Ann Oper Res

DOI 10.1007/s10479-014-1763-7

Strategic manipulability of self-selective social choice rules

Mostapha Diss © Springer Science+Business Media New York 2014

Abstract We provide exact relations giving the probability of individual and coalitional manipulation of three specific social choice functions (Borda rule, Copeland rule, Plurality rule) in three-alternative elections when the notion of self-selectivity is imposed. We use each type of tie-breaking rule in the case of three-candidate election in order to make the results more robust. Analyzing our probabilities, we can point out that the probability of individual and coalitional manipulation tend to vanish significantly when the notion of self-selectivity is imposed.

Keywords Voting rules · Self-selectivity · Stability · Manipulability · Probability

JEL Classification D72 1 Introduction

Voting rules aggregating individual preferences into a collective choice differ in their vulnerability to manipulation. Manipulating an election is for voters (individual voter or a coalition of voters) a way to announce non sincere individual preferences in order to achieve better voting result for themselves. We know from the well-known theorem of Gibbard–Satterthwaite (Gibbard (1973) and Satterthwaite (1975)) that all voting rules, when choosing a single candidate for at least three candidates and without a dictator, are vulnerable to this type of strategic

M. Diss (B)

Université de Lyon, 69007 Lyon, France e-mail: diss@gate.cnrs.fr

M. Diss

CNRS, GATE Lyon Saint-Etienne, 69130 Ecully, France

M. Diss

Université Jean Monnet, 42000 Saint-Etienne, France 123

Ann Oper Res behavior. These studies have given rise to a number of extensions and generalizations of

Gibbard–Satterthwaite theorem.

Many papers dealing with the probability of different voting rules to be manipulable under various definitions of manipulability was presented in the literature. This framework stipulates to introduce a certain measure of manipulability of a voting rule and a certain assumptions on the distribution of voter preferences. These manipulability measures were first introduced by Nitzan (1985) for the case of individual manipulation and Lepelley and Mbih (1987) for the case of coalitional manipulation. Other interesting manipulability measures were also presented by Aleskerov and Kurbanov (1999), Kelly (1988, 1993) and Smith (1999).

The question of the probability of manipulability was widely investigated by

Favardin et al. (2002), Lepelley and Mbih (1994, 1996), Favardin and Lepelley (2006),

Lepelley and Valognes (2003) and Gehrlein and Lepelley (2003). The mean idea of these papers is to evaluate the proportion of preference profiles or voting situations that are not equilibria, that is the voting situations at which the voting rule taken into consideration is manipulable by an individual or a coalition of individuals. So, the vulnerability to individual or coalitional manipulability is used in order to compare voting rules according to their sensitivity. However, no paper has tried to evaluate the probability of strategic manipulability of a self-selective voting rule.

Self-selectivity is a new principle introduced in the social choice literature. It is a desirable property of voting rules considered when individuals have to choose a voting rule in a set of voting rules. A voting rule is self-selective at some profile if, given this profile, there is no alternative rule that beats the given voting rule if the given voting rule is used to choose between the rules in the set. If a rule, which is not self-selective is used, then there will be a large enough group of voters who all prefer another voting rule to be successful in changing the rule. The notion of self-selectivity gives rise to another original concept when the given set of voting rules is considered. A set of voting rules is (weakly) stable if it always contains at least one self-selective rule at any profile or voting situation. In this case, if none of the voting rules in the set is self selective, the society could never be able to vote how to vote.

These questions were first considered by Koray (2000), Koray and Unel (2003), Barberà and

Jackson (2004), Barberà and Beviá (2002) and Houy (2003, 2004, 2005). Considering some probabilistic models, these questions were also investigated by Diss and Merlin (2010) and

Diss et al. (2012).

The concepts of individual and coalitional manipulation can be extended quite naturally to the notion of self-selective voting rules. The vote will be on voting rules and, as in the basic framework of manipulation, it can be profitable for some voter at some voting situation to misrepresent his preferences on voting rules in order to have an outcome (voting rule) preferred to that resulting in the voting situation in which his vote reflects his true preferences.

As a consequence, manipulating a self-selective voting rule means that this rule becomes notselective after manipulation. This situation can lead to a serious paradox of instability if all the other considered voting rules are not self-selective after manipulation. This is equivalent to a situation in which it is impossible to directly or indirectly reach a self-selective voting rule if implementing any voting rule of the considered set. This problem can be solved by considering self-selectivity before and after manipulation. That is, when we consider the manipulation of a self-selective voting rule, the considered set of voting rules must contain at least one self-selective voting rule after manipulation. In other words, the set of voting rules must be stable after manipulation.

We reconsider in this paper the question of the probability of both individual and coalitional manipulability for three most commonly used voting rules in the literature: Borda rule, Copeland rule and Plurality rule. The main idea will be first to characterize the vot123

Ann Oper Res ing situations at which each voting rule is self-selective and vulnerable to manipulation by an individual or a coalition of individuals given that the set of the considered voting rules remains stable after manipulation. Secondly, the probabilities of the vulnerability are found using the Impartial and Anonymous Culture (IAC) assumption and the three voting rules