Winning probabilities in a pairwise lottery system with three alternatives

Summary. The pairwise lottery system is a multiple round voting procedure which chooses by lot a winner from a pair of alternatives to advance to the next round where in each round the odds of selection are based on each alternatives majority rule votes. We develop a framework for determining the asymptotic relative likelihood of the lottery selecting in the final round the Borda winner, Condorcet winner, and Condorcet loser for the three alternative case. We also show the procedure is equivalent to a Borda lottery when only a single round of voting is conducted. Finally, we present an alternative voting rule which yields the same winning probabilities as the pairwise lottery in the limiting case as the number of rounds of the pairwise lottery becomes large.Received: 5 June 2003, Revised: 17 June 2004, JEL Classification Numbers: D71. Correspondence to: Jac C. HeckelmanWe thank Keith Dougherty and Andrew Yates for their comments.     

The Costs of Implementing the Majority Principle: The Golden Voting Rule

In a context of constitutional choice of a voting rule, this paper presents an economic analysis of scoring rules that identifies the golden voting rule under the impartial culture assumption. This golden rule depends on the weights β and (1−β) assigned to two types of costs: the cost of majority decisiveness (‘tyranny’) and the cost of the ‘erosion’ in the majority principle. Our first main result establishes that in voting contexts where the number of voters n is typically considerably larger than the number of candidates k, the golden voting rule is the inverse plurality rule for almost any positive β. Irrespective of n and k, the golden voting rule is the inverse plurality rule if β ≥ 1/2 .. This hitherto almost unnoticed rule outperforms any other scoring rule in eliminating majority decisiveness. The golden voting rule is, however, the plurality rule, the most widely used voting rule that does not allow even the slightest ‘erosion’ in the majority principle, when β=0. Our second main result establishes that for sufficiently “small size” voting bodies, the set of potential golden rules consists at most of just three rules: the plurality rule, the Borda rule and the inverse plurality rule. On the one hand, this finding provides a new rationalization to the central role the former two rules play in practice and in the voting theory literature. On the other hand, it provides further support to the inverse plurality rule; not only that it is the golden rule in voting contexts, it also belongs, together with the plurality rule and the Borda method of counts, to the “exclusive” set of potential golden voting rules in small committees. We are indebted to Jim Buchanan, Amichai Glazer, Noa Nitzan, Ken Shepsle, and an anonymous referee for their useful comments.     

A strategy-proofness characterization of majority rule

Summary. A feasible alternative x is a strong Condorcet winner if for every other feasible alternative y there is some majority coalition that prefers x to y. Let (resp., denote the set of all profiles of linear (resp., merely asymmetric) individual preference relations for which a strong Condorcet winner exists. Majority rule is the only non-dictatorial and strategy-proof social choice rule with domain , and majority rule is the only strategy-proof rule with domain . Received: August 29, 2000; revised version: November 13, 2002 RID="*" ID="*"We are grateful to Wulf Gaertner and our two referees for insightful comments on a previous draft. Correspondence to: D. E. Campbell     

The Borda method is most likely to respect the Condorcet principle

Summary We prove that in the class of weighted voting systems the Borda Count maximizes the probability that a Condorcet candidate is ranked first in a group election. A direct result is that the Borda Count maximizes the probability that a transitive, binary ranking of the candidates is preserved in a group election. A preliminary result, but one of independent interest, is that the Borda Count maximizes the probability that a majority outcome betweenany two candidates is reflected by the group election. All theorems are valid when there is a uniform probability distribution on the voter profiles and can be generalized to other uniform-like probability distributions. This work extends previous results of Fishburn and Gehrlein from three candidates to any number of candidates.This work is a portion of my doctoral dissertation The Geometric Investigation of Voting Techniques: A Comparison of Approval Voting, Positional Voting Techniques and the Borda Count written under Don Saari at Northwestern University, Evanston, Illinois.     

An Incentive‐Compatible Condorcet Jury Theorem*

We consider a group of individuals who face a binary collective decision. Each group member holds some private information, and all agree about what decision should be taken in each state of nature. However, the state is unknown, and members can differ in their valuations of the two types of mistakes that might occur, and in their prior beliefs about the true state. For a slightly randomized majority rule, we show that informative voting by all voters is the unique Nash equilibrium, that this equilibrium is strict, and that the Condorcet asymptotic efficiency result holds in this setting.     

Condorcet cycles in bipartite populations

Summary. Simple majority voting between pairs of alternatives is used to aggregate individual preferences. The occurence of Condorcet cycles is limited thanks to a principle of homogeneity on individual preferences. The restrictions induced on the domain of the latters are weak: among the n! possible orderings of n alternatives, more than one half are admissible within a domain. The resulting aggregated preference has then a neglectable probability of showing up cycles. We show moreover that the set of individual preferences can be `naturally' partitioned into two such domains. Received: June 17, 1996; revised version: April 15, 1997     

Approval voting reconsidered

Summary. The merit of approval voting has been widely discussed in the past 25 years. The distinct property of this rule is the extent of flexibility it allows; any voter can approve as many alternatives/candidates as he wishes. Nevertheless, this advantage is the very reason for two drawbacks of approval voting: its extreme vulnerability to majority decisiveness (Theorem 1) and its extreme vulnerability to erosion of the majority principle (Theorem 2). On the one hand, under some feasible voting strategies any majority of more than 1/2 of the voters can guarantee the selection of its most favorable candidate, regardless of the preferences of the other voters. On the other hand, under alternative voting strategies even the largest majority cannot impose its common most preferred candidate. A simultaneous resolution of the two problems is possible by restricted approval voting (RAV), a voting rule that allows partial voter flexibility by restricting the minimal and maximal number of candidates that can be approved. Our main result (Theorem 3) clarifies how the foregone flexibility in voters sovereignty mitigates the above mentioned drawbacks under sincere and insincere coordinated voting. Our findings suggest a new possible justification of a particular voting rule which is based on the significance assigned to three considerations: the advantages of voters flexibility, immunity to majority decisiveness and immunity to erosion of the majority principle. Such justification can provide a possible explanation to the prevalent use of some special cases of RAV, notably, of the plurality rule and of approval voting.Received: 8 October 2003, Revised: 17 June 2004, JEL Classification Numbers: D71, D72. Correspondence to: Shmuel NitzanWe are indebted to Steve Brams, Remzi Sanver and an anonymous referee for their very useful comments and suggestions.     

Condorcet Jury Theorem or Rational Ignorance

We analyze a symmetric model of an election in which voters are uncertain about which of two alternatives is desirable for them. Each voter must incur some cost to acquire information about the alternatives. We show that by focusing on unbiased voting strategies, general symmetric signal structures can be degenerated to a two‐signal model. In addition, we show that for any sequence of unbiased voting equilibria, if the second‐order derivative of the information cost function at no information is zero, then the probability of electing the desirable alternative converges to one, that is, the Condorcet Jury Theorem is valid. Otherwise, this probability converges to some value less than one; that is, the “rational ignorance” hypothesis is valid.     

Mathematical structure of voting paradoxes

Summary. A theory is developed to identify, characterize, and explain all possible positional and pairwise voting outcomes that can occur for any number of alternatives and any profile. This paper describes pairwise voting where new results include explanations for all paradoxes, cycles, conflict between Borda and Condorcet rankings, differences among procedures using pairwise votes (such as the Borda Count, Kemeny's method, and the Arrow-Raynaud rule), and discrepancies among the societal rankings as candidates are dropped or added. Other new results include new relationships among the Borda and Condorcet "winners" and "losers." The theory also shows how to construct all supporting profiles. The following companion paper does the same for positional methods.     

Malthus versus Condorcet revisited

The article poses some counter-factual questions concerning the intellectual relationship between Condorcet and Malthus in the light of the latter's attack on Condorcet's Esquisse, bearing in mind the common ‘Smithian’ basis of their thinking and their background as ‘social Newtonains’ with an interest in placing the science of morals and politics on a rational basis. Malthus's reasons for opposing Condorcet's solutions to the population problem and their divergence over redistribution of income through social insurance are considered. Some stereotypical and ideological interpretations of Malthus as a bioeconomic determinist with demoralizing and reactionary tendencies are also discussed.     

Multi-stage voting, sequential elimination and Condorcet consistency

A class of voting procedures based on repeated ballots and elimination of one candidate in each round is shown to always induce an outcome in the top cycle and is thus Condorcet consistent, when voters behave strategically. This is an important class as it covers multi-stage, sequential elimination extensions of all standard one-shot voting rules (with the exception of negative voting), the same one-shot rules that would fail Condorcet consistency. The necessity of repeated ballots and sequential elimination are demonstrated by further showing that Condorcet consistency would fail in all standard voting rules that violate one or both of these conditions.     

Condorcet Jury Theorem: An example in which informative voting is rational but leads to inefficient information aggregation

Recent research on the Condorcet Jury Theorem has proven that informative voting (that is, voting according to one’s signal) is not necessarily rational. With two alternatives, rational voting typically leads to the election of the correct alternative, in spite of the fact that not all voters vote informatively. We prove that with three alternatives, there are cases in which informative voting is rational and yet leads to the election of a wrong alternative.     

The dynamics of distributive politics

We study dynamic committee bargaining over an infinite horizon with discounting. In each period, a committee proposal is generated by a random recognition rule, the committee chooses between the proposal and a status quo by majority rule, and the voting outcome in period t becomes the status quo in period t + 1. We study symmetric Markov equilibria of the resulting game and conduct an experiment to test hypotheses generated by the theory for pure distributional (divide-the-dollar) environments. In particular, we investigate the effects of concavity in the utility functions, the existence of a Condorcet winning alternative, and the discount factor (committee “impatience”). We report several new findings. Voting behavior is selfish and myopic. Status quo outcomes have great inertia. There are strong treatment effects that are in the direction predicted by the Markov equilibrium. We find significant evidence of concave utility functions.     

A comparison of Dodgson's method and the Borda count

Summary. In an election without a Condorcet winner, Dodgson's Method is designed to find the candidate that is “closest” to being a Condorcet winner. In this paper, we show that the winner from Dodgson's Method can occur at any position in the ranking obtained from the Borda Count, the plurality method, or any other positional voting procedure. In addition, we demonstrate that Dodgson's Method does not satisfy the Independence of Irrelevant Alternatives axiom. Received: January 12, 2001; revised version: June 7, 2001     

Rational Choice and the Condorcet Jury Theorem

In a recent paper, Austen-Smith and Banks (1996,Amer. Polit. Sci. Rev.90, 34–45), criticize the current literature on the Condorcet jury theorem as neglecting the behavioral underpinnings of decision-making. They leave open the question whether allowing mixed strategies would sustain the conclusions of the Condorcet jury theorem. In this paper, it is shown that these conclusions can hold in equilibrium. In other words, “a rational choice foundation for the claim that majorities invariably ‘do better’ than individuals” is derived. (Journal of Economic LiteratureClassification Number: D72.     

Scoring rules: an alternative parameterization

This note presents an alternative parameterization of any scoring rule that satisfies the score-expansion property. This parameterization is based on the vector that specifies, for every number of alternatives k, k ≥ 3, the minimal size of a coalition that can veto an alternative which is preferred by everybody outside the coalition. Our result sheds new light on the commonly used plurality and Borda rules, as well as the inverse plurality rule and any “vote for t alternatives rule”.     

Dynamic stability and reform of political institutions

This paper examines endogenous institutional change in a class of dynamic political games. The political aggregation rules used at date t+1 are instrumental choices under rules at date t. Effectively, rules are “players” who can strategically delegate future policy-making authority to different rules. A political rule is stable if it selects itself. A reform occurs when an alternative rule is selected. The stability of a political rule is shown to depend on whether its choices are dynamically consistent. For instance, simple majority rules can be shown to be dynamically consistent in many common environments where wealth-weighted voting rules are not. The result extends to political rules that incorporate private activities such as extra-legal protests, threats, or private investment. The approach is one way of understanding various explanations of institutional change proposed in the literature. A parametric model of public goods provision gives an illustration.     

A Bayesian Model of Voting in Juries

We model voting in juries as a game of incomplete information, allowing jurors to receive a continuum of signals. We characterize the unique symmetric equilibrium of the game, and give a condition under which no asymmetric equilibria exist under unanimity rule. We offer a condition under which unanimity rule exhibits a bias toward convicting the innocent, regardless of the size of the jury, and give an example showing that this bias can be reversed. We prove a “jury theorem” for our general model: As the size of the jury increases, the probability of a mistaken judgment goes to zero for every voting rule except unanimity rule. For unanimity rule, the probability of making a mistake is bounded strictly above zero if and only if there do not exist arbitrarily strong signals of innocence. Our results explain the asymptotic inefficiency of unanimity rule in finite models and establishes the possibility of asymptotic efficiency, a property that could emerge only in a continuous model. Journal of Economic Literature Classification Numbers: C72, D72.     

Transcendental vs. comparative approaches to justice: a reappraisal of Sen's dichotomy

In The Idea of Justice, Sen describes two competing approaches to theorising about justice: “transcendental institutionalism”, in which he includes Rawls, and “realisation-focused comparison”, in which he includes Condorcet and himself. The objective of this paper is to demonstrate that a comparative approach cannot exist without a transcendental dimension. Contrary to Sen, who claims that a transcendental theory is neither necessary nor sufficient in order to frame comparative judgments, it is shown that a transcendental dimension is a necessary, albeit not sufficient, condition of any comparative approach. To illustrate our thesis, we refer to the works of three great authors: Condorcet, Sen himself and the later Rawls.     

Approval voting with endogenous candidates

We present a formal model of political competition under approval voting which allows for endogenous candidate entry. Our analysis yields a number of novel insights. First, we develop a notion of sincere voting behavior under approval voting, called relative sincerity. We then show that the relatively sincere voting behavior is consistent with the strategic calculus of voting. Second, we show that in a one-dimensional model with distance preferences, equilibria in relatively sincere strategies and without spoiler candidates always generate outcomes close to the median voter. Moreover, approval voting satisfies Duverger's Law in the sense that there are at most two winning positions! Finally, we extend our analysis to arbitrary policy spaces. In the general setting, approval voting is shown to be susceptible to the same kinds of problems as the plurality rule, such as the possibility of non-majoritarian outcomes, failure to elect the Condorcet winner and existence of spoiler candidates.