How many candidates are needed to make elections hard to manipulate?

TL;DR

This paper investigates the minimum number of candidates needed to make various voting protocols computationally hard to manipulate, providing specific thresholds for multiple voting systems.

Contribution

It determines the exact candidate counts at which several voting protocols transition from easy to hard manipulation, filling a key gap in computational social choice theory.

Findings

Plurality, Borda, and others become hard to manipulate at specific candidate counts.

Veto and plurality with runoff thresholds are identified for the first time.

Some protocols remain easy to manipulate regardless of candidate number.

Abstract

In multiagent settings where the agents have different preferences, preference aggregation is a central issue. Voting is a general method for preference aggregation, but seminal results have shown that all general voting protocols are manipulable. One could try to avoid manipulation by using voting protocols where determining a beneficial manipulation is hard computationally. The complexity of manipulating realistic elections where the number of candidates is a small constant was recently studied (Conitzer 2002), but the emphasis was on the question of whether or not a protocol becomes hard to manipulate for some constant number of candidates. That work, in many cases, left open the question: How many candidates are needed to make elections hard to manipulate? This is a crucial question when comparing the relative manipulability of different voting protocols. In this paper we answer that question for the voting protocols of the earlier study: plurality, Borda, STV, Copeland, maximin, regular cup, and randomized cup. We also answer that question for two voting protocols for which no results on the complexity of manipulation have been derived before: veto and plurality with runoff. It turns out that the voting protocols under study become hard to manipulate at 3 candidates, 4 candidates, 7 candidates, or never.