Toward a 50%-majority equilibrium when voters are symmetrically distributed
Introduction
Consider a society comprising a potentially small number 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 of individuals having preferences over a set of alternatives . An alternative is an equilibrium for the rate if there is no alternative that rallies the vote of more than individuals against . Let denote the smallest rate for which there exists an equilibrium, called the min–max rate of society (Simpson, 1969, Kramer, 1973).
The social choice literature offers several upper bounds on . 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 : 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 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 is a convex and compact set of dimension , and individuals have convex and continuous preferences over , then . 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 . 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 and . The set of alternatives is the plane: and each voter has a most preferred alternative , with (Euclidean) indifference curves being circles around . 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: . In case of a convex quadrilateral (see Fig. 1(a)), the point at the intersection between the two diagonals ( and ) 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 ( 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 over (the mass of on any line in is 0), which is furthermore sign-invariant(a weak symmetry property): for all , ; as in Tovey (1992). A first step is to compute the minimum rate for which is stable. Since , we then obtain an additional upper bound to .
For a given -sample, there is no guarantee that 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 . A strength of our contribution is borrowed from a way, due to Wendel (1962), to account for -samples drawn from a sign-invariant distribution; hence the probabilities of various scores for are independent of the shape of the distribution as long as it remains sign-invariant.
Section snippets
The model
There are measurable criteria of social choice, so that a social alternative can be represented as a -dimensional vector: . There are voters in a set . Each voter is endowed with a Euclidean preference relation on : voter , , has an ideal point in the space of social choice, , and his/her utility function over the space of social choices is decreasing with the Euclidean distance from his/her ideal point: A society is a -tuple .
We measure the
The probabilities of all possible scores of
Consider the following process which is due to Wendel (1962): choose random points on a disk centered at : . For each , , we set equal to or to with equal probability 1/2 (without loss of generality, we can choose the ’s on the same side of a hyperplane through as in Fig. 2(a) below; Fig. 2(b) corresponds to the configuration: for and for ). The points are again i.i.d. random points in the disk.
The original question answered by
References (26)
Singularity theory and core existence in the spatial model
J. Math. Econom.
(1995)- et al.
Social choice and electoral competition in the general spatial model
J. Econom. Theory
(2006) - et al.
Probability and convergence for supra-majority rule with Euclidean preferences
Math. Comput. Modelling
(1992) A critique of distributional analysis in the spatial model
Math. Social Sci.
(2010)A note of Sylvester’s four-point theorem
Studia Sci. Math. Hungar.
(2001)The Theory of Committees and Elections
(1953)Uber affine Geometrie XI: Losung des “Vierpunktproblems” von Sylvester aus der Theorie der geometrischen Wahrscheinlichkeiten
Leipz. Ber.
(1917)- et al.
On 64-majority rule
Econometrica
(1988) Advanced Combinatorics
(1974)An Introduction to Probability Theory and its Applications. Vol. 1
(1968)