A REPRESENTATION THEOREM FOR VOTING WITH LOGICAL CONSEQUENCES (original) (raw)

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.

Some notes on voting schemes and the will of the majority

Public Choice, 1969

This paper is a study in the theory of committees and elections. By a committee we will mean any group of people who arrive at a decision by means of voting. By a voting scheme I we will mean any method by which individual voting decisions are aggregated into committee decisions. Given various voting schemes we shall examine three techniques by which members may seek to manipulate committee decisions to their advantage: a) additions or deletions to the alternatives to be considered b) deliberate distortions of one's own voting preferences c) manipulation of the order in .which alternatives are voted upon, and shall prove some theorems about rational voting behavior when preferences are unidimensionally scalable.

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.

An impossibility theorem concerning positive involvement in voting

Economics Letters, 2024

In social choice theory with ordinal preferences, a voting method satisfies the axiom of positive involvement if adding to a preference profile a voter who ranks an alternative uniquely first cannot cause that alternative to go from winning to losing. In this note, we prove a new impossibility theorem concerning this axiom: there is no ordinal voting method satisfying positive involvement that also satisfies the Condorcet winner and loser criteria, resolvability, and a common invariance property for Condorcet methods, namely that the choice of winners depends only on the ordering of majority margins by size.

Impossibility theorems involving weakenings of expansion consistency and resoluteness in voting

Mathematical Analyses of Decisions, Voting, and Games, eds. M. A. Jones, D. McCune, and J. Wilson, Contemporary Mathematics, American Mathematical Society, 2023, 2023

A fundamental principle of individual rational choice is Sen's γ axiom, also known as expansion consistency, stating that any alternative chosen from each of two menus must be chosen from the union of the menus. Expansion consistency can also be formulated in the setting of social choice. In voting theory, it states that any candidate chosen from two fields of candidates must be chosen from the combined field of candidates. An important special case of the axiom is binary expansion consistency, which states that any candidate chosen from an initial field of candidates and chosen in a head-to-head match with a new candidate must also be chosen when the new candidate is added to the field, thereby ruling out spoiler effects. In this paper, we study the tension between this weakening of expansion consistency and weakenings of resoluteness, an axiom demanding the choice of a single candidate in any election. As is well known, resoluteness is inconsistent with basic fairness conditions on social choice, namely anonymity and neutrality. Here we prove that even significant weakenings of resoluteness, which are consistent with anonymity and neutrality, are inconsistent with binary expansion consistency. The proofs make use of SAT solving, with the correctness of a SAT encoding formally verified in the Lean Theorem Prover, as well as a strategy for generalizing impossibility theorems obtained for special types of voting methods (namely majoritarian and pairwise voting methods) to impossibility theorems for arbitrary voting methods. This proof strategy may be of independent interest for its potential applicability to other impossibility theorems in social choice.

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.

On the Properties of Voting Systems

Scandinavian Political Studies, 1981

The article focuses on the problem of choosing the ‘best’ voting procedure for making collective decisions. The procedures discussed are simple majority rule, Borda count, approval voting, and maximin method. The first three have been axiomatized while the maximin method has not yet been given an axiomatic characterization. The properties, in terms of which the goodness of the procedures is assessed, are dictatorship, consistency, path independence, weak axiom of revealed preference, Pareto optimality, and manipulability. It turns out that the picture emerging from the comparison of the procedures in terms of these properties is most favorable to the approval voting.

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.

Nonmanipulable voting schemes when participants' interests are partially decomposable

Social Choice and Welfare, 1991

Recent papers by Barber~i and Peleg and by Zhou have established that the Gibbard-Satterthwaite Theorem remains valid when individuals are restricted to reporting only "reasonable" preferences. We present a theorem that covers situations in which, as in Barber~i-and-Peleg and Zhou, preferences may be restricted to reasonable ones, but in which, additionally, it may be known in advance that some dimensions of the social decision do not affect all the participants-i.e., in which the social decisions are partially decomposable into decisions that affect only subsets of the participants. As in the previous theorems, the conclusion of this new theorem is that nonmanipulable voting schemes must be dictatorial.