On the Indecisiveness of Kelly-Strategyproof Social Choice Functions

TL;DR

This paper demonstrates that under weak preferences, only indecisive social choice functions can be Kelly-strategyproof, revealing significant limitations for strategyproofness in set-valued social choice mechanisms.

Contribution

It establishes fundamental impossibility results for Kelly-strategyproof social choice functions under weak preferences, highlighting the indecisiveness constraint.

Findings

Strategyproof rank-based SCFs violate Pareto-optimality.

Support-based SCFs satisfying Pareto return at least one top choice per voter.

Non-imposing SCFs necessarily select the Condorcet loser in some profiles.

Abstract

Social choice functions (SCFs) map the preferences of a group of agents over some set of alternatives to a non-empty subset of alternatives. The Gibbard-Satterthwaite theorem has shown that only extremely restrictive SCFs are strategyproof when there are more than two alternatives. For set-valued SCFs, or so-called social choice correspondences, the situation is less clear. There are miscellaneous - mostly negative - results using a variety of strategyproofness notions and additional requirements. The simple and intuitive notion of Kelly-strategyproofness has turned out to be particularly compelling because it is weak enough to still allow for positive results. For example, the Pareto rule is strategyproof even when preferences are weak, and a number of attractive SCFs (such as the top cycle, the uncovered set, and the essential set) are strategyproof for strict preferences. In this paper, we show that, for weak preferences, only indecisive SCFs can satisfy strategyproofness. In particular, (i) every strategyproof rank-based SCF violates Pareto-optimality, (ii) every strategyproof support-based SCF (which generalize Fishburn's C2 SCFs) that satisfies Pareto-optimality returns at least one most preferred alternative of every voter, and (iii) every strategyproof non-imposing SCF returns the Condorcet loser in at least one profile. We also discuss the consequences of these results for randomized social choice.