On the Computational Hardness of Manipulating Pairwise Voting Rules Rohit Vaish Neeldhara Misra Indian Institute of Science IIT Gandhinagar India India [email protected] [email protected] Shivani Agarwal Avrim Blum Indian Institute of Science Carnegie Mellon University India USA [email protected] [email protected] ABSTRACT impossibility suggested by this theorem, there has been sub- Standard voting rules usually assume that the preferences stantial work concerning a finer analysis of the situation and of voters are provided in the form of complete rankings over finding possible workarounds. A prominent example of this a fixed set of alternatives. This assumption does not hold in is the seminal work of Bartholdi, Tovey and Trick [4] who applications like recommendation systems where the set of proposed the possibility of using computational hardness as alternatives is extremely large and only partial preferences a barrier against manipulation of voting rules. They argue can be elicited. In this paper, we study the problem of that although opportunities for manipulation always exist in strategic manipulation of voting rules that aggregate voter principle, there might not exist any efficient general purpose preferences provided in the form of pairwise comparisons be- algorithm for finding them in practice. tween alternatives. Our contributions are twofold: first, we Since then, a considerable body of work has developed show that any onto pairwise voting rule is manipulable in around the computational study of manipulation (see [5] for principle. Next, we analyze how the computational complex- a survey on this topic). Much of this work models voter ity of manipulation of such rules varies with the structure of preferences as complete rankings over a fixed set of alterna- the graph induced by the pairs of alternatives that the ma- tives (or candidates). While this is a reasonable choice for nipulator is allowed to vote over and the type of the prefer- certain situations, many large-scale settings prevalent today ence relation. Building on natural connections between the (like recommendation systems) involve extremely large can- pairwise manipulation and sports elimination problems (in- didate sets (such as movies, products, webpages etc.). It is cluding a mixed-elimination variant that we introduce in this therefore unreasonable, and often impractical, to elicit pref- paper), we show that manipulating pairwise voting rules can erences from users in the form of complete rankings of these be computationally hard even in the single-manipulator set- alternatives. For these settings, it is decidedly more natural ting, a setting where most standard voting rules are known to aggregate partial user preferences to arrive at an out- to be easy to manipulate. come (e.g. top-k preferences [6,7], partial orders [8] etc.). Similarly, some situations call for relaxing the requirement of transitivity among the preferences. Indeed, when alter- General Terms natives are compared using not one but multiple quality Algorithms, Economics, Theory parameters, it is natural to permit possibly cyclic prefer- ences [9]. In this work, we consider the model of pairwise prefer- Keywords ences, where every vote is simply a collection of pairwise Social Choice Theory; Voting; Manipulation; Pairwise Pref- comparisons between alternatives with no constraints other erences than anti-symmetry (i.e. AB ) B6A). The preferences provided by voters in this setting can be incomplete (i.e. not all pairs of candidates are compared by a voter) and 1. INTRODUCTION can contain cycles (AB; BC; CA). Thus, all the situa- A central result in social choice theory [1] is the Gibbard- tions described above are subsumed by this model. This is a Satterthwaite theorem [2,3] which states that any non- simple model that offers substantial generality, and as a re- dictatorial voting rule over at least three candidates under sult there has been growing interest in designing preference which each candidate has some chance of winning is sus- aggregation algorithms that elicit pairwise preferences from ceptible to strategic voting i.e. is manipulable. Given the the users [10, 11, 12, 13, 14]. Although these studies explore the statistical properties of aggregation algorithms for pair- Appears in: Proceedings of the 15th International Conference wise preferences, the question of whether these algorithms on Autonomous Agents and Multiagent Systems (AAMAS 2016), (or pairwise voting rules) are resistant to strategic user be- J. Thangarajah, K. Tuyls, C. Jonker, S. Marsella (eds.), havior remains to be answered. The focus of the present May 9{13, 2016, Singapore. work is to address these questions from both axiomatic and Copyright c 2016, International Foundation for Autonomous Agents and computational perspectives. Multiagent Systems (www.ifaamas.org). All rights reserved. pref-type pref-type strict + acyclic strict unrestricted strict + acyclic strict unrestricted acyclic acyclic tree/ tree/ max deg=2 P max deg=2 A complete P A complete bipartite P NP-complete bipartite general general Table 2: The complexity of Copeland0-Manipulation and Table 1: The complexity of pBorda-Manipulation under Copeland1-Manipulation. the conditions specified by the corresponding action space A and type of preference relation pref-type (refer Defini- pref-type tion 2.1 in Section2). strict + acyclic strict unrstd. acyclic tree/ Contributions max deg=2 P Our contributions can be summarized as follows: A complete bipartite P NP- P 1. We show that any pairwise voting rule that is onto general complete (i.e. under which each candidate has some chance of winning) must be manipulable (Theorem 3.1). Table 3: The complexity of Copeland0:5-Manipulation. 2. Given the above impossibility, we study settings un- pref-type der which computational difficulty acts as a worst-case strict + acyclic strict unrestricted barrier against manipulation (Sections 3.2 and 3.3) acyclic Specifically, we study the problem of manipulation of tree/ pairwise voting rules along the following two dimen- max deg=2 P sions: (i) the structure of the graph induced by the set A complete of candidate pairs that the manipulator is allowed to bipartite P NP-complete vote over (e.g. tree, bipartite, complete) and (ii) the general type of preference relation (e.g. strict, acyclic). Our analysis is focused on the pairwise Borda (pBorda) rule Table 4: The complexity of Copelandα-Manipulation α and the Copeland family of voting rules. Tables1 to4 for α 2 Q \ (0; 1) n f0:5g. summarize our results for these rules. The essence of our results is that the structure of the induced graph and the type of revealed preferences (and the parame- 2. PRELIMINARIES ter α in case of Copelandα) can shape the complexity Let [n] = f1; 2; : : : ; ng denote the set of candidates and landscape in important ways. We remark that while U = fu ; u ; : : : ; u g denote the set of voters in an election. the manipulation problem in the context of rankings 1 2 m is polynomially solvable for most voting rules [4], we already encounter hardness results in the setting of Pairwise preferences and pairwise voting rules. pairwise comparisons with one manipulator for the rel- Let u ⊆ [n] × [n] denote the binary relation indicat- α atively simple pBorda and Copeland rules, which we ing the preferences of voter u, so that i u j indicates that consider an important contrast1. voter u prefers candidate i over candidate j. For each pair of candidates i; j and each voter u, we can have exactly one of i u j, j u i or neither (i.e. voter u skips the compari- Organization of this paper. son between i and j). We let R denote the set of all such anti-symmetric and irreflexive binary relations on [n]; and After setting the requisite notation and definitions in let Π = ( ; ; ··· ; ) 2 Rm denote the pairwise Section2, we describe the main ideas and proof tech- u1 u2 um preference profile of the voters. niques involved in the axiomatic (Section 3.1) and classical A pairwise voting rule r maps a pairwise preference complexity-theoretic results for pairwise Borda (Section 3.2) profile Π 2 [1 Rk to a unique candidate r(Π) 2 [n]. and Copelandα (Section 3.3) rules. A survey of the related k=1 Given a preference profile Π 2 Rm and a pair of can- literature is provided in Section4. We conclude by identi- didates i; j, let m (Π) denote the number of vot- fying further implications of our results and directions for ij ers who strictly prefer candidate i over candidate j, future research in Section5. Due to space constraints, we i.e. m (Π) = Pm 1(i j) where 1(:) is the indicator defer the detailed proofs of all the results presented here to ij k=1 uk function. A score-based pairwise voting rule is any pair- the full version of the paper. wise voting rule r for which there exists a (natural) scoring function s : [1 Rk ! n such that r(Π) is the highest- 1Even in the context of rankings, however, voting rules like k=1 R the second-order Copeland rule [4] and many elimination- scoring candidate according to s(Π) under some fixed tie- style rules [15, 16, 17, 18] are known to be computationally breaking rule. That is, r(Π) = T (arg maxi si(Π)) for some resistant to manipulation by a single voter. tie-breaking rule T : 2[n] n f;g ! [n] satisfying T (S) 2 S for (a) Before manipulation (b) After manipulation (c) After restricted manipulation Figure 1: An illustration of the election instance in Example 2.1. (a) Each vertex of the multigraph represents a candidate and each dashed edge represents the number of voters with that preference (e.g. two voters prefer AC). (b) The pairwise comparisons made by the manipulator are represented by solid edges and the pBorda score of the winning candidate is indicated in boldface.