Manipulability of consular election rules

TL;DR

This paper explores the manipulability of election rules that select committees of size two, showing that strategy-proofness implies dictatorship unless the rule can be decomposed into simpler functions, extending classical social choice theorems.

Contribution

It extends the Gibbard-Satterthwaite and Duggan-Schwartz theorems to two-member committee elections, identifying conditions under which strategy-proofness leads to dictatorship.

Findings

Strategy-proofness implies dictatorship unless the election rule is decomposable.

The result applies specifically to committee size two, such as consular elections.

Obstacles to generalizing the result to larger committees are discussed.

Abstract

The Gibbard-Satterthwaite theorem is a cornerstone of social choice theory, stating that an onto social choice function cannot be both strategy-proof and non-dictatorial if the number of alternatives is at least three. The Duggan-Schwartz theorem proves an analogue in the case of set-valued elections: if the function is onto with respect to singletons, and can be manipulated by neither an optimist nor a pessimist, it must have a weak dictator. However, the assumption that the function is onto with respect to singletons makes the Duggan-Schwartz theorem inapplicable to elections which necessarily select a committee with multiple members. In this paper we make a start on this problem by considering elections which elect a committee of size two (such as the consulship of ancient Rome). We establish that if such a consular election rule cannot be expressed as the union of two disjoint social choice functions, then strategy-proofness implies the existence of a dictator. Although we suspect that a similar result holds for larger sized committees, there appear to be many obstacles to proving it, which we discuss in detail.