Equilibria of Plurality Voting with Abstentions

In the traditional voting manipulation literature, it is assumed that a group of manipulators jointly misrepresent their preferences to get a certain candidate elected, while the remaining voters are truthful. In this paper, we depart from this assumption, and consider the setting where all voters are strategic. In this case, the election can be viewed as a game, and the election outcomes correspond to Nash equilibria of this game. We use this framework to analyze two variants of Plurality voting, namely, simultaneous voting, where all voters submit their ballots at the same time, and sequential voting, where the voters express their preferences one by one. For simultaneous voting, we characterize the preference profiles that admit a pure Nash equilibrium, but show that it is computationally hard to check if a given profile fits our criterion. For sequential voting, we provide a complete analysis of the setting with two candidates, and show that for three or more candidates the equilibria of sequential voting may behave in a counterintuitive manner.

💡 Research Summary

This paper departs from the conventional manipulation literature, which assumes a coalition of manipulators acting against a truthful majority, and instead treats every voter as a strategic agent. By modeling an election as a strategic game, the authors investigate the set of pure Nash equilibria (PNE) that can arise under two variants of plurality voting: simultaneous voting, where all ballots are cast at once, and sequential voting, where voters act one after another.

In the simultaneous setting, the authors first extend the standard plurality rule with an abstention option, allowing each voter to either vote for a candidate or submit a null ballot. They prove that a PNE exists if and only if two structural conditions hold: (1) every candidate that could possibly win must have at least one voter who is willing to vote for it, and (2) the sets of voters supporting different candidates must not be nested (i.e., no candidate’s supporter set completely contains another’s). When these conditions are satisfied, each voter can safely either vote for his most-preferred viable candidate or abstain without giving any other voter an incentive to deviate.

However, checking whether a given preference profile satisfies these conditions is computationally intractable. By a reduction from Exact‑Cover‑by‑3‑Sets, the authors show that the decision problem “does a PNE exist?” is NP‑hard. Consequently, while the characterization is theoretically clean, it offers limited practical guidance for large elections unless the profile has special structure (e.g., a constant number of candidates or hierarchical preferences).

The sequential case is analyzed in two parts. With only two candidates, the game can be solved by backward induction, yielding a complete description of all PNE. The equilibrium is uniquely determined by the order of voters and their initial rankings: each voter either reinforces the current leader or abstains, and the final winner is simply the candidate who survives this cascade of best‑response moves.

When three or more candidates are present, the equilibrium landscape becomes far richer and sometimes counter‑intuitive. The authors construct explicit examples where a candidate who is initially ahead can lose because later voters strategically abstain, allowing a less‑preferred candidate to win. Conversely, a voter may abstain even though his top choice is still viable, simply to prevent a later voter from overturning the outcome in his favor. These “reverse‑turn” and “strategic‑abstention” equilibria demonstrate that sequential plurality with abstention does not guarantee that the majority’s most preferred candidate will prevail, and that the order of moves can dramatically affect the result.

Overall, the paper contributes a game‑theoretic framework for analyzing strategic voting with abstention, provides a precise structural characterization of equilibria for simultaneous plurality, proves the associated computational hardness, and delivers a full equilibrium analysis for sequential plurality with two candidates while exposing surprising phenomena for three or more candidates. The findings highlight the importance of considering both abstention and voting order in the design of electoral mechanisms, and they open avenues for future work on mixed timing protocols, other voting rules, and the impact of information structures on strategic behavior.