Existence of majority equilibria with non-ordered preferences

Majority voting is widely observed to produce stable policy outcomes, despite theoretical predictions of instability in multidimensional policy spaces. The present paper shows that stability can arise because voters have non-ordered preferences. We model preferences as correspondences within d-dimensional policy spaces and introduce a geometric measure of orderedness based on the angular spread α of strictly preferred alternatives. Our main result is that majority equilibria exist provided α < arcsin √(d+1)/(2d) with a dimension-free bound at α < π/4 = 45◦. We use Euclidean preferences to show that for random samples of voters with probability one for d tending to infinity there exist majority equilibria provided α < arccos √1/(m+1). Our findings suggest that modest deviations from fully ordered preferences can ensure stability of collective decisions under the majority rule.