On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, w _j, to the alternative she ranks in j ^th position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P ^* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P ^* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases. Received: 30 April 1999/Accepted: 14 September 2000     

Some startling inconsistencies when electing committees

We generalize the idea of a Condorcet winner to committee elections to select a Condorcet committee of size m. As in the case of a Condorcet winner, the Condorcet committee need not exist. We adapt two methods to measure how far a set of m candidates is from being the Condorcet committee. In particular, we generalize a procedure proposed by Lewis Carroll for selecting the candidate that is closest to being the Condorcet winner to allow the selection of a committee. We also generalize Kemenys method, which gives a complete transitive ranking, to the selection of committees and show that it is closely related to the first method.We show that these methods lead to some surprising inconsistencies. For example, the committee of size k may be disjoint from the committee of size j or they may overlap in any manner, the committee arising from Carrolls method may appear at any locations in the Kemeny ranking, and except for two highly restrictive cases, the members of the committee arising from Kemenys method may appear at any location in the Kemeny ranking. The author wishes to express his thanks to Don Saari for his conversations and suggestions for the paper and to the Institute for Mathematical Behavioral Sciences at the University of California at Irvine where the initial work for this paper took place. He also wishes to thank the anonymous reviewer whose careful reading and suggestions greatly improved the paper.     

Condorcet efficiencies under the maximal culture condition

The Condorcet winner in an election is a candidate that could defeat each other candidate in a series of pairwise majority rule elections. The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that such a winner exists. The study considers the Condorcet efficiency of basic voting rules under various assumptions about how voter preference rankings are obtained. Particular attention is given to situations in which the maximal culture condition is used as a basis for obtaining voter preferences. Received: 4 February 1998/Accepted: 13 April 1998     

Positional information and preference aggregation

A new version of independence (I⁺) is proposed for social welfare functions based on the following notion of agreement. Two weak orders R and R’ on a finite set S agree on a pair {x,y}, denoted byif R|_{x,y} = R’|_{x,y} and [z R*x and z R*y for some z∈S] if and only if [z’ (R’)* x and z’(R’)*y for some z’∈S]. The last part says that x and y are strictly under z with respect to R exactly when x and y are strictly under z’ with respect to R’. Some examples and results on social welfare functions that satisfy (I⁺), Pareto, and nondictatorship are given.I am grateful for the comments and suggestions made by an anonymous referee on an earlier version of this paper.     

Permutation cycles and manipulation of choice functions

Let be a social preference function, and let v() be the Nakamura number of . If W is a finite set of cardinality at least v() then it is shown that there exists an acyclic profile P on W such that (P) is cyclic. Any choice function which is compatible with can then be manipulated. A similar result holds if W is a manifold (or a subset of Euclidean space) with dimension at least v()-1.Presented at the Fifth World Congress of the Econometric Society, MIT, Cambridge, Mass., August 17–24, 1985. This material is based on work supported by NSF Grant SES-84-18295 to the School of Social Sciences, University of California at Irvine. Particular thanks are due to David Grether, Dick McKelvey and Jeff Strnad for helpful discussion and for making available their unpublished work.     

Condorcet efficiency: A preference for indifference

The Condorcet winner in an election is the candidate who would be able to defeat all other candidates in a series of pairwise elections. The Condorcet efficiency of a voting procedure is the conditional probability that it will elect the Condorcet winner, given that a Condorcet winner exists. The study considers the Condorcet efficiency of weighted scoring rules (WSR's) on three candidates for large electorates when voter indifference between candidates is allowed. It is shown that increasing the proportion of voters who have partial indifference will increase the probability that a Condorcet winner exists, and will also increase the Condorcet efficiency of all WSR's. The same observation is observed when the proportion of voters with complete preferences on candidates is reduced. Borda Rule is shown to be the WSR with maximum Condorcet efficiency over a broad range of assumptions related to voter preferences. The result of forcing voters to completely rank all candidates, by randomly breaking ties on candidates that are viewed as indifferent, leads to a reduction in the probability that a Condorcet winner exists and to a reduction in the Condorcet efficiency of all WSR's. Received: 31 July 1999/Accepted: 11 February 2000     

Modeling large electorates with Fourier series, with applications to Nash equilibria in proximity and directional models of spatial competition

 In this paper we introduce harmonic analysis (Fourier series) as a tool for characterizing the existence of Nash equilibria in two-dimensional spatial majority rule voting games with large electorates. We apply our methods both to traditional proximity models and to directional models. In the latter voters exhibit preferences over directions rather than over alternatives, per se. A directional equilibrium can be characterized as a Condorcet direction, in analogy to the Condorcet (majority) winner in the usual voting models, i.e., a direction which is preferred by a majority to (or at least is not beaten by) any other direction. We provide a parallel treatment of the total median condition for equilibrium under proximity voting and equilibrium conditions for directional voting that shows that the former result is in terms of a strict equality (a knife-edge result very unlikely to hold) while the latter is in terms of an inequality which is relatively easy to satisfy. For the Matthews [3] directional model and a variant of the Rabinowitz and Macdonald [7] directional model, we present a sufficiency condition for the existence of a Condorcet directional vector in terms of the odd-numbered components of the Fourier series representing the density distribution of the voter points. We interpret our theoretical results by looking at real-world voter distributions and direction fields among voter points derived from U.S. and Norwegian survey data. Received: 7 July 1995 / Accepted: 14 May 1996     

Classification theorem for smooth social choice on a manifold

A classification theorem for voting rules on a smooth choice space W of dimension w is presented. It is shown that, for any non-collegial voting rule, σ, there exist integers v ^*(σ), w ^*(σ) (with v ^*(σ)<w ^*(σ)) such that

1. structurally stable σ-voting cycles may always be constructed when w ? v ^*(σ) + 1
2. a structurally stable σ-core (or voting equilibrium) may be constructed when w ? v ^*(σ) ? 1

Finally, it is shown that for an anonymous q-rule, a structurally stable core exists in dimension \(\frac{{n - 2}}{{n - q}}\) , where n is the cardinality of the society.     

Simple manipulation-resistant voting systems designed to elect Condorcet candidates and suitable for large-scale public elections

This article focuses on voting systems that (i) aim to select the Condorcet candidate in the common case where one exists and (ii) impede manipulation by exploiting voter knowledge of electorate preferences. The systems are relatively simple, both mathematically and for voter understanding, and are fully workable for large-scale elections. Their designated equilibrium strategies, under which voters vote sincerely, involve discerning the top one or two candidates in the preference ordering of the electorate. One set of systems uses its ballot to obtain voters’ preference rankings plus approval votes, and tallies the latter if no Condorcet winner exists. It offers solid advantages vis-à-vis instant-runoff voting, which uses a kindred ballot and has attracted recent reformers. Another set of systems uses only approval voting, which is examined from a new angle.     

The sensitivity of weight selection on the Condorcet efficiency of weighted scoring rules

A weighted scoring rule, Rule λ, on three alternative elections selects the winner by awarding 1 point to each voter's first ranked candidate, λ points to the second ranked candidate, and zero to the third ranked candidate. The Condorcet winner is the candidate that would defeat each other candidate in a series of pairwise elections by majority rule. The Condorcet efficiency of Rule λ is the conditional probability that Rule λ selects the Condorcet winner, given that a Condorcet winner exists. Borda rule (λ=1/2) is the weighted scoring rule that maximizes Condorcet efficiency. The current study considers the conditional probability that Borda rule selects the Rule λ winner, given that Rule λ elects the Condorcet winner with a large electorate. Received: 21 August 1996 / Accepted: 7 January 1997     

The sensitivity of weight selection for scoring rules to profile proximity to single-peaked preferences

Niemi (Am Polit Sci Rev 63:488–497, 1969) proposed a simple measure of the cohesiveness of a group of n voters’ preferences that reflects the proximity of their preferences to single-peakedness. For three-candidate elections, this measure, k, reduces to the minimum number of voters who rank one of the candidates as being least preferred. The current study develops closed form representations for the conditional probability, PASW(n,IAC|k), that all weighted scoring rules will elect the Condorcet winner in an election, given a specified value of k. Results show a very strong relationship between PASW(n,IAC|k) and k, such that the determination of the voting rule to be used in an election becomes significantly less critical relative to the likelihood of electing the Condorcet winner as voters in a society have more structured preferences. As voters’ preferences become more unstructured as measured by their distance from single-peakedness, it becomes much more likely that different voting rules will select different winners.A preliminary version of this paper was presented at the European Public Choice Society Conference in Berlin, Germany, April 15–18, 2004.     

The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences

The No Show Paradox (there is a voter who would rather not vote) is known to affect every Condorcet voting function. This paper analyses two strong versions of this paradox in the context of Condorcet voting correspondences. The first says that there is a voter whose favorite candidate loses the election if she votes honestly, but gets elected if she abstains. The second says that there is a voter whose least preferred candidate gets elected if she votes honestly, but loses the election if she abstains. All Condorcet correspondences satisfying some weak domination properties are shown to be affected by these strong forms of the paradox. On the other hand, with the exception of the Simpson-Cramer Minmax and the Young rule, all the Condorcet correspondences that (to the best of our knowledge) are proposed in the literature suffer from these two paradoxes. Received: 30 November 1999/Accepted: 27 March 2000     

Generalized condorcet-winners for single peaked and single-plateau preferences

When preferences are single peaked the choice functions that are independent of irrelevant alternatives both in Nash's and in Arrow's sense are characterized. They take the Condorcet winner of the n individual peaks plus at most n-1 fixed ballots (phantom voters). These choice functions are also coalitionally strategy-proof.Next the domain of individual preferences is enlarged to allow for singleplateau preferences: again, Nash's IIA and Arrow's IIA uniquely characterize a class of generalized Condorcet winners choice functions. These are, again, coalitionally strategy-proof.     

Existence of homogeneous representations of interval orders on a cone in a topological vector space

Necessary and sufficient conditions are presented for the existence of a pair <u,v> of positively homogeneous of degree one real functions representing an interval order on a real cone K in a topological vector space E (in the sense that, for every x,yK, xy if and only if u(x)v(y)), with u lower semicontinuous, v upper semicontinuous, and u and v utility functions for two complete preorders intimately connected with . We conclude presenting a new approach to get such kind of representations, based on the concept of a biorder.This research has been supported by the Integrated Action of Research HI2000-0116 (Spain-Italy). Also, the work of coauthors Candeal and Induráin has been partially supported by the research project PB98-551 Estructuras ordenadas y aplicaciones (M.E.C. Spain, December 1999).     

A quantitative theory of preferences: Some results on transition functions

We investigate a general theory of combining individual preferences into collective choice. The preferences are treated quantitatively, by means of preference functions (a,b), where 0(a,b) expresses the degree of preference of a to b. A transition function is a function (x,y) which computes (a,c) from (a,b) and (b,c), namely (a,c)=((a,b),(b,c)). We prove that given certain (reasonable) conditions on how individual preferences are aggregated, there is only one transition function that satisfies these conditions, namely the function (x,y)=x·y (multiplication of odds). We also formulate a property of transition functions called invariance, and prove that there is no invariant transition function; this impossibility theorem shows limitations of the quantitative method.Research supported in part by the National Science Foundation.     

A nail-biting election

In the first competitive election for President of the Social Choice and Welfare Society, the (official) approval-voting winner differed from the (hypothetical) Borda count winner, who was also the Condorcet winner. But because the election was essentially a toss-up, it is impossible to say who should have won. The election for Council was more true to form of other professional-society elections, with the winners identical, and even their rankings almost duplicative, under both voting systems. Received: 11 April 2000/Accepted: 26 March 2001     

Smallest tournaments not realizable by $${\frac{2}{3}}$$-majority voting

Define the predictability number α(T) of a tournament T to be the largest supermajority threshold for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability ; Gilboa has found a tournament on 54 alternatives (i.e. vertices) that has predictability less than , and has asked whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability , and prove that it is the smallest tournament with predictability <  . Our methodology is to formulate the problem as a finite set of two-person zero-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic. D. Shepardson was supported by a NSF Graduate Research Fellowship during the course of this work.     

Powers of subgroups in voting bodies

This paper discusses the power p _n of an n-member subgroup B _n of an N-member voting body, N odd and 1 n N. In contrast to bloc voting, we assume that the members vote independently with equal probability for and against a given issue. Power p _n is defined as the probability that the outcome of a vote changes if all members of B _n reverse their votes. Theorems: p _{n + 1} = _n for odd n < N; p _n + p _{N – n} = 1; P _m + p _n > p _{m + n} if m + n < N; p _{n + 1}/p _n (n + 1)/n as N for fixed even n; for rational 0 > > 1, p _N 2^–1 sin^{–1 1/2} as N . A simple summation formula is given for p _n .     

A self-psychological study of a shared gambling fantasy in Eugene O'Neill's Hughie

This paper presents an applied psychoanalytic study of Eugene O'Neill's two-character play, Hughie. Applying the constructs of self psychology, the play illustrates both the narcissistic features and the emotional and behavioral characteristics of compulsive gamblers. The study focuses particular attention on the role of narcissistic fantasies—with both grandiose and megalomaniacal features—in affecting, temporarily, the mood of the characters. Moreover, it is shown that a shared gambling fantasy—a winner among winners—enables them to experience a sense of camaraderie, humanness, and the illusion of kinship.     

A law of large numbers for weighted plurality

Consider an election between $k$ candidates in which each voter votes randomly (but not necessarily independently) for a single candidate, and suppose that there is a single candidate that every voter prefers (in the sense that each voter is more likely to vote for this special candidate than any other candidate). Suppose we have a voting rule that takes all of the votes and produces a single outcome and suppose that each individual voter has little effect on the outcome of the voting rule. If the voting rule is a weighted plurality, then we show that with high probability, the preferred candidate will win the election. Conversely, we show that this statement fails for all other reasonable voting rules. This result is an extension of one by Häggström, Kalai and Mossel, who proved the above in the case $k=2$ .