreeable and Th eorem 1 applies. F or an example of Theorem 2, see Figure 3, which shows a (2 , 4)-agree able linear so ciet y , and indeed there are candidates that receiv e at least 1 / 3 of the vote s (in this case ⌈ 7 / 3 ⌉ = 3). W e shall b egin with some definitions, and explain connections to classical con v exit y the- orems, graph colorings, an d m aximal cliques in graphs. Then we pr o ve Th eorem 2, discuss extensions to h igher-d imensional sp ect ra, and conclude with some qu estions for furth er stu dy . 2. Definitions In this section, we fix terminology and explain the b asic concepts up on wh ic h our resu lts rely . Let us supp ose th at th e set of all p ossible preferen ces is mo deled by a set X , called the sp e ctrum . Eac h element of the sp ect rum is a platform . Assu me th at there is a finite s et V of voters , and eac h vo ter v has an appr oval set A v of platforms. VOTING IN A GREEABLE SOCIETIES 3 W e defin e a so ci ety S to b e a triple ( X , V , A ) consisting of a sp ectrum X , a set of v oters V , and a collec tion A of appro v al sets for all the v oters. Of particular interest to us will b e the case of a line ar so ciety , in w hic h X is a closed su bset of R and approv al sets in A are of th e form X ∩ I where I is either empt y or a closed b oun ded in terv al in R . In general, how ev er, X could b e an y set and the collection A of appro v al sets could b e an y class of su bsets of X . In Figure 2 w e illus trate a linear so ciet y , where for ease of disp la y we ha ve separated the app ro v al sets v ertically so that they can b e distinguished. 1 2 3 4 6 5 7 Figure 2. A linear so ciet y with infinite s p ectrum: eac h interv al (sho wn here displaced ab o ve the sp ectrum) corresp onds to the appr o v al set of a vote r. The shaded region indicates platforms with agreement num b er 4. This is a (2 , 3)- agreeable so ciet y . Our motiv ation for considering inte rv als as appr ov al sets arises from imagining th at voters ha v e an “ideal” platform along a lin ear scale (similar to Co om bs’ J -scale [7]), an d that vot ers are willing to app ro v e “nearby” platforms, yielding app ro v al sets that are connected interv als. Unlik e the Co ombs scaling theory , h o wev er, w e are not concerned w ith the order of pr eference of appro ve d platforms; all platforms within a vote r’s appro v al set h av e equiv alen t status as “appro v ed” by that v oter. W e also note that w hile we mo d el our linear scale as a su b set of R , none of our results ab out linear so cieties d ep ends on the m etric; we only app eal to the ordin al prop erties of R . W e hav e seen that p olitics pro vides natur al examples of linear so cieties. F or a different example, X could rep r esen t a temp erature scale, V a set of p eople that liv e in a hou s e, and eac h A v a range of temp eratures that p erson v finds comfortable. Th en one may ask: at what temp erature should the thermostat b e set so as to satisfy the largest num b er of p eople? 1 2 3 4 6 5 7 ^ ^ Figure 3. A linear so ciet y with a sp ectrum of tw o candidates (at p latforms mark ed by carats): tak e the approv al sets of the so ciet y of Figure 2 and int ersect with these candidates. It is a (2 , 4)-agreeable linear so ciet y . Tw o sp ecial cases of linear so ciet ies are worth ment ioning. When X = R w e sh ould think of X as an in fi nite sp ectrum of platforms that p ote nti al cand id ates m igh t adopt. Ho w ev er, in practice ther e are n orm ally only finitely many candid ates. W e mo del that situation by letting 4 BERG , NORINE, SU, THOMAS, WOLLAN X b e the set of p latforms adopted by actual candidates. Thus one could th in k of X as either the set of all p latforms, or the set of (platforms adopted by) candidates. See Figures 2 and 3 . Let 1 ≤ k ≤ m b e integ ers. Call a so ciet y ( k , m )- agr e e able if it has at least m v oters, and for an y su bset of m v oters, there is at least one platform that at least k of them can agree up on, i.e., th er e is a p oint common to at least k of the v oters’ appr o v al sets. T h us to b e (2 , 3)-agreea ble is the same as to b e agr e e able , and to b e (2 , 2)-agreeable is the s ame as to b e sup er-agr e e able , as d efined earlier. One ma y c heck that the s o ciet y of Figure 2 is (2 , 3)-agreea ble. I t is not (3 , 4)-agreeable, ho w ev er, b eca use among v oters 1 , 2 , 4 , 7 no thr ee of them share a common p latform. The same so ciet y , after restricting the sp ectrum to a set of candidates, is the lin ear so ciet y sh o wn in Figure 3. It is not (2 , 3)-agreea ble, b ecause among v oters 2 , 4 , 7 th ere is n o pair that can agree on a candidate (in fact, vo ter 7 do es n ot appr o ve an y candidate). Ho wev er, one ma y v erify that this linear so ciet y is (2 , 4)-agreeable. F or a so ciet y S , the agr e ement numb er of a platform, a ( p ), is the n umber of v oters in S wh o appro v e of platform p . Th e agr e ement numb er a ( S ) of a so ciet y S is the maximum agreemen t n umber o v er all p latforms in the sp ectrum, i.e., a ( S ) = max p ∈ X a ( p ) . The agr e ement pr op ortion of S is simply the agreemen t n umber of S divided by the n umber of v oters of S . This concept is u s eful w hen we are interested in p ercen tages of the p opulation rather th an th e num b er of v oters. T he so ciet y of Figure 2 has agreemen t num b er 4, whic h can b e seen where the shaded rectangle co vers the set of platforms that ha v e maximum agreemen t n umber. 3. Hell y’s Theor em and Supe r-A greeable S ocieties Let us sa y that a so ciet y is R d - c onvex if th e sp ectrum is R d and eac h appr o v al set is a closed conv ex sub set of R d . Note th at an R 1 -con vex s o ciet y is a linear so ciet y with sp ectrum R . An R d -con vex so ciet y can arise w h en considering a multi-dimensional sp ectrum , suc h as when ev aluating p olitical p latforms o v er seve ral axes (e.g., conserv ativ e vs . lib eral, pacifist vs. militant , in terv en tionist vs. isolatio nist). Or, the sp ectrum might b e array ed o v er more p ersonal dimensions: the dating website eHarmony claims to u se u p to 29 of them [9]. In s u c h situations, the conv exit y of app r o v al sets migh t, for instance, follo w from an in d ep end en ce-of- axes assumption and con v exit y of appro v al sets along eac h axis. T o fin d the agreement pr op ortion of an R d -con vex s o ciet y , we turn to w ork concerning in tersections of con v ex sets. T h e most w ell-kno wn result in this area is Helly’s theorem. Th is theorem w as pro v en by Helly in 1913, b ut the r esu lt was not pub lish ed unt il 1921, by Rad on [17]. Theorem 3 (Helly) . Given n c onvex sets in R d wher e n > d , if every d + 1 of them interse ct at a c ommon p oint, then they al l interse ct at a c ommon p oint. Helly’s th eorem has a nice in terpretation for R d -con vex so cieties: Corollary 4. F or every d ≥ 1 , a ( d + 1 , d + 1) -agr e e able R d -c onvex so c iety must c ontain at le ast one platform that is appr ove d by al l voters. Notice that for the corollary to hold for d > 1 it is imp ortan t th at the sp ectrum of an R d - c onvex so ciet y b e all of R d . Ho wev er, for d = 1 that is not necessary , as we now sho w. VOTING IN A GREEABLE SOCIETIES 5 Theorem 5 (T h e Sup er-Agreeable Linear So ciet y Theorem) . A sup er-agr e e able line ar so ciety must c ontain at le ast one platform that is appr ove d by al l voters. W e provide a simple pr o of of this theorem, since th e result will b e needed later. Wh en the sp ectrum is all of R , this theorem is ju st Helly’s theorem for d = 1; a pro of of Helly’s theorem for general d ma y b e foun d in [15]. Pr o of. Let X ⊆ R denote the sp ectrum. Since eac h v oter v agrees on at least one p latform with ev er y other v oter, we see that the approv al sets A v m ust b e nonempty . Let L v = min A v , R v = max A v , and let x = max v { L v } and y = min v { R v } . The first tw o minima/maxima exist b ecause eac h A v is compact; the last t w o exist b ecause the num b er of vo ters is fi nite. ^ ^ ^ ^ y x Figure 4. A s u p er-agreeable linear so ciet y of 6 vote rs and 4 candidates, with agreemen t num b er 6. W e claim that x ≤ y . Why? S ince every pair of appro v al sets inte rsect in some platform, w e see that L i ≤ R j for ev ery pair of v oters i, j . In particular, let i b e the vote r w h ose L i is maximal and let j b e the v oter whose R j is minimal. Hence x ≤ y and ev ery app ro v al set con tains all platforms of X that are in the n onempt y inte rv al [ x, y ], and in particular, the platform x .  The idea of th is p ro of can b e easily extended to furn ish a pro of of Theorem 1. Pr o of of The or em 1. Using the same notations as in the prior p ro of, if x ≤ y then that pr o of sho ws that every appro v al set conta ins th e platform x . Otherwise x > y implies L i > R j so that A j and A i do not con tain a common platform. W e claim that for an y other v oter v , the appro v al set A v con tains either platform x or y (or b oth). After all, the so ciet y is agreeable, so some pair of A i , A j , A v m ust con tain a common platform; by the remarks ab ov e it must b e th at A v in tersects one of A i or A j . If A v do es not con tain x = L i then since L v ≤ L i (b y defin ition of x ), we must ha ve that R v < L i and A v ∩ A i do es not conta in a p latform. Then A v ∩ A j m ust con tain a platform; h ence L v ≤ R j . Since R j ≤ R v (b y defin ition of y ), the p latform y = R j m ust b e in A v . Th us ev ery ap p ro v al set con tains either x or y , and b y the pigeonhole p rinciple one of them m ust b e con tained in at least h alf the app ro v al sets.  Pro ving the more general Theorem 2 will tak e a little more work. 4. The A greement Graph of Linear Societies is Per fect T o und erstand ( k , m )-agreeabilit y , it will b e h elpful to use a graph to represen t th e inte rsec- tion relation on approv al s ets. Re call that a gr aph G consists of a fi nite set V ( G ) of vertic es and a set E ( G ) of 2-elemen t su bsets of V ( G ), called e dges . If e = { u, v } is an edge, then we 6 BERG , NORINE, SU, THOMAS, WOLLAN sa y that u, v are the ends of e , and that u and v are adjac ent in G . W e use uv as shorthand notation for the edge e . Giv en a so ciet y S , w e construct the agr e ement gr aph G of S by letting the v ertices V ( G ) b e the vo ters of S and the ed ges E ( G ) b e all p airs of v oters u, v whose appr o v al sets in tersect eac h other. Th us u and v are connected by an edge if there is a platform that b oth u and v w ould approv e. Note that the agreemen t grap h of a so ciet y with agreemen t n umber equal to the n umber of vo ters is a complete graph (b ut the con verse is false in higher dimens ions, as w e discuss later). Also note that a v ertex v is isolated if A v is empt y or disjoin t from other appro v al sets. 6 5 4 7 2 1 3 Figure 5. The agreement graph for the so ciet y in Figure 2. Note that vote rs 4 , 5 , 6 , 7 f orm a maximal clique th at corresp onds to the maximal agreement n umber in Figure 2. The clique numb er of G , wr itten ω ( G ), is the greatest intege r q suc h that G has a set of q pairwise adjacen t v ertices, called a clique of size q . By restricting our atten tion to members of a clique, and app lying the Su p er-Agreeable Linear So ciet y Theorem, we see that there is a platform th at h as the appro v al of eve ry mem b er of a clique, and hence: F act 1. F or the agr e ement gr aph of a line ar so ciety, the clique nu mb er of the gr aph is the agr e ement numb er of the so ciety. This fact do es not necessarily hold if the so ciet y is not linear. F or instance, it is easy to construct an R 2 -con vex so ciet y with three voters suc h that ev er y t w o vo ters agree on a platform, bu t all th ree of them do not. It d o es, how ev er hold in R d for b ox so cieties , to b e discussed in Section 6. No w, to get a h andle on the clique num b er, we shall make a connection b et w een the clique n umber and colorings of the agreemen t graph. Th e chr omatic numb e r of G , wr itten χ ( G ), is the minimum num b er of colors necessary to color the vertices of G suc h that no tw o adjacen t v ertices hav e the same color. Thus tw o v oters may ha ve the same color as long as they do not agree on a platform. Note that in all cases, χ ( G ) ≥ ω ( G ). A graph G is called an interval gr aph if we can assign to eve ry ve rtex x a closed interv al or an emp t y set I x ⊆ R s u c h that xy ∈ E ( G ) if and only if I x ∩ I y 6 = ∅ . W e hav e: F act 2. The agreemen t graph of a linear so ciet y is an inte rv al graph. T o see that F act 2 holds let th e linear so ciet y b e ( X , V , A ), and let the v oter app ro v al sets b e A v = X ∩ I v , wh ere I v is a closed b ounded interv al or empty . W e ma y assume that eac h I v is a minimal closed interv al satisfying A v = X ∩ I v ; then the inte rv als { I v : v ∈ V } pr o vid e an in terv al repr esen tation of the agreement graph , as d esired. An i nduc e d sub gr aph of a graph G is a graph H such that V ( H ) ⊆ V ( G ) and the edges of H are the edges of G that ha v e b oth ends in V ( H ). If every ind u ced sub grap h H of a graph VOTING IN A GREEABLE SOCIETIES 7 G satisfies χ ( H ) = ω ( H ), then G is called a p erfe ct g r aph ; see, e.g., [18]. T h e follo wing is a standard fact [20] ab out interv al graphs: Theorem 6. Interval gr aphs ar e p erfe c t. Pr o of. Let G b e an in terv al graph, and f or v ∈ V ( G ), let I v b e the interv al repr esen ting the v ertex v . S in ce eve ry in d uced su bgraph of an in terv al graph is an in terv al graph, it su ffices to sho w that χ ( G ) = ω , wh er e ω = ω ( G ). W e pro ceed by induction on | V ( G ) | . T he assertion holds for the null graph, and so w e m ay assume th at | V ( G ) | ≥ 1, and that the s tatemen t holds for all sm aller grap h s. Let us select a v ertex v ∈ V ( G ) suc h that th e righ t end of I v is as s m all as p ossible. It follo ws that the elemen ts of N , the set of neigh b ors of v in V ( G ), are p airwise adjacen t b ecause their in terv als must all con tain th e righ t end of I v , and hence | N | ≤ ω − 1. See Figure 6. By the inductive hyp othesis, the graph G \{ v } obtained from G by d eleting v can b e colored using ω colors, and since v has at most ω − 1 n eigh b ors, this coloring can b e extended to a coloring of G , as desired.  v I I w Figure 6. If I v , I w in tersect and the right end of I v is smaller than the right end of I w , then I w m ust con tain the r igh t end of I v . The p erfect graph prop ert y w ill allo w us, in the n ext section, to make a crucial connection b et we en the ( k , m )-agreeabilit y condition and the agreemen t num b er of the so ciet y . Giv en its imp ortance in our setting, it is wo rth making a few comments ab out ho w p er f ect graph s app ear in other con texts in mathematics, theoretical computer science, and op erations r esearc h. Th e concept was in tro du ced in 1961 b y Berge [1], w ho was motiv ated by a question in comm un i- cation theory , sp ecifica lly , the determination of the Shannon capacit y of a graph [19]. Chv´ atal later d isco vered that a certain class of linear p rograms alw a ys ha v e an in tegral solution if and only if the corresp ondin g matrix arises from a p erfect graph in a sp ecified wa y [5, 18, 3 ]. As p ointe d out in [18], algorithms to solv e semi-definite programs grew out of the theory of p erfect graphs. It h as b een prov en recen tly [4] that a graph is p erfect if and only if it has no indu ced subgraph isomorphic to a cycle of o dd length at least fi ve, or a complement of such a cycle. 5. ( k , m ) -A gre eable Linear Societies W e no w use the connection b etw een p erfect graphs , the clique num b er, and the c hromatic n umber to obtain a lo w er b ou n d for the agreemen t num b er of a ( k , m )-agreeable linear so ciet y (Theorem 8). W e fi rst need a lemma th at sa ys that in the corresp onding agreemen t graph , the ( k , m )-agreeable condition preven ts an y coloring of the graph fr om ha ving to o many v ertices of the same color. T h us, th ere m ust b e many colors and, since the graph is p erfect, the clique n umber m ust b e large as w ell. Lemma 7. G iven inte gers m ≥ k ≥ 2 , let p ositive inte gers q , ρ b e define d by the division with r emainder: m − 1 = ( k − 1) q + ρ , wher e 0 ≤ ρ ≤ k − 2 . L et G b e a gr aph on n ≥ m vertic es with chr omatic numb er χ such that every subset of V ( G ) of size m includes a cliqu e of size k . Then n ≤ χq + ρ , or χ ≥ ( n − ρ ) /q . 8 BERG , NORINE, SU, THOMAS, WOLLAN Pr o of. Let the graph b e colored using the colors 1 , 2 , . . . , χ , and f or i = 1 , 2 , . . . , χ let C i b e the set of vertices of G colored i . W e may assume, b y p erm uting the colors, that | C 1 | ≥ | C 2 | ≥ · · · ≥ | C χ | . Since C 1 ∪ C 2 ∪ · · · ∪ C k − 1 is colored using k − 1 colors, it includes no clique of size k , and hence, | C 1 ∪ C 2 ∪ · · · ∪ C k − 1 | ≤ m − 1. It follo ws that | C k − 1 | ≤ q , for otherwise | C 1 ∪ C 2 ∪ · · · ∪ C k − 1 | ≥ ( k − 1 )( q + 1) ≥ ( k − 1 ) q + ρ + 1 = m , a con tradiction. Thus | C i | ≤ q for eac h i ≥ k and n = k − 1 X i =1 | C i | + χ X i = k | C i | ≤ m − 1 + ( χ − k + 1) q = ( k − 1) q + ρ + ( χ − k + 1) q = χq + ρ, as desired.  Theorem 8. L et 2 ≤ k ≤ m . If G is the agr e ement gr aph of a ( k, m ) -agr e e able line ar so c i ety and q , ρ ar e define d by the divi si on with r emainder: m − 1 = ( k − 1) q + ρ , ρ ≤ k − 2 , then the clique numb er satisfies: ω ( G ) ≥ ⌈ ( n − ρ ) /q ⌉ , and this b ound is b est p ossible. It fol lows that this is also a lower b ound on the agr e ement numb er, and henc e every line ar ( k , m ) -agr e e able so ciety has agr e ement pr op ortion at le ast ( k − 1) / ( m − 1) . Pr o of. By F act 2 and T h eorem 6 the graph G is p erfect. Thus the chromatic num b er of G is equal to ω ( G ), and hence ω ( G ) ≥ ⌈ ( n − ρ ) /q ⌉ by Lemma 7, as desired. The second assertion follo w s from F act 1 and n oting th at ( n − ρ )( m − 1) = n ( k − 1) q + nρ − ρ ( m − 1) = n ( k − 1) q + ρ ( n − m + 1) ≥ n ( k − 1) q , fr om wh ic h we see that ( n − ρ ) /q ≥ n ( k − 1) / ( m − 1). Let us observ e that the b ound ⌈ ( n − ρ ) /q ⌉ in T heorem 8 is b est p ossible. Indeed, let I 1 , I 2 , . . . , I q b e disjoint in terv als, for i = q + 1 , q + 2 , . . . , n − ρ let I i = I i − q , and let I n − ρ +1 , I n − ρ +2 , . . . , I n b e pairwise disjoint and disjoint from all the p revious interv als, e.g., see Figure 7. Then the so ciet y with approv al sets I 1 , I 2 , . . . , I n is ( k , m )-agreeable and its agreemen t graph has clique num b er ⌈ ( n − ρ ) /q ⌉ .  Figure 7. A linear (4 , 15)-so ciet y with n = 21 vo ters. Here q = 4 and ρ = 2, so the clique n umber is at least ⌈ ( n − ρ ) /q ⌉ = 5. The Agreeable Linear So ciet y Theorem (Th eorem 2) n ow follo w s as a corollary of Theorem 8. 6. R d -convex a nd d -bo x Societies In this section w e pr o ve a h igher-dimensional analogue of Theorem 8 by giving a lo wer b ound on the agreemen t p rop ortion of a ( k , m )-agreeable R d -con vex s o ciet y . W e need a d ifferen t metho d than our metho d for d = 1, b ecause for d ≥ 2, neither F act 1 n or F act 2 h olds. Also, we remark that, unlike our r esults on linear so cieties, our results in this section ab out the agreemen t prop ortio n for platforms will n ot necessarily hold when restricting the sp ect rum to a finite s et of candidates in R d . VOTING IN A GREEABLE SOCIETIES 9 W e shall use the follo wing generalization of Helly’s theorem, d ue to Kalai [11]. Theorem 9 (The F ractional Helly’s Theorem) . L et d ≥ 1 and n ≥ d + 1 b e inte gers, let α ∈ [0 , 1] b e a r e al numb er, and let β = 1 − (1 − α ) 1 / ( d +1) . L et F 1 , F 2 , . . . , F n b e c onvex sets in R d and assume that for at le ast α  n d +1  of the ( d + 1) -element index sets I ⊆ { 1 , 2 , . . . , n } we have T i ∈ I F i 6 = ∅ . Then ther e exists a p oint i n R d c ontaine d i n at le ast β n of the sets F 1 , F 2 , . . . , F n . The follo wing is the promised analogue of Theorem 8. Theorem 10. L et d ≥ 1 , k ≥ 2 and m ≥ k b e inte g ers, wher e m > d . Then eve ry ( k , m ) - agr e e able R d -c onvex so ciety has agr e ement pr op ortion at le ast 1 −  1 −  k d +1  m d +1   1 / ( d +1) . Pr o of. Let S b e a ( k , m )-agreeable R d -con vex so ciet y , and let A 1 , A 2 , . . . , A n b e its v oter ap- pro v al sets. Let us call a set I ⊆ { 1 , 2 , . . . , n } go o d if | I | = d + 1 and T i ∈ I A i 6 = ∅ . By T h eorem 9 it suffices to show that there are at least  k d +1  n d +1  m d +1  go o d sets. W e will d o th is b y count - ing in t wo differen t w a ys the n umber N of all p airs ( I , J ), where I ⊆ J ⊆ { 1 , 2 , . . . , n } , I is go o d, and | J | = m . Let g b e the n umber of go o d s ets. Since ev ery go o d s et is of size d + 1 and extends to an m -elemen t s ubset of { 1 , 2 , . . . , n } in  n − d − 1 m − d − 1  w a y s , we ha v e N = g  n − d − 1 m − d − 1  . On the other hand, every m -elemen t set J ⊆ { 1 , 2 , . . . , n } includes at least one k -element set K with T i ∈ K A i 6 = ∅ (b eca use S is ( k , m )-agreeable) , an d K in turn includ es  k d +1  go o d sets. Th us N ≥  k d +1  n m  , and h ence g ≥  k d +1  n d +1  m d +1  , as d esired.  F or d = 1, Theorem 10 giv es a worse b ound th an Th eorem 8, and hence for d ≥ 2, the b ound is m ost lik ely n ot b est p ossible. Ho w ev er, a p ossible improv emen t must use a d ifferent metho d, b ecause the b ound in Theorem 9 is b est p ossible. A b ox in R d is the C artesian pro duct of d closed interv als, and we sa y that a societ y is a d - b ox so ciety if eac h of its appr o v al sets is a b o x in R d . By pro jection onto eac h axis, it follo ws from Theorem 5 that d -b o x so cieties satisfy the conclusion of F act 1 (namely , that the clique n umb er equals the agreemen t n umber), and hence their agreemen t graph s capture all the essential information ab out the so ciet y . Unfortu n ately , agreemen t graphs of d -b ox so cieties are, in general, n ot p erfect. F or in stance, th er e is a 2-b ox so ciet y (Figure 8) wh ose agreemen t graph is th e cycle on fi v e vertic es; h ence its chromatic num b er is 3 bu t its clique num b er is 2. F or k ≤ m ≤ 2 k − 2, the f ollo wing theorem will resolv e the agreement prop ortion problem for all ( k , m )-agreea ble so cieties satisfying the conclusion of F act 1, an d hen ce for all ( k , m )- agreeable d -b o x so cieties w h ere d ≥ 1 (Theorem 13). Theorem 11. L et m, k ≥ 2 b e inte ge rs with k ≤ m ≤ 2 k − 2 , and let G b e a gr aph on n ≥ m vertic es such that every subset of V ( G ) of size m includes a clique of size k . Then ω ( G ) ≥ n − m + k . Before we embark on a pro of let us make a few comments. First of all, the b ou n d n − m + k is b est p ossible, as sho wn b y the graph consisting of a clique of size n − m + k and m − k isolated v ertices. Second, the conclusion ω ( G ) ≥ n − m + k im p lies that ev ery sub set of V ( G ) of s ize m includes a clique of size k , and so the t wo statemen ts are equiv alent und er the hypothesis that k ≤ m ≤ 2 k − 2. Finally , this hyp othesis is necessary , b ecause if m ≥ 2 k − 1, then for n ≥ 2( m − k + 1), th e disj oin t union of cliques of sizes ⌊ n/ 2 ⌋ and ⌈ n/ 2 ⌉ satisfies th e hypothesis of Theorem 11, but not its conclusion. 10 BERG , NORINE, SU, THOMAS, WOLLAN Figure 8. A 2-b ox so ciet y w hose agreement graph is a 5-cycle. A vertex c over of a graph G is a set Z ⊆ V ( G ) such that ev ery edge of G h as at least one end in Z . W e say a set S ⊆ V ( G ) is stable if no edge of G has b oth ends in S . W e dedu ce Theorem 11 f rom th e follo win g lemma. Lemma 12. L et G b e a gr aph with vertex c over of size z and none of size z − 1 such that G \{ v } has a vertex c over of size at most z − 1 for al l v ∈ V ( G ) . Then | V ( G ) | ≤ 2 z . Pr o of. Let Z b e a ve rtex co v er of G of size z . F or ev ery v ∈ V ( G ) − Z let Z v b e a vertex cov er of G \{ v } of size z − 1, and let X v = Z − Z v . Then X v is a stable set. F or X ⊆ Z let N ( X ) denote the set of neigh b ors of X outside Z . W e h a ve v ∈ N ( X v ) and N ( X v ) − { v } ⊆ Z v − Z , and so | X v | = | Z − Z v | = | Z | − | Z ∩ Z v | = | Z v | + 1 − | Z ∩ Z v | = | Z v − Z | + 1 ≥ | N ( X v ) | . On the other hand, if X ⊆ Z is stable, then | N ( X ) | ≥ | X | , for otherwise ( Z − X ) ∪ N ( X ) is a v ertex co v er of G of size at most z − 1, a cont radiction. W e ha v e (1) | Z | ≥ | [ X v | ≥ | [ N ( X v ) | ≥ | V ( G ) | − | Z | , where b oth unions are o ver all v ∈ V ( G ) − Z , and hence | V ( G ) | ≤ 2 z , as requir ed . T o see that the second inequalit y h olds let u, v ∈ V ( G ) − Z . Th en | X u ∪ X v | = | X u | + | X v | − | X u ∩ X v | ≥ | N ( X u ) | + | N ( X v ) | − | N ( X u ∩ X v ) | ≥ | N ( X u ) | + | N ( X v ) | − | N ( X u ) ∩ N ( X v ) | = | N ( X u ) ∪ N ( X v ) | , and, in general, the second inequalit y of (1) follo ws b y induction on | V ( G ) − Z | .  Pr o of of The or em 11. W e pro ceed by ind u ction on n . I f n = m , then the conclusion certainly holds, and so we ma y assume that n ≥ m + 1 and that the th eorem holds for graphs on few er than n v ertices. W e may assume th at m > k , for otherwise the h yp othesis implies that G is the complete graph. W e ma y also assume that G has tw o n onadjacen t vertice s, sa y x and y , for otherw ise the conclusion holds. Then in G , ev ery clique conta ins at most one of x, y , so in the graph G \{ x, y } ev ery set of vertic es of size m − 2 includes a clique of size k − 1. S ince k − 1 ≤ m − 2 ≤ 2( k − 1) − 2 we deduce by th e inductiv e hypothesis that ω ( G ) ≥ ω ( G \ { x, y } ) ≥ n − 2 − ( m − 2) + k − 1 = n − m + k − 1. W e may assume in the last statemen t that equalit y holds through ou t, b ecause otherwise G satisfies the conclusion of the theorem. Let ¯ G den ote the complement of G ; that is, the grap h w ith v ertex set V ( G ) and VOTING IN A GREEABLE SOCIETIES 11 edge set consisting of precisely those p airs of distinct vertice s of G th at are not adjacen t in G . Let us notice that a set Q is a clique in G if and only if V ( G ) − Q is a ve rtex co v er in ¯ G . Th u s ¯ G has a ve rtex co v er of size m − k + 1 and none of size m − k . Let t b e the least in teger such that t ≥ m and ¯ G has an indu ced su bgraph H on t v ertices with no v ertex co v er of size m − k . W e claim that t = m . In d eed, if t > m , then the minimalit y of t implies that H \{ v } has a ve rtex co ver of size at most m − k for ev ery v ∈ V ( H ). Th us by L emma 12 t = | V ( H ) | ≤ 2( m − k + 1) ≤ m < t , a contradicti on. Thus t = m . By h y p othesis, the graph ¯ H has a clique Q of size k , b ut V ( H ) − Q is a v ertex co v er of H of size m − k , a con tradiction.  Theorem 13. L et d ≥ 1 and m, k ≥ 2 b e i nte g ers with k ≤ m ≤ 2 k − 2 , and let S b e a ( k , m ) - agr e e able d -b ox so ciety with n voters. Then the agr e ement numb er of S is at le ast n − m + k , and this b ound is b est p ossible. Pr o of. The agreemen t graph G of S s atisfies the hyp othesis of Theorem 11, and h en ce it has a clique of size at least n − m + k b y th at th eorem. Since d -b o x so cieties satisfy the conclusion of F act 1, the fi rst assertion follo ws. The b ound is b est p ossible, b ecause the graph consisting of a clique of size n − m + k an d m − k isolate d v ertices is an interv al graph.  7. Discussion As we h a ve seen, set in tersection theorems can pro vide a usefu l framework to mo del and understand the relationships b et we en sets of p references in v oting, and this con text leads to new mathematical questions. W e su ggest several directions which the reader may wish to explore. Recen t r esults in discrete geometry ha v e so cial in terpretations. T he piercing num b er [12] of approv al sets can b e int erpreted as the minimum n um b er of platforms that are necessary suc h that ev ery one h as some p latform of w hic h he or she app ro v es. Set intersectio n th eorems on other spaces (su ch as tr ees an d cycles) are derived in [16] as an extension of b oth Helly’s theorem and the KKM lemma [13]; as an application the authors sho w that in a su p er-agreeable so ciet y with a circular p olitical s p ectrum, there m u st b e a platform that has the appr o v al of a strict ma jorit y of v oters (in contrast with Theorem 5). Ch r is Hardin [10] has recen tly pro vided a generalizati on to ( k , m )-agreeable s o cieties with a circular p olitical sp ectrum. What results can b e ob tained for other sp ectra? The most natural pr oblem seems to b e to determine the agreemen t p rop ortion for R d -con vex and d -b ox ( k , m )-agreeable so ciet ies. The smallest case wh ere w e do n ot know the answer is d = 2, k = 2, and m = 3. Ra jneesh Hegde (priv ate comm unication) found an examp le of a (2 , 3)-agreeable 2-b ox so ciet y with agreemen t prop ortion 3 / 8. There ma y v ery well b e a nice formula, b ecause for eve ry fi xed inte ger d the agreemen t num b er of a d -b o x so ciet y can b e computed in p olynomial time. This is b ecause the clique n umber of the corresp onding agreemen t graph (also known as a graph of b oxicity at most d ) can b e determined by an easy p olynomial-time algorithm. On the other h and, for ev ery d ≥ 2 it is NP-hard to decide w h ether an in put graph has b o xicit y at most d [14, 21]. (F or d = 1 this is the same as testing w h ether a grap h is an in terv al graph, and that can b e done efficiently .) P assing fr om r esu lts ab out platforms in so cieties to r esults ab out a finite s et of candidates app ears to b e tric ky in dimensions greater than 1. Are th ere tec hn iques or additional h yp othe- ses that wo uld giv e u seful results ab out th e existence of candidates w h o h a ve strong app ro v al in so cieties with multi -dimensional sp ectra? W e may also question our assump tions. While con v exit y seems to b e a rational assu mption for appro v al sets in the linear case, in m ultiple dimens ions additional considerations ma y 12 BERG , NORINE, SU, THOMAS, WOLLAN b ecome imp ortant . One migh t also explore the p ossibilit y of disconnected approv al sets: w hat is the agreemen t pr op ortion of a ( k, m )-agreeable so ciet y in wh ich ev ery approv al set has at most t w o comp onents? One might also consider v aryin g lev els of agreemen t. F or instance, in a d -b o x so ciet y , t wo v oters might n ot agree on every axis, so th eir approv al sets d o not intersect, b u t it migh t b e the case that many of th e pro jections of their app ro v al sets d o. In this case, one may wish to consider an agreemen t graph with weigh ted edges. Finally , w e might wonder ab out the agreemen t parameters k and m f or v arious r eal-wo rld issues. F or ins tance, a so ciet y consid ering outlawing murder wo uld probably b e m uc h more agreeable than that same so ciet y considering tax reform. C urrently , we can empir ically measure these parameters only by surveyi ng large num b ers of p eople ab out th eir preferences. It is in teresting to sp eculate ab out metho d s f or estimating suitable k and m from limited data. This article grew out of the observ ation that Helly’s theorem, a classical result in con vex ge- ometry , has an interesting voting interpretation. This led to the deve lopment of mathematical questions and theorems whose int erpretations yield desirab le conclusions in th e v oting cont ext, e.g., Th eorems 2, 8, 10, and 13. It is nice to see that when classical theorems h a ve interesting so cial interpretations, the so cial con text can also motiv ate the study of n ew mathematical questions. 8. Ackno wledgemen ts The authors wish to thank a referee for many v aluable su ggestions, and to th e Monthl y Editor for careful reading and detailed comments that im p ro v ed the presentat ion of the p ap er. The au th ors gratefully ackno wledge su pp ort b y these NS F Grants: Berg by DMS-0301129, Norine b y DMS-0200595, S u by DMS-03011 29 and DMS-0701308, T h omas by DMS-020059 5 and 035474 2. References [1] C. Berge, F¨ arbung von Graphen deren s¨ am tlic he bzw. deren un gerade Kreise starr sind (Zusam- menfassung), Wissenschaftliche Zeitschrift, Martin Luther Universit¨ at Hal le-Wittenb er g, Mathematisch- Naturwissensch aftliche R ei he (1961) 114–115. [2] S . J. Brams and P . C. Fishburn, Going from theory to practice: the mixed success of approv al voting, So cial Choic e and Welfar e 25 (2005) 457–474 . [3] M. Chudno vsky , N . Rob ertson, P . D . Seymour and R. Thomas, Progress on p erfect graphs, Math. Pr o gr am . Ser. B 97 (2003) 405–42 2. [4] M. Chudno vsky , N . 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Nashtir, A Helly typ e conjecture, Discr ete Comput. Ge om. 21 ( 1999), 37–43 . [13] B. Knaster, C. Kuratowski, an d S . Mazurkiewicz, Ein Beweis des Fixpun ktsatzes f¨ ur n -d imensionale Sim- plexe, F und. Math. 14 (1929) 132–137. VOTING IN A GREEABLE SOCIETIES 13 [14] J. Kratoch vil, A special planar satisfiability problem and some consequences of its NP-completeness, Dis- cr ete Appl. Math. 52 (1994) 233-252. [15] J. Matousek, Le ctur es on Discr ete Ge ometry , S pringer-V erlag, N ew Y ork, 2002. [16] A. Niedermaier, D . R izzolo, and F. E. Su, A combinatori al app roac h to K KM th eorems on metric trees and cycles, preprint, 2006. [17] J. Radon, Mengen kon vexer K¨ orp er, die einen gemeinsamen Punkt enthalten, Math. Ann. 83 (1921) 113– 115. [18] J. L. Ram ´ ırez Alfons ´ ın and B. A. R eed, Perfe ct gr aphs , J. Wiley and Sons, Chic hester, 2001. [19] C. E. Shannon, The zero error capacity of a noisy channel, I.R .E. T r ans. Inform. T he ory IT-2 ( 1956) 8–19. [20] D. B. W est, Intr o duction to Gr aph The ory , Prentice-Hall, Upp er Saddle River, NJ, 2001. [21] M. Y annak akis, The complexity of the partial order d imension problem, SIAM J. Algebr ai c Discr ete Metho ds 3 (1982) 351–358. DEBORAH E. BERG received a B.S. in Mathematics from Harvey Mudd College in 2006 and an M.S. in Mathematics from th e Universit y of Nebrask a-Lin coln in 2008. Sh e is currently working tow ards her Ph.D. in Mathematics Edu cation. When not busy stud ying, doing graph theory , or t eaching, she sp ends her t ime practicing Shotok an kara te. Dep artment of Mathematics, Uni versity of Nebr aska, Linc oln NE 68588 debbie.ber g@gmail.com SER GUEI NORINE receive d h is M.Sc. in mathematics from St. Petersburg State Universit y , Ru ssia in 2001, and Ph.D. from A lgorithms, Combinatorics and O p timization program at Georgia Institute of T echnology in 2005. After a brief stint in finance follow ing his graduation, he return ed to academia and is now an instructor at Princeton Universit y . His interests include graph th eory and combinatori cs. Dep artment of Mathematics, Fi ne Hal l , Washington R o ad, Princ eton NJ, 08544 snorin@mat h.princeton.e du FRANCIS EDW ARD SU received h is Ph.D. at Harv ard in 1995, and is now a Professor of Mathematics at Harvey Mudd College. F rom the MAA, he receiv ed the 2001 Merten M. Hasse Prize for h is writing and the 2004 Henry L. Alder Award for h is teac hing, and he wa s the 2006 James R.C. Leitzel Lecturer. H e enjo ys mak ing connections b etw een his research in geometric combinatorics and applications to t h e so cial sciences. He also authors the p opular Math F un F acts website. Dep artment of Mathematics, Harvey Mudd Col le ge, Clar emont CA 91711 su@math.hm c.edu R OBIN THOMAS received his Ph.D. from Charles Universit y in Prague, formerly Czec h oslo vakia, now the Czec h Republic. He h as work ed at t he Georgia Institu te of T echnology since 1989. Currently h e is Professor of Mathematics and Director of the multidisciplinary Ph.D. p rogram in A lgorithms, Com binatorics and Optimiza- tion. In 1994 he won, jointly with Neil Rob ertson and Paul Sey mour, th e D. Ray F ulkerson prize in Discrete Mathematics. Scho ol of Mathematics, Ge or gia I nstitute of T e chnolo gy, Atlanta, Ge or gia 30332 thomas@mat h.gatech.edu P AUL WO LLAN received his Ph.D. in Georgia T ec h’s interdisciplinary A lgorithms, Com binatorics, and Optimization program in 2005. He sp ent a year as a p ost do c at t h e Universit y of W aterloo, and he is currently a Humboldt Research F ello w at the Universi ty of Hamburg. Mathematisches Seminar der Universit¨ at Ham bur g, Bundesstr. 55, D-20146 Hambur g Germany wollan@mat h.uni-hamburg .de