The Implementability of Liberalism

TL;DR

This paper characterizes the conditions under which choice liberal and Nash implementable social choice rules exist, focusing on the size of the outcome set relative to the number of players and the structure of preferences.

Contribution

It provides a precise characterization of when choice liberal and Nash implementable social choice rules are possible under unrestricted domain, including a constructive proof of the mechanism.

Findings

Existence of social choice rules depends on at least three players and outcome set size.

A social choice rule is choice liberal if it allows for outcomes aligned with individual preferences.

A Nash implementation mechanism can be constructed to realize socially desirable outcomes.

Abstract

This note shows that under the unrestricted domain, there exists a choice liberal and Nash implementable social choice rule if and only if there are at least three players and the outcome set is at least twice as large as the player set. A social choice rule is choice liberal if and only if for every player, there exists at least one pair of outcomes such that if this player strictly prefers one over the other, the one he prefers is socially desirable and the other one is not. A social choice rule is Nash implementable if and only if there exists a mechanism such that at every preference profile, the set of Nash equilibrium outcomes coincides with the set of socially desirable ones. The proof constructs an intuitive Nash implementing mechanism.