Distance rationalization of voting rules (original) (raw)

Distance Rationalization of Voting Rules 1

2010

The concept of distance rationalizability allows one to define new voting rules or “rationalize” existing ones via a consensus class of elections and a distance. A consensus class consists of elections in which there is a consensus in the society who should win. A distance measures the deviation of the actual election from consensus elections. Together, a consensus class and a distance define a voting rule: a candidate is declared an election winner if she is the consensus candidate in one of the nearest consensus elections. It is known that many classic voting rules are defined in this way or can be represented via a consensus class and a distance, i.e., distance-rationalized. In this paper, we focus on the power and the limits of the distance rationalizability approach. We first show that if we do not place any restrictions on the class of possible distances then essentially all voting rules are distance-rationalizable. Thus, to make the concept of distance ratioanalizability mean...

On Distance Rationalizability of Some Voting Rules

SSRN Electronic Journal, 2000

The concept of distance rationalizability has several applications within social choice. In the context of voting, it allows one to define ("rationalize") voting rules via a consensus class (roughly, a set of elections in which it is obvious who should win) and a distance function: namely, a candidate is said to be an election winner if it 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 represented in this manner. In this paper, we provide new results on distance rationalizability of several well-known voting rules such as all scoring rules, Approval, Young's rule and Maximin. We also show that a previously published proof of distance rationalizability of Young's rule is incorrect: the consensus notion and the distance function used in that proof give rise to a voting rule that is similar to-but distinct from-the Young's rule. Finally, we demonstrate that some voting rules cannot be rationalized via certain notions of consensus. To the best of our knowledge, these are the first non-distance-rationalizability results for voting rules.

On the role of distances in defining voting rules

2010

A voting rule is an algorithm for determining the winner in an election, and there are several approaches that have been used to justify the proposed rules. One justification is to show that a rule satisfies a set of desirable axioms that uniquely identify it. Another is to show that the calculation that it performs is actually maximum likelihood estimation relative to a certain model of noise that affects voters (MLE approach). The third approach, which has been recently actively investigated, is the so-called distance rationalizability framework. In it, a voting rule is defined via a class of consensus elections (i.e., a class of elections that have a clear winner) and a distance function. A candidate c is a winner of an election E if c wins in one of the consensus elections that are closest to E relative to the given distance. In this paper, we show that essentially any voting rule is distance-rationalizable if we do not restrict the two ingredients of the rule: the consensus class and the distance. Thus distance rationalizability of a rule does not by itself guarantee that the voting rule has any desirable properties. However, we demonstrate that the distance used to rationalize a given rule may provide useful information about this rule's behavior. Specifically, we identify a large class of distances, which we call votewise distances, and show that if a rule is rationalized via a distance from this class, many important properties of this rule can be easily expressed in terms of the underlying distance. This enables us to provide a new characterization of scoring rules and to establish a connection with the MLE framework. We also give bounds on the complexity of the winner determination problem for distance-rationalizable rules.

Homogeneity and monotonicity of distance-rationalizable voting rules

2011

Distance rationalizability is a framework for classifying voting rules by interpreting them in terms of distances and consensus classes. It also allows to design new voting rules with desired properties. A particularly natural and versatile class of distances that can be used for this purpose is that of votewise distances [12], which "lift" distances over individual votes to distances over entire elections using a suitable norm. In this paper, we continue the investigation of the properties of votewise distance-rationalizable rules initiated in . We describe a number of general conditions on distances and consensus classes that ensure that the resulting voting rule is homogeneous or monotone. This complements the results of , where the authors focus on anonymity, neutrality and consistency. We also introduce a new class of voting rules, that can be viewed as "majority variants" of classic scoring rules, and have a natural interpretation in the context of distance rationalizability.

Good rationalizations of voting rules

2010

We explore the relationship between two approaches to rationalizing voting rules: the maximum likelihood estimation (MLE) framework originally suggested by Condorcet and recently studied in and the distance rationalizability (DR) framework (Meskanen and Nurmi 2008; Elkind, Faliszewski, and Slinko 2009). The former views voting as an attempt to reconstruct the correct ordering of the candidates given noisy estimates (i.e., votes), while the latter explains voting as search for the nearest consensus outcome. We provide conditions under which an MLE interpretation of a voting rule coincides with its DR interpretation, and classify a number of classic voting rules, such as Kemeny, Plurality, Borda and Single Transferable Vote (STV), according to how well they fit each of these frameworks. The classification we obtain is more precise than the ones that result from using MLE or DR alone: indeed, we show that the MLE approach can be used to guide our search for a more refined notion of distance rationalizability and vice versa.

Rationalizations of Condorcet-consistent rules via distances of hamming type

Social Choice and Welfare, 2012

The main idea of the distance rationalizability approach to view the voters' preferences as an imperfect approximation to some kind of consensus is deeply rooted in social choice literature. It 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 paper, we provide new results on distance rationalizability of several Condorcet-consistent voting rules. In particular, we distance rationalize Young's rule and Maximin rule using distances similar to the Hamming distance. We show that the claim that Young's rule can be rationalized by the Condorcet consensus class and the Hamming distance is incorrect; in fact, these consensus class and distance yield a new rule which has not been studied before. We prove that, similarly to Young's rule, this new rule has a computationally hard winner determination problem.

On the Properties of Voting Systems

Scandinavian Political Studies, 1981

The article focuses on the problem of choosing the ‘best’ voting procedure for making collective decisions. The procedures discussed are simple majority rule, Borda count, approval voting, and maximin method. The first three have been axiomatized while the maximin method has not yet been given an axiomatic characterization. The properties, in terms of which the goodness of the procedures is assessed, are dictatorship, consistency, path independence, weak axiom of revealed preference, Pareto optimality, and manipulability. It turns out that the picture emerging from the comparison of the procedures in terms of these properties is most favorable to the approval voting.

On swap-distance geometry of voting rules

2013

Axioms that govern our choice of voting rules are usually defined by imposing constraints on the rule's behavior under various transformations of the preference profile. In this paper we adopt a different approach, and view a voting rule as a (multi-)coloring of the election graph - the graph whose vertices are elections over a given set of candidates, and two vertices are adjacent if they can be obtained from each other by swapping adjacent candidates in one of the votes. Given this perspective, a voting rule F is characterized by the shapes of its "monochromatic components", i.e., sets of elections that have the same winner under F. In particular, it would be natural to expect each monochromatic component to be convex, or, at the very least, connected. We formalize the notions of connectivity and (weak) convexity for monochromatic components, and say that a voting rule is connected/(weakly) convex if each of its monochromatic components is connected/(weakly) convex. ...

A Dynamic Rationalization of Distance Rationalizability

2012

Abstract Distance rationalizability is an intuitive paradigm for developing and studying voting rules: given a notion of consensus and a distance function on preference profiles, a rationalizable voting rule selects an alternative that is closest to being a consensus winner. Despite its appeal, distance rationalizability faces the challenge of connecting the chosen distance measure and consensus notion to an operational measure of social desirability.

A REPRESENTATION THEOREM FOR VOTING WITH LOGICAL CONSEQUENCES

Economics and Philosophy, 2006

This paper concerns voting with logical consequences, which means that anybody voting for an alternative x should vote for the logical consequences of x as well. Similarly, the social choice set is also supposed to be closed under logical consequences. The central result of the paper is that, given a set of fairly natural conditions, the only social choice functions that satisfy social logical closure are oligarchic (where a subset of the voters are decisive for the social choice). The set of conditions needed for the proof include a version of Independence of Irrelevant Alternatives that also plays a central role in Arrow s impossibility theorem.