Consistent Probabilistic Social Choice

TL;DR

This paper demonstrates that in probabilistic social choice, two traditionally incompatible axioms are uniquely satisfied by Fishburn's maximal lotteries, which are optimal mixed strategies computable via linear programming.

Contribution

It characterizes Fishburn's maximal lotteries as the unique solution satisfying both axioms in probabilistic social choice, extending classical axioms to the probabilistic setting.

Findings

Maximal lotteries always exist due to von Neumann's Minimax Theorem.

Maximal lotteries are almost always unique.

Maximal lotteries can be efficiently computed using linear programming.

Abstract

Two fundamental axioms in social choice theory are consistency with respect to a variable electorate and consistency with respect to components of similar alternatives. In the context of traditional non-probabilistic social choice, these axioms are incompatible with each other. We show that in the context of probabilistic social choice, these axioms uniquely characterize a function proposed by Fishburn (Rev. Econ. Stud., 51(4), 683--692, 1984). Fishburn's function returns so-called maximal lotteries, i.e., lotteries that correspond to optimal mixed strategies of the underlying plurality game. Maximal lotteries are guaranteed to exist due to von Neumann's Minimax Theorem, are almost always unique, and can be efficiently computed using linear programming.