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This paper provides three versions of May's theorem on majority rule, adapted to the one-dimensional model common in formal political modeling applications. The key contribution is that single peakedness of voter preferences allows us to drop May's restrictive positive responsiveness axiom. The simplest statement of the result holds when voter preferences are single peaked and linear (no indifferences), in which case a voting rule satisfies anonymity, neutrality, Pareto, and transitivity of weak social preference if and only if the number of individuals is odd and the rule is majority rule.

May's theorem famously shows that, in social decisions between two options, simple majority rule uniquely satisfies four appealing conditions. Although this result is often cited in support of majority rule, it has never been extended beyond decisions based on pairwise comparisons of options. We generalize May's theorem to many‐option decisions where voters each cast one vote. Surprisingly, plurality rule uniquely satisfies May's conditions. This suggests a conditional defense of plurality rule: If a society's balloting procedure collects only a single vote from each voter, then plurality rule is the uniquely compelling electoral procedure. To illustrate the conditional nature of this claim, we also identify a richer informational environment in which approval voting, not plurality rule, is supported by a May‐style argument.

There are four standard normative defenses for majority rule on two alternatives: fairness (i.e., May's theorem), epistemic (i.e., Condorcet's jury theorem), utilitarian (i.e., Rae-Taylor theorem), and contractarian (i.e., maximization of the number of self-determined voters). There are many ways to generalize majority rule to multiple alternatives, but the standard generalization is majority preference (i.e., Condorcet method). Unfortunately, these arguments fail to generalize to multiple alternatives due to Condorcet's paradox. In this paper, I generalize each of those four defenses to multiple alternatives using the consent of the majority generalization (e.g., approval voting) of majority rule. For example, I show that among Arrovian voting systems, an Arrovian version of approval voting is the voting system with the least restrictive domain which satisfies May's theorem's four conditions, and independence of irrelevant alternatives. The findings suggest that we should normatively and formally explore multiple interpretations of majority rule, beyond majority preference.

May's celebrated theorem (1952) shows that, if a group of individuals wants to make a choice between two alternatives (say x and y), then majority voting is the unique decision procedure satisfying a set of attractive minimal conditions. The conditions are (i) universal domain: the decision procedure should produce an outcome (x, y or tie) for any logically possible combination of individual votes for x and y; (ii) anonymity: the collective choice should be invariant under permutations of the individual votes, i.e. all individual votes should have equal weight; (iii) neutrality: if the individual votes for x and y are swapped, then the outcome should be swapped in the same way, i.e. the labels of the alternatives should not matter; (iv) positive responsiveness: supposing all other votes remain the same, if one individual changes his or her vote in favour of a winning alternative, then this alternative should remain the outcome; if there was previously a tie, a change of one individual vote should break the tie in the direction of that change. May's theorem is often interpreted as a vindication of majoritarian democracy when a collective decision between two alternatives is to be made. Many collective decision problems are, however, more complex. They may not be confined to a binary choice between two alternatives, or between the acceptance or rejection of a single proposition. Suppose there are three or more alternatives (say x, y and z). In that case, it may seem natural to determine an overall collective preference ranking of these alternatives by applying majority voting to each pair of alternatives. But, unfortunately, pairwise majority voting may lead to cyclical collective preferences. Suppose person 1 prefers x to y to z, person 2 prefers y to z to x, and person 3 prefers z to x to y. Then there are majorities of two out of three for x against y, for y against z, and for z against x, a cycle. This is Condorcet's paradox. But a greater number of alternatives is not the only way in which a collective decision problem may deviate from the single binary choice framework of May's theorem. A collective decision problem may also involve simultaneous decisions on the acceptance or rejection of multiple interconnected propositions. For instance, a policy package or a legal decision may consist of multiple propositions which mutually constrain each other. To ensure consistency, the acceptance or rejection of some of these propositions may constrain the acceptance or rejection of others. Once again, a natural suggestion would be to apply majority voting to each proposition separately. As we will see in detail below, however, this method also generates a paradox, sometimes called the 'doctrinal paradox' or 'discursive dilemma': propositionwise majority voting over multiple interconnected propositions may lead to inconsistent collective sets of judgments on these propositions. We have thus identified two dimensions along which a collective decision problem may deviate from the single binary choice framework of May's theorem: (a) the number of alternatives, and (b) the number of interconnected propositions on which simultaneous decisions are to be made. Deviations along each of these dimensions lead to a breakdown of the attractive properties of majority voting highlighted by May's theorem. Deviations along dimension (a) can generate Condorcet's paradox: pairwise majority voting over multiple alternatives may lead to cyclical collective preferences. And deviations along dimension (b) can generate the 'doctrinal paradox' or 'discursive dilemma': propositionwise majority voting over multiple interconnected propositions may lead to inconsistent collective sets of judgments on these propositions. In each case, we can ask whether the paradox is just an artefact of majority voting in special contrived circumstances, or whether it actually illustrates a more general problem. Arrow's impossibility theorem (1951/1963) famously affirms the latter for dimension (a): Condorcet's paradox brings to the surface a more general impossibility problem of collective decision making between three or more alternatives. But Arrow's theorem does not apply straightforwardly to the case of dimension (b). List and Pettit (2001) have shown that the 'doctrinal paradox' or 'discursive dilemma' also illustrates a more general impossibility problem, this time regarding simultaneous collective decisions on multiple interconnected propositions. The two impossibility theorems are related, but not identical. Arrow's result makes it less surprising to find that an impossibility problem pertains to the latter decision problem too, and yet the two theorems are not trivial corollaries of each other. The aim of this paper is to compare these two impossibility results and to explore their connections and dissimilarities. Sections 2 and 3 briefly introduce, respectively, Arrow's theorem and the new theorem on the aggregation of sets of judgments. Section 4 addresses the question of whether the two generalizations of May's single binary choice framework -- the framework of preferences over three or more options and the framework of sets of judgments over multiple connected propositions -- can somehow be mapped into each other. Reinterpreting preferences as ranking judgments, section 5 derives a simple impossibility theorem on the aggregation of preferences from the theorem on the aggregation of sets of judgments, and compares the result with Arrow's theorem. A formal proof of the result is given in an appendix. Section 6 discusses escape-routes from the two impossibility results, and indicates their parallels. Section 7, finally, explores the role of two crucial conditions underlying the two impossibility theorems -- independence of irrelevant alternatives and systematicity --, and identifies a unifying mechanism generating both impossibility problems.

This article provides a justification of proportional representation (PR) in strictly liberal terms. Previous justifications of proportional representation have tended to be based either on its intuitive fairness to political parties, or on its being fair to social groups. The arguments of critics of PR, we argue, likewise rely on fairness to group identities. In contrast, our result shows that proportionality is logically implied by liberal equality, that is, by the requirement that all individual voters be treated equally. Thus we provide a justification for PR in terms of the theory of voting, similar to May's theorem for majority rule.

The aim of this paper is to find normative foundations of Approval Voting. In order to show that Approval Voting is the only social choice function that satisfies anonymity, neutrality, strategy-proofness and strict monotonicity we rely on an intermediate result which relates strategy-proofness of a social choice function to the properties of Independence of Irrelevant Alternatives and monotonicity of the corresponding social welfare function. Afterwards we characterize Approval Voting by means of strict symmetry, neutrality and strict monotonicity and relate this result to May's Theorem. Finally, we show that it is possible to substitute the property of strict monotonicity by the one efficiency of in the second characterization. ; This research was undertaken with support from the fellowship 2001FI-00451 of the Generalitat de Catalunya and from the research grant BEC2002-02130 of the Ministerio de Ciencia y Tecnología of Spain.

Cover -- Frontmatter -- Contents -- Preface -- Prologue -- 1 The problem of rule selection -- 2 Plan of the book -- 3 Concluding remarks -- Appendix: Elementary properties of binary relations -- Rational choice and revealed preference -- 0 Introduction -- 1 Choice functions and revealed preference -- 2 Characterization of rational choice functions -- 3 Some weaker variants of the choice-consistency conditions -- 4 Concluding remarks -- Appendix A: A generalized theory of rationalizability -- Appendix B: Counterexamples -- Arrovian impossibility theorems -- 0 Introduction -- 1 Collective choice rule -- 2 Arrow's impossibility theorem -- 3 Refinements on Arrow's impossibility theorem -- 4 Impossibility theorems without collective rationality -- 5 Concluding remarks -- Simple majority rule and extensions -- 0 Introduction -- 1 Simple majority decision rule: May's axiomatization -- 2 Extensions of the SMD rule -- 3 Transitive closure of the SMD rule -- 4 Concluding observations: two defects of the TCM rule -- Appendix: Transitive closure choice functions -- The fairness-as-no-envy approach in social choice theory -- 0 Introduction -- 1 Framework of analysis -- 2 Fairness-as-no-envy and Pareto efficiency: I. Some impossibility theorems -- 3 Fairness-as-no-envy and Pareto efficiency: II. Goldman-Sussangkarn extended collective choice rule -- 4 Fairness-as-no-envy and Pareto efficiency: III. Leximin envy rule -- 5 Concluding remarks -- Impartiality and extended sympathy -- 0 Introduction -- 1 Framework of analysis -- 2 Impartiality and principle of justice -- 3 Constrained majoritarian rules -- 4 Discussion and remarks on the literature -- Individual rights and libertarian claims -- 0 Introduction -- 1 Inviolable rights, Pareto principle, and an impossibility theorem -- 2 Coherent rights system and liberal individual: a possibility theorem.

An original defense of the unique value of voting in a democracyVoting is only one of the many ways that citizens can participate in public decision making, so why does it occupy such a central place in the democratic imagination? In Election Day, political theorist Emilee Booth Chapman provides an original answer to that question, showing precisely what is so special about how we vote in today's democracies. By presenting a holistic account of popular voting practices and where they fit into complex democratic systems, she defends popular attitudes toward voting against radical critics and offers much-needed guidance for voting reform.Elections embody a distinctive constellation of democratic values and perform essential functions in democratic communities. Election day dramatizes the nature of democracy as a collective and individual undertaking, makes equal citizenship and individual dignity concrete and transparent, and socializes citizens into their roles as equal political agents. Chapman shows that fully realizing these ends depends not only on the widespread opportunity to vote but also on consistently high levels of actual turnout, and that citizens' experiences of voting matters as much as the formal properties of a voting system. And these insights are also essential for crafting and evaluating electoral reform proposals.By rethinking what citizens experience when they go to the polls, Election Day recovers the full value of democratic voting today