Results 1  10
of
14
Majority Voting on Restricted Domains
 Journal of Economic Theory
, 2010
"... In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments su ¢ cient for consistent majority judgments. Some are based on global orders of propositions or in ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
(Show Context)
In judgment aggregation, unlike preference aggregation, not much is known about domain restrictions that guarantee consistent majority outcomes. We introduce several conditions on individual judgments su ¢ cient for consistent majority judgments. Some are based on global orders of propositions or individuals, others on local orders, still others not on orders at all. Some generalize classic socialchoicetheoretic domain conditions, others have no counterpart. Our most general condition generalizes Sen’s triplewise valuerestriction, itself the most general classic condition. We also prove a new characterization theorem: for a large class of domains, if there exists any aggregation function satisfying some democratic conditions, then majority voting is the unique such function. Taken together, our results support the robustness of majority rule. 1
Fundamentals of Social Choice Theory
 Northwestern University
, 1996
"... Abstract. This paper offers a short introduction to some of the fundamental results of social choice theory. Topices include: Nash implementability and the MullerSatterthwaite impossibility theorem, anonymous and neutral social choice correspondences, twoparty competition in tournaments, binary a ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
(Show Context)
Abstract. This paper offers a short introduction to some of the fundamental results of social choice theory. Topices include: Nash implementability and the MullerSatterthwaite impossibility theorem, anonymous and neutral social choice correspondences, twoparty competition in tournaments, binary agendas and the top cycle, and median voter theorems. The paper begins with a simple example to illustrate the importance of multiple equilibria in gametheoretic models of political institutions. An introductory model of political institutions Mathematical models in social science are like fables or myths that we read to get insights into the social world in which we live. Our mathematical models are told in a specialized technical language that allows very precise descriptions of the motivations and choices of the various individuals in these stories. When we prove theorems in mathematical social science, we are making general statements about whole classes of such stories all at once. Here we focus on gametheoretic models of political institutions. So let us begin our study of political institutions by a simple gametheoretic model that tells a story of how political institutions may arise. Consider first the simple twoperson "Battle of Sexes" game shown in The two players in this game, who may be called player 1 and player 2, must independently choose one of two possible strategies: to defer (f ) or to grab (g ). If the players i i both grab or both defer then neither player gets anything; but if exactly one player grabs then he 2 gets payoff 6 while the deferential player gets payoff 3. This game has three equilibria. There is an equilibrium in which player 1 grabs while player 2 defers, giving payoffs (6,3). There is another equilibrium in which player 1 defers while player 2 grabs, giving payoffs (3,6). There is also a symmetric randomized equilibrium in which each player independently randomizes between grabbing, with probability 2/3, and deferring, with probability 1/3. In this randomized equilibrium, the expected payoffs are (2,2), which is worse for both players than either of the nonsymmetric equilibria. Now think of an island with a large population of individuals. Every morning, these islanders assemble in the center of their island, to talk and watch the sun rise. Then the islanders scatter for the day. During the day, the islanders are randomly matched into pairs who meet at random locations around the island, and each of these matched pairs plays the simple Battle of Sexes game once. This process repeats every day. Each player's objective is to maximize a longrun discounted average of his (or her) sequence of payoffs from these daily Battle of Sexes matches. One longrun equilibrium of this process is for everyone to play the symmetric randomized equilibrium in his match each day. But rising up from this primitive anarchy, the players could develop cultural expectations which break the symmetry among matched players, so that they will share an understanding of who should grab and who should defer. One possibility is that the islanders might develop an understanding that each player has a special "ownership" relationship with some region of the island, such that a player is expected to grab whenever he is in the region that he owns. Notice that this system of ownership rights is a selfenforcing equilibrium, because the other player does better by deferring (getting 3 rather than 0) when he expects the "owner" to grab, and so the owner should indeed grab confidently. But such a system of traditional grabbing rights might fail to cover many matching situations where no one has clear "ownership." To avoid the costly symmetric equilibrium in such cases, other ways of breaking the players' symmetry are needed. A system of leadership can be used to solve this problem. That is, the islanders might appoint one of their population to serve as a leader, who will announce each morning a set of instructions that specify which one of the two players should 3 grab in each of the daily matches. As long as the leader's instructions are clear and comprehensive, the understanding that every player will obey these instructions is a selfenforcing equilibrium. A player who grabbed when he was instructed to defer would only lower his expected payoff from 3 to 0, given that the other player is expected to follow his instruction to grab here. To make this system of government work on our island, the islanders only need a shared understanding as to who is the leader. The leader might be the eldest among the islanders, or the tallest, or the one with the loudest voice. Or the islanders might determine their leader by some contest, such as a chess tournament, or by an annual election in which all the islanders vote. Any method of selection that the islanders understand can be used, because everyone wants to obey the selected leader's instructions as long as everyone else is expected to obey him. Thus, selfenforcing rules for a political system can be constructed arbitrarily from the equilibrium selection problem in this game. The islanders could impose limits to a leader's authority in this political system. For example, there might be one leader whose instructions are obeyed on the northern half of the island, and another leader whose instructions are obeyed on the southern half. The islanders may even have ways to remove a leader, such as when he loses some reelection contest or when he issues instructions that violate some perceived limits. If a former leader tries to make an announcement in his former domain of authority, every player would be expected to ignore this announcement as irrelevant cheap talk. Of course, the real world is very different from the simple island of this fable. But as in this island, coordination games with multiple equilibria are pervasive in any real society. Thus, any successful society must develop leadership structures that can coordinate people's expectations in situations of multiple equilibria. So the first point of this fable is the basic social need for leadership and for political institutions that can provide it. The second point of this fable is that the effectiveness of any political institution may be derived simply from a shared understanding that it is in effect, as Hardin (1989) has argued. Thus, any political system may be one of many possible equilibria of a more fundamental coordination game of constitutional selection. That is, the process of selecting a constitution can 4 be viewed as an equilibriumselection problem, but it is the equilibriumselection problem to solve all other equilibriumselection problems. The arbitrariness of political structures from this gametheoretic perspective validates our treating them as exogenous explanatory parameters in political economics. A question might be posed, for example, as to whether one form of democracy might generate higher economic welfare than some other forms of government. Such a question would be untestable or even meaningless if the form of government were itself determined by the level of economic welfare. But our fable suggests that the crucial necessary condition for democracy is not wealth or literacy, but is simply a shared understanding that democracy will function in this society (so that an officer who waves his pistol in the legislative chamber should be perceived as a madman in need of psychiatric treatment, not as the new leader of the country). A general impossibility theorem There is an enormous diversity of democratic political institutions that could exist. Social choice theory is a branch of mathematical social science that tries to make general statements about all such institutions. Given the diversity of potential institutions, the power of social choice theory may be quite limited, and indeed its most famous results are negative impossibility theorems. But it is good to start with the general perspective of social choice theory and see what can be said at this level. Later we can turn to formal political theory, where we will focus on narrower models that enable us to say more about the specific kinds of political institutions that exist in the real world. Modern social choice theory begins with the great theorem of Let N denote a given set of individual voters, and let Y denote a given set of alternatives Maskin (1985) showed that any social choice correspondence that is constructed as the set of Nash equilibrium outcomes of a fixed game form must be monotonic in this sense. A game form is a function of the form H:× S 6Y where each S is a nonempty strategy set for i. The pure Nash equilibrium outcomes of the game form H with preferences u is the set E(H,u) = {H(s)* s 0 × S , and, oei0N, oer 0S , u (H(s)) $ u (H(s ,r ))}. That is, x is a Nash equilibrium outcome in E(H,u) iff there exists a profile of strategies s such that x=H(s) and no individual i could get an outcome of H that he would strictly prefer under the preferences u by unilaterally deviating from s to another strategy r . Condition (1) above says that the set of outcomes that are strictly better than x for any player is the same or smaller when the preferences change from u to v, and so x=H(s) must still be an equilibrium outcome under v. So if x0E(H,u) and the preference profiles u and v satisfy condition (1) for x, then we must have x0E(H,v). Thus, for any game form H, the social choice correspondence E(H,C) is monotonic. That is, T is decisive for (x,y) iff x can be chosen by F when the individuals in T all prefer x over y but everyone else prefers y over x. Lemma 1 asserts that if T is decisive for (x,y) then y is never chosen by F when everyone in T prefers x over y. Lemma 3. Suppose that #X > 2. If the set T is decisive for some pair of distinct alternatives in X×X, then T is decisive for every such pair. Proof of Lemma 3. Suppose that T is decisive for (x,y), where x 0 X, y 0 X, and x = / y. Choose any other alternative z such that z 0 X and x = / z = / y. T is also decisive for (x,z). So decisiveness for (x,y) implies decisiveness for (x,z) and decisiveness for (z,y). The general statement of Lemma 3 can be derived directly from repeated applications of this fact. Q.E.D. To complete the proof of the MullerSatterthwaite theorem, let T be a set of minimal size among all sets that are decisive for distinct pairs of alternatives in X. Lemma 2 tells us that T cannot be the empty set, so #T = / 0. Suppose that #T > 1. Select an individual h in T, and select alternatives x, y, and z in X, and let u be a preference profile such that and everyone prefers x, y, and z over all other alternatives. Decisiveness of T implies that F(u) = / y. If F(u) were x then {h} would be decisive for (x,z), which would contradict minimality of T. If F(u) were z then T\{h} would be decisive for (z,y), which would also contradict minimality of T. But Lemma 2 implies F(u)0{x,y,z}. This contradiction implies that #T must equal 1. So there is some individual h such that {h} is a decisive set for all pairs of alternatives. That is, for any pair (x,y) of distinct alternatives in X, there exists a preference profile u such that This theorem tells us that the only way to design a game that always has a unique Nash equilibrium is to give one individual all the power, or to restrict the possible outcomes to two. In fact, many institutions of government actually fit one of these two categories. Decisionmaking in the executive branch is often made by a single decisionmaker, who may be the president or the minister with responsibility for a given domain of social alternatives. On the other hand, when a vote is called in a legislative assembly, there are usually only two possible outcomes: to approve or to reject some specific proposal that is on the floor. (Of course, the current vote may be just one stage in a longer agenda, as when the assembly considers a proposal to amend another proposal that is to scheduled be considered later. We consider sequential voting in Sections 1.4 and 1.5. But the MullerSatterthwaite theorem also leaves us another way out. The crucial assumption in the MullerSatterthwaite theorem is that F is a social choice function, not a multivalued social choice correspondence. Dropping this assumption just means admitting that political processes might be games that sometimes have multiple equilibria. As Anonymity and neutrality Having a dictatorship as a social choice function is disturbing to us because it is manifestly unfair to the other individuals. But nondictatorship is only the weakest equity requirement. In the theory of democracy, we should aspire to much higher forms of equity than We may call this example the ABC paradox (where "ABC" stands for Arrow, Black, and Condorcet, who drew attention to such examples); it is also known as the Condorcet cycle. An example like this appeared at the heart of the proof of the impossibility theorem in the preceding section. Any alternative in this example can be mapped to any other alternative by a permutation of Y such that an appropriate permutation of N can then return the original preference profile. Thus, an anonymous neutral social choice correspondence must choose either the empty set or the set of all three alternatives for this ABC paradox, and so an anonymous neutral social choice function cannot be defined. This argument could be also formulated as a statement about implementation by Nash equilibria. Under any voting procedure that treats the voters anonymously and is neutral to the various alternatives, the set of equilibrium outcomes for this example must be symmetric around the three alternatives {a,b,c}. Thus, an anonymous neutral voting game cannot have a unique purestrategy equilibrium that selects only one out of the three alternatives for the ABC paradox. This argument does not generalize to randomizedstrategy equilibria. The symmetry of this example could be satisfied by a unique equilibrium in randomized strategies such that each alternative is selected with probability 1/3. The MullerSatterthwaite theorem does not consider randomized social choice functions, but Tournaments and binary agendas When there are only two alternatives, majority rule is a simple and compelling social choice procedure. K. When there are more than two alternatives, we might still try to apply the principle of majority voting by dividing the decision problem into a sequence of binary questions. For example, one simple binary agenda for choosing among three alternatives {a,b,c} is as follows.
Top Monotonicity: A Common Root For Single Peakedness, Single Crossing and the Median Voter Result
, 2008
"... When the members of a voting body exhibit single peaked preferences, majority winners exist. Moreover, the median(s) of the preferred alternatives of voters is (are) indeed the majority (Condorcet) winner(s). This important result of Duncan Black (1958) has been crucial in the development of public ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
When the members of a voting body exhibit single peaked preferences, majority winners exist. Moreover, the median(s) of the preferred alternatives of voters is (are) indeed the majority (Condorcet) winner(s). This important result of Duncan Black (1958) has been crucial in the development of public economics and political economy, even if it only provides a sufficient condition. Yet, there are many examples in the literature of environments where voting equilibria exist and alternative versions of the median voter results are satisfied while single peakedness does not hold. Some of them correspond to instances
Approximating optimal social choice under metric preferences
 In Proceedings of the TwentyNinth AAAI Conference on Artificial Intelligence
, 2015
"... We examine the quality of social choice mechanisms using a utilitarian view, in which all of the agents have costs for each of the possible alternatives. While these underlying costs determine what the optimal alternative is, they may be unknown to the social choice mechanism; instead the mechanism ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We examine the quality of social choice mechanisms using a utilitarian view, in which all of the agents have costs for each of the possible alternatives. While these underlying costs determine what the optimal alternative is, they may be unknown to the social choice mechanism; instead the mechanism must decide on a good alternative based only on the ordinal preferences of the agents which are induced by the underlying costs. Due to its limited information, such a social choice mechanism cannot simply select the alternative that minimizes the total social cost (or minimizes some other objective function). Thus, we seek to bound the distortion: the worstcase ratio between the social cost of the alternative selected and the optimal alternative. Distortion measures how good a mechanism is at approximating the alternative with minimum social cost, while using only ordinal preference information. The underlying costs can be arbitrary, implicit, and unknown; our only assumption is that the agent costs form a metric space, which is a natural assumption in many settings. We quantify the distortion of many wellknown social choice mechanisms. We show that for both total social cost and median agent cost, many positional scoring rules have large distortion, while on the other hand Copeland and similar mechanisms perform optimally or nearoptimally, always obtaining a distortion of at most 5. We also give lower bounds on the distortion that could be obtained by any deterministic social choice mechanism, and extend our results on median agent cost to more general objective functions. 1
2015): A characterization of singlepeaked preferences via random social choice functions,forthcoming in Theoretical Economics
"... 2014 ..."
MOVE, UAB and Barcelona GSE
, 2010
"... When members of a voting body exhibit single peaked preferences, pairwise majority voting equilibria (Condorcet winners) always exist. Moreover, they coincide with the median(s) of the voters’most preferred alternatives. This important result, known as the median voter result, has been crucial in t ..."
Abstract
 Add to MetaCart
(Show Context)
When members of a voting body exhibit single peaked preferences, pairwise majority voting equilibria (Condorcet winners) always exist. Moreover, they coincide with the median(s) of the voters’most preferred alternatives. This important result, known as the median voter result, has been crucial in the development of public economics and political economy. Yet, there are many examples in the literature where
Ordinal Bayesian Incentive Compatibility in Restricted Domains ∗
, 2015
"... We study deterministic voting mechanisms by considering an ordinal notion of Bayesian incentive compatibility (OBIC). If the beliefs of agents are independent and generic, we show that a mechanism is OBIC and satisfies an additional condition called elementary monotonicity if and only if it is a dom ..."
Abstract
 Add to MetaCart
We study deterministic voting mechanisms by considering an ordinal notion of Bayesian incentive compatibility (OBIC). If the beliefs of agents are independent and generic, we show that a mechanism is OBIC and satisfies an additional condition called elementary monotonicity if and only if it is a dominant strategy incentive compatible mechanism. Our result works in a large class of preference domains (that include the unrestricted domain, the single peaked domain, the single dipped domain, and some single crossing domains). We can significantly weaken elementary monotonicity in our result in the single peaked domain if we assume unanimity and in a large class of domains if we assume unanimity and topsonlyness.
Local Incentive Compatibility with Transfers ∗
, 2015
"... We consider locally incentive compatible mechanisms with deterministic allocation rules and transfers with quasilinear utility. We identify a rich class of type spaces, which includes the single peaked type space, where local incentive compatibility does not imply incentive compatibility. Our main r ..."
Abstract
 Add to MetaCart
We consider locally incentive compatible mechanisms with deterministic allocation rules and transfers with quasilinear utility. We identify a rich class of type spaces, which includes the single peaked type space, where local incentive compatibility does not imply incentive compatibility. Our main result shows that in such type spaces, a mechanism is locally incentive compatible and paymentonly incentive compatible if and only if it is incentive compatible. Paymentonly incentive compatibility requires that a mechanism that generates the same allocation at two types must have the same payment at those two types. Our result works on a class of ordinal type spaces, which are generated by considering a set of ordinal preferences over alternatives and then considering all nonnegative type vectors representing such preferences.