Condorcet Consistent Rules Research Articles

Compromising as an equal loss principle

A social choice rule aggregates the preferences of a group of individuals over a set of alternatives into a collective choice. The literature admits several social choice rules whose recommendations are supposed to reflect a compromise among individuals. We observe that all these compromise rules can be better described as procedural compromises, i.e., they impose over individuals a willingness to compromise but they do not ensure an outcome where everyone has effectively compromised. We revisit the concept of a compromise in a collective choice environment with at least three individuals having strict preferences over a finite set of alternatives. Referring to a large class of spread measures, we view the concept of compromise from an equal loss perspective, favoring an outcome where every voter concedes as equally as possible. As such, being a compromise may fail Pareto efficiency, which we ensure by asking voters to concede as equally as possible among the Pareto efficient alternatives. We show that Condorcet consistent rules, scoring rules (except antiplurality) and Brams-Kilgour compromises (except fallback bargaining) all fail to ascertain an outcome which is a compromise. A slight restriction on acceptable spread measures suffices to extend the negative result to antiplurality and fallback bargaining. This failure also prevails for social choice problems with two individuals: all well-known two-person social choice rules of the literature, namely, fallback bargaining, Pareto and veto rules, short listing and veto rank, fail to pick ex-post compromises. We conclude that there is a need to propose and study rules that satisfy this equal loss, or outcome oriented, notion of a compromise.

Pairwise Justifiable Changes in Collective Choices

Consider the following principle regarding the performance of collective choice rules. If a rule selects alternative x in situation 1, and alternative y in situation 2, there must be an alternative z, and some member of society whose appreciation of z relative to x has increased when going from situation 1 to situation 2. This principle requires a minimal justification for the fall of x in the consideration of society: someone must have decreased its appreciation relative to some other possible alternative. We study the consequences of imposing this requirement of pairwise justifiability on a large class of collective choice rules that includes social choice and social welfare functions as particular cases. When preference profiles are unrestricted, it implies dictatorship, and both Arrow's and the Gibbard-Satterthwaite theorems become corollaries of our general result. On appropriately restricted domains, pairwise justifiability, along with anonymity and neutrality, characterize Condorcet consistent rules, thus providing a foundation for the choice of the alternatives that win by majority over all others in pairwise comparisons, when they exist.

Frequency based analysis of collective aggregation rules

A new property for collective aggregation rules called positive discrimination is introduced. This property is satisfied by many anonymous and neutral collective aggregation rules. We discuss unimodal profiles (or distributions) for which this property on its own determines the outcome as the mode of a unimodal distribution. We investigate relaxations of this unimodal distribution condition and show that the Condorcet consistent rules as well as the Borda rule and the plurality rule have the mode as the outcome at such profiles. Further, we show, by an example, how the addition of unimodal profiles and the property of positive discrimination can help to determine an outcome in the case of a large number of voters.

A new approach for identifying the Kemeny median ranking

Condorcet consistent rules were originally developed for preference aggregation in the theory of social choice. Nowadays these rules are applied in a variety of fields such as discrete multi-criteria analysis, defence and security decision support, composite indicators, machine learning, artificial intelligence, queries in databases or internet multiple search engines and theoretical computer science. The cycle issue, known also as Condorcet's paradox, is the most serious problem inherent in this type of rules. Solutions for dealing with the cycle issue properly already exist in the literature; the most important one being the identification of the median ranking, often called the Kemeny ranking. Unfortunately its identification is a NP-hard problem. This article has three main objectives: (1) to clarify that the Kemeny median order has to be framed in the context of Condorcet consistent rules; this is important since in the current practice sometimes even the Borda count is used as a proxy for the Kemeny ranking. (2) To present a new exact algorithm, this identifies the Kemeny median ranking by providing a searching time guarantee. (3) To present a new heuristic algorithm identifying the Kemeny median ranking with an optimal trade-off between convergence and approximation.

Stable Rules for Electing Committees and Divergence on Outcomes

For three-candidate elections, this paper focuses on the relationships that exist between three stable rules for committee elections and the classical rules from which each of these stable rules are adapted. When selecting committees, a voting rule is said to be stable if it always elects a fixed-size subset of candidates such that there is no candidate in this set that is majority dominated by a candidate outside (Barbera and Coelho in Soc Choice Welfare 31:79–96, 2008; Coelho in Understanding, evaluating and selecting voting rules through games and axioms, 2004). There are some cases where a committee selected by a stable rule may differ from the committee made by the best candidates of the corresponding classical rule from which this stable rule is adapted. We call this the divergence on outcomes. We characterize all the voting situations under which this event is likely to occur. We also evaluate the likelihood of this event using the impartial anonymous culture assumption. As a consequence of our analysis, we highlight a strong connection between three Condorcet consistent rules: the Dodgson rule, the Maximin rule and the Young rule.

Intensity of preference and related uncertainty in non-compensatory aggregation rules

Non-compensatory aggregation rules are applied in a variety of problems such as voting theory, multi-criteria analysis, composite indicators, web ranking algorithms and so on. A major open problem is the fact that non-compensability implies the analytical cost of loosing all available information about intensity of preference, i.e. if some variables are measured on interval or ratio scales, they have to be treated as measured on an ordinal scale. Here this problem has been tackled in its most general formulation, that is when mixed measurement scales (interval, ratio and ordinal) are used and both stochastic and fuzzy uncertainties are present. Objectives of this article are first to present a comprehensive review of useful solutions already proposed in the literature and second to advance the state of the art mainly in the theoretical guarantee that weights have the meaning of importance coefficients and they can be summarized in a voting matrix. This is a key result for using non-compensatory Condorcet consistent rules. A proof on the probability of existence of ties in the voting matrix is also developed.

Rationalizations of Condorcet-consistent rules via distances of hamming type

In voting, the main idea of the distance rationalizability framework is to view the voters’ preferences as an imperfect approximation to some kind of consensus. This approach, which is deeply rooted in the social choice literature, allows one to define (“rationalize”) voting rules via a consensus class of elections and a distance: a candidate is said to be an election winner if she is ranked first in one of the nearest (with respect to the given distance) consensus elections. It is known that many classic voting rules can be distance-rationalized. In this article, we provide new results on distance rationalizability of several Condorcet-consistent voting rules. In particular, we distance-rationalize the Young rule and Maximin using distances similar to the Hamming distance. It has been claimed that the Young rule can be rationalized by the Condorcet consensus class and the Hamming distance; we show that this claim is incorrect and, in fact, this consensus class and distance yield a new rule, which has not been studied before. We prove that, similarly to the Young rule, this new rule has a computationally hard winner determination problem.

Compilation Complexity of Common Voting Rules

In computational social choice, one important problem is to take the votes of a subelectorate (subset of the voters), and summarize them using a small number of bits. This needs to be done in such a way that, if all that we know is the summary, as well as the votes of voters outside the subelectorate, we can conclude which of the m alternatives wins. This corresponds to the notion of compilation complexity, the minimum number of bits required to summarize the votes for a particular rule, which was introduced by Chevaleyre et al. [IJCAI-09]. We study three different types of compilation complexity. The first, studied by Chevaleyre et al., depends on the size of the subelectorate but not on the size of the complement (the voters outside the subelectorate). The second depends on the size of the complement but not on the size of the subelectorate. The third depends on both. We first investigate the relations among the three types of compilation complexity. Then, we give upper and lower bounds on all three types of compilation complexity for the most prominent voting rules. We show that for l-approval (when l ≤ m/2), Borda, and Bucklin, the bounds for all three types are asymptotically tight, up to a multiplicative constant; for l-approval (when l > m/2), plurality with runoff, all Condorcet consistent rules that are based on unweighted majority graphs (including Copeland and voting trees), and all Condorcet consistent rules that are based on the order of pairwise elections (including ranked pairs and maximin), the bounds for all three types are asymptotically tight up to a multiplicative constant when the sizes of the subelectorate and its complement are both larger than m1+ε for some ε > 0.

Subgame perfect implementation of voting rules via randomized mechanisms

We characterize completely ordinal and onto choice rules that are subgame perfect of Nash equilibrium (SPE) implementable via randomized mechanisms under strict preferences. The characterization is very operationalizable, and allows us to analyse SPE implementability of voting rules. We show that no scoring rule is SPE implementable. However, the top-cycle and the uncovered correspondences as well as plurality with runoff and any strongly Condorcet consistent voting rule can be SPE implemented. Therefore our results are favourable to majority based voting rules over scoring rules. Nevertheless, we show that many interesting Condorcet consistent but not strongly Condorcet consistent rules, such as the Copeland rule, the Kramer rule and the Simpson rule, cannot be SPE implemented.

Efficiency in the Degree of Compromise: A New Axiom for Social Choice

We introduce a social choice axiom called efficiency in the degree of compromise. Our axiom is based on the trade-off between the quantity and quality of support that an alternative receives. What we mean by the quantity of support is the number of voters behind an alternative, while the quality of support is about the definition of “being behind” depending on the rank of an alternative in voters' preference orderings. Naturally, one can increase the quantity of support of an alternative to the expense of giving up from its quality. We say that an alternative is an efficient compromise if there exists no other alternative with at least an equal quantity of support with a higher quality. Our efficient compromise axiom is based on not choosing inefficient compromises. We introduce it and show that many standard social choice rules of the literature, such as Condorcet-consistent rules, plurality with a runoff, the Borda count and the single transferable vote, may choose inefficient compromises.

The asymptotic strategyproofness of scoring and Condorcet consistent rules

We calculate the proportion of preference profiles where “small” coalitions of agents may successfully manipulate any given scoring rule and show that it decreases to zero at a rate proportional to \(\frac{1}{\sqrt{n}}\) with the number of agents. If agents have to incur a small cost in order to decide how to manipulate the voting rule, our results imply that scoring rules are robust to such manipulation in large groups of agents. We present examples of asymptotically strategyproof and non strategyproof Condorcet consistent rules.