Which Scoring Rule Maximizes Condorcet Efficiency Under Iac?

Consider an election in which each of the n voters casts a vote consisting of a strict preference ranking of the three candidates A, B, and C. In the limit as n→∞, which scoring rule maximizes, under the assumption of Impartial Anonymous Culture (uniform probability distribution over profiles), the probability that the Condorcet candidate wins the election, given that a Condorcet candidate exists? We produce an analytic solution, which is not the Borda Count. Our result agrees with recent numerical results from two independent studies, and contradicts a published result of Van Newenhizen (Economic Theory 2, 69–83. (1992)).     

On the (Sample) Condorcet Efficiency of Majority Rule: An alternative view of majority cycles and social homogeneity

The Condorcet efficiency of a social choice procedure is usually defined as the probability that this procedure coincides with the majority winner (or majority ordering) in random samples, given a majority winner exists (or given the majority ordering is transitive). Consequently, it is in effect a conditional probability that two sample statistics coincide, given certain side conditions. We raise a different issue of Condorcet efficiencies: What is the probability that a social choice procedure applied to a sample matches with the majority preferences of the population from which the sample was drawn? We investigate the canonical case where the sample statistic is itself also majority rule and the samples are drawn from real world distributions gathered from national election surveys in Germany, France, and the United States. We relate the results to the existing literature on majority cycles and social homogeneity. We find that these samples rarely display majority cycles, whereas the probability that a sample misrepresents the majority preferences of the underlying population varies dramatically and always exceeds the probability that the sample displays cyclic majority preferences. Social homogeneity plays a fundamental role in the type of Condorcet efficiency investigated here.

     

Plurality Rule Works In Three-Candidate Elections

In the citizen–candidate approach each citizen chooses whether or not to run as candidate. In a single-peaked preference domain, we find that the strategic entry decision of the candidates eliminates one of the most undesirable properties of Plurality rule, namely to elect a poor candidate in three-candidate elections since as we show, the Condorcet winner among the self-declared candidates is always elected. We find that the equilibria with three candidates are basically 2-fold, either there are two right-wing candidates and a left-wing candidate who wins the elections (or its symmetric), or there is a right-wing candidate, a left-wing candidate, and a candidate located in between the two others who becomes winner. We also show that when four or more candidates enter the contest, Plurality rule can elect the Condorcet-loser among the self-declared candidates.      

Positionalist voting functions

Positionalist voting functions are those social choice functions where the positions of the alternatives in the voter's preference orders crucially influence the social ordering of the alternatives. An important subclass consists of those voting functions where numbers are assigned to the alternatives in the preference orders and the social ordering is computed from these numbers. Such voting functions are called representable. Various well-known conditions for voting functions are introduced and it is investigated which representable voting functions satisfy these conditions. It is shown that no representable voting function satisfies the Condorcet criterion. This condition and Arrow's independence condition, which are typical non-positionalist conditions, are shown to be incompatible. The Borda function, which is a well-known positionalist voting function, is studied extensively, conditions uniquely characterizing it are given and some modifications of the function are investigated.My thanks are due to professor Bengt Hansson for encouragement and several helpful suggestions.     

Some measures of closeness to unanimity and their implications

This note demonstrates how certain collective decision methods ensure the selection of alternatives which are the closest to win unanimously. By using four different functions for measuring the distance between preference profiles, we obtain the equivalence between the closeness to unanimity procedure (CUP) and the Borda method, the plurality rule, the probabilistic Borda rule and the L-procedure respectively.     

Sophisticated voting under the plurality procedure: A test of a new definition

The Niemi-Frank definition of sophisticated voting can now be evaluated on two grounds. First, we can compare our definition to Farquharson's. For the most part, the two definitions yield identical outcomes. Both pickCondorcet winners a very high proportion of the time and prevent the selection of Condorcet losers. The major differences are in the logic underlying the two definitions and in the rate of determinacy of outcomes. Here there is a tradeoff. The logic underlying the Farquharson model is especially persuasive, although it is our feeling that the Niemi-Frank definition comes closer to mirroring the way in which voters might actually analyze a plurality situation. In any case, the price paid by the Farquharson definition for its ironclad logic is a much higher rate of indeterminacy. In over half of the cases, the Farquharson logic fails to lead to any conclusion whatsoever. The Niemi-Frank definition yields many more determinate situations, with mostly Condorcet winners and with strategies that make good, if not completely unassailable sense. A second way of evaluating the Niemi-Frank definition is in comparison with sincere voting. A commonly-cited shortcoming of plurality voting is that often fails to choose a Condorcet winner. As we notedearlier, sophisticated plurality voting, unlike binary voting, is imperfect in this respect. Nonetheless, even taking account of the indeterminacy thatremains in the Niemi-Frank definition, sophisticated voting picked a Condorcet winner about 10 percent more frequently than did sincere voting as well as eliminating the possibility of a Condorcet loser being chosen. By this measure, the Niemi-Frank definition is not only acceptable but suggests that this form of strategic behavior actually leads tobetter outcomes. By proposing and now by testing a new definition of sophisticated voting under plurality rule, we have begun to make some headway on understanding strategic behavior and its effects in an outwardly simple yet deceptively complex voting system. We are, of course, far from finished. Most significantly, our definition applies to only three alternatives, and Farquharson's (even if one is willing to live with its high indeterminacy) becomes extraordinarily cumbersome with more than three alternatives? In any event, the results of this foray into sophisticated nonbinary voting suggests once again that strategic behavior, rather than making things worse, improves the chances that the outcome will be the one most favored by the majority criterion.     

The Unexpected Behavior of Plurality Rule

When voters’ preferences on candidates are mutually coherent, in the sense that they are at all close to being perfectly single-peaked, perfectly single-troughed, or perfectly polarized, there is a large probability that a Condorcet Winner exists in elections with a small number of candidates. Given this fact, the study develops representations for Condorcet Efficiency of plurality rule as a function of the proximity of voters’ preferences on candidates to being perfectly single-peaked, perfectly single-troughed or perfectly polarized. We find that the widely used plurality rule has Condorcet Efficiency values that behave in very different ways under each of these three models of mutual coherence.     

On the stability of a triplet of scoring rules

When choosing a voting rule to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this article the menu constituted by the three well-known scoring rules {Borda, Plurality, and Antiplurality}. Under the Impartial Culture assumption, which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules.     

Opinion leaders,independence, and Condorcet's Jury Theorem

Condorcet's Jury Theorem shows that on a dichotomous choice, individuals who all have the same competence above 0.5, can make collective decisions under majority rule with a competence that approaches 1 as either the size of the group or the individual competence goes up. The theorem assumes that the probability of each voter's being correct is independent of the probability of any other voter being correct. Contrary to several authors, the presence of mutual or common influences such as opinion leaders does not easily rule independence either in or out. Indeed, and this ought to be surprising,under certain conditions deference to opinion leaders can improve individual competence without violating independence, and so can raise group competence as well.     

The Pareto rule and strategic voting

This paper uses a particular choice rule over sets of alternatives under the Pareto rule. Starting from the sincere situation every strategic misrevelation of preference is shown to be an improvement for all voters. The existence of an equilibrium under successive misrepresentations by sincere voters is demonstrated.     

Condorcet’s principle and the strong no-show paradoxes

We consider two no-show paradoxes, in which a voter obtains a preferable outcome by abstaining from a vote. One arises when the casting of a ballot that ranks a candidate in first place causes that candidate to lose the election, superseded by a lower-ranked candidate. The other arises when a ballot that ranks a candidate in last place causes that candidate to win, superseding a higher-ranked candidate. We show that when there are at least four candidates and when voters may express indifference, every voting rule satisfying Condorcet’s principle must generate both of these paradoxes.     

Best-of-two contests with psychological effects

We study a best-of-two contest with two homogeneous players (teams). The winner is the player who wins in both stages. In the case of one victory for each team, a draw determines the identity of the winner. Each team has a value of winning the contest as well as a value of winning a single game. A team’s value of winning a game at its home field is higher than its value of winning a game away from home. We show that if winning in the first stage provides a psychological advantage in the second stage, each team would prefer to play the second stage at its home field.     

Condorcet's paradox and the likelihood of its occurrence: different perspectives on balanced preferences*

Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, and the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections.     

Communication compatible voting rules

We reassess the possibility of full information pooling in a Condorcet jury environment featuring heterogeneous and privately known preference types. We find that in general, with uncorrelated preference types, only very limited heterogeneity is compatible with full pooling. We provide a sufficient condition, based on a simple measure of preference misalignment, under which the set of voting rules compatible with full pooling is at most a singleton. As a caveat to any simplistic conclusions, we identify a case in which an increase in heterogeneity (i.e. polarization) systematically generates the possibility of full pooling. Increased jury size, in contrast, is shown to always render full pooling more difficult.     

An application of social choice theory to U.S.D.A. forest service decision making

To meet the requirements of the National Environmental Policy Act of 1969, the U.S.D.A. Forest Service has changed the way forest plans are developed. The focus of this paper is to address the group-decision problem using social choice theory, specifically the voting models of Condorcet and Borda. The elements of a social choice problem are voters, alternatives, preferences, and aggregation. A case study from the Shoshone National Forest is used to demonstrate the use of the voting models from social choice theory. The solutions derived from the analysis are strategy and coalitional strategy proof implying that behaviors intended to influence the outcome, such as vote trading, would be unsuccessful.     

Individual rationality and the concept of social welfare

This paper, written in October 1974, deals with some game aspects of the social choice problem. The question asked is whether there exists a social decision rule satisfying the conditions imposed by Arrow over all the preference profiles that may logically arise under it (in the sense of being compatible with individual rationality). This question is answered in the affirmative. The meaning of this result is that if Arrow's condition of unrestricted domain is modified so as to exclude any profile which contradicts individual rationality, then an Arrovian social welfare function can be shown to exist (subject to the assumption that whenever the social outcome is in doubt, individuals use the maximin criterion in order to choose their voting strategy).This research was written up in October 1974 at Northwestern University and was partially supported by the National Science Foundation.     

Conditions for the optimality of simple majority decisions in pairwise choice situations

Recently it has been proved in a number of studies, that, under a proper set of assumptions, the optimal group decision rule in pairwise choice situations is a weighted majority rule, with weights that are proportional to the logarithms of the decision makers' odds of choosing the correct alternative.The purpose of the present note is to specify the necessary and sufficient conditions for this rule to coincide with the simple majority rule, and with restricted simple majority rules (which are defined as rules of simple majority, based on some subset of the most competent group members). These conditions are formulated in terms of inequalities between the group members' weights, thereby permitting easy verification of the optimality of the above mentioned rules.     

Condorcet vs. Borda in light of a dual majoritarian approach

Many voting rules and, in particular, the plurality rule and Condorcet-consistent voting rules satisfy the simple-majority decisiveness property. The problem implied by such decisiveness, namely, the universal disregard of the preferences of the minority, can be ameliorated by applying unbiased scoring rules such as the classical Borda rule, but such amelioration has a price; it implies erosion in the implementation of the widely accepted “majority principle”. Furthermore, the problems of majority decisiveness and of the erosion in the majority principle are not necessarily severe when one takes into account the likelihood of their occurrence. This paper focuses on the evaluation of the severity of the two problems, comparing simple-majoritarian voting rules that allow the decisiveness of the smallest majority larger than 1/2 and the classical Borda method of counts. Our analysis culminates in the derivation of the conditions that determine, in terms of the number of alternatives k, the number of voters n, and the relative (subjective) weight assigned to the severity of the two problems, which of these rules is superior in light of the dual majoritarian approach.     

How relevant are ‘Irrelevant’ Alternatives?

Arrow's Independence of Irrelevant Alternatives Condition is examined. It is shown why the standard rationale for (or against) the condition tends to be inconclusive as it fails to consider the basic game issue in social choice. Specifically it is explained how some recent results (Gibbard-Satterthwaite) on the general non-existence of strategy-proof voting procedures provide the strongest rationale for the independence condition. Also, it is shown that this rationale was exactly the one used by Condorcet in his work on decision rules for juries and elections.