p.787

On 64%-Majority Rule

Andrew Caplin
Barry Nalebuff

Abstract

Many electoral rules (such as those governing the U.S. Constitution) require a super-majority vote to change the status quo. It is well known that without some restriction on preferences, super-majority rules have paradoxical properties. For example, electoral cycles are possible with anything other than 100%-majority rule. Can these problems still arise if there is sufficient similarity of attitudes among the voting population? We introduce a definition of social consensus which involves two restrictions ondomain: one on individual preferences, the other on the distribution of preferences. Individuals vote for the proposal closest (in Euclidean distance) to ther most preferred point. The density of voters' ideal points is concave over its support in R''. Under these conditions, there exists an unbeatable proposal according to 64%-majority rule. In addition, no electoral cycles are possible. For n-dimensional decision problems, the precise majority size necessary to avoid cycles is 1 - (n/n + 1))^n which rises monotonically to 1 - (1/e), just below 64%. Our approach is based on the Simpson-Kramer min-max rule. We compare this rule with Condorcet's original proposal for an electoral system immune to his paradox of votiing. We conclude by considering the properties of a voting constitution based on 64%-majority rule.