Elsevier

Mathematical Social Sciences

Volume 90, November 2017, Pages 145-149
Mathematical Social Sciences

Toward a 50%-majority equilibrium when voters are symmetrically distributed

https://doi.org/10.1016/j.mathsocsci.2016.08.006Get rights and content

Abstract

Consider a two-dimensional spatial voting model. A finite number m of voters are randomly drawn from a (weakly) symmetric distribution centered at O. We compute the exact probabilities of all possible Simpson–Kramer scores of O. The computations are independent of the shape of the distribution. The resulting expected score of O is an upper bound of the expected min–max score.

Introduction

Consider a society comprising a potentially small number m of individuals, what is the smallest rate of (super) majority for which there exists an equilibrium? This question is important for anyone interested in the governance of such institutions as corporations or partnership, political parties and associations, if only to optimize the design of the institution’s charter or constitution.

Let us be more precise. We consider a society I of m(=I) individuals having preferences over a set of alternatives X. An alternative xX is an equilibrium for the rate ρ[0,1] if there is no alternative xX that rallies the vote of more than ρm individuals against x. Let ρ denote the smallest rate for which there exists an equilibrium, called the min–max rate of society I (Simpson, 1969, Kramer, 1973).

The social choice literature offers several upper bounds on ρ. Black’s (1953) median voter theorem states that if alternatives can be ordered along a one-dimensional space, and preferences are single-peaked over this space, then ρ0.5: there is an equilibrium for the (simple) majority rule. However, we know since the seminal work of Plott (1967) that this does not extend to higher dimensions: McKelvey and Wendell (1976), McKelvey (1979), Rubinstein (1979), Schofield (1983), McKelvey and Schofield (1987), Banks (1995), Saari (1997), Banks et al. (2006) have shown that the set of configurations for which ρ0.5 has measure 0 in three or more dimensions, and in two dimensions when the number of voters is odd. Tovey (2010a) completes the study of the two-dimensional case by proving that, when the number of voters is even and voters are sampled i.i.d. from any nonsingular distribution, the measure of the latter set converges to 0 exponentially rapidly.

Among other important contributions, Greenberg (1979) shows that if X is a convex and compact set of dimension d, and individuals have convex and continuous preferences over X, then ρ11d+1. Caplin and Nalebuff (1988) show that when voters have Euclidean preferences, and the distribution of their ideal points is concave over a convex support, then ρ0.64. One of the beautiful attributes of the latter result (compared with Greenberg’s) is that it is independent of the dimension of the space of alternatives, another one is that the super majority rate is not too high. One of the strengths of Greenberg’s result is that it is independent of the distribution of ideal points.

Our study is geared toward societies with potentially small number of individuals. As an illustration, consider m=4 and d=2. The set of alternatives is the plane: X =R2 and each voter i{1,2,3,4} has a most preferred alternative xiR2, with (Euclidean) indifference curves being circles around xi. There can be only two geometric configurations,1 depending on whether the quadrilateral that they form is convex or reentrant, as Fig. 1 shows.

As far as the opening question is concerned, the two configurations lead to the same result: ρ0.5. In case of a convex quadrilateral (see Fig. 1(a)), the point at the intersection between the two diagonals ([x1,x3] and [x2,x4]) is stable with respect to a 0.5-majority rule. In case of a reentrant quadrilateral, the point in the convex hull of the three others (x4 in Fig. 1(b)) is also stable with respect to a 0.5-majority rule.2

What more general can be said? The problem being highly complex, we restrict ourselves to a two-dimensional space of issues and moreover we consider that voters’ ideal points are selected independently from a nonsingular distribution f over R2 (the mass of f on any line in R2 is 0), which is furthermore sign-invariant(a weak symmetry property): for all xR2, f(x)=f(x); as in Tovey (1992). A first step is to compute the minimum rate ρ(O) for which O is stable. Since ρρ(O), we then obtain an additional upper bound to ρ.

For a given m-sample, there is no guarantee that O be stable for any rate below 1.3 So our strategy is to compute the exact probability of the whole range of scores, and consequently the expected value of ρ(O). A strength of our contribution is borrowed from a way, due to Wendel (1962), to account for m-samples drawn from a sign-invariant distribution; hence the probabilities of various scores for O are independent of the shape of the distribution as long as it remains sign-invariant.

Section snippets

The model

There are d measurable criteria of social choice, so that a social alternative can be represented as a d-dimensional vector: xRd. There are m voters in a set I. Each voter is endowed with a Euclidean preference relation on Rd: voter i, 1im, has an ideal point in the space of social choice, xiRn, and his/her utility function over the space of social choices is decreasing with the Euclidean distance from his/her ideal point: xRdui(x)=xix. A society is a m-tuple X=(xi)i=1m.

We measure the

The probabilities of all possible scores of O

Consider the following process which is due to Wendel (1962): choose m random points on a disk centered at O: Q1,Q2,,Qm. For each i, 1im, we set Pi equal to Qi or to Qi with equal probability 1/2 (without loss of generality, we can choose the Qi’s on the same side of a hyperplane through O as in Fig. 2(a) below; Fig. 2(b) corresponds to the configuration: Pi=Qi for i=1,3,4 and Pi=Qi for i=2,5). The points P1,,Pm are again i.i.d. random points in the disk.

The original question answered by

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