Implementation of Democratic Social Choice Functions

A social choice function is said to be implementable if and only if there exists a game form such that for all preference profiles an equilibrium strategy n-tuple exists and any equilibrium strategy n-tuples of the game yield outcomes in the social choice set. A social choice function is defined to be minimally democratic if and only if whenever there exists an alternative which is ranked first by n − 1 voters and is no lower than second for the last voter, then the social choice must be uniquely that alternative. No constraints are placed on the social choice function for other preference profiles. Using the classical definitions of equilibria for n-person games—namely Nash and strong equilibria, it is shown here that over unrestricted preference domains, as long as there are at least as many alternatives as individuals, no minimally democratic social choice function is implementable. A similar result holds in certain restricted domains of the type assumed by economists over public goods spaces. We then show that a different notion of equilibrium—namely that of sophisticated equilibrium—allows for implementation of democratic social choice functions also having further appealing properties.