Condorcet Winner Research Articles

Was Bolsonaro’s 2018 electoral victory an institutional accident?

Presidential Elections are critical moments for polyarchical systems, particularly in contexts of high social tension. In this regard, the 2018 presidential election in Brazil, which used a two-round system, is a significant case study. Intriguingly, the most divisive candidates went to the second round. Was this an institutional accident? Pairwise and positional voting procedures embody different generalizations of a majoritarian credo that underpins such elections. The paper mobilizes both perspectives and, using representative survey data, reconstructs the top four preferences of the Brazilian electorate a week before the election. The analysis reveals that the electoral winner, Jair Messias Bolsonaro – despite being a divisive candidate – was a Condorcet Winner and a Borda Winner. Conversely, the second-round loser, Fernando Haddad, was a Condorcet Loser. Thus, Bolsonaro’s victory in 2018 was not an institutional accident. The paper also explores possible alternative scenarios under different feasible sets of candidates through simulations, contributing to a deeper understanding of the role of decision procedures in critical junctures.

Voter coordination in elections: A case for approval voting

We study how voting rules shape voter coordination in large three-candidate elections. We consider three rules, that differ on the number of candidates that voters can support: Plurality (one), Anti-Plurality (two) and Approval Voting (one or two). We show that the Condorcet winner is always elected at some equilibrium under Approval Voting, and that this rule provides better welfare guarantees than Plurality. We then numerically study a dynamic process of political tâtonnement which delivers rich insights. The Condorcet winner is virtually always elected under Approval Voting, but not under the other rules. The dominance of Approval Voting is robust to several alternative welfare criteria and the introduction of expressive voters.

The Borda and Condorcet winners coincide for lexicographic preferences

We show that, for any lexicographic preference profile, the sets of Borda and Condorcet winners coincide. Moreover, the sets of Borda and Condorcet losers also coincide. However the binary relations induced by the Borda Count and the majority pairwise comparisons are not the same.

Implementing direct democracy via representation

Today, several social movements in western democracies argue that traditional representative democracy has failed to adequately represent the will of the “people”, and instead support direct democracy as the only political system to restore the will of the majority. We analyze under what conditions the policy – a vector of decisions on every issue – implemented by the winner of a bipartisan electoral competition coincides with the policy that citizens would choose by means of direct democracy. We find necessary and sufficient conditions for this equivalence to hold, implying that, as long as at least one of them is not fulfilled, a divergence of outcomes between direct and representative democracy arises. The first condition requires that the outcome of majority voting issue-by-issue is the Condorcet winner relative to the voters’ preference profile over the set of policies. The second requires that either that outcome is the preferred policy for at least one of the candidates, or that candidates’ preferred policies differ on every single issue. We reinterpret some findings in the literature in the light of our model and present them as potential reasons why the equivalence between direct and representative democracy may fail.

Relaxed notions of Condorcet-consistency and efficiency for strategyproof social decision schemes

Social decision schemes (SDSs) map the preferences of a group of voters over some set of m alternatives to a probability distribution over the alternatives. A seminal characterization of strategyproof SDSs by Gibbard (Econometrica 45(3):665–681, 1977) implies that there are no strategyproof Condorcet extensions and that only random dictatorships satisfy ex post efficiency and strategyproofness. The latter is known as the random dictatorship theorem. We relax Condorcet-consistency and ex post efficiency by introducing a lower bound on the probability of Condorcet winners and an upper bound on the probability of Pareto-dominated alternatives, respectively. We then show that the randomized Copeland rule is the only anonymous, neutral, and strategyproof SDS that guarantees the Condorcet winner a probability of at least 2/m. Secondly, we prove a continuous strengthening of Gibbard’s random dictatorship theorem: the less probability we put on Pareto-dominated alternatives, the closer to a random dictatorship is the resulting SDS. Finally, we show that the only anonymous, neutral, and strategyproof SDSs that maximize the probability of Condorcet winners while minimizing the probability of Pareto-dominated alternatives are mixtures of the uniform random dictatorship and the randomized Copeland rule.

Voting with interdependent values: The Condorcet winner

We generalize the standard, private values voting model with single-peaked preferences and incomplete information by introducing interdependent preferences. Our main results show how standard mechanisms that are outcome-equivalent and implement the Condorcet winner under complete information or under private values yield starkly different outcomes if values are interdependent. We also propose a new notion of Condorcet winner under incomplete information and interdependent preferences, and discuss its implementation. The new phenomena in this paper arise because different voting rules (including dynamic ones) induce different processes of information aggregation and learning.

Semi-random Impossibilities of Condorcet Criterion

The Condorcet criterion (CC) is a classical and well-accepted criterion for voting. Unfortunately, it is incompatible with many other desiderata including participation (PAR), half-way monotonicity (HM), Maskin monotonicity (MM), and strategy-proofness (SP). Such incompatibilities are often known as impossibility theorems, and are proved by worst-case analysis. Previous work has investigated the likelihood for these impossibilities to occur under certain models, which are often criticized of being unrealistic. We strengthen previous work by proving the first set of semi-random impossibilities for voting rules to satisfy CC and the more general, group versions of the four desiderata: for any sufficiently large number of voters n, any size of the group 1

Properties of Position Matrices and Their Elections

We study the properties of elections that have a given position matrix (in such elections each candidate is ranked on each position by a number of voters specified in the matrix). We show that counting elections that generate a given position matrix is #P-complete. Consequently, sampling such elections uniformly at random seems challenging and we propose a simpler algorithm, without hard guarantees. Next, we consider the problem of testing if a given matrix can be implemented by an election with a certain structure (such as single-peakedness or group-separability). Finally, we consider the problem of checking if a given position matrix can be implemented by an election with a Condorcet winner. We complement our theoretical findings with experiments.

Differentially Private Condorcet Voting

Designing private voting rules is an important and pressing problem for trustworthy democracy. In this paper, under the framework of differential privacy, we propose a novel famliy of randomized voting rules based on the well-known Condorcet method, and focus on three classes of voting rules in this family: Laplacian Condorcet method (CMLAP), exponential Condorcet method (CMEXP), and randomized response Condorcet method (CMRR), where λ represents the level of noise. We prove that all of our rules satisfy absolute monotonicity, lexi-participation, probabilistic Pareto efficiency, approximate probabilistic Condorcet criterion, and approximate SD-strategyproofness. In addition, CMRR satisfies (non-approximate) probabilistic Condorcet criterion, while CMLAP and CMEXP satisfy strong lexi-participation. Finally, we regard differential privacy as a voting axiom, and discuss its relations to other axioms.

An Examination of Ranked-Choice Voting in the United States, 2004–2022

ABSTRACT From the perspective of social choice theory, ranked-choice voting (RCV) is known to have many flaws. RCV can fail to elect a Condorcet winner and is susceptible to monotonicity paradoxes and the spoiler effect, for example. We use a database of 185 American ranked-choice elections for political office from the years 2004 to 2022 to investigate empirically how frequently RCV’s deficiencies manifest in practice. Our general finding is that RCV’s weaknesses are rarely observed in real-world elections, with the exception that ballot exhaustion frequently causes majoritarian failures.

Impact of a priori positive information on the results of voting methods

The aim of this paper is to present the results of experiments relating to voting methods based on the bounded rationality theory. The research demonstrated that a positive nudge changes the voting results. The study focused on three methods of voting: the Borda Count method, the Condorcet winner method and the anti-manipulation method. In a laboratory experiment, the subjects were asked to select the best musician. They were to manipulate their voting so that a predetermined winner is chosen. In the first voting, the subjects did not receive any a priori information, while in the second voting, some a priori information was provided, i.e. the true, objective ranking of the musicians. What followed was another voting. It was initially assumed that the participants would manipulate their voting the same way as in the first voting. The results, however, were different. The obtained second ranking of musicians was closest to the true, objective ranking, thus proving that the manipulation effect was neutralised by the a priori positive information about the true, objective order.

Beyond kemeny rank aggregation: A parameterizable-penalty framework for robust ranking aggregation with ties

Rank Aggregation has ubiquitous applications in operations research, artificial intelligence, computational social choice, and various other fields. Interest in this problem has increased due in part to the need to consolidate lists of rankings and scores output by different decision-making processes and algorithms. Although most attention has focused on the variant of this problem induced by the Kemeny-Snell distance, other robust rank aggregation problems have been proposed. This work delves into the rank aggregation problem under the generalized Kendall-tau distance —a parameterizable-penalty distance measure for comparing rankings with ties— which contains Kemeny aggregation as a special case. First, it derives exact and heuristic solution methods. Second, it introduces a social choice property (GXCC) that encloses existing variations of the Condorcet criterion as special cases, thereby expanding this seminal social choice concept beyond Kemeny aggregation for the first time. GXCC offers both computational and theoretical advantages. In particular, GXCC may help to divide the original problem into smaller subproblems, while still ensuring that solving them independently yields the optimal solution to the original problem. Experiments on two benchmark datasets conducted herein show that the featured exact and heuristic solution methods can benefit from GXCC. Finally, this work derives new theoretical insights into the effects of the generalized Kendall-tau distance penalty parameter on the optimal ranking and on the proposed social choice property.

A Learning Framework for Distribution-Based Game-Theoretic Solution Concepts

The past few years have seen several works exploring learning economic solutions from data, including optimal auction design, function optimization, stable payoffs in cooperative games, and more. In this work, we provide a unified learning-theoretic methodology for modeling such problems and establish tools for determining whether a given solution concept can be efficiently learned from data. Our learning-theoretic framework generalizes a notion of function space dimension—the graph dimension—adapting it to the solution concept learning domain. We identify sufficient conditions for efficient solution learnability and show that results in existing works can be immediately derived using our methodology. Finally, we apply our methods in other economic domains, yielding learning variants of competitive equilibria and Condorcet winners.

When ties are possible: Weak Condorcet winners and Arrovian rationality

We use Ehrhart polynomials to estimate the likelihood of each three-candidate social ranking produced by pairwise majority rule assuming an even number of voters and the Impartial Anonymous Culture condition. We then calculate the probability the ranking is transitive and the probability of a weak Condorcet winner. Finally, we determine the weak Condorcet efficiency of various voting rules. We prove that Baldwin, Nanson, Copeland, and ranked pairs are weak Condorcet efficient, as is Borda unless there are no ties. Simulations show that among the rest, Dowdall is typically the most efficient rule for small voter groups and anti-plurality the least.

A Dodgson-Hare synthesis

In 1876, Charles Dodgson (better known as Lewis Carroll) proposed a committee election procedure that chooses the Condorcet winner when one exists and otherwise eliminates candidates outside the Smith set, then allows for re-votes until a Condorcet winner emerges. The present paper discusses Dodgson’s work in the context of strategic election behavior and suggests a “Dodgson-Hare” method: a variation on Dodgson’s procedure, for use in public elections. This method allows for candidate withdrawal and employs Hare’s plurality-loser-elimination method to resolve persistent cycles. Given plausible assumptions about how candidates decide whether to withdraw when there is a cycle, Dodgson-Hare outperforms Hare, Condorcet-Hare, and 12 other voting rules in a series of spatial-model simulations that count how often each rule is vulnerable to coalitional manipulation. In the case of a one-dimensional spatial model, all coalitional voting strategies that are possible under Condorcet-Hare can be undone in Dodgson-Hare, by the withdrawal of candidates who have incentive to withdraw.

Approximate Condorcet Partitioning: Solving large-scale rank aggregation problems

Rank aggregation has ubiquitous applications in computer science, operations research, and various other fields. Most attention on this problem has focused on an NP-hard variant known as Kemeny aggregation, for which solution approaches with provable guarantees that can handle difficult high-dimensional instances remain elusive. This work introduces exact and approximate methodologies inspired by the social choice foundations of the problem, namely the Condorcet Criterion. We formalize the concept of the finest-Condorcet partition for rankings that may contain ties and specify its required conditions. We prove that this partition is unique and devise an efficient algorithm to obtain it. To deal with instances where it does not yield many subsets, we propose Approximate Condorcet Partitioning (ACP), with which larger subsets can be further broken down and more easily solved. ACP is a scalable solution technique capable of handling large instances while still providing provable guarantees. Although ACP approximation factors are instance-specific, their values were lower than those offered by all known constant-factor approximation schemes — inexact algorithms whose resulting objective values are guaranteed to be within a specified fixed percent of the optimal objective value — for all 113 instances tested herein (containing up to 2,820 items). What is more, ACP obtained solutions that deviated by at most two percent from the optimal objective function values for a large majority of these instances.

Intermediate Condorcet Winners and Losers

Intermediate Condorcet Winners and Losers

Transformation into anti-manipulation method in voting. Changes in properties

Abstract This paper examines the properties of the anti-manipulation method in voting. Such a method can be used by committees and similar bodies to ensure that votes reflect genuine preferences. The anti-manipulation method is based on the Borda Count and discourages strategic voting by excluding scores that deviate excessively from the mean. The method does not eliminate strategic voting but diminishes the motivation to apply it. We compare the properties of the Borda Count and the anti-manipulation method. The properties, which are most often found in the literature, were chosen for comparison. Thus, the following properties are examined: consistency, vulnerability to the no-show paradox, vulnerability to the subset choice condition, homogeneity, monotonicity, and vulnerability to the reversal bias paradox as well as the Condorcet winner and loser paradoxes. The anti-manipulation method fails to satisfy most of these properties. A real data example, the voting of a certain jury, is used as a counterexample in most cases.

Radial Symmetry Does Not Preclude Condorcet Cycles If Different Voters Weight the Issues Differently

Radial symmetry, by our definition, is a precise condition on continuous ideal-point distributions, rarely if ever found exactly in practice, that is similar to the classical 1967 symmetry condition of Plott but pertains to an infinite electorate; the bivariate normal distribution provides an example. A Condorcet cycle exists if the electorate prefers alternative X to Y, Y to Z, and Z to X. An alternative K is a Condorcet winner if there is no alternative that the electorate prefers to K. Lack of a Condorcet winner may engender turmoil. The nonexistence of a Condorcet winner implies that a Condorcet cycle exists. Radial symmetry precludes the existence of Condorcet cycles and thus guarantees a Condorcet winner; but this result assumes that all voters weight the dimensions alike. Our counterexamples show that a Condorcet cycle can arise, even under radial symmetry, if the weighting of issues varies across voters. This finding may be of more than theoretical value: It may suggest that in an empirical setting (without radial symmetry), a Condorcet cycle may be more frequent if voters differ as to how they weight the dimensions. We examine, for illustration based on two dimensions (left–right, linguistic), a Condorcet preference cycle in Finland’s 1931 presidential election.

Implications of Bounds in Approval Voting

We analyze and show the implications of required limitations on common voting system criteria for the three different versions of voting restrictions in the election website proposed for the Saudi Chamber of Commerce Business Awards. Electoral systems properties might be considered as performance indicators of a voting system. When presented with the same underlying voter preferences, different voting systems constraints might produce different results. Those voting systems may violate or satisfy voting properties. Distinct ways meet different requirements, and it was shown that no particular method can meet all of them.The studied voting systems mechanisms on the aforementioned website are called: Single Casting, Limited Casting, and Unlimited Casting methods.The first version permits the voter to vote for one unique candidate, and corresponds to the common Single Plurality Vote mechanism. The second version permits the voter to vote for three unique candidates at a time, and it is a new method that can be described as the Approval Voting with bounded ballot size. The third version permits the voter to vote for all candidates, and corresponds to the common Approval Voting mechanism.In the long version analysis, we proved that one of the new voting schemes in the election website proposed for the SaudiChamber of Commerce Business Awards, a version of Approval Voting where the size of the ballot is bounded, a set of the properties of Approval Voting are lost, in particular Pareto efficiency, Condorcet criteria winner support, Condorcet criteria loser, and Independence to Irrelevant Alternatives.