A model of electoral competition with abstaining voters

This work studies a two-candidate single-dimensional Downsian voting model with variable voter turnout. I consider a one-shot game between two candidates, where the payoffs of the candidates are their vote shares, and the strategy set of each candidate is a one-dimensional set of policy positions. The utility that a voter attributes to a candidate is linear in both the absolute distance between the candidate’s policy position and the voter’s preferred policy, and in the electorate’s bias toward the candidate (known as the candidate’s valence). I assume that the voter decides to abstain if either they are indifferent between the two candidates, or significantly dissatisfied with both candidates; otherwise, the voter votes for the candidate they prefer. I show that under any of the two assumptions about voting behavior, a pure-strategy Nash equilibrium exists under a broad range of conditions, even if the valence of one of the candidates is nonzero. In either case, the candidates do not always choose identical policy positions. I then study how the positions of the candidates, their relative shares of vote, and turnout depend on the distribution of voter preferences and candidate valence.