Peter C. Fishburn

Condorcet's paradox and anonymous preference profiles

Condorcets paradox [6] of simple majority voting occurs in a voting situation with n voters and m candidates or alternatives if for every alternative there is a second alternative which more voters prefer to the first alternative than conversely. The paradox can arise only if the strict simple majority relation on the alternatives is cyclic, provided that m is finite. Studies of the paradox are usually based either on profiles or A-profiles (anonymous preference profiles). A profile is a function that assigns a preference order on the alternatives to each voter. An A-profile, which has also been called a return [28], profile [31] and pattern [20], is a function that assigns a nonnegative number of voters to each potential preference order on the alternatives such that the sum of the assigned integers equals n. In general, many different profiles which retain voter identities map into the same A-profile, and any two profiles that map into the same A-profile bear the same simple majority relation on the alternatives. Hence, it may appear that it is purely a matter of personal taste or analytical convenience whether one works with profiles or with A-profiles in studying Condorcets paradox. Although this is true in one sense, there are important differences between the two bases that will be explored in the present paper. Of special concern will be the fact that some A-profiles correspond to very few profiles (consider an A-profile that assigns all n voters to the same preference

Even-chance lotteries in social choice theory

This paper discusses aspects of the theory of social choice when a nonempty choice set is to be determined for each situation, which consists of a feasible set of alternatives and a preference order for each voter on the set of nonempty subsets of alternatives. The individual preference assumptions include ordering properties and averaging conditions, the latter of which are motivated by the interpretation that subset A is preferred to subset B if and only if the individual prefers an even-chance lottery over the basic alternatives in A to an even-chance lottery over the basic alternatives in B. Corresponding to this interpretation, a choice set with two or more alternatives is resolved by an even-chance lottery over these alternatives. Thus, from the traditional no-lottery social choice theory viewpoint, ties are resolved by even-chance lotteries on the tied alternatives. Compared to the approach which allows all lotteries to compete along with the basic alternatives, the present approach is a contraction which allows only even-chance lotteries.After discussing individual preference axioms, the paper examines Pareto optimality for nonempty subsets of a feasible set in a social choice context with n voters. Aspects of simple-majority comparisons in the even-chance context follow, including an analysis of single-peaked preferences. The paper concludes with an Arrowian type impossibility theorem that is designed for the even-chance setting.

Approval voting, Condorcet's principle, and runoff elections

Approval voting allows each voter to vote for as many candidates as he wishes in an election but not cast more than one vote for each candidate of whom he approves. If there is a strict Condorcet candidate — a candidate who defeats all others in pairwise contests — approval voting is shown to be the only nonranked voting system that is always able to elect the strict Condorcet candidate when voters use sincere admissible strategies. Moreover, if a strict Condorcet candidate must be elected under ordinary plurality voting when voters use admissible strategies, then he must also be elected under approval voting when voters use admissible strategies, but the converse does not hold.The widely used plurality runoff method can also elect a strict Condorcet candidate when voters use admissible strategies on the first ballot, but some of these may have to be insincere to get the strict Condorcet candidate onto the runoff ballot. Furthermore, there is no case in which the strict Condorcet candidate is invariably elected under the plurality runoff method when voters use admissible first-ballot strategies. Thus, approval voting is superior to the plurality runoff method with respect to the Condorcet principle in its ability to elect the strict Condorcet candidate by sincere voting and in its ability to guarantee the election of the strict Condorcet candidate when voters use admissible strategies. In addition, approval voting is more efficient since it requires only one election and is probably less subject to strategic manipulation.

Representations of Binary Decision Rules by Generalized Decisiveness Structures

This paper is motivated by two apparently dissimilar deficiencies in the theory of social choice and the theory of cooperative games. Both deficiencies stem from what we regard as an inadequate conception of decisiveness or coalitional power. Our main purpose will be to present a more general concept of decisiveness and to show that this notion allows us to characterize broad classes of games and social choice procedures.

On the family of linear extensions of a partial order

Given a partial order P defined on a finite set X, a binary relation ≻P may be defined on X by setting x ≻P y for elements x and y in X just when more linear extensions L of P on X have xLy than yLx. A linear extension L of P on X is a linear order on X with P ⊆ L. There exist partial orders P such that ≻P includes cycles. Thus, in a voting situation in which voters are unanimous in their preferences on the pairs in P and express all possible linearly ordered preferences on X which are consistent with P, with no two voters having the same preference order, strict simple majorities as given by ≻P can cycle.

Choice probabilities and choice functions

Abstract Choice probabilities are basic to much of the theory of individual choice behavior in mathematical psychology. On the other hand, consumer economics has relied primarily on preference relations and choice functions for its theories of individual choice. Although there are sizable literatures on the connections between choice probabilities and preference relations, and between preference relations and choice functions, little has been done—apart from their common ties to preference relations—to connect choice probabilities and choice functions. The latter connection is studied in this paper. A family of choice functions that depends on a threshold parameter is defined from a choice probability function. It is then shown what must be true of the choice probability function so that the choice functions satisfy three traditional rationality conditions. Conversely, it is shown what must be true of the choice functions so that the choice probability function satisfies a version of Luces axiom for individual choice probabilities.

Convex stochastic dominance with finite consequence sets

Stochastic dominance is a notion in expected-utility decision theory which has been developed to facilitate the analysis of risky or uncertain decision alternatives when the full form of the decision makers von Neumann-Morgenstern utility function on the consequence space X is not completely specified. For example, if f and g are probability functions on X which correspond to two risky alternatives, then f first-degree stochastically dominates g if, for every consequence x in X, the chance of getting a consequence that is preferred to x is as great under f as under g. When this is true, the expected utility of f must be as great as the expected utility of g.Most work in stochastic dominance has been based on increasing utility functions on X with X an interval on the real line. The present paper, following [1], formulates appropriate notions of first-degree and second-degree stochastic dominance when X is an arbitrary finite set. The only ‘structure’ imposed on X arises from the decision makers preferences. It is shown how typical analyses with stochastic dominance can be enriched by applying the notion to convex combinations of probability functions. The potential applications of convex stochastic dominance include analyses of simple-majority voting on risky alternatives when voters have similar preference orders on the consequences.

Approximations of multiattribute utility functions

Abstract : This report extends previously developed approximation theory for two-attribute cardinal utility functions to utility functions defined on three or more attributes. If focuses on the simple additive and multiplicative approximations along with several other elementary forms. The purpose of the report is to assess how well simple approximations can describe more complex utility functions on multiattribute outcome spaces. (Author)

On the foundations of mean-variance analyses

Let (μ, σ) and (μ′, σ′) be mean-standard deviation pairs of two probability distributions on the real line. Mean-variance analyses presume that the preferred distribution depends solely on these pairs, with primary preference given to larger mean and smaller variance. This presumption, in conjunction with the assumption that one distribution is better than a second distribution if the mass of the first is completely to the right of the mass of the second, implies that (μ, σ) is preferred to (μ′, σ′) if and only if either μ > μ′ or (μ = μ′ and σ < σ′), provided that the set of distributions is sufficiently rich. The latter provision fails if the outcomes of all distributions lie in a finite interval, but then it is still possible to arrive at more liberal dominance conclusions between (μ, σ) and (μ′, σ′).