Finding Strategyproof Social Choice Functions via SAT Solving ∗ Felix Brandt Christian Geist Technische Universität München Technische Universität München Munich, Germany Munich, Germany [email protected] [email protected] ABSTRACT due to Tang and Lin [19], who reduce well-known impos- A promising direction in computational social choice is to sibility results such as Arrow's theorem to finite instances, address open research problems using computer-aided prov- which can then be checked by a satisfiability (SAT) solver ing techniques. In conjunction with SAT solving, this ap- [3]. Geist and Endriss [13] were able to extend this method proach has been shown to be viable in the context of classic to a fully-automatic search algorithm for impossibility the- impossibility theorems such as Arrow's impossibility as well orems in the context of preference relations over sets of al- as axiomatizations of preference extensions. In this paper, ternatives. In this paper, we apply these techniques to im- we demonstrate that it can also be applied to the more com- prove our understanding of strategyproofness in the context plex notion of strategyproofness for irresolute social choice of set-valued (or so-called irresolute) social choice functions. functions. These types of problems, however, require a more These types of problems, however, are more complex and evolved encoding as otherwise the search space rapidly be- require an evolved encoding as otherwise the search space comes much too large. We present an efficient encoding for rapidly becomes too large. Table 1 illustrates how quickly translating such problems to SAT and leverage this encoding the number of involved objects grows and that, therefore, to prove new results about strategyproofness with respect to an exhaustive search is doomed to fail. Kelly's and Fishburn's preference extensions. For example, Formally, a social choice function (SCF) is defined as a we show that no Pareto-optimal majoritarian social choice function that maps individual preferences over a set of al- function satisfies Fishburn-strategyproofness. ternatives to a set of socially most preferred alternatives. An SCF is strategyproof if no agent can obtain a more pre- ferred outcome by misrepresenting her preferences. It is Categories and Subject Descriptors well-known from the Gibbard-Satterthwaite theorem that, [Applied computing]: Economics; [Computing when restricting attention to SCFs that always return a sin- 1 methodologies]: Artificial intelligence gle alternative, only trivial SCFs can be strategyproof. A proper definition of strategyproofness for the more general setting of irresolute SCFs requires the specification of pref- General Terms erences over sets of alternatives. Rather than asking the Economics, Theory, Verification, Algorithms agents to specify their preferences over all sets (which would be bound to various rationality constraints), it is typically assumed that the preferences over single alternatives can Keywords be extended to preferences over sets. Of course, there are Computer-aided theorem proving, strategyproofness, tour- various ways how to extend preferences to sets (see, e.g., nament solutions, set extensions, SAT [12, 2, 20]), each of which leads to a different class of strat- egyproof SCFs. A function that yields a preference relation 1. INTRODUCTION over subsets of alternatives when given a preference relation over single alternatives is called a set extension or preference Ever since the famous Four Color Problem was solved us- extension. In this paper, we focus on two very natural set ing a computer-assisted approach, it is clear that computers extensions due to Kelly [14] and Fishburn [9]2. can contribute significantly to finding and proving formal statements. Due to its rigorous axiomatic foundation, so- 1The assumption of single-valuedness has been criticized for cial choice theory appears to be a field in which computer- being restrictive and unreasonable (see, e.g., [11, 14, 20]). aided theorem proving is a particularly promising line of 2G¨ardenfors [12] attributed this extension to Fishburn be- research. Perhaps the best known result in this context is Alternatives 4 5 6 7 ∗Corresponding author Choice sets 15 31 63 127 Tournaments 64 1,024 32,768 ∼ 2 · 106 Appears in: Alessio Lomuscio, Paul Scerri, Ana Bazzan, Canonical tourn. 4 12 56 456 and Michael Huhns (eds.), Proceedings of the 13th Inter- Maj. SCFs 50; 625 ∼ 1018 ∼ 10101 ∼ 10959 national Conference on Autonomous Agents and Multiagent Systems (AAMAS 2014), May 5-9, 2014, Paris, France. Table 1: Number of objects involved in problems Copyright c 2014, International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org). All rights reserved. with irresolute majoritarian SCFs While strategyproofness for Kelly's extension (henceforth Definition 1. A social choice function (SCF) is a func- Kelly-strategyproofness) is known to be a rather restrictive tion f : R∗(A) ! 2A n ;. condition [14, 1, 18], some SCFs such as the Pareto rule, ∗ the omninomination rule, the top cycle, the minimal cov- An SCF is resolute if jf(R)j = 1 for all R 2 R (A), otherwise ering set, and the bipartisan set were shown to be Kelly- it is irresolute. strategyproof [6]. Interestingly, the more discriminating of We restrict our attention to majoritarian SCFs (or tour- these SCFs are majoritarian, i.e., they are based on the pair- nament solutions), which are defined using the majority re- wise majority relation only, and, moreover, can be ordered lation. The majority relation (or: majority graph) RM of a with respect to set inclusion. In particular, these results preference profile R is the relation on A × A defined by suggest that the bipartisan set may be the finest Kelly- (a; b) 2 RM iff w(a; b) ≥ 0, for all alternatives a; b 2 A. strategyproof majoritarian SCF. In this paper, we show that this not the case by automatically generating a Kelly- An SCF f is said to be majoritarian if it is neutral5 and its strategyproof SCF which is strictly contained in the bipar- outcome only depends on the (unweighted) majority com- tisan set. parisons between pairs of alternatives, i.e., f(R) = f(R0) 0 For the more demanding notion of Fishburn- whenever RM = RM . strategyproofness, existing results suggested that it may We will now introduce the majoritarian SCFs considered only be satisfied by rather indiscriminating SCFs such as in this paper (see [15, 5], for more information). the top cycle [7, 8].3 Using our computer-aided proving technique we were able to confirm this suspicion by proving Top Cycle The top cycle rule TC (also known as weak that, within the domain of majoritarian SCFs, Fishburn- closure maximality, GETCHA, or the Smith set) returns the strategyproofness is incompatible with Pareto-optimality. maximal elements of RM . In order to achieve this impossibility, we manually proved Uncovered Set Let C denote the covering relation on a novel characterization of Pareto-optimal majoritarian A × A, i.e., a C b (\a covers b") if and only if b RM b and SCFs and an induction step, which allows us to generalize b RM x implies a RM x for all x 2 A. The uncovered set UC the computer-verified impossibility to larger numbers of returns the maximal elements of C, i.e., those alternatives alternatives. that are not covered by any other alternative. The universality of our method and its ease of adaptation suggests that it could be applied to similar open questions Bipartisan Set Consider the symmetric two-player zero- in the future. sum game in which the set of actions for both players is given by A and payoffs are defined as follows. If the first player chooses a and the second player chooses b, the payoff 2. MATHEMATICAL FRAMEWORK OF for the first player is 1 if a RM b, −1 if b RM a, and 0 oth- STRATEGYPROOFNESS erwise. The bipartisan set BP contains all alternatives that In this section, we provide the terminology and notation are played with positive probability in some Nash equilib- required for our results and introduce notions of strate- rium of this game. gyproofness for majoritarian SCFs that allow us to abstract An SCF f is called a refinement of another SCF g if away any reference to preference profiles. f(R) ⊆ g(R) for all preference profiles R. In short, we write f ⊆ g in this case. It can be shown for the above 2.1 Social Choice Functions that BP ⊆ UC ⊆ TC (see, e.g., Laslier [15]). Let N = f1; : : : ; ng be a set of at least 3 voters with preferences over a finite set A of m alternatives. For conve- 2.2 Strategyproofness 4 nience, we assume that n is odd. The preferences of each Even though our investigation of strategyproof SCFs is voter i 2 N are represented by a complete, anti-symmetric, universal in the sense that it can be applied to any set exten- and transitive preference relation Ri ⊆ A × A. The inter- sion, in this paper we will concentrate on two well-known set pretation of (a; b) 2 Ri, usually denoted by a Ri b, is that extensions due to Kelly [14] and Fishburn [9], respectively. voter i values alternative a at least as much as alternative b. They are defined as follows: Let Ri be a preference relation The set of all preference relations over A will be denoted over A and X; Y ⊆ A two nonempty subsets of A. by R(A). The set of preference profiles, i.e., finite vectors of ∗ K preference relations, is then given by R (A). The typical ele- XRi Y iff x Ri y for all x 2 X and all y 2 Y [14] ∗ ment of R (A) will be R = (R1;:::;Rn). In accordance with One interpretation of this extension is that voters are com- conventional notation, we write Pi for the strict part of Ri, pletely unaware of the mechanism (e.g., a lottery) that will i.e., a Pi b if a Ri b but not b Ri a.