Positively Responsive Collective Choice Rules and Majority Rule: A Generalization of May's Theorem to Many Alternatives (original) (raw)
Related papers
2018
A collective choice rule selects a set of alternatives for each collective choice problem. Suppose that the alternative ’x’, is in the set selected by a collective choice rule for some collective choice problem. Now suppose that ‘x’ rises above another selected alternative ‘y’ in some individual’s preferences. If the collective choice rule is “positively responsive”, ‘x’ remains selected but ‘y’ is no longer selected. If the set of alternatives contains two members, an anonymous and neutral collective choice rule is positively responsive if and only if it is majority rule (May 1952). If the set of alternatives contains three or more members, a large set of collective choice rules satisfy these three conditions. We show, however, that in this case only the rule that assigns to every problem its strict Condorcet winner satisfies the three conditions plus Nash’s version of “independence of irrelevant alternatives” for the domain of problems that have strict Condorcet winners. Further, ...
An extension of May's Theorem to three alternatives: axiomatizing Minimax voting
2023
May’s Theorem [K. O. May, Econometrica 20 (1952) 680-684] characterizes majority voting on two alternatives as the unique preferential voting method satisfying several simple axioms. Here we show that by adding some desirable axioms to May’s axioms, we can uniquely determine how to vote on three alternatives (setting aside tiebreaking). In particular, we add two axioms stating that the voting method should mitigate spoiler effects and avoid the so-called strong no show paradox. We prove a theorem stating that any preferential voting method satisfying our enlarged set of axioms, which includes some weak homogeneity and preservation axioms, must choose from among the Minimax winners in all three- alternative elections. When applied to more than three alternatives, our axioms also distinguish Minimax from other known voting methods that coincide with or refine Minimax for three alternatives.
Collectively rational voting rules for simple preferences
Journal of Mathematical Economics, 2011
Collective rationality of voting rules, requiring transitivity of social preferences (or quasi-transitivity, acyclicity for weaker notions), has been known to be incompatible with other standard conditions for voting rules when there is no prior information, thus no restriction, on individual preferences Sen, 1970). proposes two restricted domains of individual preferences where majority voting generates transitive social preferences; they are the domain consisting of preferences that have at most two indifference classes, and the domain where any set of three alternatives is partitioned into two non-empty subsets and alternatives in one set are strictly preferred to alternatives in the other set. On these two domains, we investigate whether majority voting is the unique way of generating transitive, quasi-transitive, or acyclic social preferences. First of all, we rule out non-standard voting rules by imposing monotonicity, anonymity, and neutrality. Our main results show that majority rule is the unique voting rule satisfying transitivity, yet all other voting rules satisfy acyclicity (also quasi-transitivity on the second domain). Thus we find a very thin border dividing majority and other voting rules, namely, the gap between transitivity and acyclicity.
A Note on A. D. Taylor’s Property of Independence of Irrelevant Alternatives for Voting Rules
Studies in Microeconomics, 2017
In a widely used textbook on mathematics and politics, Taylor introduced an interesting property of social choice procedures, which we call 'Taylor's Independence of Irrelevant Alternatives (TIIA)'. Taylor proved a result showing that every voting procedure belonging to a certain class of voting procedures violates TIIA. The purpose of this note is to supplement Taylor's result by showing that a large number of voting rules, which do not belong to the class of voting procedures figuring in Taylor's result, also violate TIIA.
Collective choice under dichotomous preferences
Journal of Economic Theory, 2005
Agents partition deterministic outcomes into good or bad. A direct revelation mechanism selects a lottery over outcomes -also interpreted as time-shares. Under such dichotomous preferences, the probability that the lottery outcome be a good one is a canonical utility representation.
Triple-consistent social choice and the majority rule
TOP, 2013
A society has to choose within a set X of programs, each defining a decision regqrding a finite number D of yes-no issues. An X-profile associates with every program x in X a finite number of voters who support x. We prove that the outcome of the issue-wise simple majority rule Maj is an element of X at any X-profile where Maj is well-defined if and only if this is true when Maj is applied to any profile involving only 3 elements of X, each being supported by exactly one voter. We call this property triple-consistency. Moreover, we characterize the class of anonymous issue-wise choice functions that are triple-consistent. We discuss three applications of the results. First, interpreting X as a domain of preference relations over a finite set of alternatives, we argue that they generalize a well-known consequence of the value-restriction propery . Second, we can characterize the sets of approval ballots for which the strong version of paradox of multiple elections never occurs. Third,we can provide some new insights to the dynamics of club formation.
A general concept of majority rule
Mathematical Social Sciences, 2002
We develop a general concept of majority rule for finitely many choice alternatives that is consistent with arbitrary binary preference relations, real-valued utility functions, probability distributions over binary preference relations, and random utility representations. The underlying framework is applicable to virtually any type of choice, rating, or ranking data, not just the linear orders or paired comparisons assumed by classic majority rule social welfare functions. Our general definition of majority rule for arbitrary binary relations contains the standard definition for linear orders as a special case.