Incentives in Social Decision Schemes with Pairwise Comparison Preferences

TL;DR

This paper investigates the axiomatic properties of social decision schemes with pairwise comparison preferences, establishing key impossibility results and clarifying conditions under which certain desirable properties can or cannot coexist.

Contribution

It resolves three open questions about the incompatibility of Condorcet-consistency, efficiency, and strategyproofness in SDSs with pairwise comparison preferences, and clarifies conditions for possibility.

Findings

No Condorcet-consistent SDS satisfies PC-strategyproofness for four or more alternatives.

No anonymous, neutral SDS satisfies both PC-efficiency and PC-strategyproofness for four or more alternatives.

No anonymous, neutral SDS satisfies both PC-efficiency and strict PC-participation for four or more alternatives.

Abstract

Social decision schemes (SDSs) map the ordinal preferences of individual voters over multiple alternatives to a probability distribution over the alternatives. In order to study the axiomatic properties of SDSs, we lift preferences over alternatives to preferences over lotteries using the natural -- but little understood -- pairwise comparison (PC) preference extension. This extension postulates that one lottery is preferred to another if the former is more likely to return a preferred outcome. We settle three open questions raised by Brandt (2017): (i) there is no Condorcet-consistent SDS that satisfies PC-strategyproofness; (ii) there is no anonymous and neutral SDS that satisfies PC-efficiency and PC-strategyproofness; and (iii) there is no anonymous and neutral SDS that satisfies PC-efficiency and strict PC-participation. All three impossibilities require $m\geq 4$ alternatives and turn into possibilities when $m\leq 3$. We furthermore settle an open problem raised by Aziz et al. (2015) by showing that no path of PC-improvements originating from an inefficient lottery may lead to a PC-efficient lottery.