Sincere-Strategy Preference-Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control

Sincere-S trategy Preferen ce-Based Ap prov al V oting Fully Resists Constru cti ve Co ntrol and Broadly Resists Destructi ve Control ∗ G ´ abor Erd ´ elyi † and Markus Nowa k and J ¨ or g Rothe ‡ Institut f ¨ ur Informatik Heinrich-Heine-Univ ersit ¨ at D ¨ usseldorf 40225 D ¨ usseldorf Germany June 12, 2008 Abstract W e study sincere-strategy preferen ce-based approv al v o ting (SP-A V), a system proposed by Brams and San ver [1] and here ad justed so as to co er ce a dmissibility o f the votes (rather than excluding inad missible votes a priori ), with r espect to p rocedu ral control. In such co n- trol scenario s, an external agent seeks to chang e the outcome of an electio n via actions such as adding/d eleting/par titioning either candidates or voters. SP- A V combines the voters’ p reference ranking s with their appr ovals of candidates, where in elections with at least two cand idates the voters’ ap proval st rategies are adjusted—if need ed—to app rove of their most-pref erred cand i- date and to disap prove of their least-pref erred candidate. This r ule coerc es admissibility of the votes ev en in the pr esence of con trol actions, and hy bridizes, in effect, approval with pluralitiy voting. W e prove that this system is computation ally resistant (i.e., the cor respond ing control prob- lems are NP-hard) to 19 o ut of 2 2 types of co nstructive and destruc ti ve con trol. Thus, SP- A V has m ore resistances to co ntrol th an is currently k nown fo r any oth er natur al voting system with a poly nomial-time winner problem . In particular, SP-A V is ( after Cop eland voting, see Falisze wski et al. [2, 3]) the second n atural voting system with an easy winner-determination proced ure that is k nown to have fu ll resistance to constru ctiv e contro l, an d unlike Copeland voting it in addition displays broad resistance to destructi ve control. ∗ Supported in p art by the DFG un der grants R O 120 2/12-1 (within th e Europ ean Science Fo undation’ s EUR OCORE S program LogICCC: “Computational Found ations of Social Choice”) and R O 1202/11 -1 and by the Alexande r von Hum- boldt Foundation’ s Tran sCoop program. Pr eliminary version s of this paper have been presented at the 33rd International Symposium on Mathematical Foundations of C omputer Science (MFC S-08) [4] and at the 2nd International W orkshop on Computational Social Choice (COMSOC-08). W ork done in part while the ﬁrst author was visiti ng Universit ¨ at Trier and while the third author was visiting the Uni versity of Rochester . † URL: ccc.cs.uni -duesseldor f.de/ ∼ erdel yi. ‡ URL: ccc.cs.uni -duesseldor f.de/ ∼ rothe . 1 1 Introd uction V ot ing prov ides a particularly useful m ethod for preference aggrega tion and collecti ve decisi on- making. While voting systems were originally used in politica l scienc e, econo mics, and operat ions researc h, the y are now also of central importanc e in v arious areas of computer science, such as artiﬁcial inte lligence (in particular , within multiagent systems). In automat ed, large- scale comput er setting s, voti ng systems ha ve b een applied , e.g., for plan ning [5] and simila rity search [6], and ha ve also been used in the design of recommender systems [7] and ranki ng algorithms [8] (where they help to lessen the spam in meta-sear ch web-pag e rankings). For such applicatio ns, it is cruci al to exp lore the computational propert ies of voti ng systems and, in particula r , to study the complexity of prob lems relate d to votin g (see, e.g., the surve y by Falisze w ski et al. [9]). The study o f v oting sys tems from a compl exit y-theor etic perspe cti ve was initiated by Bartho ldi, T ove y , and Trick ’ s series of seminal papers about t he complex ity of winner determination [ 10], manipula tion [11], an d proced ural control [12] in election s. This paper contri b utes to the study of e lectora l control, where an exte rnal agent—trad itionall y called the cha ir —seeks to inﬂu- ence th e outcome of an election via procedural changes to the election’ s structu re, namely via adding /deleti ng/partitioning either can didates or v oters (see Sectio n 2.2 for the for mal deﬁnit ions of our control problems). W e consider both constructive control (introduce d by Barthol di, T ov ey , and T rick [12]), where the chair’ s goal is to make a gi ve n candi date the unique winner , and destru ctive contro l (introduced by Hemaspaand ra, Hemaspaan dra, and Rothe [13]), where the chair’ s goal is to pre vent a giv en candida te from bei ng a unique winner . W e in vesti gate the same twenty types of constr ucti ve and destructi ve contr ol that were studied for approv al voti ng [13] and two additional contro l types introduced by Falisz e wski et al. [14] (see also [2]), and we do so for a varian t o f a v oting system th at was prop osed by Brams an d San ver [1] as a combina tion of preferen ce-bas ed and approv al v oting . Appro v al voting was introd uced by Brams and Fishb urn [15] as follo ws: Every vote r either appro ves or disap prov es of each candida te, and e very cand idate with the lar gest number of appro vals is a winner . One of the simplest prefere nce- based vot ing s ystems is p lurality : All vot ers re port their pref erence r anking s of the ca ndidat es, and the winners are the cand idates that are rank ed ﬁrst-plac e by the large st number of v oters. The purpo se of this paper is to sho w that Brams and San ver’ s combined system (adapted here so as to kee p its useful features e ven in the prese nce of control actions) combine s the strengths, in terms of computa tional resis tance to control, of plurality and approv al v oting . Some v oting systems are immune to ce rtain types of contro l in the sense t hat it is ne ver possib le for the chair to reach his or her goal via the correspond ing control action. Immunity to any type of contro l unconditi onally shield s the voti ng system against this p articul ar control type. Howe ver , like most v oting systems approv al v oting is susceptibl e (i.e., not immune) to many types of co ntrol, and plural ity v oting is suscep tible to all types of contro l. 1 Ho wev er , and this was Bartholdi , T ove y , and T rick’ s brillian t insight [12 ], e ve n for systems susceptible to control, the chair ’ s task of contro lling a gi ven e lection may be t oo hard co mputatio nally (namely , N P-hard) fo r him o r her to succe ed. T he 1 A related line o f research has sho wn that, in principle, all natural v oting sy stems can be man ipulated by strateg ic v ot- ers. Most notable among such results is the classical w ork of Gibbard [16] and Satterthwaite [17]. The study of strategy- proofness is still an extremely active and interesting area in social choice theory (see, e.g., Duggan and Schwartz [18]) and in artiﬁcial intelligence (see, e.g., Everaere et al. [19]). 2 Number of Condorc et Appro v al Llull Copelan d Plurality SP-A V resista nces 3 4 14 15 16 19 immunities 4 9 0 0 0 0 vulner abilitie s 7 9 8 7 6 3 Referenc es [12, 13] [12, 13] [2, 14, 3] [2, 14, 3] [12, 13, 2, 14] The orem 3.1 T able 1: Number of res istance s, imm unities , and vulnerab ilities to our 22 c ontrol type s. (Regarding the “Condorc et” column, see Footnote 3.) v oting system is then said to be re sistant to this control type. If a vo ting system is suscepti ble to some type of control, bu t the chair’ s task can be solv ed in polynomial time, the system is said to be vulner able to this control type. The quest for a natu ral v oting syst em with an easy winner -determinat ion procedu re that is uni- ver sally resist ant to contro l has lasted for more than 15 years now . A mong the vo ting syste ms that ha ve been stud ied with respe ct to control are plur ality , Condorcet, appro v al, cumulati ve, Llull, and (v ariant s of) Copeland v oting [12, 13, 20, 21, 14, 3, 22, 2]. Among these systems, plurali ty and Copeland v oting (denoted Copeland 0 . 5 in [2, 3]) display the broades t resista nce to control, yet eve n the y are no t uni versal ly control-r esistan t. The o nly syst em currentl y kno wn to be fu lly resis tant—to the 20 types of construc ti ve and destru cti ve control stud ied in [13, 20]—is a highly artiﬁcial sys- tem construc ted via hybridiz ation [20]. (W e mention that this syste m was not designed for direct, real-w orld use as a “natur al” system but rather was inten ded to rule out the exis tence of a certain impossib ility theor em [20].) While appr ov al vo ting nicely disting uishes between each vo ter’ s acceptable and inaccep table candid ates, it ignores the prefer ence rankin gs the voter s may hav e about their approv ed (or disap- pro ved) c andida tes. T his sh ortcomin g motiv ated Brams and San ver [ 1] to in troduc e a v oting s ystem that comb ines appro v al and prefe rence-b ased vo ting, and the y deﬁned the relate d notions of sinc ere and admissib le appro val strategi es, which are quite natu ral requirements. W e adapt their sincere- strate gy preferenc e-based appro v al voti ng system in a natural way such that, for electio ns w ith at least tw o c andida tes, admissibility of app rov al strategi es (see Deﬁnition 2 .1) can be ensu red ev en in the presence of control actions such as deleting candidate s and partitionin g candidates or vo ters. 2 The purpos e of this paper is to study if, and to w hat extent, this system inherit s the control resis- tances of plural ity (which is perhaps the simplest preferen ce-base d sys tem) and approv al v oting. Denoting this system by SP -A V , we sho w that SP-A V does combin e all the resistan ces of plura lity and appro val voti ng. More s peciﬁcally , w e prove that sin cere-st rateg y preference-b ased appro v al v oting is resistan t to 19 and vuln erable to only three of the 22 types of con trol considered here. For compa rison, T able 1 sho w s the number of resis tances, immunities, and vulner abilitie s to our 22 control types that are 2 Note that in control by partition of voters (see Section 2.2) the run-of f may hav e a reduced number of candidates. 3 kno wn for each of Condorc et, 3 appro val, Llull, pluralit y , 4 and C opelan d vo ting (see [12 , 13, 2, 14, 3]), and for SP-A V (see Theorem 3.1 and T able 2 in Sectio n 3). This paper is or ganized as follo ws. In Section 2, we deﬁne and discu ss sincere-s trate gy prefere nce-ba sed appr ov al vot ing, the types of control stu died in this paper , and t he no tions of immunity , suscept ibility , vuln erabilit y , and resistan ce. In Section 3, we p rov e our resul ts on SP-A V. Finally , in S ection 4 we gi ve our conclusio ns and state some open probl ems. 2 Pr eliminaries 2.1 Pr efer ence-Based Appr oval V oting An electio n E = ( C , V ) is speciﬁed by a ﬁnite set C of candida tes and a ﬁ nite col lection V of v oters who expr ess their preference s over the candidates in C , where distin ct v oters may , of course, hav e the same preferen ces. How the vote r preferences are represe nted depends on the v oting system used. In appro v al v oting (A V , for short), eve ry voter draws a line betwee n his or her acceptable and inacce ptable candid ates (by specifyin g a 0-1 approv al vector , where 0 repre sents disa ppro v al and 1 repres ents a ppro v al), yet does not rank them. In co ntrast, many other important v oting syst ems ( e.g., Condorc et v oting, Copeland votin g, all scoring protocols , inc luding plurality , B orda count, veto, etc.) are based on vo ter pref erences that are speciﬁed as tie-free line ar order ings of the candida tes. As is most common in the literature, v otes w ill here be represen ted nons uccinct ly: one ballot per v oter . Note that some papers (e.g., [23, 2, 14, 3]) also consider succ inct input represen tations for electio ns where multipl icities of votes are gi ven in binary . Brams and San ver [1] introd uced a v oting system that combin es approv al and prefe rence-b ased v oting. T o disti nguish this syste m from other systems that these author s introdu ced with the same purpo se of combining appro v al and prefere nce-ba sed v oting [24], w e call the va riant considere d here (incl uding the assumption of sincer ity as expl ained belo w and includ ing Rule 1 belo w , which will coerce admissib ility) sinc er e-strate gy pr efer ence-based appr oval voting (S P-A V , for sho rt). Deﬁnition 2.1 (Brams and San ver [1 ]) . Let ( C , V ) be an election, wher e the voters both indicate appr ovals/di sappr ovals of the candid ates and pr ovid e a tie-fr ee line ar ord ering of all candidates . F or each voter v ∈ V , an A V strate gy of v is a subset S v ⊆ C suc h that v appr oves of all candidate s in S v and disappr oves of all candidate s in C − S v . The list of A V str ate gies for all voters in V is 3 Note that T able 1 li sts only 14 instead of 22 types of control for Co ndorcet. The reason is that, as in [13], we consider two types of control by partition of candidates (namely , with and without run-off) and one type of control by partition of voters, and for each parti tion case we use the rules TE (“ties eliminate”) and TP (“ties promote”) for handling ties that may occur in t he corresponding subelections (see Section 2.2). Howe ver , since C ondorcet winners are alw ays unique when they exist, the disti nction between TE and TP is not made for the partition cases wi thin C ondorcet voting. Note further that the t wo additional control types in S ection 2.2.1 (namely , constructiv e and destructi ve control by adding a limited number of candidates [2, 14]) hav e not been considered for Condorcet voting [12, 1 3]. 4 Regarding the references given in T able 1 for plurality , Faliszewsk i et al. [2, 14] note t hat plurality is resistant to constructiv e and destructiv e control by adding a limited number of candidates (see Section 2.2 for the deﬁniti on of this problem). Hemaspaandra et al. [13] obtained all other results for destructi ve control within plurality , and for the constructi ve partitioning control cases in models TE and TP. The remaining results for plurality are due to Bart holdi et al. [12]. 4 called an A V strate gy proﬁle for ( C , V ) . (W e sometimes also speak of V ’ s A V strate gy proﬁle for C .) F or eac h c ∈ C , let sc or e ( C , V ) ( c ) = k{ v ∈ V | c ∈ S v }k denote the numbe r of c’ s app r ovals . Every candid ate c with the lar ges t scor e ( C , V ) ( c ) is a winner of electi on ( C , V ) . An A V strate gy S v of a voter v ∈ V is said to be admissible if S v contai ns v’ s most-pr eferr ed candid ate and does not contain v’ s least-pr eferr ed candidat e. 5 S v is said to be sincere if for each c ∈ C, if v appr oves of c then v also appr oves of each candidate ran ked higher than c (i.e., ther e ar e no gaps allo wed in sincer e appr ova l strate gies). An A V strate gy pr oﬁle for ( C , V ) is admiss ible (r espectively , sincere ) if the A V strate gies of all voter s in V ar e admissible (r espect ively , sincer e). Admissibil ity and sincerity are quite natural requiremen ts. In particular , requiring the vote rs to be since re ensures that their preference rankings and their appro vals/ disappr ov als are not contra- dictor y . Note that sincere strategie s for at least two candidat es are alwa ys admissible if vote rs are neithe r allowed to app rov e of e verybod y nor to d isappro ve of e verybody (i.e., if we requ ire vote rs v to ha ve only A V strateg ies S v with / 0 6 = S v 6 = C ), an assumption adop ted by B rams and San ver [1]. 6 Hencefor th, we will assume that only since re A V strategy proﬁle s a re c onside red, which—as suming that the tri vial cas es S v = / 0 and S v = C are ex cluded—n ecessa rily are ad missible whe ne ver the re are at least two cand idates. 7 A v ote with an insincere strateg y w ill be con sidered void. The follo w ing notation w as used by B rams and San ver for a dif ferent e lection system [24], b ut is usefu l for SP -A V as w ell: Preferences are represent ed by a left-to- right ranking (separated by a space) of the candidates (e.g., a b c ), with the leftmost candid ate being the most-prefe rred one, and approv al strategi es are denoted by insert ing a straight line into such a ranking , where all candid ates left of this line are appro ved of and all candida tes right of this line are disap prov ed of (e.g., “ a | b c ” mea ns tha t a is ap pro ved o f, whil e b oth b and c are disappro ved of by this v oter). In our constructio ns, we sometimes also inse rt a subs et B ⊆ C into such approv al rankings, where we assume some arbitra ry , ﬁxed order of the can didate s in B (e.g., “ a | B c ” means that a is appr ov ed of, while all b ∈ B an d c are disappro ved of by this vo ter). 2.2 Contr ol Problems f or Prefer ence-Based Appr oval V oting The con trol problems consid ered here w ere i ntroduc ed by Barth oldi, T ove y , and T rick [12] for co n- structi ve control and by H emaspaan dra, Hemaspaandr a, and Rothe [13] for destructi ve control. In constr ucti ve control scen arios th e c hair’ s goal is to mak e a fa vor ite candi date win, and in de structi ve contro l scen arios the chair’ s goal is to ensure th at a desp ised c andida te does not win . As is common, the chair is assumed to ha ve complete kno wledge of the vo ters’ prefere nce rankings and appro val strate gies, 8 and as in most pa pers on elec toral control we deﬁne t he contro l problems in the un ique- 5 Brams and Sanv er [1] deﬁne an A V strategy t o be admissible if it is not dominated in a game-theoretic sense [15], and no te that “admissible strategies under A V in volve always v oting for a most-preferred candid ate and n e ver voting for a least-preferred candidate. ” S ince we do not focus on the gam e-theoretic aspects of A V strategies, we deﬁne admissibility as in Deﬁnition 2.1. 6 Brams and Sanv er actually preclude only the case S v = C for sincere voters v by stating that “sincere strat egies are alway s admissible if we exclude ‘vote for e verybo dy’ ” [1]. Ho we ver , an A V st rategy that disapprov es of all candidates obvio usly is sincere, yet not admissible according to Deﬁnition 2.1, which is why we also exclude the case of S v = / 0. 7 Note that an A V strategy is ne ver admissible for less than tw o candidates. 8 A detailed discussion of this assumption can be found i n [13]. In a nutshell, one justiﬁcation of this assumption is t hat it is realistic in many (though certainly not in all) situations, particularly in those in volving small-scale priv ate 5 winner model. 9 In this model, the chair seeks to, via the control action, either make a designa ted candid ate the unique winner (in the constru cti ve case) or to prev ent a designated candidate from being a unique winner (in the destruc ti ve case). T o achie ve his or her goal, the chair modiﬁes the structur e of a giv en election via adding /deleti ng/partitioning eith er candidates or vo ters. Such control actions —speciﬁcall y those with respect to control via deleting or partitionin g candid ates or via partitio ning vo ters—may ha ve an unde sirable impact on the resu lting election in that the y might turn admissi ble A V str ateg ies into inadmiss ible ones. That is why we deﬁne the follo wing rule that coerces admissib ility (e ven under such contr ol actio ns): Rule 1 (A V Stra tegy Re write Rule) . If in a n electio n ( C , V ) with k C k ≥ 2 we hav e S v = / 0 o r S v = C for some voter v ∈ V , then each such voter’ s A V str ate gy is adjuste d to appr ove of his or her top candid ate and to disappr ove of his or her bottom candidate . This rule coerces / 0 6 = S v 6 = C for each v ∈ V whene ver there are at least two candi dates. That is, though it is legal for a voter to cast an inad missible vote, the SP-A V system w ill re w rite this v ote to make it admissibl e. In Section 2.3 below , we w ill brieﬂy discus s the SP-A V system and, in particu lar , some subtle points regard ing Rule 1. W e no w formally deﬁne our control problems, where each problem is deﬁned by stating the proble m instance together w ith two ques tions, one for the constr ucti ve and one for the destructi ve case. These control problems are tailore d to sincer e-strate gy prefer ence-ba sed appro v al vo ting by requir ing e very electio n occurrin g in these cont rol problems (be it before, durin g, or after a contr ol action —so, in particular , this also applies to the subelect ions in the partit ioning cases) to ha ve a sincer e A V strate gy proﬁle. Note that w hen the number of candidates is reduced (due to deleting candid ates or pa rtitioni ng candidates or v oters), approv al lines may ha ve to be mo ve d in ac cordan ce with Rule 1. T o av oid unneces sary repetition , w hen deﬁning the 22 contro l scenari os and problems consid- ered in this paper , we will o mit (or only v ery brieﬂy sk etch) the moti v ation of these control s cenario s. Note, ho w e ver , that each scen ario considere d has a natur al real-world interpre tation—ra nging from “get-o ut-the-v ote” dri ves (contro l by adding vo ters) ove r v ote suppres sion or disenfra nchise ment (contr ol by deleting vot ers) to gerrymand ering (c ontrol by partitio ning v oters) for v oter control, and similarly natural real-world interpretat ions hav e been discussed in detail for the single cases of candid ate contr ol. These real-world interpretat ions and moti v ating exampl es hav e been described at leng th in a number of pre vious papers on control, such as [12, 13, 2, 20]. (Note that the journal ver sion of [20] appears in the same special issue as the present paper .) elections and in those inv olving large-scale elections among software agents that cooperate in a multiagent en vironment and have an incen tiv e to reve al their preference s o ver the gi ven a lternativ es. Another justiﬁcation is that this pape r focu ses on proving control resistances of (i.e., NP-hardness results for) SP-A V , and an NP-hardness result in the more restrictive setting o f comp lete kno wledge clearly implies the correspon ding NP-hardness res ult in the more ﬂ exible setting of partial kno wl edge (see [13] for more discussion of this point). 9 Exceptions are, e.g., [21, 2, 3 , 14], where [2, 3, 14] co nsider b oth the unique-winner mod el an d the no nunique-winne r model. 6 2.2.1 Contr ol by Add ing Candidates In this contr ol scenario, the chair seeks to reach his or her goal by adding to the election, which origin ally in volv es only “qualiﬁed” candi dates, some ne w candid ates who are chosen from a gi ven pool of spoiler candid ates. In their study of control for plurality , Condo rcet, and approv al votin g, Hemaspaan dra, Hemaspaan dra, and Rothe [13] considere d only the case of adding an unlimited number of spoiler candidat es (which is the original varian t of this prob lem as deﬁned by B arthold i, T ove y , an d Tri ck [12]). W e consider the same vari ant of this problem here to make our results comparab le with tho se establish ed in [13], b ut for completeness we in addition conside r the cas e of adding a limited number of spoil er candidate s, where the prespec iﬁed limit is part of the problem instan ce. This v ariant of this proble m was introd uced by Falisze wski et al. [2, 14, 3 ] in analogy with the deﬁnition s of control by deleting candidate s and of control by adding or deleti ng v oters. They sho wed that, for the elect ion system Copeland α the y in vest igate, the complexity of these two proble ms can drastically change depending on the parameter α , see [2, 3]. W e ﬁrst deﬁne the unlimited v ariant of contro l by addi ng candidates . Name Contr ol by Adding an Unlimited Number of Candidates . Instance An elec tion ( C ∪ D , V ) and a desig nated candidate c ∈ C , w here the set C of qualiﬁed candid ates and the set D of spoi ler candidates are disjoint. Question (construct iv e) Is it possible to choose a subset D ′ ⊆ D such that c is the uniq ue winner of election ( C ∪ D ′ , V ) ? Question (destructi ve ) Is it possible to choose a subset D ′ ⊆ D such that c is not a unique w inner of election ( C ∪ D ′ , V ) ? The problem Control by Adding a Limited Number of Candidat es is deﬁned analogous ly , with the only dif ference being that the chair seeks to reach his or her goal by adding at most ℓ spoiler candid ates, w here ℓ is part of the prob lem instance. 2.2.2 Contr ol by Deleting Candidates In this control scenario, the chair seeks to reach his or her goal by deletin g (up to a giv en number of) cand idates . Here it m ay ha ppen th at i nadmissi ble A V strate gies are create d by the c ontrol actio n, b ut Rule 1 will coerc e admissibilit y again (by mov ing the lin e bet ween some v oter’ s acceptable and inacce ptable candidate s to behind the top candidate or to before the bottom candidate whene ver necess ary). Name Contr ol by Deleting Candidates. Instance An elec tion ( C , V ) , a desig nated cand idate c ∈ C , and a nonnega ti ve integ er ℓ . Question (construct iv e) Is it possible to delete up to ℓ candidates from C such that c is the uniq ue winner of the resultin g electi on? Question (destructi ve ) Is it possible to delete up to ℓ candid ates (other than c ) from C such that c is not a unique winner of the resulti ng elec tion? 7 2.2.3 Contr ol by Partition and Run-Off Parti tion of Cand idates There are two parti tion-of -candidates control scenar ios. In both scenarios, the chair seeks to reach his or her goal b y partit ioning the candidate set C into two subse ts, C 1 and C 2 , after which the electio n is condu cted in two stages. In control by partition of candi dates, the electio n’ s ﬁrst stage is held within only one group, say C 1 , and this group’ s winners that surviv e the tie-handling rule used (se e the n ext para graph) run ag ainst all me mbers of C 2 in th e seco nd and ﬁnal stag e. In con trol by run- of f partition of candida tes, the ele ction’ s ﬁrst stage is hel d separatel y within bo th groups, C 1 and C 2 , an d the winne rs of bo th su belectio ns that survi ve the t ie-hand ling rule use d run a gainst each other in the second and ﬁnal stage . W e use the two tie-handlin g rules proposed by Hemaspaan dra, Hemaspaand ra, and Rothe [13]: ties-pr omote (TP ) and ties-eli minate (TE). In the TP model, all the ﬁrst-stage winners of a subelec- tion, ( C 1 , V ) or ( C 2 , V ) , are promoted to the ﬁ nal round. In the TE model, a ﬁrst-stage winner of a subele ction, ( C 1 , V ) or ( C 2 , V ) , is promoted to the ﬁnal rou nd exactly if he or she is that subelect ion’ s uniqu e w inner . Note that partitio ning the candid ate set C into C 1 and C 2 is, in some sense , similar to deleting C 2 from C to obtai n subelec tion ( C 1 , V ) and to deleting C 1 from C to obtain subelect ion ( C 2 , V ) . Also, the ﬁ nal stage of the election may ha ve a reduced number of candida tes (which depends on the tie-handli ng rule used) . S o, in the partition ing cases, it may again happ en that inadmissi ble A V strate gies are created by the control action, but Rule 1 will coerc e admissibility again. Name Contr ol by Partition of Candida tes. Instance An elec tion ( C , V ) and a designated candidate c ∈ C . Question (construct iv e) Is it po ssible to par tition C into C 1 and C 2 such that c is th e uniq ue winner of the ﬁnal stage of the two-stage election in which the w inners of subel ection ( C 1 , V ) that survi ve the tie-handling rule run against all candidat es in C 2 (with respect to the v otes in V )? Question (destructi ve ) Is it possib le to p artition C into C 1 and C 2 such th at c is not a unique winn er of the ﬁnal stage of the two-stage election in which the w inners of subel ection ( C 1 , V ) that survi ve the tie-handling rule run against all candidat es in C 2 (with respect to the v otes in V )? Name Contr ol by Run-Off Pa rtition of Candida tes. Instance An elec tion ( C , V ) and a designated candidate c ∈ C . Question (construct iv e) Is it possible to partition C into C 1 and C 2 such that c is the uniq ue win- ner of the ﬁnal stage of the two- stage electio n in which the winner s of subelection ( C 1 , V ) that survi ve the tie-handlin g rule run (with respect to the v otes in V ) agains t the winners of subele ction ( C 2 , V ) that survi ve the tie-hand ling rule? Question (destructi ve ) Is it possible to parti tion C into C 1 and C 2 such th at c is not a unique winner of the ﬁnal stage o f th e two-s tage elec tion in which th e winner s of sub electio n ( C 1 , V ) that survi ve the tie-handlin g rule run (with respect to the v otes in V ) agains t the winners of subele ction ( C 2 , V ) that survi ve the tie-hand ling rule? 8 2.2.4 Contr ol by Add ing V oters In this contr ol scenar io, the chair seeks to reach his or her goal by intr oducin g ne w v oters into a gi ven election . T hese additional v oters are chosen from a giv en pool of v oters w hose preference s and app rov al strate gies o ver the can didates f rom t he origi nal elect ion are kn o wn. Again, the number of v oters that can be added is prespec iﬁed. Name Contr ol by Adding V oters . Instance An election ( C , V ) , a collectio n W of addition al v oters w ith kno wn preferences and ap- pro v al strategies ov er C , a designated candidate c ∈ C , and a nonnega ti ve integer ℓ . Question (construct iv e) Is it pos sible to choose a subs et W ′ ⊆ W with k W ′ k ≤ ℓ such that c is the uniqu e w inner of electio n ( C , V ∪ W ′ ) ? Question (destructi ve ) Is it possibl e to choose a sub set W ′ ⊆ W with k W ′ k ≤ ℓ such tha t c is not a uniqu e w inner of electio n ( C , V ∪ W ′ ) ? 2.2.5 Contr ol by Deleting V oters The cha ir here seeks to reach his or h er goa l by su ppress ing (up to a presp eciﬁed number of) v oters . Name Contr ol by Deleting V oters. Instance An elec tion ( C , V ) , a desig nated cand idate c ∈ C , and a nonnega ti ve integ er ℓ . Question (construct iv e) Is it possible to delete up to ℓ vo ters from V such that c is the unique winner of the resultin g electi on? Question (destructi ve ) Is it possible to delete up to ℓ v oters from V such that c is not a unique winner of the resultin g electi on? 2.2.6 Contr ol by Partition of V oters In this scenari o, the election again is condu cted in two stage s, and the chair now seeks to reach his or her goal by partition ing the v oters V into two subcommitte es, V 1 and V 2 . In the ﬁrst stage, the subele ctions ( C , V 1 ) and ( C , V 2 ) are held sepa rately in parallel, and the winners of each subelection who survi ve the tie-h andlin g rule mov e forwar d to the seco nd and ﬁnal stag e in which the y co mpete agains t each other . As in the cand idate-d eletion and the candid ate-par tition c ases, also in contro l by partitio n of v oters it may happen that inadmissible A V strat egie s are created by the contro l action, since the ﬁnal stage of the election may hav e a redu ced number of candidates. H o wev er , if that happ ens then Rule 1 will again coerc e admissibility . Name Contr ol by Partition of V oters. Instance An elec tion ( C , V ) and a designated candidate c ∈ C . Question (construct iv e) Is it possi ble to partition V into V 1 and V 2 such that c is the unique w in- ner of the ﬁnal stage of the two-stage election in which the winners of sube lection ( C , V 1 ) that survi ve the tie-handlin g rule run (with respect to the v otes in V ) agains t the winners of subele ction ( C , V 2 ) that surviv e the tie-handlin g rule? 9 Question (destructi ve ) Is it possi ble to part ition V in to V 1 and V 2 such that c is not a unique w in- ner of the ﬁnal stage of the two-stage election in which the winners of sube lection ( C , V 1 ) that survi ve the tie-handlin g rule run (with respect to the v otes in V ) agains t the winners of subele ction ( C , V 2 ) that surviv e the tie-handlin g rule? 2.3 A Brief Discussion of SP-A V The notion of SP -A V , as deﬁned here, slightl y dif fers from the deﬁnition propos ed in this paper’ s precur sors [4, 25]. For examp le, [4] speciﬁcally required for single-ca ndidat e elections that each v oter must appro ve of this candidate. In the present paper , we drop this requi rement, as it in fac t is not neede d (because the one candida te in a s ingle-c andida te electio n w ill al ways win—e ve n with zero appro vals, i.e., SP -A V is a “v oiced” voti ng system). The other deﬁnitiona l change is m ore subtle. In [4, 25], we adopted Brams and San ver’ s as- sumption that v oters v are requi red to h a ve admiss ible A V strat egie s (i.e., only A V stra tegie s S v with / 0 6 = S v 6 = C were allo wed). 10 By this assumption, any vot e with an inadmissib le A V strate gy was consid ered void, and we app lied our rule of re writing inadmissi ble A V stra tegie s to coerce admissi- bility only when a contro l action had turned an orig inally admissible vote into an inadmissib le one. One pro blem with this a pproac h was that th is rule dep ended on (and co uld be vi e wed as red eﬁning) contro l rather than being an integral part of the v oting system itself. In contrast, we no w allo w vot- ers to cast inadmissi ble v otes, and Rule 1 w ill turn them into admissible votes the same way it will coerce admissibi lity for vote s that became inadmissible in the course of a contro l acti on. The in- prepar ation book chapte r [26] elab orates on this point and on other points reg arding the deﬁnitional chang es S P-A V has under gone in the course of its dev elopment up to its ﬁnal form in the presen t paper . W e str ess that none of the two changes mention ed above has a se vere impact on our ﬁndings or their proofs . Another issue to be addre ssed is that the choice of Rule 1 m ight seem to be purel y a matter of taste, at ﬁrst glanc e. For example, gi ven an inadmissible A V strate gy of the form | a b c d (re- specti vely , a b c d | ), why do n’ t we change it into an admis sible vo te of the form, sa y , a b | c d rather th an, acc ording to Rule 1, int o a | b c d (resp ecti vely , a b c | d )? T he reaso n is that, once we ha ve agreed that it is desi rable to coerce admissib ility , our choice of Rule 1 is the m ost sensibl e way , as t his is the minimal ly in v asi ve r ule to coe rce admissibi lity among all poss ible such rul es: W e do change the v oters’ appro v al strategie s, b ut we wish to do this in the least harmful way . 2.4 Immunity , Susceptibility , V ulnerability , and Resistance The follo wing notions—whi ch are due to Barthol di, T ov ey , and Trick [12] (see also [13, 20, 2, 3, 14])—will be centra l to our complexity analys is of the con trol problems for SP -A V. Deﬁnition 2.2. Let E be an elect ion system and let Φ be some given type of contr ol. 1. E is said to be immune to Φ -control if 10 Except that [1] exclude s only t he case S v 6 = C , see Footnote 6. 10 (a) Φ is a const ructive c ontr ol type and it is never pos sible for the c hair to turn a designat ed candid ate fr om being not a unique winner into being the unique winner via e xerting Φ - contr ol, or (b) Φ is a destru ctive contr ol type and it is ne ver possible for the c hair to t urn a des ignated candid ate fr om being the unique winner into being not a unique winner via e xerting Φ -cont r ol. 2. E is said to be suscep tible to Φ -cont rol if it is n ot immune to Φ -contr ol. 3. E is said to be vul nerable to Φ -contr ol if E is susce ptible to Φ -contr ol and the con tr ol p r oblem associ ated with Φ is solva ble in polynomial time. 4. E is said to be resistant to Φ -control if E is suscepti ble to Φ -contr ol and the contr ol pr oblem associ ated with Φ is NP -har d. For example, appr ov al v oting is known to be immune to eight of the twelve type s of candid ate contro l considered in [13]. The proofs of these results crucia lly employ the equiv alences and im- plicati ons between immunity/susc eptibil ity for v arious control types sho wn in [13] and the f act that appro val v oting satisﬁes the unique version of the W eak Axiom of R e veal ed P referen ce (deno ted by Unique-W ARP , see [13, 12]): If a candidate c is the unique winner in a set C of candi dates, then c is the uniqu e winner in ev ery subset of C that includes c . In contras t with appro val v oting, sincer e-strate gy preferen ce-bas ed approv al voti ng does not satisfy Unique- W ARP , and we will see later in Section 3.1 that it indeed is susce ptible to each typ e of control considered here. Pro position 2.3. Sincer e-strate gy pr efer ence-base d appr oval vot ing do es not sa tisfy Unique- W ARP . Pro of. Con sider the electio n ( C , V ) with candidate set C = { a , b , c , d } and v oter collection V = { v 1 , v 2 , v 3 , v 4 } . Removin g candidate d chang es the proﬁle as foll o ws according to Rule 1: v 1 : b c a | d b c | a v 2 : c | a d b is changed to c | a b v 3 : a b c | d (by remo ving d ): a b | c v 4 : b a c | d b a | c Note that the approv al/disappro val line has been mov ed in vote rs v 1 , v 3 , and v 4 accord ing to Rule 1. A lthoug h c was the unique winner of ( C , V ) , c is not a winner of ( { a , b , c } , V ) (in fac t, b is the unique winner of ( { a , b , c } , V ) ). Thus, SP-A V doe s not satisfy U nique- W ARP . ❑ 3 Results f or Sincer e-Strategy Prefer ence-Based Ap pr oval V oting Theorem 3.1 belo w (see also T able 2) sho ws the complex ity results reg arding control of electio ns for SP-A V. As men tioned in the int roducti on, with 19 resi stances and o nly t hree vuln erabili ties, this system has more resista nces and fe wer vulnera bilitie s to contro l (for our 22 control types) than is curren tly kno wn for any other natural v oting system with a polynomial -time winner proble m. 11 Plurali ty SP-A V A V Control by Constr . Destr . Constr . Destr . Constr . Destr . Adding an Unlimite d Num ber of Candi dates R R R R I V Adding a Limited Number of Candida tes R R R R I V Deleti ng Candidates R R R R V I Parti tion of Candidate s TE: R TE: R TE: R TE: R T E: V TE: I TP: R TP: R TP: R TP: R TP: I TP: I Run-of f Pa rtitio n of Candidates TE: R TE: R TE: R TE: R T E: V TE: I TP: R TP: R TP: R TP: R TP: I TP: I Adding V oters V V R V R V Deleti ng V oters V V R V R V Parti tion of V oters TE: V TE: V TE : R TE: V TE: R TE: V TP: R TP: R TP: R TP: R TP: R TP: V T able 2: Overvie w of results. Key : I means immune, R means resistant, V means vulnerable , TE means ties-eliminate , and TP means tie s-promot e. Results for SP-A V are ne w; th eir pr oofs are either ne w or draw on proo fs from [13]. Results for plur ality and A V , stated here to allo w comparison, are due to Bartholdi, T ove y , and T rick [12] and to Hemaspaandra , Hemaspaandra , and Rothe [13]. (The resul ts for control by adding a limited number of candidates for plurality and approv al v oting, thoug h not stated ex plicitl y in [12, 13], follo w immediately from the proofs of the corresp onding results for the “unlimite d” v ariant of the problem, see Footnote 4.) Theor em 3.1. Sincer e-strate gy pr efer ence-based appr oval voting has the r esista nces and vulner a- bilitie s to the 22 types of contr ol deﬁned in Section 2.2 that ar e shown in T able 2. 3.1 Susceptibility By deﬁnitio n, all resistance and vulner ability results in particular require susceptib ility . In the fol- lo wing two lemmas, w e pro ve that sincere- strate gy preference-b ased approv al v oting is susceptible to the 22 types of control deﬁned in Section 2.2. T o this end, we will make use of Theorems 4.1, 4.2, and 4.3 of Hemaspaandra, Hemaspaandra, and Rothe [13 ] that provi de susceptibil ity eq ui v- alence s and implicatio ns for vari ous control type s. 11 For the sak e of self-contai nment, we gi ve these results belo w , stated essentially word-for -word as in [13]. In particular , Theorem 3.2 (which is [13, Thm. 4.1] ) gi ves four e qui valenc es between susce ptibilit y to constru cti ve /destru ctiv e control by addin g/dele ting candidates/ v oters; Theorem 3.3 (which is [13 , Thm. 4.2]) giv es four implic a- tions that li nk suscepti bility to control by (run-of f) partit ion of candidat es/v oters w ith susce ptibili ty to control by dele ting candidates/ v oters; and Theorem 3.4 (which is [13, Thm. 4.3]) states t hat e very “vo iced” v oting syst em is susc eptible to constructi ve control by dele ting candidate s and to destru cti ve cont rol by adding candid ates, and that for each v oiced vot ing syste m suscep tibilit y to destru cti ve control by partitio n of v oters (in model TE or TP ) implies susceptibili ty to destructi ve contro l by deleting vot ers. A votin g system is said to be voic ed if in e very one-can didate election, this candid ate w ins. 11 Although [13] does not consider t he case of control by adding a limited number of candidates explicitly , it is imme- diate that all proofs for the “unlimited” case in [13] work also for this “limited” case. 12 Theor em 3.2 (Thm. 4.1 of [13]) . 1. A voting syst em is suscepti ble to co nstruct ive contr ol by addin g (either a limited or an unlimited number of) candidates if and only if it is suscep tible to destruc tive contr ol by deleti ng cand idates . 2. A voting system is susce ptible to const ructive contr ol by deleting candida tes if and only if it is suscepti ble to destructiv e contr ol by addi ng (eithe r a limited or an unli mited number of) candid ates. 3. A voting system is suscep tible to constructi ve contr ol by adding voters if and only if it is suscep tible to des tructiv e contr ol by deleting voters. 4. A voting system is susceptib le to constructi ve contr ol by delet ing voter s if and only if it is suscep tible to des tructiv e contr ol by adding voters. Theor em 3.3 (Thm. 4.2 of [13]) . 1. If a voting system is susce ptible to constru ctive contr ol by partit ion of voters (in model TE o r TP), then it is su sceptib le to con structi ve cont r ol by delet- ing candid ates. 2. If a voting system is suscep tible to constru ctive contr ol by partition or run-o f f partition of candid ates (in model TE or TP), then it is susceptible to constructiv e contr ol by deleting candid ates. 3. If a voting syste m is suscepti ble to con structiv e contr ol by partiti on of vote rs in model TE, then it is suscep tible to con structi ve contr ol by deleting voters . 4. If a vo ting system is suscep tible to destru ctive co ntr ol by partition or run -of f pa rtition of candid ates (in model TE or T P), then it is susceptib le to destruct ive contr ol by delet ing can- didate s. Theor em 3.4 (Thm. 4.3 of [13]) . 1. If a voiced voting system is suscep tible to dest ructive con- tr ol by partition of vote rs (in m odel TE or TP ), then it is suscept ible to destruc tive contr ol by deletin g voter s. 2. Each voiced voting system is suscept ible to cons tructiv e contr ol by deleting candidates. 3. Each voiced voting syste m is susc eptible to destru ctive contr ol by adding (either a limited or an unlimited number of) candid ates. W e start with suscep tibilit y to candidate control for S P-A V . Lemma 3.5. SP-A V is susceptible to constr uctive and destructi ve contr ol by adding candidat es (in both the “limited” and the “unlimited” variant of the pr oblem), by deleting candidat es, and by partit ion of candid ates (with or w ithout run-of f and for each in both tie-hand ling m odels, TE and TP). 13 Pro of. From Theorem 3.4 and the obvious fact that SP-A V is a voic ed v oting syst em, it i m- mediatel y follo ws that SP-A V is suscepti ble to constructi ve control by deleting candid ates and to destru cti ve control by adding cand idates (in both the “limited ” and the “unl imited” varian t of the proble m). Consider the elect ion ( C , V ) with candidate set C = { a , b , c , d , e , f } and vo ter collectio n V = { v 1 , v 2 , . . . , v 6 } and the followin g partition of C into C 1 = { a , c , d } and C 2 = { b , e , f } : ( C , V ) is partition ed into ( C 1 , V ) and ( C 2 , V ) v 1 : a b c | d e f a c | d b | e f v 2 : b c | a d e f c | a d b | e f v 3 : a c | b d e f a c | d b | e f v 4 : b a c | d e f a c | d b | e f v 5 : a b d e c | f a d | c b e | f v 6 : a e d f c | b a d | c e f | b W ith six approv als, c is the unique winner of ( C , V ) . H o wev er , a is the unique winner of ( C 1 , V ) , which implies that c is not promoted to the ﬁnal st age, re gardless of whether we use t he TE or TP tie- handli ng rule an d reg ardless of w hether w e emplo y a p artition o f can didates with or witho ut run-o f f. Thus, SP-A V is sus ceptibl e to destruc ti ve contr ol by part ition of cand idates (with or withou t run-of f and for each in both tie-ha ndling models, TE and T P). By Theorem 3.3, SP-A V is also susce ptible to destructi ve control by dele ting candi dates. By Theorem 3.2 in turn, SP -A V is also susce ptible to constr ucti ve control by addin g candidates (in both the “limited” and the “unlimited” varian t of the proble m). Note th at a is no t the u nique winner o f ( C , V ) , a s a l oses to c by 5 to 6. Howe ver , if we pa rtition C into C 1 = { a , c , d } and C 2 = { b , e , f } , then a is the unique winner of ( C 1 , V ) and b is the uniq ue winner of ( C 2 , V ) . Since both subelect ions ha ve a unique winner , it does not matter whethe r the TE rule or the TP rule is appli ed. The ﬁnal-st age election is ( { a , b } , V ) in the case of run-of f partition of candida tes, and it is ( { a , b , e , f } , V ) in the case of part ition of candidates. Since a wins against b in the former case b y 4 to 2 and in th e latter case by 5 to 4 (an d e and f do ev en wo rse than b in th is case), a is the unique winner in both cases. Thus, S P-A V is suscep tible to constructi ve control by partiti on of can didates (with or w ithout run -of f and for each in both models, TE and TP ). ❑ W e no w turn to suscep tibility to v oter con trol. Lemma 3 .6. SP-A V is susceptib le to constructive and destructiv e contr ol by ad ding vo ters , by d elet- ing voter s, and by partition of voters in both tie-handl ing models, TE and TP. Pro of. Consider the electi on ( C , V ) with candidate set C = { a , b , c , d , e , f } and v oter collecti on V = { v 1 , v 2 , . . . , v 8 } and par tition V into V 1 = { v 1 , v 2 , v 3 , v 4 } and V 2 = { v 5 , v 6 , v 7 , v 8 } . Thus, we 14 chang e: ( C , V ) into ( C , V 1 ) and ( C , V 2 ) v 1 : a b c | d e f a b c | d e f v 2 : a c | b d e f a c | b d e f v 3 : c b a d | e f c b a d | e f v 4 : a b | d e c f a b | d e c f v 5 : a d c | b e f a d c | b e f v 6 : e b c d | a f e b c d | a f v 7 : d e c f | b a d e c f | b a v 8 : d f | b a c e d f | b a c e W ith six ap prov als, c is the uniqu e winner of ( C , V ) . Ho wev er , a is the uni que winner o f ( C , V 1 ) and d is the unique winner of ( C , V 2 ) , which implies that c is not promoted to the ﬁ nal stage, regard less of whether we use the TE or TP tie-hand ling rule. (In the ﬁ nal-sta ge election ( { a , d } , V ) , a wins by 5 to 3.) Thus, SP-A V is suscep tible to destru cti ve control by partitio n of vote rs in models TE and TP. B y Theorem 3.4 and since S P-A V is a v oiced system, SP -A V is also susc eptible to destructi ve contro l by deleting v oters. F inally , by Theorem 3.2 , SP -A V is also susceptible to constructi ve c ontrol by adding v oters. No w , if we l et a and c chan ge their roles i n the a bov e election and ar gument, we see tha t SP-A V is also susceptible to construc ti ve control by partiti on of v oters in models TE and TP. By Theo- rem 3.3, susce ptibili ty to construc ti ve contr ol by partition of voters in model TE implies suscep ti- bility to cons tructi ve contro l by deleting v oters. A gain, by Theorem 3.2, SP-A V is also susceptible to destr ucti ve contro l by addi ng v oters. ❑ 3.2 Candidate Contr ol Theorems 3.7 and 3.10 belo w sho w that sincere-str ateg y pref erence- based approv al vot ing is fully resista nt to candidate contro l. This result should be contra sted with that of Hemaspaa ndra, Hema- spaand ra, and Rothe [13], who pro ved immunit y and vuln erabili ty for all c ases of ca ndidat e control within appro val votin g (see T able 2). In fa ct, S P-A V has the same resistanc es to candidate control as plura lity , and we wil l s ho w th at the con structio n presented in [1 3] to prove plu rality resist ant also works for since re-strat egy preferenc e-based approv al vo ting in all cases of candidate control ex- cept one—na mely , except for c onstruc ti ve contr ol by del eting candid ates. Theorem 3 .10 e stablis hes resista nce for this one missing case. All resistance results in this section follow via a reducti on from the NP-complete problem Hit- ting Set (see, e.g., Garey an d Johnson [27]): Name Hittin g Set. Instance A set B = { b 1 , b 2 , . . . , b m } , a nonempty collectio n S = { S 1 , S 2 , . . . , S n } of subsets S i ⊆ B , 12 and a positi ve integer k ≤ m . 12 Our as sumption that S be nonempty (i.e., that n ≥ 1) is not explicitly speciﬁed in Garey and Johnson [27 ]. Howe ver , it is clear that requiring n ≥ 1 does not change the complex ity of the problem. 15 Question D oes S ha ve a hitting set of size at most k , i.e., is there a set B ′ ⊆ B with k B ′ k ≤ k such that for each i , S i ∩ B ′ 6 = / 0? Note th at so me o f our proofs for SP-A V are b ased on constr uctions pre sented in [13] to prov e the corres pondin g results for ap prov al v oting or plurality , whereas so me ot her of our res ults r equire ne w insigh ts to make the proo f work for SP-A V . For completeness , we w ill present each con structio n here (e ven if the modiﬁcatio n of a pre vious constru ction is rath er straigh tforwa rd), explicitl y stating whether it is base d on a previo us const ruction from [13], and if so, we will state in each case on which constr uction it is based and what the differe nces to the related previ ous constructio n are. Theor em 3.7. SP-A V is re sistant to all types of constr uctive and destruct ive candidate contr ol de- ﬁned in Section 2.2 exce pt for cons tructiv e contr ol by deleting candidat es (which will be handled separ ately in Theor em 3.10). Resistan ce of SP-A V to const ructi ve control by deleting candidate s, which is the missing case in Theorem 3.7, will be sho wn as Theor em 3.10 below . The proof of Theore m 3.7 is based on a constructio n for plurality in [13], exce pt th at only the ar guments for des tructiv e candidate con trol a re gi ven ther e (simpl y b ecause p luralit y was sho wn resista nt to all cases of constructi ve can didate control alr eady by B arthold i, T o ve y , and T rick [12 ] via dif ferent construct ions). W e now prov ide a short proof sketch of Theorem 3.7 and the constructio n from [13 ] (slightly modiﬁed so as to fo rmally con form with the SP-A V voter representati on) in order to (i) sho w that the same const ruction can be used to establish all b ut one resistan ces of S P-A V to constr uctive candid ate control , and (ii) expla in why construct iv e control b y del eting ca ndidat es (which is missing in Theorem 3.7) does not follo w from this constructi on. Pro of Sketch of Th eor em 3.7. Suscepti bility holds by L emma 3.5 in each case. The resistance proofs are base d on a reduction from Hitting Set and emplo y Constructi on 3.8 belo w , slightly mod- iﬁed so as to formally conform with the SP-A V v oter represent ation. Construction 3.8 (Hemas paandr a et al. [ 13]) . Let ( B , S , k ) be a given ins tance of Hitting Set, wher e B = { b 1 , b 2 , . . . , b m } is a s et, S = { S 1 , S 2 , . . . , S n } is a n onempty colle ction of subset s S i ⊆ B, and k ≤ m is a pos itive inte ger . Deﬁne the ele ction ( C , V ) , wher e C = B ∪ { c , w } is the can didate set and wher e V consists of the following voters: 1. Ther e ar e 2 ( m − k ) + 2 n ( k + 1 ) + 4 voter s of the form: c | w B . 2. Ther e ar e 2 n ( k + 1 ) + 5 voters of the form: w | c B . 3. F or eac h i, 1 ≤ i ≤ n, ther e ar e 2 ( k + 1 ) voters of the form: S i | c w ( B − S i ) . 4. F or eac h j, 1 ≤ j ≤ m, ther e ar e two voters of the form: b j | w c ( B − { b j } ) . Since scor e ( { c , w } , V ) ( c ) − scor e ( { c , w } , V ) ( w ) = ( 2 ( m − k ) + 2 n ( k + 1 ) + 4 + 2 n ( k + 1 )) − ( 2 n ( k + 1 ) + 5 + 2 m ) = 2 k ( n − 1 ) + 2 n − 1 16 is posit iv e (because of n ≥ 1), c is the un ique winner of elec tion ( { c , w } , V ) . The ke y obser v ation is the follo wing propo sition , which can be prov en as in [13]. Pro position 3.9 (Hemaspaandr a et al. [13]) . 1. If S has a hitting set B ′ of size k , then w is the uniqu e SP-A V winner of election ( B ′ ∪ { c , w } , V ) . 2. Let D ⊆ B ∪ { w } . If c is not the uniqu e SP-A V winner of election ( D ∪ { c } , V ) , then ther e e xists a set B ′ ⊆ B suc h that (a) D = B ′ ∪ { w } , (b) w is the unique SP-A V winner of ele ction ( B ′ ∪ { c , w } , V ) , and (c) B ′ is a hitting set of S of size less tha n or equal to k . As a n e xample, the res istance of SP-A V to co nstruct iv e and destru cti ve cont rol by add ing can di- dates (both in the limited and the unlimit ed ve rsion of the problem) now follo w s immediately from Proposit ion 3.9, via mapping the Hitting S et instance ( B , S , k ) to the set { c , w } of qualiﬁed candi- dates and t he set B of spo iler cand idates , to th e v oter co llection V , an d by hav ing c be the de signat ed candid ate in the destruc ti ve case and by ha ving w be the designat ed candid ate in the constru cti ve case. The other cases of Theorem 3.7 can be pro ven similarly . ❑ Theorem 3.7 T urning now to the missing case m ention ed in Theorem 3.7 abov e: Why do es C onstru ction 3.8 not work for construct iv e cont rol by deleting candidates ? Inf ormally put, the reas on is that c is the only serious riv al of w in the electio n ( C , V ) of Construct ion 3.8, so by simply delet ing c the chair could m ake w the unique SP -A V winner , regardl ess of whether S has a hitting set of size k . Ho wev er , via a dif ferent constru ction, we can pro ve resis tance also in this case. Theor em 3.10. SP-A V is r esistant to constructi ve contr ol by deleting candi dates. Pro of. Suscepti bility holds by L emma 3.5. T o pro ve resistan ce, w e provid e a reducti on from Hitting Set. 13 Let ( B , S , k ) be a gi ven instance of Hitting Set, where B = { b 1 , b 2 , . . . , b m } is a set, S = { S 1 , S 2 , . . . , S n } is a nonempty collection of subsets S i ⊆ B , and k < m is a positi ve integ er . 14 Deﬁne the elect ion ( C , V ) , where C = B ∪ { w } is the cand idate set and V is the collectio n of v oters. W e assume that the candidate s in B are in an arbitrary but ﬁxed order , and for each voter belo w , this order is also used in each subset of B . For exa mple, if B = { b 1 , b 2 , b 3 , b 4 } (where the elements of B are ordered as b 1 , b 2 , b 3 , b 4 ) and some subset S i = { b 1 , b 3 } of B occurs in some voter then this v oter prefers b 1 to b 3 , and so does an y other voter whose pref erence list contains S i . V consists of the follo w ing 4 n ( k + 1 ) + 4 m − 2 k + 3 vote rs: 1. For each i , 1 ≤ i ≤ n , there are 2 ( k + 1 ) vo ters of the form: S i | ( B − S i ) w . 2. For each i , 1 ≤ i ≤ n , there are 2 ( k + 1 ) vo ters of the form: ( B − S i ) w | S i . 13 In contrast, Bartholdi, T ove y , and T rick [12] gav e a reduction from Exact Cov er by Three-Sets ( which is deﬁned i n the proof of Theorem 3.11) to prov e that plurality is resistant to constructi ve control by deleting candidates. 14 Note t hat if k = m then B i s always a hitting set of size at most k (provide d that S contains only nonempty sets—a requirement that doesn’ t affect the NP-completene ss of the problem), and we thus may require t hat k < m . 17 3. For each j , 1 ≤ j ≤ m , there are two v oters of the fo rm: b j | w ( B − { b j } ) . 4. There are 2 ( m − k ) vote rs of the form: B | w . 5. There are three voter s of the form: w | B . Since for each b j ∈ B , th e diffe rence scor e ( C , V ) ( w ) − scor e ( C , V ) ( b j ) = 2 n ( k + 1 ) + 3 − ( 2 n ( k + 1 ) + 2 + 2 ( m − k )) = 1 − 2 ( m − k ) is ne gati ve (due to k < m ), w loses to each member of B and so does not win election ( C , V ) . W e claim that S has a hitting set B ′ of size k if and only if w can be m ade the unique SP-A V winner by deletin g at most m − k candidates . From left to right: S uppos e S has a hitting set B ′ of size k . Then, for each b j ∈ B ′ , scor e ( B ′ ∪{ w } , V ) ( w ) − scor e ( B ′ ∪{ w } , V ) ( b j ) = 2 n ( k + 1 ) + 2 ( m − k ) + 3 − ( 2 n ( k + 1 ) + 2 + 2 ( m − k )) = 1 , since the approv al line is mov ed for 2 ( m − k ) v oters of the third group acco rding to Rule 1, thus transfe rring the ir ap prov als f rom members of B − B ′ to w . It is easy to see th at th e approv al line is n ot mov ed in any of the other v oters accord ing to Rule 1; in particu lar , the approv al line is not mov ed in any of the vo ters from the ﬁrst and second group, since B ′ ∩ S i 6 = / 0 for each i , 1 ≤ i ≤ n . So w is the unique SP-A V winner of election ( B ′ ∪ { w } , V ) . Since B ′ ∪ { w } = C − ( B − B ′ ) , it follo ws from k B k = m and k B ′ k = k that deletin g m − k candidate s from C makes w the uniq ue SP -A V winner . From right to left: Let D ⊆ B be any set such that k D k ≤ m − k and w is the unique SP -A V winner of electi on ( C − D , V ) . Let B ′ = ( C − D ) − { w } . Note that B ′ ⊆ B and that we ha ve the follo wing score s in ( B ′ ∪ { w } , V ) : scor e ( B ′ ∪{ w } , V ) ( w ) = 2 ( n − ℓ )( k + 1 ) + 2 ( m − k B ′ k ) + 3 , scor e ( B ′ ∪{ w } , V ) ( b j ) ≤ 2 n ( k + 1 ) + 2 ℓ ( k + 1 ) + 2 + 2 ( m − k ) for each b j ∈ B ′ , where ℓ is the number of sets S i ∈ S that are not hit by B ′ , i.e., B ′ ∩ S i = / 0. Recall that for each i , 1 ≤ i ≤ n , all of the 2 ( k + 1 ) v oters of the fo rm S i | ( B − S i ) w in the ﬁ rst v oter gr oup ha ve rank ed the cand idates in the s ame ord er . Thus, for each i , 1 ≤ i ≤ n , whene ver B ′ ∩ S i = / 0 o ne and th e same candid ate in B ′ beneﬁts from movi ng the appro val line according to Rule 1 , namely the candidate occurr ing ﬁ rst in our ﬁxed ord ering of B ′ . Call this cand idate b and note that scor e ( B ′ ∪{ w } , V ) ( b ) = 2 n ( k + 1 ) + 2 ℓ ( k + 1 ) + 2 + 2 ( m − k ) . Since w is the unique SP-A V winner of ( B ′ ∪ { w } , V ) , w has more appro v als than any candidate in B ′ and in parti cular m ore than b . Thus, w e ha ve scor e ( B ′ ∪{ w } , V ) ( w ) − scor e ( B ′ ∪{ w } , V ) ( b ) = 2 ( n − ℓ )( k + 1 ) + 2 ( m − k B ′ k ) + 3 − 2 n ( k + 1 ) − 2 ℓ ( k + 1 ) − 2 − 2 ( m − k ) = 1 + 2 ( k − k B ′ k ) − 4 ℓ ( k + 1 ) > 0 . 18 Solving this inequa lity for ℓ , we obtain 0 ≤ ℓ < 1 + 2 ( k − k B ′ k ) 4 ( k + 1 ) < 4 + 4 k 4 ( k + 1 ) = 1 . Thus ℓ = 0. It follo ws that 1 + 2 ( k − k B ′ k ) > 0, which implies k B ′ k ≤ k . Thus, B ′ is a hitting set of size at most k . ❑ 3.3 V oter Contr ol T urning no w to contr ol by adding and b y del eting vo ters, it is k no w n from [1 3] that appro val v oting is resist ant to constru cti ve control and is vulner able to destructi ve control (see T able 2). 15 Their proofs can be modiﬁed s o as to a lso appl y to sin cere-st rateg y prefe rence-b ased appro val v oting . W e here pro vide only proof sk etches ; more details of the proof s are pro vided in the technical report ver sion [25]. Theor em 3.11. SP-A V is re sistan t to constructiv e con tr ol by adding voter s and by deleting voter s and is vulne rab le to destru ctive contr ol by addin g voters and by deleting voters . Pro of Sketch of Th eor em 3.11. Susceptibil ity holds by Lemma 3.6 in all cases . T o prov e resis- tance to construct iv e control by adding vo ters (respecti vely , by deleti ng v oters), the constru ction of [13, Thm. 4.43] (respecti vely , of [13, T hm. 4.44]) works, modiﬁed only by specifying voter prefer - ences consisten tly with the voter s’ approv al strategi es (and, in the deleting -v oters case, by adding a dummy candidate who is disappro ved of and ranked last by ev ery v oter in the construc tion to ensure an admiss ible A V st rateg y proﬁle). T hese con structi ons pro vide polynomial -time redu ctions from the NP-complet e proble m Exact Cov er by Three-Sets (denote d by X3C; see, e.g., G arey and Johns on [27]), which is deﬁne d as follo ws: Name Exact Co ve r by Three-Sets (X3C). Instance A set B = { b 1 , b 2 , . . . , b 3 m } , m > 1, 16 and a collec tion S = { S 1 , S 2 , . . . , S n } of subsets S i ⊆ B with k S i k = 3 f or each i . Question D oes S ha ve an exact cov er for B , i.e., is there a subcol lection S ′ ⊆ S such that eve ry element of B occurs in exac tly one set in S ′ ? The polynomial -time algorithms sho wing that approv al vo ting is vulner able to destructi ve con- trol by adding voter s and by deleting v oters [13, Thm. 4.24] can be straightfo rwardl y adapted to also work for sincere- strate gy preference-b ased approv al v oting, since no approv al lines are move d accord ing to R ule 1 in these con trol scenarios. ❑ 15 Meir et al. [21] proved in their interesting “multi-winner” model (which generalizes Bartholdi, T ovey , and Trick’ s model [12] by adding a u tility fun ction and some other parameters) that approv al v oting is resistant to construc tiv e c ontrol by adding voters. According to Footnote 13 of [13], t his resist ance result immediately follows from the corresponding resistance result in [28, 13], essentially due to the fact that lo wer bou nds in more ﬂex ible models are in herited from mo re restrictiv e models. 16 Our assumption that m > 1 is not explicitly speciﬁed in Garey and Johnson [27 ]. Ho wev er, it is clear that requiring m > 1 does not change the comple xity of the problem. 19 W e now prov e that, just like plural ity , sincere -strate gy prefere nce-bas ed appro v al v oting is re- sistan t to constructi ve and destructi ve contro l by partition of v oters in model T P. In fact, the proof presen ted in [13] for plural ity in these two cases also works for SP-A V with minor modiﬁcations. In contr ast, appr ov al v oting is vulnerab le to the dest ructi ve v arian t of this con trol type [13]. Theor em 3.12. SP-A V is r esistant to constru ctive and destruc tive contr ol by partiti on of voters in model TP. Pro of S ketch of Theore m 3.12. The proof is again based on Construc tion 3.8, but the reduction is no w from Restricted Hitting Set, which is d eﬁned just as Hittin g Set (see Section 3.2) e xcept that n ( k + 1 ) + 1 ≤ m − k is requir ed in addition . Restricted Hitting S et is also N P-compl ete [13]. Now , the ke y observ ation is the follo wing proposition , which can be prov en as in [13]. Pro position 3.13 (Hemaspaa ndra et al. [13 ]) . Let ( B , S , k ) be a given Restricted Hitting Set in- stance , wher e B = { b 1 , b 2 , . . . , b m } is a s et, S = { S 1 , S 2 , . . . , S n } is a n onempty colle ction of sub sets S i ⊆ B, and k ≤ m is a positi ve inte ger such that n ( k + 1 ) + 1 ≤ m − k . If ( C , V ) is the electio n r esulting fr om ( B , S , k ) via Construct ion 3.8, then the followin g thr ee statements ar e equivale nt: 1. S has a hitting set of size less than or equal to k . 2. V can be par titione d such that w is the uniqu e SP-A V winner in model TP. 3. V can be par titione d such that c is not the uni que SP-A V w inner in model TP. The theore m no w follows immediat ely from P roposit ion 3.13. ❑ Theorem 3.12 Finally , we turn to control by partitio n of vote rs in model T E. For this control type, Hema- spaand ra et al. [13] proved appro val votin g resista nt in the construc ti ve case and vulnerab le in the destru cti ve case. W e hav e the same results for sincere-s trate gy preference- based appro v al v oting. Our resist ance proof in the constru cti ve case (see the proof of Theorem 3.14) is similar to the cor - respon ding proof of resistan ce in [13]. H o we ver , while our polynomial-t ime algorithm sho wing vulner ability for SP-A V in the destruct iv e case (see the proof of T heorem 3.15) is based on the corres pondin g poly nomial-ti me algorith m for appro val vo ting in [13], it extends their algo rithm in a nontri vial way . Theor em 3.14. SP-A V is r esistant to constructi ve contr ol by partitio n of vote rs in model TE. Pro of. S uscept ibility holds by L emma 3.6. The proof of resistance is based on the constructio n of [13, Thm. 4.46] with only minor chang es. Let an X3C instance ( B , S ) be gi ven , where B = { b 1 , b 2 , . . . , b 3 m } , m > 1, is a set and S = { S 1 , S 2 , . . . , S n } is a collectio n of subsets S i ⊆ B with k S i k = 3 for each i , 1 ≤ i ≤ n . W ithout loss of generalit y , we may assume that n ≥ m . Deﬁne the v alue ℓ j = k{ S i ∈ S | b j ∈ S i }k for each j , 1 ≤ j ≤ 3 m . Deﬁne the election ( C , V ) , where C = B ∪ { w , x , y } ∪ Z is the candidate se t wit h the distinguishe d candid ate w , Z = { z 1 , z 2 , . . . , z n } , and where V is deﬁned to consist of the follo wing 4 n + m vote rs: 1. For each i , 1 ≤ i ≤ n , there is one vot er of the form: y S i | w (( B − S i ) ∪ { x } ∪ Z ) . 20 2. For each i , 1 ≤ i ≤ n , there is one vot er of the form: y z i | w ( B ∪ { x } ∪ ( Z − { z i } )) . 3. For each i , 1 ≤ i ≤ n , there is one vote r of the form: w ( Z − { z i } ) B i | x y z i ( B − B i ) , where B i = { b j ∈ B | i ≤ n − ℓ j } . 4. There are n + m v oters of the form: x | y ( B ∪ { w } ∪ Z ) . Note that sco r e ( C , V ) ( b j ) = n for each b j ∈ B . Since the ab ov e co nstruc tion is only sli ghtly modiﬁed from the proof of [13, T hm. 4.46], so as to formally con form w ith the SP-A V vote r repre- sentat ion, the same ar gument as in th at pro of sho ws that S h as an e xact cov er for B if and on ly if w can be made the uniqu e SP-A V winner by partition of v oters in m odel TE. Note that, in the present contro l scenario, appr ov al voting and SP-A V can differ only in the run-of f, but the construc tion ensure s that they don’ t dif fer there. From left to right, if S has an exact cov er for B then partition the set of voter s as follo w s: V 1 consis ts of the m v oters of the form y S i | w (( B − S i ) ∪ { x } ∪ Z ) that correspond to the sets in the exa ct cov er , of the n + m vo ters who appro ve of only x , and of the n v oters who approv e of y and z i , 1 ≤ i ≤ n . Let V 2 = V − V 1 . It follo ws that w is the un ique SP-A V winne r of both sub electio n ( C , V 2 ) and the run-of f, simply becau se no candid ate proceed s to the run-of f from the othe r subelec tion, ( C , V 1 ) , in which x and y tie for winner with a score of n + m each. From right to left, suppose w can be made the uniqu e SP-A V winner by partition of voters in model TE. Let ( V 1 , V 2 ) be a partit ion of V such that w is the unique SP-A V winner o f the run- of f. According to model TE, w must also be the unique SP-A V winner of one subele ction, say of ( C , V 1 ) . Note that each vo ter of the form y z i | w ( B ∪ { x } ∪ ( Z − { z i } )) has to be in V 2 (oth- erwise, we would hav e scor e ( C , V 1 ) ( w ) = scor e ( C , V 1 ) ( z i ) for at least one i , and so w would not be the unique SP-A V winner of ( C , V 1 ) anymore). Howe ver , if there were more than m voters of the form y S i | w (( B − S i ) ∪ { x } ∪ Z ) in V 2 then scor e ( C , V 2 ) ( y ) > n + m , and so y would be the uniqu e S P-A V winner of the oth er subelecti on, ( C , V 2 ) . B ut then, also in the SP-A V model , y would win the run-of f again st w becau se scor e ( { w , y } , V ) ( y ) = 3 n + m > n = scor e ( { w , y } , V ) ( w ) , which con- tradict s the assump tion that w has been made the unique SP-A V winner by the partition ( V 1 , V 2 ) . Hence, t here are at most m voters of the form y S i | w (( B − S i ) ∪ { x } ∪ Z ) in V 2 , an d these m v oters corres pond to an exa ct cov er of B , since otherwise there would be at least one b j ∈ B with scor e ( C , V 1 ) ( b j ) = n = scor e ( C , V 1 ) ( w ) . ❑ Theor em 3.15. SP-A V is vulne rabl e to dest ructive contr ol by partition of voters in model TE . Pro of. Suscepti bility holds by Lemma 3.6. T o prove vulnerabi lity , we describ e a polyn omial- time algori thm showing that (and how) the chai r can ex ert destruct iv e control by partition of v ot- ers in model T E for sincere-str ateg y preference-b ased appro v al voting . Our algorith m exten ds the polyn omial-time algorithm designe d by Hemaspaand ra et al. [1 3] to p rov e appr ov al votin g vulnera- ble to thi s type of co ntrol. Speciﬁcally , our algo rithm adds Loop 2 belo w to their al gorithm, and we will exp lain belo w why it is necessar y to add this second loop. Let ( C , V ) be an election, and for each v oter v ∈ V , let S v ⊆ C denote v ’ s A V strateg y . In each iterati on of Loo p 1 i n t he alg orithm belo w , we will c onside r three can didates , a , b , an d c . Deﬁne th e 21 follo wing ﬁ ve number s: 17 W c = k{ v ∈ V | a 6∈ S v , b 6∈ S v , c ∈ S v }k , L c = k{ v ∈ V | a ∈ S v , b ∈ S v , c 6∈ S v }k , D a = k{ v ∈ V | a ∈ S v , b 6∈ S v , c 6∈ S v }k , D b = k{ v ∈ V | a 6∈ S v , b ∈ S v , c 6∈ S v }k , and D ac = k{ v ∈ V | a ∈ S v , b 6∈ S v , c ∈ S v }k . In additio n, we introduce th e follo wing notation. Gi ven an election ( C , V ) and two distin ct candid ates x , y ∈ C , let diff ( x , y ) d enote the number of v oters in V who prefer x to y minus the number of vo ters in V who prefer y to x . Deﬁne B c to be the set of candidate s y 6 = c in C such that dif f ( y , c ) ≥ 0. The input to our algo rithm is an election ( C , V ) , where each v oter v ∈ V has a since re A V strate gy S v (other wise, the inpu t is co nsidere d malformed and outr ight r ejected ), and a distingui shed candid ate c ∈ C . On this input, our algorithm works as follo ws. 1. Ch ecking the tri vial cases: can be done as in the case of approv al v oting, see the proof of [13, Thm. 4.21]. In particular , if C = { c } then outpu t “contro l impossibl e” and halt, since c canno t help b ut win. If C contains more cand idates than only c b ut c al ready is n ot the uniq ue SP-A V winner of ( C , V ) then output the (succe ssful) partition ( V , / 0 ) and halt. Otherwise, if k C k = 2 the n output “ contro l impossible” and hal t, as c is th e uniqu e SP-A V winne r of ( C , V ) in the curren t case and so, ho wev er the voter s are partitioned, c must win—agains t the one ri v alling candidate —at least one subelec tion and also the ru n-of f. 2. Loop 1: For each a , b ∈ C such that k{ a , b , c } k = 3, chec k whether V can be partitione d into V 1 and V 2 such that scor e ( C , V 1 ) ( a ) ≥ sc or e ( C , V 1 ) ( c ) and scor e ( C , V 2 ) ( b ) ≥ scor e ( C , V 2 ) ( c ) . As sho w n in the proof of [13, Thm. 4.21], this is equi v alent to checking W c − L c ≤ D a + D b . (3.1) If (3.1) fails, this a and b cannot prev ent c from bei ng the un ique winner o f at le ast one subele ction and thus also of the run-of f, so w e mov e on to test the next a and b in this loop. If (3.1) holds, ho wev er , output the partition ( V 1 , V 2 ) and halt, where V 1 consis ts of the v oters contri bu ting to D a , of the v oters contrib uting to D ac , and of min ( W c , D a ) v oters contri b uting to W c , and where V 2 = V − V 1 . 3. Loop 2: For eac h d ∈ B c , parti tion V as follo ws. Let V 1 consis t of all v oters in V who app rov e of d , a nd let V 2 = V − V 1 . If d is the unique winner of ( C , V 1 ) , then output ( V 1 , V 2 ) as a succes sful partit ion and halt. Otherwise, go to the next d ∈ B c . 4. T ermination: If in no iteration of either Loop 1 or Loop 2 a succe ssful partiti on of V was found , then outp ut “control impossible” and halt. 17 This no tation is adopted from [13] and adjusted h ere to the SP-A V sy stem. W c is the number of votes in which c w ins one approv al against both a and b , L c is the number of votes in which c l oses one approv al against both a and b , and D a , D b , and D ac are the numbers of votes in which the candidate(s) in the subscript gain o ne approv al against the candidate(s) not in the subscript (thus decreasing their d eﬁcit). 22 Let us giv e a short expla nation of why Loop 2 is need ed for SP-A V by stressin g the diffe rence with appro val vot ing. As shown in the pr oof of [13, Thm. 4.21], if none of the trivi al cases applied, then cond ition (3.1) hol ds for so me a , b ∈ C with k{ a , b , c }k = 3 if and on ly if de structi ve cont rol by partiti on of vote rs in model TE is pos sible for approv al voti ng. Thus, for approv al voti ng, if Loop 1 was not suc cessful for any such a and b , we may immediatel y jump to the termination stage, where the algo rithm outp uts “ control i mpossible ” and h alts. In co ntrast, if n one of th e t ri vial c ases a pplied, then the exis tence of candida tes a and b with k{ a , b , c }k = 3 who satisfy (3.1) is not equi vale nt to destru cti ve co ntrol by partition of v oters in model TE be ing po ssible f or SP-A V : It is a sufﬁcie nt, yet not a necessary co ndition . The re ason i s tha t e ven if th ere are no candidates a and b who can pre vent c from winning one subele ction (in some partition of v oters) and from proceeding to the run-of f, it might still be possibl e that c loses or ties the run-of f due to moving the appro val line accor ding to Rule 1. Indeed , if Loop 1 was not suc cessful , c will lose or t ie the run -of f exactl y if there exists a candid ate d 6 = c suc h tha t d if f ( d , c ) ≥ 0 a nd d can win one s ubelect ion (for some pa rtition o f v oters). This is precisely what is being checked in Loop 2. Indeed, note that the partitio n ( V 1 , V 2 ) chosen in Loop 2 for d ∈ B c is the best possib le partition for d in the follo wing sense : If d is not a unique SP-A V w inner of subelectio n ( C , V 1 ) then, for each W ⊆ V , d is not a unique SP -A V winner of subele ction ( C , W ) . T o see this, simply note that if d is not a unique SP-A V winner of ( C , V 1 ) , then there is some candid ate x with scor e ( C , V 1 ) ( x ) = scor e ( C , V 1 ) ( d ) = k V 1 k , which by our choice of V 1 implies scor e ( C , W ) ( x ) ≥ sco r e ( C , W ) ( d ) for eac h subset W ⊆ V . ❑ 4 Conclusions and Open Questions W e ha ve sho w n that Brams a nd San ver’ s sincer e-strate gy prefere nce-ba sed approv al v oting sys - tem [1], when adjusted so as to coer ce admissibili ty (rather than excludin g inadmissib le votes a priori ), combine s the resistances of appro v al and plurality v oting to procedur al control: SP-A V is resista nt to 19 of the 22 prev iously studied types of control. On the one hand, like Copeland v oting [2, 3], SP-A V is fully resistan t to cons tructi ve control, yet unli ke Cope land it addit ionally is br oadly resista nt to destructi ve control. On the other hand, lik e plurality [2, 12, 13, 14], SP-A V is fully resista nt to ca ndidat e control, yet unli ke plu rality it add itional ly is broadly r esistan t to v oter contro l. In co nclusi on, for th ese 2 2 typ es of control, SP-A V ha s mor e resi stances and fewer vulnerabilit ies to contro l than is currently kno wn for any other natural v oting system with a polynomia l-time winner proble m (see T able 1). H o wev er , when comparing appr ov al vot ing and SP-A V , it should also be noted that the former is eve n immune to nine of these 22 control types, wherea s the latter has no immunities at al l. Since immuni ty may be seen as a pe rfect p rotecti on against con trol and resis tance pro vides protectio n to contro l only in a computatio nal sense, one shoul d carefully ev aluate the pros and con s of both sys tems. T he resu lt of such an ev aluation will ce rtainly dep end on whic h partic ular types of contro l one wishes to be prot ected against. As an interesting tas k f or fut ure re search, we propose t o e xpand the study o f SP-A V with re spect to other computational prope rties than its beha vior rega rding proc edural control (see, e.g., [9, 26]), and to in ves tigate also its social cho ice properties in more detail. In additi on, we propose as an 23 interes ting and extremely ambitious ta sk fo r futu re wo rk th e study of SP -A V (and o ther votin g systems as well) beyond the worst-cas e—as we hav e done here—and to wards an appropri ate typi cal- case complexit y model; see, e.g., [29, 30, 31, 32 , 33] for interestin g resu lts and discus sion in this directi on. Acknowledgments: W e are grateful to Edith and Lane A. H emaspaan dra for he lpful co mments an d interes ting discus sions that are reﬂected in part s of Section 2.3. W e thank the anon ymous ML Q, MFCS-08, and C OMSOC-08 referees for their helpful comments on prelimin ary versio ns of this paper . Refer ences [1] S. Brams and R. Sanver, Critical strategies under approval voting: Who gets ru led in and ruled o ut, Electoral Studies 25 (2 ), 28 7–30 5 (200 6). [2] P . Falisze wski, E. Hemaspaan dra, L. Hemaspaandra, and J . Rothe, Llull and Copeland v oting compu- tationally re sist br ibery an d constru ctiv e co ntrol, Journ al of Artiﬁcial I ntelligence Researc h. T o appear . Full version a vailable as [34]. [3] P . Faliszewski, E. Hem aspaandra , L. Hemaspaand ra, and J. Rothe, Copeland voting fully resists con- structive contro l, in: Proc eedings of th e 4th Internatio nal Con ference o n Algo rithmic Aspects in In- formation and Management, (Sprin ger-V erlag Lecture Notes in Computer Science #50 34 , June 20 08), pp. 1 65–17 6. [4] G. E rd ´ elyi, M. N ow ak, an d J. Rothe, Sincere-strategy p referen ce-based approval voting broadly re- sists contro l, in: Pro ceedings of the 33rd Inter national Sy mposium on Ma thematical Foundation s of Computer Science, ( Springer-V er lag Lectur e Notes in Computer Science #5162 , Au gust 2008) , pp. 3 11–32 2. [5] E . Ephr ati and J. Rosensch ein, Multi- agent p lanning as a dy namic search for so cial consensu s, in: Proceeding s of the 13th Internationa l Joint Confer ence on Artiﬁcial In telligence , (Morgan Kaufmann , 1993) , pp . 423 –429 . [6] R. Fagin, R. Kumar, and D. Si vakumar, E fﬁcient sim ilarity search and classiﬁcation via ran k aggrega- tion, in: Proceeding s of the 2003 ACM SIGMOD In ternational Conf erence on Ma nagemen t of Data, (A CM Press, 2003), pp. 301 –312. [7] S. Gh osh, M. Mu ndhe, K. Hernandez, and S. Sen, V oting for movies: The anatomy of recommen der systems, in: Pro ceedings of the 3rd Ann ual Conferen ce on Autonomous Agents, (A CM Press, 19 99), pp. 4 34–43 5. [8] C. Dwork, R. Kumar , M. Naor, an d D. Siv ak umar, Ran k agg regation methods for the web, in: Pro- ceedings of the 10th Interna tional W orld Wide W eb Conferen ce, (ACM P ress, 200 1), pp. 613–6 22. [9] P . F alisze wski, E. Hemaspaan dra, L. Hemaspaandra, and J. Rothe, A richer understandin g of the com- plexity of election sy stems, in : Fun damental Problem s in Computing : Essays in Hono r of Profe ssor Daniel J. Rosenkrantz, edited by S. Ravi and S. Shukla, (Springer , 2 009), chap. 14, pp. 375– 406. [10] J. Bartholdi II I, C. T ovey, and M. Trick, V oting sch emes for w hich it can b e difﬁcult to tell who won the election, Social Choice and W elfare 6 (2), 157–1 65 (19 89). 24 [11] J. Barth oldi II I, C. T ovey, and M. Trick, The co mputatio nal difﬁculty of man ipulating an election, Social Choice and W elfare 6 (3), 227–24 1 (1989). [12] J. Barthold i III, C. T ove y, and M. T rick, How ha rd i s it to control an election?, Mathematical C omput. Modelling 16 (8 /9), 27–40 (1 992). [13] E. Hemasp aandra, L. Hemaspaandr a, and J. Rothe, Anyone but him: The complexity o f pr ecluding an alternative, Artiﬁcial Intelligenc e 171 (5–6), 255–285 (2007). [14] P . Falis zewski, E. Hemaspaandra, L. Hemaspaandra, and J. Rothe , Llull and Copeland voting broadly resist briber y an d control, in: Proceeding s of the 2 2nd AAAI Conference on Artiﬁcial Intelligence , (AAAI Press, July 2007 ), pp . 72 4–730 . [15] S. Brams and P . Fishburn, Approval voting, American Political Science Review 72 (3), 831 –847 (1978 ). [16] A. Gibbard, Manipulation of v oting schemes, Econometr ica 41 (4), 587–601 (1973). [17] M. Satterthwaite, Strategy-p roofn ess and Arrow’ s cond itions: Existence an d corr esponden ce theo- rems for voting procedure s and social welfare fun ctions, Journal of Economic Theory 10 ( 2), 18 7–217 (1975 ). [18] J. Duggan and T . Schwartz, Strategic manip ulability without resoluteness or shared beliefs: Gibbard– Satterthwaite generalized, Social Choice and W e lfare 17 (1) , 85 –93 (2000). [19] P . Everaere, S. Konieczny, an d P . Marquis, The strategy-p roofn ess landscape of m erging, Journ al of Artiﬁcial Intelligen ce Research 28 , 49–105 (2007) . [20] E. Hemaspaan dra, L. Hemaspaan dra, and J. Rothe, Hybr id elections broaden com plexity-theore tic resistance to co ntrol, in: Proceed ings o f the 2 0th Inter national Join t Confer ence on Artiﬁcial I ntelli- gence, (AAAI Press, January 2007 ), p p. 13 08–13 14. [21] R. Meir, A. Pro caccia, J. Rosenschein, and A. Zohar , Comp lexity of strategic behavior in multi-winn er elections, Journal of Artiﬁcial Intelligence Research 33 , 149– 178 (2 008) . [22] N. Betzler and J. Uhlman n, Parameter ized comp lexity o f cand idate contr ol in electio ns and related digraph problems, in: Proc eedings of the 2nd Annual International Con ference on C ombina torial Op - timization and Applicatio ns, (Springer-V erlag Lectur e Notes in Comp uter Science #516 5 , J uly 20 08), pp. 4 3–53. [23] P . Falisze wski, E. He maspaandr a, an d L. Hemaspaandr a, The com plexity o f brib ery in elections, in: Proceeding s of the 21 st National Confer ence on Artiﬁcial Intelligence, (AAAI Press, July 200 6), pp. 6 41–64 6. [24] S. Brams an d R. Sanver, V oting systems that c ombine app roval and prefer ence, in: The Ma thematics of Preference, Choice, and Order : Essays in Ho nor of Peter C. Fishburn, edited by S. Brams, W . Gehr lein, and F . Rober ts, (Springer, to ap pear). [25] G. Erd ´ elyi, M. Nowak, and J. Rothe, Sin cere-strategy preferen ce-based a pproval voting fully resists constructive control and bro adly resists destructive c ontrol, T ech. Rep . arXiv:0806.05 35 v4 [cs.GT], A CM Co mputing Research Repository (CoRR), June 2008, re vised September 2008. [26] D. Baumeister, G. Erd ´ elyi, E. Hemaspaandra, L. Hemaspaan dra, and J. Rothe, Computational aspec ts of ap proval v oting, in: Hand book of Ap proval V oting, e dited b y J. Laslier and R. San ver , ( Springer, manuscrip t, to appear). 25 [27] M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Comp leteness, (W . H. Freeman and Company, Ne w Y ork, 1979) . [28] E. Hemasp aandra, L. Hemaspaandr a, and J. Rothe, Anyone but him: The complexity o f pr ecluding an alternative, in: Pro ceedings o f th e 20th Natio nal Con ference o n Ar tiﬁcial Intelligence, (AAAI Press, 2005) , pp . 95– 101. [29] J. M cCabe-Dansted, G. Pritchard, an d A. Slinko, Ap prox imability of Do dgson’ s rule, Social Choice and W elfare 31 (2), 311– 330 (2 008). [30] A. Procac cia and J. Rosenschein, Junta distrib utions and the average-case complexity of manipulating elections, Journal of Artiﬁcial Intelligence Research 28 , 157– 181 (2 007) . [31] V . Conitzer an d T . Sandh olm, No nexistence of voting rules th at are usually hard to man ipulate, in: Proceeding s of the 21 st National Confer ence on Artiﬁcial Intelligence, (AAAI Press, July 200 6), pp. 6 27–63 4. [32] C. Homan and L. Hemaspaand ra, Guaran tees for the success frequen cy of an algorithm fo r ﬁnding Dodgson -election winner s, Journ al of Heuristics. T o appear . Full version available as Department of Computer Science, University of Rochester T e ch. Rep. TR-881, September 2005, revised J une 2007 . [33] G. Erd ´ ely i, L. Hemaspaan dra, J. Rothe, and H. Spa kows ki, O n ap proxim ating optimal weighted lob- bying, and frequency of correctness versus a verage-case po lynom ial time, in: Pr oceeding s o f the 16 th Internatio nal Sym posium on Fundamen tals of Computation Theor y , (Springer-V er lag Lectu r e Note s in Computer Science #463 9 , August 20 07), pp. 300 –311. [34] P . Falisze wsk i, E . Hemaspaand ra, L. Hemaspaandra, and J. Rothe, Llull an d Copeland voting co m- putationally resist br ibery and control, T ech. Rep. ar Xiv:0809.4484 v2 [cs.GT], AC M Computin g Re- search Repository (CoRR), September 2008. 26

Sincere-Strategy Preference-Based Approval Voting Fully Resists Constructive Control and Broadly Resists Destructive Control

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment