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On Anscombe’s paradox and vote trading
Louis-Aimé Fono?, Boniface Mbih?? and Aristide Valeu Address(es) of author(s) should be given
The date of receipt and acceptance will be inserted by the editor
Contents
1 Introduction . . . 1 Abstract The aim of this paper is to study the circumstances at which vote trading is suscep- tible to arise when individual votes are dichotomous. We provide an answer to some questions about the size of the society, the majority quotas, and some domain restriction of individual votes. Conclusions are also obtained for logrolling and Anscombe’s paradox
Key words vote trading, logrolling, Anscombe’s paradox, majority, successive votes.
JEL Classi…cationD71.
1 Introduction
One usual way to vote on motions in committees consists in yes-no voting. A motion is accepted if more than a (possibly quali…ed) majority of individuals vote yes; otherwise, it is rejected.
Further, it often happens that a committee has to vote on several independent motions, in the sense that the social decision on some motion does not have any in‡uence on the decision on some other motion: all motions can be accepted or all of them can be rejected. For example, without any budget constraint, a city council may have to decide whether to (a) build a bridge, (b) build a school and/or (c) build a retirement house. Majority votes are then organized suc- cessively on these di¤erent motions. Dummett (1984, Chap. 1) extensively studies this process
? Université de Douala, Cameroun
?? Université de Caen, France
2 Louis-Aimé Fono, Boniface Mbih and Aristide Valeu
and speci…cally, he points out that after all successive votes, the con…guration of the votes may lead to a somewhat paradoxical situation:
“Suppose that the composition of a committee is …xed for a certain period, during which it has to take a number of decisions. The following proposition, at …rst glance surprising, belongs to the dynamic theory of voting: a majority of the committee members may be in the minority on a majority of occasions.”
Situations exhibiting such a feature are known to describe Anscombe’s paradox (Anscombe 1976). It also appears that this paradox is closely related to the possibility for some group of voters to secure a better outcome for each of its members: with three motions (or more), it is easy to construct an example showing that “if they form a party, and agree always to vote for the outcome preferred by the majority within the party... each of them obtains the outcome he prefers in two out of three occasions...” (Dummett 1984, pp 15-16). More precisely, such a behavior amounts to vote trading, or to logrolling, that is a behavior by which voters - usually committee or legislative members - act in order to obtain passage of motions of interest for each of them (see precisions in Section 2).
Vote trading and logrolling have been widely studied in modern public choice literature (see for example Wilson 1969, Riker and Brams 1973, Stratmann 1997 or Miller 2001 among others), especially according to their relationships with voting paradoxes; Miller (1975, 1977) studies the relation with the Condorcet’s paradox with reference to parliamentary contexts, while Lagerspetz (1996) also examines the relation with other paradoxes, such as the Ostrogorski’s paradox and the Anscombe’s paradox. More recently, empirical works examine the evidence of vote trading agreements in Congressional voting (e.g. Stratman 1992, or Wiseman 2004) or in international institutions (Eldar 2008).
The aim of this paper is to provide an answer to some unsolved questions related to vote trading, logrolling and Anscombe’s paradox under dichotomous votes, that is yes-no successive (possibly quali…ed) majority voting. More precisely, we start by determining, depending on the number of motions, the minimum size of a committee for those phenomena to be possible. We then tackle questions about what quali…ed majority is necessary to avoid vote trading opportu- nities? How large can coalitions involved in vote trading behavior be? An …nally, can we …nd a way to restrict the domain of individual votes in order to avoid vote trading?
The paper is organized as follows: Section 2 introduces the notations and basic de…nitions;
Section 3 is concerned with preliminary results; then Section 4 studies the characterization of
On Anscombe’s paradox and vote trading 3
vote trading situations and other types of behavior are analyzed in Section 5. Finally Section 7 concludes the paper.
