Mathematical Social Sciences 40 2000 215–226 www.elsevier.nl locate econbase
Voting power in an ideological spectrum ´
The Markov-Polya index
Thommy Perlinger
Lund University , Department of Statistics, POB 7008, 220 07 Lund, Sweden
Received December 1997; received in revised form December 1998; accepted August 1999
Abstract
Power distributions of voting games in which only certain coalitions are allowed, due to the fact that the players are spread across an ideological spectrum, are considered. The paper extends
Edelman’s model [Edelman, P., 1997. A note on voting. Math. Soc. Sci. 34, 37–50] concerning ´
allowable coalitions. The Markov-Polya index is introduced as a parametrized family of power indices which has Edelman’s extension of the Shapley-Shubik index as a special case. Unlike the
extended Shapley-Shubik index, it puts different weights on the allowable coalitions. For particular parameter values, simple explicit power formulas are derived.
2000 Elsevier
Science B.V. All rights reserved.
´ Keywords
: Voting power index; Connected coalitions; Initiator; Markov-Polya distribution
1. Introduction
Game theoretic solutions to voting situations, in the form of common power indices, do not take into account that the voters are often spread across an ideological spectrum.
Voting power indices in common use are, for example, the Banzhaf-Coleman index, Banzhaf, 1965 which treats all coalitions as equally probable and the Shapley- Shubik
index, Shapley and Shubik, 1954 where all permutations i.e. vote sequences are treated as equally probable. In these two power indices, voters are assumed anonymous
and treated symmetrically.
When in a voting situation, a coalition comes into being, this procedure can be regarded as the formation of a series of subcoalitions, where one voter at a time joins the
coalition. When ideological positions matter, the problem arises of how to model this
Tel.: 146-46-222-3653; fax.: 146-46-222-4220. E-mail address
: thommy.perlingerstat.lu.se T. Perlinger 0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved.
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. Perlinger Mathematical Social Sciences 40 2000 215 –226
process in a reasonably realistic way. Should, for instance, a coalition that consists of two voters from opposite extremes be as likely as a coalition that consists of two voters
who are ideological neighbors? Is it at all reasonable to believe that the former coalition will arise? These questions suggest that in ideological voting situations, all coalitions
should not be treated as equally probable to occur. The problem then is: which are the least likely coalitions and how do we restrict them?
An interesting approach to this kind of problem is suggested by Edelman 1997. His model of coalition formation is based on the concept of convex sets, and only a certain
type of coalitions are allowed to occur. A specific application of the model is when the players are spread across an ideological spectrum, such as the usual left–right ideology
scale. The idea is that a coalition is permissible if and only if there are no ideological gaps between members of the coalition; in other words, there can be no non-member
who is ideologically in an intermediate position between any two coalition members. Edelman shows that the Shapley-Shubik index very naturally extends to accommodate
these restricted games, which henceforth will be referred to as spectrum games.
In voting situations modeled by spectrum games, one would expect the player in the median position to be the strongest in terms of voting power, since he is connected to
both flanks and thus have more possibilities to form and join coalitions. This is not the case, however, when voting power in spectrum games is measured by the extended
Shapley-Shubik index. Somewhat surprisingly, it is instead the players in quartile positions who are the strongest. Only the outer players in the most extreme positions on
the ideology scale are weaker than the median player.
The purpose of the present paper is to study voting power in spectrum games by combining Edelman’s ideas with those of Berg 1997. To do this, we first introduce the
concept of an initiator as the first member of a coalition. Edelman’s model determines implicitly an initiator distribution, which favors or rather dis-favors the player in the
median position. Taking the cue from Berg, we then introduce a model for coalition formation, in which each grand coalition is seen as the realization of a sequence of
´ Markov-Polya binary random variables. In this manner, we derive a parametrized family
of Shapley-Shubik type indices, and in the process we also obtain a simple and flexible family of initiator distributions. Edelman’s model of voting is a special case, and other
special cases have simple and appealing probabilistic interpretations.
The rest of this paper is organized as follows. To establish notation, Section 2 gives a brief overview of n-person weighted voting games and voting power indices. The
