Copeland Voting Fully Resists Constructive Control

Cop eland V oting F ully R e sists Constructive Con trol ∗ Piotr F aliszewsk i Departmen t of Compute r Science Univ ersit y of Ro c hester Ro c hester, NY 14627 Edith Hemaspaandra † Departmen t of Computer Science Ro c hester Institute of T echn ology Ro c hester, NY 14623 Lane A. Hemaspaandra Departmen t o f Computer Science Univ ersit y of Ro c hester Ro che ster, NY 14627 J¨ org Rothe ‡ Institut f ¨ ur Informatik Heinric h-Heine-Univ ersit¨ at D ¨ usseld orf 40225 D ¨ usse ldorf, German y Decem b er 9, 2007 Abstract Control and brib ery are settings in whic h a n external agent seeks to influence the outcome of an e lection. F alisze ws ki et al. [7] prov ed that Llull voting (which is here denoted by Cop eland 1 ) and a v ariant (her e denoted by Cop eland 0 ) of Cop eland voting are computationally resis tant to man y , yet not all, t y pe s of constructive control and that they also provide broad resistance to briber y . W e study a parameterize d version o f Copela nd voting, denoted b y Copela nd α , where the parameter α is a rational num b er b etw een 0 and 1 that sp ecifies how ties a re v alued in the p airwise comparis ons of candidates in Copeland elections. W e esta blish resistance o r vulnerability results, in ev er y previously studied con trol scenario, for Copeland α for each rational α , 0 < α < 1. In par ticular, we prov e tha t Cop eland 0 . 5 , the s ystem co mmonly referr ed to as “Cop eland voting,” provides full resis tance to constructive c o ntrol. Among the s ystems with a p olynomial- time winner pr o blem, this is the first natural election system proven to have full res istance to constructive control. Results on brib er y and fixed-parameter tractability of bo unded-case control prov en for Cop eland 0 and Cop eland 1 in [7] a re extended to Cop eland α for each r ational α , 0 < α < 1; we also give results in more flexible mo dels such a s microbr ibe ry and extended control. 1 In tro duction Preference aggreg ation by voti ng pro cedu res h as b een th e fo cus of muc h atten tion w ithin th e field of multia gen t systems. Agen ts (calle d v oters in the con text of vot ing) ma y hav e different , often conflicting individu al pr eferen ces ov er th e giv en alternativ es (or cand idates). V oting r ules (or, syn- on ym ously , elec tion systems) pro vid e a useful metho d for them to come to a “reasonable” decision ∗ Supp orted in part by DFG grant RO-1202/ 9-3 and RO-1202/ 11-1, NSF grants CCR-0311021 , CCF-0426761 , and I IS-0713061, t he Alexander von H umboldt F oun dation’s T ransCo op p rogram, and a F riedric h Wilhelm Bessel Researc h Award. A version of this pap er also app ears as URCS-TR-2007-923. † W ork done in p art while visiting Heinrich-Heine-Universit¨ at D ¨ ussel dorf. ‡ W ork done in p art while visiting the Universit y of R ochester. 1 on wh ic h alternativ e to c h o ose. One k ey iss ue h ere is that there m igh t b e attempts to influen ce the outcome of elections. S ettings in w h ic h su c h influ en ce on elections can b e implemented include ma- nipulation [3, 13], electoral con trol [2, 7, 8, 9, 13], and b rib ery [6, 7]. Although reasonable electio n systems typica lly are susceptible to these kinds of influ ence (for manipulation this is u niv ersally true, via the Gibbard–Satterth w aite and Duggan–Sc h w artz Theorems), computational complexit y can b e used to pr o vide some p rotection in eac h suc h setting. W e study the extent to whic h the Cop eland election system [4 ] (see also [14, 11]; a similar system w as also studied by Zermelo) resists, computationally , con trol and brib ery attempts. Cop eland electio ns are one of the classical vot ing p ro cedures that are based on pairwise compar- isons of candid ates: The w inner (b y a strict ma jorit y of v otes) of eac h suc h a head-to-head contest is a warded one p oin t and the loser receiv es n o p oint; w ho eve r collects the most p oin ts o v er all these con tests (including tie-relat ed p oints) is the electio n’s w inner. The p oin ts a w arded for ties in suc h head-to-head ma jorit y-rule conte sts are treated in v arious wa ys in th e literature. F aliszewski et al. [7] prop osed a parameterized v ersion of C op eland electio ns, denoted by Cop eland α , where the parameter α is a rational n u m b er b etw een 0 and 1 suc h that, in case of a tie, b oth candidates receiv e α p oints. So the system wid ely referred to in the literature as “Cop eland elections” is Cop eland 0 . 5 , where tied candid ates receiv e half a p oin t eac h (see, e.g., Merlin and Saari [14, 11]; th e defin ition used b y C onitzer et al. [3] can b e scaled to b e equiv alen t to Cop eland 0 . 5 ). Cop eland 0 , where tied candidates come aw a y empty-handed, h as sometimes also b een r eferred to as “Cop eland elections” (see, e.g., [12, 7]). An election system prop osed by the C atalan philosopher and theologia n Ramon Llull in the 13th cen tur y (see, e.g., the r eferences in [7]) is in this notation nothing other than Cop eland 1 , where tied candidates are a w ard ed one p oin t eac h, just like winn er s of head-to-head con tests. F aliszewski et al. [7] stud ied the systems C op eland 0 and Cop eland 1 with resp ect to their (com- putational) r esistance and vu ln erabilit y to b rib er y an d pro cedural con trol. Brib ery and con trol are settings in wh ich an external actor seeks to influ ence the outcome of an electio n. Brib ery is somewhat akin to electoral manipu lation and strategic voting in that the brib er tries to reac h his or h er goal by brib ing some vo ters to c h an ge their preferences. (The difference b etw een brib ery and manipulation is that manip ulativ e voters themselv es cast their v otes insincerely , i.e., there is no external agen t.) In contrast, the external actor in con trol scenarios (wh o by tradition is, p otenti ally confusingly , called “the c hair”) seeks to r eac h this goal via changing the election pr o cedure, namely via adding/deleting/partitio ning either candidates or v oters. Bartholdi, T o v ey , and T ric k [2] w ere th e first to stud y the computational asp ects of con trol: Ho w hard is it, computationally , for the c h air to exert cont rol? In their seminal pap er they in tro duced a num b er of f u ndamenta l con trol scenarios inv olving (what is no w called) c onstructive con trol, i.e., where the c hair’s goal is to mak e some d esignated candidate win. Other p ap ers stud ying con trol include [8, 13, 9, 7], whic h in addition to constructiv e control also consider destructive con trol, where th e c hair tries to pr eclude some designated cand idate fr om win ning. The notion of brib ery in electio ns was in tro duced b y F alisz ewski et al. [6] and was also stu d ied in [7]. A t fir st glance, one migh t b e tempted to think th at the definitional p ertur bation due to the parameter α in Cop eland α electio ns is negligible. Ho wev er, as noted in [7], “. . . it can m ake the dynamics of Llull’s system quite d ifferent fr om those of [Cop eland 0 ]. Pro ofs of results for Llull differ considerably from those for [Cop eland 0 ].” This statemen t notwithstanding, w e sho w that in most cases it is p ossible to obtain a u nified—though sometimes r ather in v olve d—construction that w orks for b oth systems, and ev en for Cop eland α with resp ect to eve ry rational α , 0 ≤ α ≤ 1. 2 In particular, w e establish r esistance or vulnerabilit y results f or Cop eland 0 . 5 (whic h is the system commonly referred to as “Cop eland”) in every previously studied con trol scenario. 1 In doing so, w e p ro vid e an example of a control pr oblem where the complexit y of Cop eland 0 . 5 differs f rom that of b oth Cop eland 0 and Cop eland 1 : While the latter t wo pr oblems are vulnerable to constructiv e con trol by adding (an unlimited num b er of ) candidates, Cop eland 0 . 5 is r esistant to this con trol typ e (see Section 2 for d efinitions and Theorem 3.7 for this result). Th us C op eland (i.e., Cop eland 0 . 5 ) is the first natural elect ion system with a p olynomial-time winner problem that is pro v en to b e r esistan t to every t yp e of constru ctiv e con trol th at has b een prop osed in the literat ure to date. Moreo ver, if one uses the hybridization method of Hemaspaandra et al. [9] to combine this full resistance of C op eland 0 . 5 to constructiv e con trol with the full resistance of Cop eland 0 . 5 to destructive voter con trol (whic h w e also pro v e here) and with the full resistance of plur alit y 2 to destructive candidate con trol (see [8, 7]), one obtains a hybrid election system that (a) is r esistan t to ev ery (constructiv e and destructive ) con trol t yp e previously considered in the literature, (b) has a p olynomial-time winner pr ob lem, and (c) has only natural election systems as its constituen ts. In con trast, one of th e constituent systems for the hybrid constru cted in [9], which is there sho w n to resist tw ent y control t yp es, is rather artificial. 2 Preliminaries An election is sp ecified by a finite set C of cand idates and a fin ite collection V of v oters, where eac h v oter has preferences o v er the candid ates. W e consider b oth rational and irrational vo ters. The preferences of a rational v oter are expressed by a preference list of the form a > b > c (assuming C = { a, b, c } ), wh ere the underlyin g r elation > is a strict linear ord er that is transitiv e. The preferences of an irrational v oter are expressed b y a p reference table that, f or any t w o distinct candidates, sp ecifies whic h of them is p referred to th e other by this vo ter. An election s y s tem is a ru le that determines the win n er(s) of eac h giv en election ( C , V ). In this p ap er, we consider a parameterized v ersion of Cop eland’s election system [4], d enoted Cop eland α , where the parameter α is a rational n um b er b et ween 0 and 1 that s p ecifies h o w ties are rew ard ed in the head-to-head ma jorit y -r ule cont ests b et ween an y t wo distinct candidates. Definition 2.1 ([7]) L et α , 0 ≤ α ≤ 1 , b e a fixe d r ational numb er. In a Cop eland α ele ction, the voters indic ate which among any two distinct c andidates they pr efer. F or e ach such he ad-to-he ad c ontest, if some c andidate is pr efe rr e d by a strict majority of voters then he or she obtains one p oint and the other c andidate obtains zer o p oints, and i f a tie o c curs then b oth c andidates obtain α p oints. L et E = ( C, V ) b e an ele ction. F or e ach c ∈ C , sc or e α E ( c ) is the sum of c ’s C op eland α p oints in E . Every c andidate c with maximum sc or e α E ( c ) wins. L et Cop eland α Irrational denote the same ele ction system but with voters al lowe d to b e irr ationa l. In the literature, th e term “Cop eland elections” is most often used for the system C op eland 0 . 5 , and is sometimes u sed for C op eland 0 . Th e system C op eland 1 w as prop osed by Llull already in the 13th cen tur y (see the r eferen ces in [7]) and so is called Llull v oting. 1 Also, our new results apply , e.g., to even ts suc h as the group stage of FIF A world-cup finals, which is, in essence, a series of Copeland α tournaments with α = 1 3 . 2 In pluralit y-rule elections, every voter gives one p oint to his or her most preferred candidate. Who ever collects the most p oints is this election’s plurality winner. 3 W e no w define the control pr oblems we consider, in b oth the constructive and the d estructiv e v ers ion. Let E b e an election system. I n our case, E w ill b e either Cop eland α or Cop eland α Irrational , where α , 0 ≤ α ≤ 1, is a fi xed rational num b er. In fact, since the types of con trol w e consider here are wel l-kno wn from the literature (see, e.g., [2, 7, 8]), w e will con tent ourselve s with the d efinition of some examples of these p roblems (in particular some of those that o ccur in the pro ofs to b e present ed in Section 3.2 b elo w). W e start with defining control via add ing candidates. Note that there are tw o v ersions of this con trol t y p e. The u nlimite d v ersion (whic h, for the constru ctiv e case, was in tro duced by Bartholdi, T o vey , and T r ic k [2]) asks whether the election chair can add (any num b er of ) candidates from a giv en p o ol of sp oiler candidates in ord er to either m ak e h is or her fav orite candidate win the electio n (in the constructiv e case), or pr ev ent his or her despised cand idate from win ning (in the destructiv e case): Name: E -CCA C u and E -DCA C u . Giv en: Disjoint cand idate sets C and D , a collect ion V of v oters repr esen ted via their p r efer- ence lists (or preference tables in the irrational case) o v er the candidates in C ∪ D , and a distinguished candidate p ∈ C . Question ( E - CCA C u ): Does ther e exist a subs et D ′ of D such that p is a winner of the E election with candidates C ∪ D ′ and v oters V ? Question ( E - DCA C u ): Does there exist a s u bset D ′ of D suc h that p is n ot a winner of the E electio n with candidates C ∪ D ′ and voters V ? The only difference in the limite d ve rsion of constructiv e and destructive control via adding candidates ( E -CCA C and E -DCAC, for sh ort) is that the chair n eeds to ac hiev e h is or h er goal b y adding at most k cand id ates from the giv en s et of sp oiler candidates. This v ers ion of con trol by adding candidates w as prop osed in [7] to syn c hr onize the defi n ition of control by add ing candidates with the definitions of con trol by d eleting candidates, adding v oters, and deleting v oters. Our second example r egards con trol via run-off partition of candidates, where we fo cus on th e constructiv e case: Name: E -CCRPC-TP (resp ectiv ely , E -CCRPC-TE). Giv en: A set C of candid ates and a collection V of vot ers represente d via their preferen ce lists (or pr eference tables in the ir r ational case) o v er C , a d istin gu ish ed candidate p ∈ C , and a nonnegativ e in teger k . Question: Is it p ossible to partition C into C 1 and C 2 suc h th at p is a winner of the t w o-stage electio n where th e winners of sub election ( C 1 , V ) that survive the tie-handling rule (TP or TE) comp ete against the winners of sub electio n ( C 2 , V ) that survive the tie-handling rule? (Sub electio ns are conducted using system E .) As one can see fr om the ab ov e examples, w e use the follo w in g naming con v entio ns for control problems. T h e name of a control problem starts with the elect ion system u sed (wh en clear from con text, it ma y b e dropp ed), follo wed by C C for “constructive con trol” or by DC for “destructiv e con trol,” follo w ed b y the acron ym of the type of con trol: A C for “adding (a limited num b er of ) 4 candidates,” AC u for “add in g (an u nlimited num b er of ) candid ates,” DC for “deleting candidates,” PC for “partitio n of candidates,” RPC for “run-off p artition of candidates,” A V for “adding v oters,” D V for “deleting v oters,” and PV for “partition of v oters,” and all the p artitioning cases (PC, RPC , and PV) are follo wed by the acron ym of the tie-handling rule used in sub elections, n amely TP for “ties promote” (i.e., all winners of a giv en sub election are p romoted to the final r ound of the electio n) and TE for “ties eliminate” (i.e., if there is more than one win ner in a giv en sub election then none of this sub election’s winners is promoted to the fi nal round of the election). W e now turn to the definition of brib ery problems (see [6]), wh ere the brib er seeks to reac h his or her goal via b ribing certain v oters to mak e th em change their p references. Name: E -brib ery. Giv en: A set C of candidates, a collection V of vo ters r epresen ted via their p r eference lists (or preference tables in the irr ational case) o ver C , a d istinguished candidate p ∈ C , and a nonnegativ e in teger k . Question: Does there exist a vo ter collectio n V ′ o ver C , w h ere V ′ results f rom V by mo difyin g at most k v oters, suc h that p wins the E election ( C , V ′ )? F or E -destructive-brib ery, the destru ctiv e brib ery problem for E , w e require p to b e not a winner. Note that the ab o ve d efinitions fo cu s on a winner , i.e., they are in the nonunique- winner mo del . The u ni q ue-winner analogs of these problems can b e defi n ed by requirin g the distingu ish ed cand i- date p to b e the unique winner (or to not b e a u nique winner in the d estructiv e case). Let E b e an election system and let Φ b e a con trol t yp e. W e sa y E is immune to Φ - c ontr ol if the c hair can neve r reac h his or her goal (of making a give n candidate win in th e constructiv e case, and of blo c king a giv en candid ate f r om winning in th e destructiv e case) via asserting Φ -con trol. E is said to b e susc eptible to Φ - c ontr ol if E is not imm une to Φ-con tr ol. E is said to b e vulner able to Φ -c ontr ol if it is susceptible to Φ-con trol an d th ere is a p olynomial-time algorithm for solving the control problem asso ciated with Φ. E is said to b e r esistant to Φ - c ontr ol if it is susceptible to Φ-con trol and the con trol problem asso ciated with Φ is NP-hard. The ab o ve notions w ere int ro duced by Bartholdi, T o vey , and T rick [2] (see also, e.g., [8, 13, 9, 7 ]). W e say E is vulner able to c onstructive (r esp e ctively, destructive) brib ery if E -brib ery (resp ectiv ely , E -d estructiv e -brib ery) is in P. W e sa y E is r esistant to c onstructive (r esp e ctiv e ly, destructive) brib e ry if E -brib ery (resp ectiv ely , E -destructiv e -brib ery) is NP-hard. Man y of our reductions in Section 3 are from the NP-complete v ertex co ver problem: Giv en an undirected graph G = ( V ( G ) , E ( G )) and a nonnegativ e integ er k , do es there exist a s et W such that W ⊆ V ( G ), k W k ≤ k , and for ev ery edge e = { u, v } , e ∈ E ( G ), it holds that e ∩ W 6 = ∅ ? The stu dy of fixed-parameter complexit y (see, e.g., [5]) has b een expanding explosiv ely since it w as paren ted as a field b y Do wney , F ello ws, and others in the late 1980s and the 1990s. Although the area has built a ric h v ariet y of complexit y classes regarding parameterized pr oblems, for the purp ose of the current pap er we need fo cu s only on one v ery im p ortant class, namely , the class FPT. Briefly p ut, a problem parameterized b y some v alue j (whic h , note, can b e viewe d as a family of problems, one p er v alue of j ) is said to b e fixe d-p ar ameter tr actable (equiv alen tly , to b elong to the class FPT) if there is an algorithm for the pr oblem whose running time is f ( k ) n O (1) . In our context , w e consider tw o p arameterizatio ns: b oundin g the num b er of candidates and b ound ing the n um b er of v oters. W e use the same notations used th roughout this pap er to describ e 5 Cop eland α α = 0 0 < α < 1 α = 1 Con trol t yp e CC DC CC DC CC DC A C u V V R V V V A C R V R V R V DC R V R V R V RPC-TP R V R V R V RPC-TE R V R V R V PC-TP R V R V R V PC-TE R V R V R V PV-TE R R R R R R PV-TP R R R R R R A V R R R R R R D V R R R R R R T able 1: Resistance (R) and vulnerability (V) of Cop eland α electio ns, for rationals α , 0 ≤ α ≤ 1 . problems, except we p ostp end a “-BV j ” to a problem name to state that the num b er of v oters may b e at most j , and we p ostp end a “-BC j ” to a p roblem name to state that the num b er of candidates ma y b e at most j . In eac h case, the b oun d app lies to the full n u m b er of such items in volv ed in the problem. F or example, in the case of control by adding v oters, the j must b oun d the total of the n um b er of vo ters in the election added together with the n um b er of v oters in the p o ol of vo ters a v ailable for adding. 3 Con trol 3.1 Ov erview of Results Our main result regarding con trol is Th eorem 3.1 b elo w. Theorem 3.1 F or e ach r ational α , 0 ≤ α ≤ 1 , C op eland α ele ctions ar e r esistant and vulner able to c ontr ol as shown in T able 1, b oth for r ational and irr ational voters and in b oth the nonunique-winner mo del and the unique - winner mo del. Boldface results in T able 1 are new to this p ap er and nonb oldf ace results are du e to F aliszewski et al. [7]. Note that the n otion wid ely referred to in the literature s im p ly as “Cop eland elections,” whic h we her e for clarit y call Cop eland 0 . 5 , p ossesses all ten of our basic typ es of constructiv e resistance and, in addition, ev en has constru ctiv e A C u resistance. These r esistances should b e compared with the resu lts kno wn for the other notion that in th e literature is o ccasionally referred to as “Cop eland elections,” namely Cop eland 0 , and w ith the results kno wn for Llull elections, whic h are h er e denoted by Cop eland 1 , see [7]. While Cop eland 0 and Cop eland 1 p ossess all ten of our b asic types of constru ctiv e resistance, they b oth are vulnerable to this elev ent h type of constructiv e con trol, the in congruous but historically resonant notion of constru ctiv e cont rol by adding an unlimited n u m b er of candidates (i.e., CCA C u ). It is kno wn that plu ralit y is resistant to the six basic t yp es of destru ctiv e candidate cont rol and also to DCA C u , see [8, 7]. Since by Th eorem 3.1, Cop eland 0 . 5 pro vides resistance for all ten b asic 6 constructiv e con trol t yp es and for CCAC u , and also for the four b asic t yp es of destructiv e v oter con trol, the h yb rid (in th e sense of [9]) of plu ralit y with Cop eland 0 . 5 is resistant to eac h b asic t yp e of constru ctive and d estructiv e control and in addition to constructiv e and destru ctive A C u con trol. This result follo ws via Theorem 3.1 and the results of Hemaspaandra et al. [9]. And, un lik e the h ybrid system constructed b y Hemaspaandra et al. [9], this h ybrid uses only natural systems as its constituen ts. Corollary 3.2 The hybrid (in the sense of [9]) of plur ality and Cop eland 0 . 5 is r esistant to e ach of the twenty b asic typ es of c onstructive and destructive c ontr ol and also to c onstructive and destructive AC u c ontr ol, and it has a p olynomial -time winner pr oblem. The next tw o sections discuss the sin gle r esults con tained in Theorem 3.1 in more detail and sk etc h some of the pro ofs. All the results stated in Sections 3.2 an d 3.3 are true b oth in the rational and irrational v oter mo del and in b oth the nonunique-winner mo del and the u nique-winner mo del. 3.2 Candidate Con t rol W e start with cand idate con trol. F aliszewski et al. [7] sho w ed that b oth C op eland 0 and Cop eland 1 are vulnerable to eac h destructiv e con trol type in T ab le 1. T o extend these resu lts to Cop eland α electio ns for eac h ratio nal α , 0 ≤ α ≤ 1, our pro ofs for destru ctiv e control by adding and deleting candidates use the follo wing ob s erv ation. Let ( C, V ) b e an elect ion, and let α b e a fixed ratio nal n umb er such th at 0 ≤ α ≤ 1. F or every candidate c ∈ C it holds that: sc or e α ( C,V ) ( c ) = P d ∈ C −{ c } sc or e α ( { c,d } ,V ) ( c ) . The candidate partition and run-off partition cases can b e shown to redu ce to the case of deleting candidates, and the vulnerabilit y results in Theorem 3.3 use greedy algorithms. Theorem 3.3 F or e ach r ational numb er α , 0 ≤ α ≤ 1 , Cop eland α is vulner able to destructive c ontr ol via (a) adding c andidates (b oth DCAC and DCAC u , i.e., b oth for a limite d and an unlimite d numb er of c andidates), (b) deleting c andidates (D CDC), (c ) p artition of c andidates (in b oth the TP and TE mo del, i.e., DCPC- TP and DCPC- TE), and (d) run-off p artition of c andidates (in b oth the TP and TE mo del, i.e., D CRPC-TP and DCRPC- TE), T urning now to constructive cand idate cont rol, our resistance pro ofs use th e follo win g t wo lemmas, whic h w e here state without pro of. Lemma 3.4 shows ho w to construct a “padded ” electio n w ith usefu l prop erties. Lemma 3.5 then sh ows h o w to build an election via com bining smaller ones. Lemma 3.4 L et α b e a r ationa l numb er such that 0 ≤ α ≤ 1 . F or e ach p ositive inte ger n , ther e is an ele ction P ad n = ( C, V ) such that k C k = 2 n + 1 and, for e ach c andidate c ∈ C , it holds that sc or e α Pa d n ( c ) = n . Lemma 3.5 L et E = ( C , V ) b e an ele ction wher e C = { c 1 , . . . , c n } , and let α b e a r ational numb er such that 0 ≤ α ≤ 1 . F or e ach c andidate c i , we denote the numb er of he ad-to-he ad ties of c i in E by t i . L et k 1 , . . . , k n b e a se quenc e of n nonne gative inte gers such that for e ach k i we have 0 ≤ k i ≤ n . Ther e is an ele ction E ′ = ( C ′ , V ′ ) such that: (a) C ′ = C ∪ D , wher e D = { d 1 , . . . , d 2 n 2 } ; (b) for e ach i , 1 ≤ i ≤ n , sc or e α E ′ ( c i ) = 2 n 2 − k i + αt i ; (c) for e ach i , 1 ≤ i ≤ 2 n 2 , sc or e α E ′ ( d i ) ≤ n 2 + 1 . 7 F aliszewski et al. [7] sho wed th at b oth Cop eland 0 and C op eland 1 are r esistan t to constructiv e con trol via adding (a limited num b er of ) candidates. This is su bsumed by the follo wing more general result. Theorem 3.6 F or e ach r ational numb er α such that 0 ≤ α ≤ 1 , Cop eland α is r esistant to c on- structive c ontr ol vi a adding c andidates (CCAC). In con trast with the kno wn result that b oth C op eland 0 and Cop eland 1 are vulnerable to con- structiv e con trol via adding an unlimited num b er of candidates [7], we sh o w that Cop eland α is resistan t to this con trol t yp e if 0 < α < 1. Notatio n: In the pro ofs of Section 3.2, we often identify an electio n with its set of candidates, since in the case of candid ate con trol th e set of v oters is fixed and cannot change . Theorem 3.7 F or e ach r ational numb er α , 0 < α < 1 , Cop eland α is r esistant to c onstructive c ontr ol via adding an unlimite d numb er of c andida tes (CCAC u ). Pro of. W e pro vide a reduction from the vertex co v er p roblem. Let ( G, k ) b e an instance of the v ertex co ver problem, wh ere G is an u ndirected graph and k is the b ound on the s ize of the ve rtex co ve r that we seek. Let E ( G ) = { e 1 , . . . , e m } b e the set of G ’s edges and V ( G ) = { 1 , . . . , n } b e the set of G ’s v ertices. Usin g Lemma 3.5, we can build an election E ′ = ( C, V ′ ) such that: (a) k C k = 2 ℓ 2 + ℓ , w h ere ℓ = 2 n + 2 m ; (b) { p, r , e 1 , . . . , e m } ⊆ C (the remaining candid ates are u sed for padding); (c) sc or e α E ′ ( p ) = 2 ℓ 2 − 2; (d) sc or e α E ′ ( r ) = 2 ℓ 2 − 2 − k + k α in th e nonunique-winner case (resp ectiv ely , sc or e α E ′ ( r ) = 2 ℓ 2 − 2 − k + ( k − 1) α in the unique-winner case); (e) for eac h e i ∈ C , sc or e α E ′ ( e i ) = 2 ℓ 2 − 2 + α in the non u nique-winn er case (resp ectiv ely , sc or e α E ′ ( e i ) = 2 ℓ 2 − 2 in the unique-winn er case); (f ) the scores of all candid ates other than p, e 1 , . . . , e m are at most 2 ℓ 2 − n − 2. (W e omit the details of the construction, bu t ment ion that one can start fr om any election with at least 2 n + 2 m candidates, add s ome ties b et w een s ome padding candidates and the e i ’s if needed, and then apply Lemma 3.5.) W e form election E = ( C ∪ D , V ) from E ′ via adding candidates D = { 1 , . . . , n } and appr opriate v oters suc h that the resu lts of head-to-head con tests are: (1) p ties with all candidates in D ; (2) for eac h e j , if e j is incident with i ∈ D then candidate i defeats candidate e j , and otherwise they tie; (3) all other candidates in C ′ defeat eac h of th e cand idates in D . Our instance of CCA C u is form ed b y th e candidate set C (the candidates already enrolled in the election), the candidate set D (the candidates that can b e ad d ed to the election), and the s et of v oters V , where eac h v oter has preferen ces ov er th e candidates in C ∪ D . Note that the candidates in D corresp ond to the v ertices of G . W e claim that there is a set D ′ (where D ′ ⊆ D ) suc h that p is a winner (resp ectiv ely , the uniqu e winner) of the Cop eland α electio n ( C ∪ D ′ , V ) if and only if G has a v ertex co v er of size at most k . It is easy to see that if D ′ corresp onds to a v ertex co ver of size at most k then p is a winn er (resp ectiv ely , th e un ique winner) of Cop eland α electio n C ∪ D ′ . Th e reason is th at add ing an y one mem b er of D ′ increases p ’s score by α , increases r ’s score by one, and for eac h e j , add ing i ∈ D ′ increases e j ’s score b y α if and only if e j is not incident with i . Th us, the n onpadding candidates in C ∪ D ′ ha ve th e follo wing scores in the resulting election E ′′ with candidates C ∪ D ′ (it is clear that none of the padding candidates has enou gh Cop eland α p oints to b ecome a winn er after addin g an y subset of candidates fr om D ): (a) sc or e α E ′′ ( p ) = 2 ℓ 2 − 2 + k α ; (b) sc or e α E ′′ ( r ) = 2 ℓ 2 − 2 + kα in the nonunique-winner case (resp ectiv ely , 2 ℓ 2 − 2 + ( k − 1) α in the unique-winn er case); (c) 8 sc or e α E ′′ ( e i ) ≤ 2 ℓ 2 − 2 + kα in th e nonunique-winner case (r esp ectiv ely , 2 ℓ 2 − 2 + ( k − 1) α in the unique-winn er case). As a result, w e see that adding all members of D ′ mak es p a winner (resp ectiv ely , the unique winner). On the other han d , assu me that p can b ecome a w inner via add ing some subset D ′ of candidates from D . Firs t, note that k D ′ k ≤ k , since otherwise r wo uld end up with more p oin ts (resp ectiv ely , at least as many p oin ts) as p and so p would not b e a winn er (resp ectiv ely , would not b e the un ique winner). W e claim that D ′ corresp onds to a vertex co ver of G . F or the sake of con tradiction assum e that there is some edge e j inciden t to v ertices u and v su c h that neither u nor v is in D ′ . How ev er, if this were the case th en candid ate e j w ould ha v e more p oin ts (resp ectiv ely , at least as man y p oin ts) as p and so p would not b e a w inner (resp ectiv ely , wo uld not b e the unique w inner). Thus, D ′ m ust form a v ertex co v er of size at most k . ❑ The follo wing resu lt extends to all rationals α , 0 ≤ α ≤ 1, the known r esult that Cop eland 0 and Cop eland 1 are resistan t to constru ctiv e con trol via deleting cand idates [7]. Theorem 3.8 L et α b e a r ational nu mb er such that 0 ≤ α ≤ 1 . Cop eland α is r esistant to c on- structive c ontr ol vi a deleting c andidates (CCDC). Pro of. The pro of follo ws via a redu ction from the vertex co ver p r oblem. W e fir s t handle the non unique-winn er case. Let ( G, k ) b e the input instance of the ve rtex co ver problem, w here G is an undir ected graph and k is the u pp er b oun d on the size of the v ertex co v er that we seek. Let V ( G ) = { 1 , . . . , n } and let E ( G ) = { e 1 , . . . , e m } . W e build election E ′ = ( C ′ , V ′ ), where C ′ = { p, r, e 1 , . . . , e m , 1 , . . . , n } and the v oter set V ′ yields th e follo win g results of head-to-head con tests (omitting the details of the construction d ue to space): (1) p defeats r ; (2) eac h candidate e i ∈ C defeats exactly those tw o candidates u, v ∈ { 1 , . . . , n } th at the edge e i is incident w ith; (3) eac h cand id ate u ∈ { 1 , . . . , n } defeats b oth p an d all candidates e i suc h that ve rtex u is not inciden t to e i ; (4) all the other contests result in a tie. Let ℓ = n + m . W e form an election E = ( C , V ) via combining election E ′ with election P ad ℓ = ( C ′′ , V ′′ ), where C ′′ = { t 0 , . . . , t 2 ℓ } and the set V ′′ of v oters is set as in L emma 3.4. W e select the follo wing results of head-to-head con test b et wee n the candidates in C ′ and the candidates in C ′′ : p and all candidates e i ∈ C defeat ev eryone in C ′′ and eac h candid ate in C ′′ defeats all candidates in C ′ − { p, e 1 , . . . , e m } . It is easy to ve rify that election E yields the follo wing Cop eland α scores: (a) sc or e α E ( p ) = mα + 1 + 2 ℓ + 1; (b ) sc or e α E ( r ) = m + n α ; (c) for eac h e i ∈ C , sc or e α E ( e i ) = mα + 2 + 2 ℓ + 1; (d) for eac h i ∈ C , sc or e α E ( i ) ≤ 1 + m + nα ; (e) for eac h t i ∈ C , sc or e α E ( t i ) = ℓ + n + 1. Th us, the set of winners of E is W = { e 1 , . . . , e m } . W e claim that p can b ecome a winner of Cop eland α electio n E via d eleting at most k candidates if and only if the graph G has a v ertex co ve r of s ize at most k . First note that if k ≥ n then G obviously has a v ertex co ver of size at most k (namely , the set of all the v ertices) and so from now on w e assu me k < n . Also, it is easy to see that if k ≥ m (i.e., if our verte x co ver can hav e more elemen ts than there are edges) then clearly a vertex cov er exists and so w e assume that k < m . Also, w e note that all candidates except p lose by at least n + 1 p oints to eac h of th e winners and s o p is th e only candidate that can p ossibly b ecome a winner via deleting at most k ≤ n candidates. Assume that p can b ecome a winner via deleting at most k < n candidates, and let D ⊆ C b e a smallest set su c h that d eleting exactly the candidates in D from election E guaran tees p ’s 9 victory . W e start b y obs er v in g that D necessarily conta ins only candidates that corresp ond to v ertices of G . F or the sake of con trad iction, assu me that D do es con tain some candid ate d su c h that d / ∈ { 1 , . . . , n } . Clearly , d cannot b e r , since d eleting r decreases p ’s score w ithout c hanging the score of an y of the candidates in W and so remo ving r from D would yield a smaller set of candidates whose d eletion guaran tees p ’s victory . Similarly , d cannot b e any other non-v ertex candidate, sin ce deleting d fr om election E w ould affect the score of p and the scores of all remaining candidates from { e 1 , . . . e m } in the same w a y . 3 Th us, again, removing d fr om D would yield a smaller set with the required prop erty . No w n ote that, in electio n E , eac h of e 1 , . . . , e m has exactly one Cop eland α p oint of adv an tage o ver p . Deleti ng any candidate u corresp ondin g to a verte x of G do es not affect p ’s score but it d o es lo wer b y one the score of all the candidates e 1 , . . . , e m that corresp ond to th e edges incident with u . Since deleting the candidates in D mak es p a winner an d since D cont ains only up to k candidates that corresp ond to v ertices in G , it must b e the case that the candid ates in D corresp ond to a v ertex co ver of G h a vin g s ize at most k . F or the con verse, it is easy to see that if G has a v ertex co ve r of size at most k then deleting the candidates that corr esp ond to this v ertex co v er guarante es p ’s victory . Th is completes the pro of for the non unique-winner case. T o obtain th e pro of for the unique-winn er case, we need to add one m ore candidate, ˆ r , that is a “clone” of r (i.e., ˆ r ties in the head-to-head contest with r and has the same resu lts as r in all other head-to-head con tests). In such a mo dified election, p h as the same Cop eland α score as eac h of the e i ’s and has to gain at least one p oint o v er eac h of them to b ecome the uniqu e winn er. The rest of the argumen t r emains the same. ❑ Theorem 3.8 will b e helpf ul in treating the (run-off ) partition-of-candidates cases. Again, it is known from [7] that b oth Cop eland 0 and Cop eland 1 are resistant to constructiv e control b y (run-off ) partition of candidates in b oth the ties-promote mo del and the ties-eliminate mo del. Theorem 3.9 L et α b e a r ational nu mb er such that 0 ≤ α ≤ 1 . Cop eland α is r esistant to c on- structive c ontr ol via run-off p artition of c andidates i n b oth the tie s-pr omote mo del (CCRPC-TP) and the ties- eliminate mo del (CCRPC- TE ). Pro of. Our pro of will, again, follo w via a reduction f rom the vertex co ve r problem. Ou r input is a graph G an d a n onnegativ e in teger k and we seek an election E wh ere our fa vo rite candidate p can b e made a winn er in the CC R P C-TP (resp ectiv ely , CCRPC-T E ) mo d el if and only if G con tains a ve rtex co ver of size at most k . Let G ha ve the edge set E ( G ) = { e 1 , . . . , e m } and the vertex set V ( G ) = { 1 , . . . , n } . The follo w ing construction is the basis of our pro of. Construction 3.10 L et F and H b e two ele ctions, with c andidate sets { f 1 , . . . , f n } and { h 1 , . . . , h q } , r esp e ctively. Define E = ( C, V ) , wher e C = { r , f 1 , . . . , f n , h 1 , . . . , h q } and voters in V ar e set so that we have the fol lowing r esults of the he ad-to-he ad c ontests: (1) for e ach f i ∈ C , f i defe ats r ; (2) for e ach h i ∈ C , r defe ats h i ; (3) for e ach h i , f j ∈ C , h i defe ats f j ; (4) al l the r emaining he ad-to-he ad c ontests ar e as in F and H , r esp e ctively. In tuitiv ely , Cons tr uction 3.10 works as f ollo ws. W e s et F to b e an election that con tains the candidate p and where p can b e made a winner (resp ectiv ely , the u nique winn er) via deleting a set 3 Keep in mind th at d could b e one of the candidates e 1 through e m . How ever, since k < m , D cannot contain all of th ese candidates. 10 D of at most k candidates (sp ecifically—and imp ortan tly—we will us e the elections b uilt in the pro of of Theorem 3.8). W e will set the election H in suc h a w a y that the only run-off partitions of candidates in E that could p ossibly r esult in p b eing a winner (resp ectiv ely , th e u nique winner) w ould form t wo sub committees suc h that the fi rst one would con tain candidates in F , p ossibly without h a ving u p to k of them, and the other sub committee would con tain r , th e candidates from H , and th e remaining candid ates in D . T h is w a y the p roblem of constructiv e con trol via run-off partition of candidate redu ces to the problem of fin ding the set D , which in Theorem 3.8 w e hav e sho wn to b e NP-complete. Let F and H b e t wo electio ns where H con tains at least t wo candidates, and let E b e the electio n ob tained from F and H using Constru ction 3.10. W e assume that our preferred candidate, p , b elongs to F , and w e assume that there are no ties in head-to-head contests b et ween candidates in H . Later on we will precisely sp ecify ho w the electio ns F and H are built, and for now we only men tion that in the TE case we will hav e H ha v e a unique winner. W e ha v e the follo wing result r egarding the p ossible structure of the su b elections in the run-off partition of candidates. Lemma 3.11 L et ( C 1 , C 2 ) b e a p artition of c andidat es in E such that p is a winner (r esp e ctively, the uniqu e winner), wher e p p articip ates in sub ele ction C 1 . It holds that C 1 = F − D and C 2 = H ∪ D ∪ { r } , wher e D ⊆ F − { p } . Lemma 3.11 f ollo ws directly from Lemma 3.12 b elo w the p ro of of whic h is omitted. Lemma 3.12 L et ( C 1 , C 2 ) b e a p artition of c andidat es in E such that p is a winner (r esp e ctively, the u nique winner). The sub c ommitte e that c ontains p do es not c ontain any memb er of H nor r . Using the ab ov e lemmas, we can s p ecify the elections F and H and complete the pr o of. W e will first hand le the nonunique-winner cases and then we will argue h o w to mo dify the pr o of to apply to the unique-winner mo del. F or the ties-promote (resp ectiv ely , ties-eliminate) case, w e set F to b e the election built in the pro of of T heorem 3.8 for the non u nique-winner mo del (resp ectiv ely , for the unique-winner mo del). F or the ties-promote case, we set H to b e an election w ith candidate set { r , h 1 , . . . , h q } suc h that, for some nonn egativ e integer ℓ , w e ha v e the follo w ing s cores: 4 (a) sc or e α H ( r ) = ℓ ; (b) sc or e α H ( h 1 ) = ℓ − k − 1; (c) sc or e α H ( h 2 ) = ℓ − k − 1; (d) for eac h i ∈ { 3 , . . . , q } , sc or e α H ( h i ) ≤ ℓ − k − 1. Suc h an election is easy to build in p olynomial time u sing Lemma 3.5. F or the ties-eliminate case, w e set H to hav e candidate set { r, h 1 , . . . , h q } with the follo win g scores: (a) sc or e α H ( r ) = ℓ ; (b) sc or e α H ( h 1 ) = ℓ − k ; (c) for eac h i ∈ { 2 , . . . , q } , sc or e α H ( h i ) < ℓ − k . Lemma 3.13 Set D in L emma 3.11 c annot c ontain mor e than k elements. The pro of of Lemma 3.13 is omitted. T o complete the pr o of of Theorem 3.9, note that since p can b ecome a winner of his or her sub committee if and only if p can b e made a w inner (r esp ectiv ely , the unique winner) of election F − D , where D ⊆ F − { p } and k D k ≤ k , we see that p can b ecome a winner (resp ectiv ely , th e un ique w inner) only if G has a vertex co ve r of size at most k . This f ollo ws b y our c h oice of F . On the other hand , if w e choose D to b e the set of vertice s that corresp ond to 4 Candidate r in Construction 3.10 w as, strictly sp eaking, not a member of H , b u t for the sake of building the election E here it is easier to consider members of H and r join tly . 11 an at-most- size- k v ertex co v er of G and partition the candidates in C as in Lemma 3.11, then p is a w inner (resp ectiv ely , the unique winner), sin ce if we use suc h a p artition and the set D then the sub committee H ∪ D ∪ { r } either has no winner or h as the u nique win n er r , an d th e sub committee F − D either h as the uniqu e winn er p (in the TE case) or has winner set { p } ∪ W (in th e TP case), where W ⊆ { e 1 , . . . , e m } . Since p and all members of W are tied, they all b ecome the winner s of electio n E . This completes the pro of for the n on unique-winner case. W e still n eed to handle the unique-winn er cases. Ho wev er, note that in the case of the T E mo del the current pro of already works just as w ell in the unique-winner mo del. It remains to h andle the unique-winn er case in the ties-promote mo del. Ho w ev er, in this case we simply need to tak e F to b e the uniqu e-winner v ersion of election E from Theorem 3.8. Ou r lemmas describing the structure of the sub committees app ly in the ties-promote case, and so if p is to b e a winner then r should b e the uniqu e winner of s u b committee H ∪ D ∪ { r } , wh ere D is a su bset of F − { p } with at most k elemen ts, and the global win ner of the election, if any , is the unique winn er of election F − D . Since we know that p can b ecome a unique win ner of F via d eleting at most k candidates if and only if G has a verte x co v er of size at most k , the pro of is complete. ❑ Theorem 3.9 Finally , we state without pro of our result for constructiv e con trol by partition of candid ates, whic h extends the corresp onding kn o wn r esult for Cop eland 0 from [7]. Unlik e in the case of ru n-off partition of candidates, ho wev er, our pro of d o es not apply to the case of α = 1 for the TE mo del, but note th at these resistances of Cop eland 1 w ere already sh o w n in [7] for b oth th e TP and the TE mo del. Theorem 3.14 1. L et α b e a r ational numb er with 0 ≤ α ≤ 1 . Cop eland α is r esistant to c onstructive c ontr ol via p artition of c andidates in the tie s- pr omote mo del (CCPC-TP). 2. L et α b e a r ational numb er with 0 ≤ α < 1 . Cop eland α is r esistant to c onstructive c ontr ol via p artition of c andidates i n the ties-eliminate mo del (CCPC-TE). 3.3 V oter Con t rol Our first result regarding v oter con trol extends to all rationals α , 0 ≤ α ≤ 1, the corresp ond ing result for Cop eland 0 and Cop eland 1 from [7]. The pro of is omitted. Theorem 3.15 L et α b e a r ational numb er such that 0 ≤ α ≤ 1 . Cop eland α is r esistant to b oth c onstructive and destructive c ontr ol via adding voters (CCA V and DCA V). Next, we state w ithout pr o of our result for con trol by d eleting vo ters, whic h extends the corre- sp ond ing known r esult for Cop eland 0 from [7]. No te that our pro of do es not apply to the case of α = 1, but we men tion that these r esistances of Cop eland 1 w ere already sho w n in [7]. Theorem 3.16 L et α b e a r ational numb er such that 0 ≤ α < 1 . Cop eland α is r esistant to b oth c onstructive and destructive c ontr ol via deleting voters (CCDV and DCDV). Finally , we state without p r o of our result for control b y (run-off ) partition of v oters, w hic h extends the corresp ondin g results for Cop eland 0 and C op eland 1 from [7]. Theorem 3.17 L et α b e a r ational numb er such that 0 ≤ α ≤ 1 . Cop eland α is r esistant to c onstructive and destructive c ontr ol via p artition of voters (in b oth the TP and TE mo del, i.e., CCPV-TP, CCPV-TE, DCP V-TP, and DCP V -TE), and to run-off p artition of voters (in b oth the TP and TE mo del, i.e., CCRPV-TP, CCRPV -TE, DCRPV-TP, and DCRPV -TE). 12 3.4 FPT Algorithm Schemes for B ounded-Case Con trol 3.4.1 Fixed-P arameter T ractabilit y Results In their semin al pap er on NP-hard winn er-determination problems, Bartholdi, T ov ey , and T ric k [1] suggested considering hard election p roblems for the cases of a b ounded num b er of candidates or a b ound ed n um b er of vo ters, and they obtained efficien t-algo rithm results for su c h cases. Within the study of electio ns, this same appr oac h—seeking efficien t fi xed-parameter algorithm f amilies—has also b een used, for example, within the stud y of brib ery [6]. F aliszewski et al. [7] sho w ed that the 16 resistance r esults for constructiv e and destructiv e vot er control within Cop eland 0 and Cop eland 1 (see T able 1) are in FPT (i.e., they eac h are fixed-p arameter tractable) if the num b er of candidates is b ounded , and also if the n u m b er of v oters is b ounded . T hey also show ed that these results hold ev en when the m u ltiplicities of preference lists in a giv en election are represent ed su ccinctly (b y a binary n umber). W e extend these results in Theorems 3.18 and 3.19 b elo w. T o state these results concisely , w e b orro w a notational app roac h from transform ational grammar, and u se square b rac kets as an “indep end en t choic e” notation. So, for example, the claim  It She He  h runs wa lks i is a sh orthand for six assertions: It runs ; S he runs; He runs; It walks; S he wa lks; and He walks. A sp ecial case is the sym b ol “ ∅ ” wh ic h , when it app ears in suc h a b r ac ket , means that wh en un w oun d it should b e viewe d as n o text at all. F or example, “  Succinct ∅  Cop eland is fun” asserts b oth “Su ccinct Cop eland is fun” and “Cop eland is fun.” Theorem 3.18 F or e ach r ational α , 0 ≤ α ≤ 1 , a nd e ach choic e fr om the indep en- dent choic e b r ackets b elow, the sp e cifie d pr oblem family (as j varies over N ) is in FPT :  succinct ∅  -  Cop eland α Cop eland α Irrational  -  C D  C     A V D V PV-TE PV-TP     -  BV j BC j  . Theorem 3.19 F or e ach r ational α , 0 ≤ α ≤ 1 , a nd e ach choic e fr om the indep en- dent choic e b r ackets b elow, the sp e cifie d pr oblem family (as j varies over N ) is in FPT :  succinct ∅  -  Cop eland α Cop eland α Irrational  -  C D  C           A C u A C DC PC-TE PC-TP RPC-TE RPC-TP           - BC j . The pro ofs of Theorems 3.18 and 3.19, wh ic h in particular employ Lenstra’s [10] algo rithm for b ound ed-v ariable-cardinalit y inte ger pr ogramming, are omitted here. 3.4.2 FPT and Extended Con t rol In this section, we introd uce and lo ok at extended con trol. By that we d o not mean changing the basic con trol n otions of add in g/deleting/partiti oning candidates/v oters. Rather, we mean generalizing past merely lo oking at the constructiv e (mak e a distinguished candidate a winner) 13 and th e destructiv e (prev ent a distinguish ed cand idate from b eing a winner) cases. In particular, w e are interested in con trol wher e the goal can b e far more flexibly sp ecified, for example (though in the partition cases w e will b e ev en more flexible than this), w e w ill allo w as our goal region an y (reasonable—there are some time-related conditions) su b collection of “Cop eland outcome tables” (sp ecifications of who w on/lost/tie d eac h h ead-to-head con test). Since from a Cop eland outcome table, in concert w ith the curr ent α , one can r ead off the Cop eland α Irrational scores of the candidates, this allo ws us a tremendous range of descriptive flex- ibilit y in sp ecifying our control goals, e.g., w e can sp ecify a linear order d esired for the candi- dates with resp ect to their Cop eland α Irrational scores, w e can sp ecify a linear-order-with-ties desired for the candidates with resp ect to their Cop eland α Irrational scores, we can sp ecify the exact de- sired Cop eland α Irrational scores for one or more cand idates, we can sp ecify that we w ant to ensu re that no candidate f rom a certain s u bgroup has a Cop eland α Irrational score that ties or b eats th e Cop eland α Irrational score of any candidate from a certain other subgroup, etc. All the FPT algorithms giv en in the pr evious section regard, on their surf ace, the stand ard con trol problem, w hic h tests whether a giv en cand idate can b e made a winner (constructiv e case) or can b e pr ecluded from b eing a win ner (destru ctive case). W e note that the general approac h es used in that section in f act yield FPT schemes even for th e far more flexible n otions of con trol men tioned ab o v e. 3.4.3 Resistance Results In contrast w ith the FPT resu lts in [7] for C op eland 0 and C op eland 1 , F ali szewski et al. [7] show ed that, for α ∈ { 0 , 1 } , Cop eland α Irrational remains resistant to all typ es of candidate con trol ev en for t wo vot ers. W e extend these results by sho wing that ev en for eac h r ational α , 0 ≤ α ≤ 1, for Cop eland α Irrational all 19 candidate-con trol cases that we sho wed earlier in th is pap er (i.e., without b ound s on the num b er of vot ers) to b e r esistan t r emain resistant even for the case of b oun ded v oters (nonsuccinct). This resistance holds even when the inpu t is not in succinct form at, and so it certainly also holds when the input is in succinct format. It r emains op en whether T able 1’s resistan t, rational-v oter, candidate-con trol cases remain resistan t f or the b ound ed-v oter case. 4 Brib ery Theorem 4.1 extends to all rationals α , 0 ≤ α ≤ 1, the corresp onding result for C op eland 0 and Cop eland 1 from [7]. Theorem 4.1 F or e ach r ational α , 0 ≤ α ≤ 1 , Cop eland α is r esistant to b oth c onstructive and destructive brib ery i n b oth the r ational-voters c ase and the irr ational-voters c ase. W e also extend another r esult for Cop eland 0 and Cop eland 1 from [7] to all rationals α , 0 ≤ α ≤ 1: Cop eland α Irrational is vuln erable to destructive microbrib ery . Informally pu t, microbrib ery means that the brib er pa ys separately for eac h preference-table en try flip of ir rational v oters. All p ro ofs of Sections 3.4 and 4 are omitted due to sp ace, bu t can b e found (along with other omitted pro ofs) in the currently 82-page, in-preparation full v ersion of this pap er. 14 5 Conclusions In this pap er w e stud ied Cop eland α electio ns with resp ect to their resistance and vulnerabilit y to con trol and b rib er y . Among the election systems wh ose winners can b e determined in p olynomial time, we iden tified the firs t natural election system, Cop eland (i.e., Cop eland 0 . 5 ), th at p ro vides full resistance to constructive control. Using this result, w e also obtained the fi r st (hybrid) election system that is resistant to eac h t yp e of constructiv e and destructiv e cont rol, has a p olynomial- time winner p roblem, and is b u ilt only from natural election systems. In addition, we extend ed previous resistance results on brib ery and fixed-parameter tractabilit y of b ounded-case cont rol to Cop eland α for eac h rational α , 0 < α < 1. Regarding the latter, questions th at remain op en concern the r ational-v oter, candidate-con trol, b ounded -vote r cases. Another op en question r egards the complexit y of constructiv e m icrobrib ery for Cop eland 0 . 5 Irrational . References [1] J. Bartholdi, I I I, C. T ov ey , and M. T rick. V o ting schemes for which it ca n b e difficult to tell who won the election. So cial Choic e and Welfar e , 6(2):15 7–16 5 , 198 9. [2] J. Bartholdi, I I I, C. T ov ey , and M. T rick. How hard is it to cont rol an electio n? Mathematic al and Computer Mo deling , 16(8/9):27 –40, 1992. [3] V. Conitzer, T. Sandho lm, and J. La ng. When a re elections with few candidates hard to manipulate? Journal of the ACM , 54(3):Article 14 , 2007 . [4] A. Co pe land. A ‘reasonable’ so cial welfare function. Mimeogr aphed notes from a Seminar on Applica- tions o f Mathematics to the So cial Sciences, University of Michigan, 1951. [5] R. Downey a nd M. F ellows. Par ameterize d Complexity . Spring er-V er lag, 1999 . [6] P . F aliszewski, E. Hemaspaandra, and L. Hemaspaa ndra. The complexity of brib ery in elections. In Pr o c. AAAI’06 , pa ges 641– 646. AAAI Pres s, 200 6. [7] P . F aliszewsk i, E. Hemaspaandr a, L. Hemaspaandr a, and J. Rothe. Llull and Cop eland voting broadly resist brib ery and control. In Pr o c. AAAI’07 , pages 724 –730 . AAAI Press , 200 7 . [8] E. Hemas paandra, L. Hemaspaa ndra, a nd J. Rothe. Any o ne but him: The complexity of prec luding a n alternative. Artificial Intel ligenc e , 171(5-6):2 55–2 8 5, 2007. [9] E. Hemaspaa ndra, L . Hemaspaa ndra, a nd J . Rothe. Hybrid elections broa den complexity-theoretic resistance to co ntrol. In Pr o c. IJ CAI’07 , pages 130 8–13 14. AAAI Press , 2007 . [10] H. Lenstra, Jr. In tege r programming with a fixed num b er of v ar iables. Mathematics of Op er ations R ese ar ch , 8 (4 ):538–5 48, 1 983. [11] V. Merlin and D. Saar i. Cop eland method I I: Ma nipulation, monotonicity , and paradoxes. Journ al of Ec onomic The ory , 72(1):14 8–17 2, 1997 . [12] A. Pro ca ccia, J. Rosenschein, and G. Ka mink a. O n the robustness of prefer ence aggre g ation in noisy environmen ts. In Pr o c. AAMAS’07 , pa g es 41 6–422 . A CM Pr ess, 20 07. [13] A. Pro c accia, J. Rosenschein, and A. Zohar. Multi-winner elections: Complexity o f manipulation, control, a nd winner -determination. In Pr o c. IJCAI’07 , pag es 1476 – 1481 . AAAI Press , 200 7 . [14] D. Saari and V. Merlin. The Cop ela nd metho d I: Relatio nships and the dictiona ry . Ec onomic The ory , 8(1):51–7 6, 1996 . 15