Condorcet Winner Research Articles

On Dynamic Aggregation Systems

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭr (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭr (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ2 (A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function fJ,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭr (A) with respect to a set of lots 𝒥, then the aggregation function fJ,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences fJ,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

Income inequality, size of government, and tax progressivity: A positive theory

I investigate the relationship between income inequality and the composition of public spending in redistributive policies. I extend the Meltzer–Richard model of voting over redistribution allowing voters to choose not only the amount of a uniform lump-sum transfer, but also the level of provision of a public good. The governmental budget is balanced; thus these two choices determine the tax rate on labor income. The multidimensionality of the policy space implies that there is no Condorcet winner. I adopt a citizen-candidate model of electoral competition to tackle this problem. I show that the progressivity of the tax system is increasing in the mean-to-median income ratio while the size of the government need not be. This means that higher income inequality implies a more progressive tax system but, in contrast with the traditional analysis, it may also result in a smaller size of government. Such results are consistent with the most recent findings in the empirical literature.

The prevalence and consequences of ballot truncation in ranked-choice elections

In ranked-choice elections, voters vote by indicating their preference orderings over the candidates. A ballot is truncated when the ordering is incomplete (called partial voting). Sometimes truncation is forced—voters are allowed to rank only a limited number of candidates—but sometimes it is voluntary. During the vote tabulating process, a truncated ballot is exhausted when all of the candidates it ranks have been eliminated. Ballot exhaustion and, therefore ballot truncation, is a concern in single-winner elections when the margin of victory in the final stage is less than the number of exhausted ballots. That concern motivates our study. We review evidence from actual single-winner ranked-choice elections and conclude that voluntary ballot truncation is very common. Moreover, it is difficult to explain strategically. To assess the significance of ballot truncation, we simulate ranked-choice elections with four, five and six candidates, using both spatial and random models of voter preferences. Does truncation change the probability that a Condorcet winner wins the election? Does the winner change as the extent of truncation increases? We find that even small amounts of truncation can alter election outcomes.

Daunou's Voting Method

Pierre Daunou, a contemporary of Borda and Condorcet during the era of the French Revolution and active debates on alternative voting rules, proposed a rule that chooses the strong Condorcet winner if there is one, otherwise eliminates Condorcet losers and uses plurality voting on the remaining alternatives. We characterize his rule which combines potentially conflicting desiderata of majoritarianism by ordering them lexicographically. This contribution serves not just to remind ourselves that a 19th-century vintage may still retain excellent aroma and taste, but also to promote a promising general approach to reconcile potentially conflicting desiderata by accommodating them lexicographically.

Why is free education so popular? A political economy explanation

AbstractThis paper analyzes the political support for different funding regimes of education in a one‐person, one‐vote democracy. We focus the analysis on four systems that have had a preponderant presence in the political debate on education: a private system, a public system that delivers the same resources to each student (universal‐free education), a public system that intends to equalize results, and a public system that aims to maximize the output of the economy. We show that a system of universal free education is the Condorcet winner. The level of income inequality and the degree to which income distribution is skewed to the right are key factors behind this conclusion. We also show that the voting outcome of public versus private funding for education depends crucially on the type of public funding under consideration.

Recognizing Top-Monotonic Preference Profiles in Polynomial Time

We provide the first polynomial-time algorithm for recognizing if a profile of (possibly weak) preference orders is top-monotonic. Top-monotonicity is a generalization of the notions of single-peakedness and single-crossingness, defined by Barbera and Moreno. Top-monotonic profiles always have weak Condorcet winners and satisfy a variant of the median voter theorem. Our algorithm proceeds by reducing the recognition problem to the SAT-2CNF problem.

Condorcet winners and social acceptability

We say that an alternative is socially acceptable if the number of individuals who rank it among their most preferred half of the alternatives is at least as large as the number of individuals who rank it among the least preferred half. A Condorcet winner may not necessarily be socially acceptable. However, if preferences are single-peaked, single-dipped, or satisfy the single-crossing property, any Condorcet winner is socially acceptable. We identify maximal families of preferences that guarantee that Condorcet winners are socially acceptable.

Condorcet efficiency of the preference approval voting and the probability of selecting the Condorcet loser

Under approval voting (AV), each voter just distinguishes the candidates he approves of from those appearing as unacceptable. The preference approval voting (PAV) is a hybrid version of the approval voting first introduced by Brams and Sanver (in: Brams, Gehrlein, Roberts (eds) The mathematics of preference, choice and order. Springer, Berlin, pp 215–237, 2009). Under PAV, each voter ranks all the candidates and then indicates the ones he approves. In this paper, we provide an analytical representation of the limiting probability that PAV elects the Condorcet winner (resp. the Condorcet loser) when she exists in three-candidate elections. We perform our analysis by assuming the assumption of the Extended Impartial Culture. The aim is to measure at which extend PAV performs better than AV both on the propensity of electing the Condorcet winner and on that of the non-election of the Condorcet loser. For this purpose, we also provide an analytical representation of the limiting probability that AV elects the Condorcet winner (resp. the Condorcet loser) when she exists in three-candidate elections. Our representation of the limiting probability that AV elects the Condorcet winner is more general than that provided by Diss et al. (in: Laslier and Sanver (eds) Handbook on approval voting. Springer, Berlin, pp 255–283, 2010) and it leads to the same figures as the representation provided by Gehrlein and Lepelley (Group Decis Negot 24:243–269, 2015).

POSITIVELY RESPONSIVE COLLECTIVE CHOICE RULES AND MAJORITY RULE: A GENERALIZATION OF MAY'S THEOREM TO MANY ALTERNATIVES

AbstractMay's theorem shows that if the set of alternatives contains two members, an anonymous and neutral collective choice rule is positively responsive if and only if it is majority rule. We show that if the set of alternatives contains three or more alternatives only the rule that assigns to every problem its strict Condorcet winner satisfies the three conditions plus Nash's version of “independence of irrelevant alternatives” for the domain of problems that have strict Condorcet winners. We show also that no rule satisfies the four conditions for domains that are more than slightly larger.

Electoral Institutions with Impressionable Voters

We use a model of impressionable voters to study multi-candidate elections under different electoral rules. Instead of maximizing expected utility, voters cast their ballots based on impressions. We show that, under each rule, there is a monotone relationship between voter preferences and vote measures. The nature of this relationship, how- ever, varies by electoral rule. Vote measures are biased upwards for socially preferred candidates under plurality rule, but biased downwards under negative plurality. There is no such bias under approval voting or Borda count. Voters always elect the socially preferred candidate in two-way races for any electoral rule. In multi candidate elections, however, the ability to elect a Condorcet winner varies by rule. The results show that multi-candidate elections can perform well even if voters follow simple behavioral rules. The relative performance of specific electoral institutions, however, depends on the assumed behavioral model of voting.

Majoritarian preference, utilitarian welfare and public information in Cournot oligopoly

Can individual preferences for public information among heterogeneous consumers be aggregated into a meaningful social preference that does not suffer from Condorcet cycles? In a Cournot model where homogeneous producers observe a public signal about an uncertain cost of production prior to taking quantity decisions, we show that the majoritarian preference of consumers for the precision of public information is fairly well behaved so that a Condorcet winner always exists. Under a monotonicity condition on the demand function, we characterize the Condorcet-winning precision in terms of the demand function and the number of firms under which the Condorcet-winning precision (i) hurts consumers' surplus and profits or (ii) remains conflict-free. These results have interesting implications on ‘collective’ Bayesian persuasion by agencies representing consumers, showing that when full transparency maximizes expected consumers' surplus, collective Bayesian persuasion can lead to full opacity, and vice versa.

Condorcet consistent scoring rules and single-peakedness

We study voting problems with an odd number of agents and single-peaked preferences. With only three alternatives, there are scoring rules that yield the Condorcet winner only for committees of three and five agents. With four or more alternatives, only committees of three agents work. In all these scoring rules, the best and worst alternatives are assigned a score of 1and 0, respectively, and any middle alternative a score between 0 and 12. For five or more alternatives, the score of any middle alternative must be the same, and we call this family semiplurality scoring rules.

Impressionable Voters

We propose a model of impressionable voters. Impressionable voters vote based on impressions rather than maximizing expected utility. We apply our model to elections with multiple candidates and solve for the stationary distributions of the implied stochastic process. In elections with two candidates, we find that impressionable voters select the candidate that maximizes utilitarian welfare. In multi-candidate elections, Condorcet winners are elected provided that a majority of voters is able to sufficiently distinguish competing candidates. In such cases, models with impressionable voters may have better normative properties than models with rational voters. (JEL D71, D72, D91)

Selection of Construction Products Suppliers According to the Condorcet Criterion

The continuous growth of competition on the construction market results in the fact that construction companies are forced to keep searching for solutions facilitating the building process and construction logistics. In the recent decades, one of the most fascinating ideas which enable a real improvement in the aspect of a company’s competitiveness on the market is supply chain management. It has undergone a transformation from fragmented actions to integrated operations intending to optimize the flow of physical resources and information. The first step towards the application of the modern logistics tools is the selection and evaluation of entities involved in a cooperation within the supply chain framework. The selection of reliable partners to supply a construction site or a building company may reduce the operational costs, minimize inventory and improve timeliness as well as the quality of offered services. Any mistakes made in the process of supplier selection may lead to issues such as supply delays, difficulties in keeping contract deadlines and even complete failure of a project or bankruptcy of a construction company. An 18th century criterion for the selection of the most preferable candidate developed by Nicolas de Condorcet is commonly accepted owing to its fair rules of choice (anyone who defeats most of their competitors in a direct confrontation wins). Due to its democratic approach, most criteria decide as to the selection of the winner. Should the choice of a winner or establishing a Condorcet ranking from the perspective of the social choice theory prove impossible, the selected candidate ought to be as much similar to a winner in the Condorcet sense as possible. The theory was developed in order to facilitate the selection of the most suitable candidate in the election, however it may also find application in the selection of a supply chain elements evaluated with the use of multiple criteria. There are a number of decision-making methods which fulfil the Condorcet criterion. They vary in their computational complexity and may provide different final order. The intention of this paper is to review the methods which may offer support while selecting the most preferable supplier of construction products.

The acclamation consensus state and an associated ranking rule

The study of conditions, under which the existence of an “absolute” best winner can be assured, is a hot topic in the field of social choice. Unanimity is an evident example of a condition under which the winner is obvious. However, many more properties weaker than unanimity have been analysed in literature: the presence of a Condorcet winner, strong stochastic transitivity, the presence of a candidate that Borda dominates all other candidates, etc. Unfortunately, one could easily find a prominent ranking rule, for which the outcome does not agree with these relaxed conditions. In this study, we aim to identify a condition weaker than unanimity, but under which the social outcome is still obvious. This condition, defined as the conjunction of three properties already studied by the present authors and hereinafter referred to as acclamation, will be proven to be a meeting point for the most prominent ranking rules in social choice theory, and will be used for introducing an intuitively appealing ranking rule.

Trading Votes for Votes. A Dynamic Theory

We develop a framework to study the dynamics of vote trading over multiple binary issues. We prove that there always exists a stable allocation of votes that is reachable in a finite number of trades, for any number of voters and issues, any separable preference profile, and any restrictions on the coalitions that may form. If at every step all blocking trades are chosen with positive probability, convergence to a stable allocation occurs in finite time with probability 1. If coalitions are unrestricted, the outcome of vote trading must be Pareto optimal, but unless there are three voters or two issues, it need not correspond to the Condorcet winner.

The Uncovered Set and the Core: Cox's Result Revisited

In this work first it is shown, in contradiction to the well-known claim in Cox (1987), that the uncovered set in a multidimensional spatial voting situation (under the usual regularity conditions) does not necessarily coincide with the core even when the core is singleton: in particular, the posited coincidence result, while true for an odd number of voters, may cease to be true when the number of voters is even. Second we provide a characterisation result for the case with an even number of voters: a singleton core is the uncovered set in this case if and only if the unique element in the core is the Condorcet winner.

The No-Show Paradox Under a Restricted Domain

The no-show paradox occurs whenever a group of identically-minded voters is better off abstaining than by voting according to its preferences. Moulin’s (J Econ Theory 45:53–64, 1988) result states that if one wants to exclude the possibility of the no-show paradox, one has to resort to procedures that do not necessarily elect the Condorcet winner when one exists. This paper examines ten Condorcet-consistent and six Condorcet-non-consistent procedures in a restricted domain, viz., one where there exists a Condorcet winner who is elected in the original profile and the profile is subsequently modified by removing a group of voters with identical preferences. The question asked is whether the no-show paradox can occur in these settings. It is found that only two of the ten Condorcet-consistent procedures investigated (Maximin and Schwartz’s procedure) are not vulnerable to the no-show paradox, whereas only two of the six non-Condorcet-consistent ranked procedures investigated (Coombs’ and the Negative Plurality Elimination Rule procedures) are vulnerable to this paradox in the restricted domain. In other words, for a no-show paradox to occur when using Condorcet-consistent procedures it is not, in general, necessary that a top Condorcet cycle exists in the original profile, while for this paradox to occur when using (ranked) non-Condorcet-consistent procedures it is, almost always, necessary that the original profile has a top cycle.

Axiomatic Foundations of a Unifying Core

We provide an axiomatic characterization of the core of games in effectiveness form. We point out that the core, whenever it applies to appropriate classes of these games, coincides with a wide variety of prominent stability concepts in social choice and game theory, such as the Condorcet winner, the Nash equilibrium, pairwise stability, and stable matchings, among others. Our characterization of the core invokes the axioms of restricted non-emptiness, coalitional unanimity, and Maskin invariance together with a principle of independence of irrelevant states, and uses in its proof a holdover property echoing the conventional ancestor property. Taking special cases of this general characterization of the core, we derive new characterizations of the previously mentioned stability concepts.

The Myth of the Condorcet Winner

There is consensus among legal scholars that, when choosing among multiple alternatives, the Condorcet winner, should it exist, is the preferred option. In this essay I will refute that claim, both normatively and positively. In addition, I will suggest that a different approach, based in behavioral economics, might be a more productive way to model the choices that legislatures make among multiple alternatives.