Condorcet Winner Research Articles

Cycles in synchronous iterative voting: general robustness and examples in Approval Voting

We consider synchronous iterative voting, where voters are given the opportunity to strategically choose their ballots depending on the outcome deduced from the previous collective choices. We propose two settings for synchronous iterative voting, one of classical flavor with a discrete space of states, and a more general continuous-space setting extending the first one. We give a general robustness result for cycles not relying on a tie-breaking rule, showing that they persist under small enough perturbations of the behavior of voters. Then we give examples in Approval Voting of electorates applying simple, sincere and consistent heuristics (namely Laslier’s Leader Rule or a modification of it) leading to cycles with bad outcomes, either not electing an existing Condorcet winner, or possibly electing a candidate ranked last by a majority of voters. Using the robustness result, it follows that those “bad cycles” persist even if only a (large enough) fraction of the electorate updates its choice of ballot at each iteration. We complete these results with examples in other voting methods, including ranking methods satisfying the Condorcet criterion; an in silico experimental study of the rarity of preference profiles exhibiting bad cycles; and an example exhibiting chaotic behavior.

The α-Groups under Condorcet Clustering

We introduce a new approach to clustering categorical data: Condorcet clustering with a fixed number of groups, denoted α-Condorcet. As k-modes, this approach is essentially based on similarity and dissimilarity measures. The paper is divided into three parts: first, we propose a new Condorcet criterion, with a fixed number of groups (to select cases into clusters). In the second part, we propose a heuristic algorithm to carry out the task. In the third part, we compare α-Condorcet clustering with k-modes clustering. The comparison is made with a quality’s index, accuracy of a measurement, and a within-cluster sum-of-squares index. Our findings are illustrated using real datasets: the feline dataset and the US Census 1990 dataset.

Voting with Interdependent Values: The Condorcet Winner

Voting with Interdependent Values: The Condorcet Winner

The Art of Compromising: Voting with Interdependent Values and the Flag of the Weimar Republic

We model a situation where ex ante opinions in a legislature are dichotomous but cross traditional left-right party lines, e.g., crucial decisions on ethical issues such as gay marriage and abortion, or joining/exiting an economic/political union. In addition to the two extreme positions on the left and on the right, we consider the effect of a compromise alternative whose location may be endogenous. We compare sequential, binary voting schemes conducted by privately informed agents with interdependent preferences: the voting process gradually reveals and aggregates relevant information about the location of preferred alternatives. The Anglo-Saxon amendment procedure (AV) always selects the (complete-information) Condorcet winner. In contrast, the continental successive procedure (SV) does not. This holds because AV allows learning about the preferences of both leftists and rightists, while SV only allows one-directional learning at each step. Moreover, under SV, the agenda that puts the alternative with ex ante higher support last elects the Condorcet winner more often than the agenda that puts that alternative first. The optimal compromise location for various goals is also shown to differ across voting procedures. We illustrate our main findings with a fascinating historical episode, the vote on the flag of the Weimar republic.

Correction to: When are committees of Condorcet winners Condorcet winning committees?

Correction to: When are committees of Condorcet winners Condorcet winning committees?

Axiomatic foundations of the core for games in effectiveness form

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 weak nonemptiness, coalitional unanimity, and Maskin monotonicity 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.

A New Binary Programming Formulation and Social Choice Property for Kemeny Rank Aggregation

Rank aggregation is widely used in group decision making and many other applications, where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem—whose ranking inputs may be complete and incomplete, with and without ties. Moreover, it leverages the equivalence of two ranking aggregation problems, namely, that of minimizing the Kemeny-Snell distance and of maximizing the Kendall-τ correlation, to compare the newly introduced binary programming formulation to a modified version of an existing integer programming formulation associated with the Kendall-τ distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the nonstrict extended Condorcet criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Condorcet criterion. Unlike its parent properties, the new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and with benchmark instances from PrefLib, a library of preference data.

When are committees of Condorcet winners Condorcet winning committees?

When are committees of Condorcet winners Condorcet winning committees?

An analytical and experimental comparison of maximal lottery schemes

Maximal lottery ( ML ) schemes constitute an interesting class of randomized voting rules that were proposed by Peter Fishburn in 1984 and have been repeatedly recommended for practical use. However, the subtle differences between different ML schemes are often overlooked. Two canonical subsets of ML schemes are schemes (which only depend on unweighted majority comparisons) and schemes (which only depend on weighted majority comparisons). We prove that schemes are the only homogeneous ML schemes that satisfy SD -efficiency and SD -participation, but are also among the most manipulable ML schemes. While all ML schemes are manipulable and even violate monotonicity, they are never manipulable when a Condorcet winner exists and satisfy a relative notion of monotonicity. We also evaluate the frequency of manipulable preference profiles and the degree of randomization of ML schemes via extensive computer simulations. In summary, ML schemes are rarely manipulable and often do not randomize at all, especially for few alternatives. The average degree of randomization of schemes is consistently lower than that of schemes.

Preserving Condorcet Winners under Strategic Manipulation

Condorcet extensions have long held a prominent place in social choice theory. A Condorcet extension will return the Condorcet winner as the unique winner whenever such an alternative exists. However, the definition of a Condorcet extension does not take into account possible manipulation by the voters. A profile where all agents vote truthfully may have a Condorcet winner, but this alternative may not end up in the set of winners if agents are acting strategically. Focusing on the class of tournament solutions, we show that many natural social choice functions in this class, such as the well-known Copeland and Slater rules, cannot guarantee the preservation of Condorcet winners when agents behave strategically. Our main result in this respect is an impossibility theorem that establishes that no tournament solution satisfying a very weak decisiveness requirement can provide such a guarantee. On the bright side, we identify several indecisive but otherwise attractive tournament solutions that do guarantee the preservation of Condorcet winners under strategic manipulation for a large class of preference extensions.

Condorcet Relaxation In Spatial Voting

Consider a set of voters V, represented by a multiset in a metric space (X,d). The voters have to reach a decision - a point in X. A choice p∈ X is called a β-plurality point for V, if for any other choice q∈ X it holds that |{v∈ V ∣ β⋅ d(p,v)≤ d(q,v)}| ≥|V|/2 . In other words, at least half of the voters ``prefer'' over q, when an extra factor of β is taken in favor of p. For β=1, this is equivalent to Condorcet winner, which rarely exists. The concept of β-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [SoCG 2020] as a relaxation of the Condorcet criterion. Denote by β*(X,d) the value sup{ β ∣ every finite multiset V in X admits a β-plurality point}}. The parameter β* determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane β*(ℝ2,\|⋅\|2)=√3/2 , and more generally, for d-dimensional Euclidean space, 1/√d ≤ β*(ℝd,\|⋅\|2)≤√3/2 . In this paper, we show that 0.557≤ β*(ℝd,\|⋅\|2) for any dimension d (notice that 1/√d

Condorcet Loser in 2016: Apparently Trump; Condorcet Winner: Not Clinton?

Using thermometer score data from the ANES, we show that while there may have been no clear-cut Condorcet winner among the 2016 US presidential candidates, there appears to have been a Condorcet loser: Donald Trump. Thus the surprise is that the electorate preferred not only Hillary Clinton, but also the two “minor” candidates, Gary Johnson and Jill Stein, to Trump. Another surprise is that Johnson may have been the Condorcet winner. A minimal normative standard for evaluating voting systems is advanced, privileging those systems that select Condorcet winners if one exists, and critiquing systems that allow the selection of Condorcet losers. A variety of voting mechanisms are evaluated using the 2016 thermometer scores: Condorcet voting, plurality, Borda, (single winner) Hare, Coombs, range voting, and approval voting. We conclude that the essential problem with the existing voting procedure—Electoral College runoff of primary winners of two major parties—is that it (demonstrably) allows the selection of a Condorcet loser.

Daunou’s voting rule and the lexicographic assignment of priorities

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 candidates. 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. Journal of Economic Literature Classification Nos.: D71, D72.

The Incidence of Some Voting Paradoxes Under Domain Restrictions

Voting paradoxes have played an important role in the theory of voting. They typically say very little about the circumstances in which they are particularly likely or unlikely to occur. They are basically existence findings. In this article we study some well known voting paradoxes under the assumption that the underlying profiles are drawn from the Condorcet domain, i.e. a set of preference profiles where a Condorcet winner exists. The motivation for this restriction is the often stated assumption that profiles with a Condorcet winner are more likely than those without it. We further restrict the profiles by assuming that the starting point of our analysis is that the Condorcet winner coincides with the choice of the voting rule under scrutiny. The reason for making this additional restriction is that—intuitively—the outcomes that coincide with the Condorcet winner make those outcomes stable and, thus, presumably less vulnerable to various voting paradoxes. It will be seen that this is, indeed, the case for some voting rules and some voting paradoxes, but not for all of them.

Information sharing in democratic mechanisms

We examine how democratic mechanisms can yield socially desirable outcomes in the presence of uncertainty about an underlying state of nature. We depart from a conventional mechanism design approach because we aim for democratic mechanisms to reflect some basic properties of decision-making in democracies. In particular, actual decisions are made by majority voting. The proposals to be voted upon are made by a selfish agenda-setter. Moreover, communication is limited to a binary message space (that is, voting Yes or No). We show how suitable democratic mechanisms can resolve uncertainty, reveal the state of nature, and implement the Condorcet winner. We demonstrate that this implementation result requires (at most) two voting stages regardless of the number of states or the number of alternatives. We also show that implementation requires a conditional privilege for a small representative subset of the population.

Majority decision and Condorcet winners

Majority decision and Condorcet winners

Gehrlein stability in committee selection: parameterized hardness and algorithms

In a multiwinner election based on the Condorcet criterion, we are given a set of candidates, and a set of voters with strict preference ranking over the candidates. A committee is \em weakly Gehrlein stable (WGS) if each committee member is preferred to each non-member by at least half of the voters. Recently, Aziz et al. [IJCAI 2017] studied the computational complexity of finding a WGS committee of size k. They show that this problem is \sf NP -hard in general and polynomial time solvable when the number of voters is odd. In this article, we initiate a systematic study of the problem in the realm of parameterized complexity. We first show that the problem is \sf W [1]-hard when parameterized by the size of the committee. To overcome this intractability result, we use a known reformulation of WGS as a problem on directed graphs and then use parameters that measure the "structure'' of these directed graphs. In particular, we consider the majority graph, defined as follows: there is a vertex corresponding to each candidate, and there is a directed arc from a candidate c to $c'$ if the number of voters that \em prefer c over $c'$ is more than those that prefer $c'$ over c. The problem of finding WGS committee of size k corresponds to finding a vertex subset X of size k in the majority graph with the following property: the set X contains no vertex outside the committee that has an in-neighbor in X. Observe that the polynomial time algorithm of Aziz et al. [IJCAI 2017] corresponds to solving the problem on a tournament (a complete graph with orientation on edges). Thus, natural parameters to study our problem are "closeness'' to being a tournament. We define closeness as the number of missing arcs in the given directed graph and the number of vertices we need to delete from the given directed graph such that the resulting graph is a tournament. We show that the problem is fixed parameter tractable (\sf FPT ) and admits linear kernels with respect to closeness parameters. Finally, we also design an exact exponential time algorithm running in time $OO(1.2207^nn^OO(1) )$. Here, n denotes the number of candidates.

The Evolution of Collective Choice Under Majority Rule

We consider a dynamic process of collective choice under majority rule in which a status quo policy evolves. The analysis is based on stochastic evolutionary game theory. The Condorcet winner is uniquely a long-run equilibrium for all (super-)majority rules. When the Condorcet winner does not exist, the long-run equilibria under all majority voting rules belong to the top cycle of policies under a simple majority. When the policy space is multidimensional and the voting quota is larger than the min-max quota, the long-run equilibrium belongs to the min-max set. Finally, the Borda winner appears as a long-run equilibrium under unanimity if voters' behavior is governed by a logit choice rule.

Strategic Voting in Two-Party Legislative Elections

It is commonly thought that in an election with two parties there can be no strategic voting - voters simply vote for their preferred candidate. In this paper, I show that strategic voting comes to the fore in legislative elections with multiple policy dimensions. In sharp contrast to single-district elections, the intensity of a voter’s preference on each dimension is irrelevant for her voting decision. Instead, she votes solely based on the dimension which is most likely to be pivotal in the legislature. Anticipating this behaviour, candidates put forward a different set of policies than they would in a single-district election. For large elections I show that the implemented policy bundle: (a) is uniquely pinned down by voter preferences, (b) is preferred by a majority of districts on each dimension, (c) is a Condorcet winner, if one exists. These properties are not guaranteed in a single-district election. Furthermore, I show that (i) parliamentary systems generate superior policies to presidential systems and (ii) voter polarisation affects outcomes in single-district elections but not legislative elections.

Borda paradox in the 2017 Iranian presidential election: empirical evidence from opinion polls

Different voting paradoxes identified by social choice theorists have rarely been documented in real-world elections. The collected data from the opinion polls in the 2017 Iranian presidential election provides substantial empirical evidence indicating that during the electoral campaigns, strong Borda paradox (the Condorcet loser wins plurality.) and weak Borda paradox (An alternative that is not the Condorcet winner wins plurality.) have occurred. At the same time, there was no evidence of a Condorcet paradox, i.e., a cycle among the top three candidates. The results support the empirical relevance of theoretical paradoxes in social choice and a call for the importance of voting procedures.