Comparing the Manipulability of Approval Voting and Borda

TL;DR

This paper compares the susceptibility to manipulation of approval voting and Borda rules, extending a case-by-case manipulability measure from matching to voting, and finds limited ability to rank these rules by manipulability.

Contribution

It introduces an extension of the manipulability comparison method to voting rules, specifically analyzing $k$-approval and truncated Borda rules.

Findings

Most voting rules cannot be meaningfully ordered by manipulability using this measure.

The case-by-case approach does not distinguish significantly between different scoring rules.

The exception to the general finding suggests some rules are more manipulable than others.

Abstract

The Gibbard-Satterthwaite theorem established that no non-trivial voting rule is strategy-proof, but that does not mean that all voting rules are equally susceptible to strategic manipulation. Over the past fifty years numerous approaches have been proposed to compare the manipulability of voting rules in terms of the probability of manipulation, the domains on which manipulation is possible, the complexity of finding such a manipulation, and others. In the closely related field of matching, Pathak and Sonmez pioneered a notion of manipulability based on case-by-case comparison of manipulable profiles. The advantage of this approach is that it is independent of the underlying statistical culture or the computational power of the agents, and it has proven fruitful in the matching literature. In this paper, we extend the notion of Pathak and Sonmez to voting, studying the families of $k$-approval and truncated Borda scoring rules. We find that, with one exception, the notion does not allow for a meaningful ordering of the manipulability of these rules.