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Stackelberg Voting Games: Computational Aspects and Paradoxes∗

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Stackelberg Voting Games: Computational Aspects and Paradoxes∗ Stackelberg Voting Games: Computational Aspects and Paradoxes∗ Lirong Xia Vincent Conitzer Department of Computer Science Department of Computer Science Duke University Duke University Durham, NC 27708, USA Durham, NC 27708, USA [email protected] [email protected] Abstract spective). Then, a voting rule is applied to the profile (vector) We consider settings in which voters vote in sequence, each of reported linear orders, producing a winning alternative. voter knows the votes of the earlier voters and the preferences An important concern is that voters may vote strategically of the later voters, and voters are strategic. This can be mod- rather than truthfully. That is, a voter may report a false vote eled as an extensive-form game of perfect information, which that does not represent her true preferences to make herself we call a Stackelberg voting game. better off. This phenomenon is called manipulation; if the We first propose a dynamic-programming algorithm for find- voting rule r is such that no voter can ever benefit from ing the backward-induction outcome for any Stackelberg vot- manipulating, then r is said to be strategy-proof. Unfortu- ing game when the rule is anonymous; this algorithm is effi- nately, there are some very minimal conditions that are not cient if the number of alternatives is no more than a constant. We show how to use compilation functions to further reduce satisfied by any strategy-proof voting rule, according to the the time and space requirements. celebrated Gibbard-Satterthwaite theorem (Gibbard 1973; Our main theoretical results are paradoxes for the backward- Satterthwaite 1975). induction outcomes of Stackelberg voting games. We show This raises the following fundamental question: if vot- that for any n ≥ 5 and any voting rule that satisfies non- ers vote strategically, what outcome can we expect? It is imposition and with a low domination index, there exists natural to turn to game theoretic solution concepts to an- a profile consisting of n voters, such that the backward- swer this question. One approach is to consider the game induction outcome is ranked somewhere in the bottom two where all voters vote at the same time, and study the equi- positions in almost every voter’s preferences. Moreover, this libria of this game. Unfortunately, even in a complete- outcome loses all but one of its pairwise elections. Further- information setting where all voters’ preferences are com- more, we show that many common voting rules have a very mon knowledge, this leads to an extremely large number of low (= 1) domination index, including all majority-consistent voting rules. For the plurality and nomination rules, we show equilibria, many of them bizarre. For example, in a plu- even stronger paradoxes. rality election (where everyone votes for a single alterna- Finally, using our dynamic-programming algorithm, we run tive), it may be the case that all voters prefer alternative a simulations to compare the backward-induction outcome of to both b and c. Nevertheless, in one equilibrium of this the Stackelberg voting game to the winner when voters vote game, all voters will vote for either b or c. This equi- truthfully, for the plurality and veto rules. Surprisingly, our librium is quite robust, because voting for a is a waste, experimental results suggest that on average, more voters pre- given that nobody else is expected to vote for a. There fer the backward-induction outcome. has been some work exploring different solution concepts in simultaneous-move voting games—e.g., (Farquharson 1969; Introduction Moulin 1979)—but in some sense, the equilibrium selection Voting is a useful methodology that allows multiple agents issue in the above example is inherent in settings where vot- to aggregate their preferences over alternatives and make a ers vote simultaneously. group decision. In the standard voting setting, each voter However, in many practical situations, the voters vote one (agent) reports a linear order over (strict ranking of) the al- after another, and the later voters know the votes cast by the ternatives; moreover, the voters are generally assumed to do earlier voters. For example, consider online systems that al- so simultaneously (or without knowledge of previously re- low users to rate movies or other products. This is the set- ported votes, which is equivalent from a game-theoretic per- ting that we consider in this paper. We assume that voters’ ∗ preferences over the alternatives are strict; we also make We thank Edith Elkind and anonymous AAAI reviewers for a complete-information assumption that the voters’ prefer- helpful comments. Lirong Xia is supported by a James B. Duke ences are common knowledge (among the voters themselves, Fellowship and Vincent Conitzer is supported by an Alfred P. Sloan though not necessarily to the election organizer).1 This re- Research Fellowship. We also thank NSF for support under award numbers IIS-0812113 and CAREER 0953756. Copyright c 2010, Association for the Advancement of Artificial 1While this is clearly a simplifying assumption, it approximates Intelligence (www.aaai.org). All rights reserved. the truth in many settings, and with this assumption we do not need sults in an extensive-form game of perfect information that ranked in the top position in at least one vote in P (the al- can be solved by backwardinduction. In sharp contrast to the ternatives that have been “nominated”). We choose the first simultaneous-move setting, this results in a unique outcome alternative in Tops(P ) according to a fixed order. That is, ∗ (winning alternative). We refer to this game as Stackelberg Nom(P )= ci∗ where i = min{i : ci ∈ Tops(P )}. voting game. There are also certain criteria that voting rules can Related work and discussion. The idea of modeling a vot- satisfy. We now give two criteria and some example rules ing process in which voters vote one after another as an that satisfy them. (We do not define these example rules extensive-form game is not new. Sloth (1993) studied elec- here because they are well known in the computational tions with two alternatives, as well as settings with more al- social choice community, and a definition is not technically ternatives where a pairwise decision between two options is necessary because the results in this paper will hold for all made at every stage. She relates the outcomes of this process rules satisfying the criterion.) to the multistage sophisticated outcomes of the game (McK- • Condorcet-consistent rules: A voting rule r is Condorcet- elvey and Niemi 1978; Moulin 1979). In the extensive-form consistent if it always selects the Condorcet winner, when- games studied by Dekel and Piccione (2000), multiple vot- ever one exists. (A Condorcet winner is an alternative ers can vote simultaneously in each stage. They compare that wins in every pairwise election—that is, for any other the equilibrium outcomes of these games to the outcomes of alternative, more voters prefer the Condorcet winner to this the symmetric equilibria of their simultaneous counterparts. alternative than vice versa.) Copeland, maximin, ranked Battaglini (2005) studies how these results are affected by pairs, Kemeny, Slater, Dodgson, and voting trees are all the possibility of abstention and a small cost of voting. Condorcet-consistent. Our approach is significantly different from the previous • Majority-consistent rules: A voting rule r is majority- approaches in several aspects. First, the prior work focuses consistent if it always selects the majority winner, whenever mostly on the case of two alternatives or, in the case of mul- one exists. (A majority winner is an alternative that is ranked tiple alternatives, on particular voting procedures; in con- first by more than half the votes.) Any Condorcet-consistent trast, we consider general (anonymous)voting rules with any rule is also majority-consistent, because a majority winner is number of alternatives, and correspondingly derive very gen- always a Condorcet winner. In addition, plurality, plurality eral paradoxes. Second, we also study how the backward- with runoff, STV, and Bucklin are all majority-consistent. induction outcome can be efficiently computed, and we use these algorithmic insights in simulations to evaluate the qual- Stackelberg voting game. We now consider the strategic ity of the Stackelberg voting game’s outcome “on average.” Stackelberg voting game. We use a complete-information Desmedt and Elkind (2010) simultaneously and indepen- assumption: all the voters’ preferences are common knowl- dently studied a similar setting in which voters vote sequen- edge. Given this assumption, for any voting rule r, the pro- tially under the plurality rule, and showed several types of cess where voters vote in sequence can be modeled as an paradoxes. In their model, voters are allowed to abstain, extensive-form game of perfect information, as follows. The and voting comes at a small cost. They assume random tie- game has n stages. In stage j (j ≤ n), voter j chooses an breaking and therefore need to consider expected utilities. action from L(X ). Each leaf of the tree is associated with Their paradoxes are significantly different from ours. an outcome, which is the winner for the profile consisting of the votes that were cast to reach this leaf. Preliminaries Because the voters’ preferences are linear orders (which implies that there are no ties), we can solve the game by Let X be the set of alternatives, |X | = m. A vote V is a backward induction, which results in a unique outcome. We linear order over X . The set of all linear orders over X is note that this requires only ordinal preferences, that is, we do denoted by L(X ). An n-profile P is a collection of n votes, not need to define utilities. The backward-induction process that is, P ∈ L(X )n.
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