Existence of Nash equilibrium in the victory-indicator-only presidential priming game

Determine whether a Nash equilibrium necessarily exists in the multi-candidate presidential priming game under the victory-indicator-only utility u_ind(c, w) = v(c, w), where each candidate allocates a fixed campaign budget across issues to linearly increase issue salience (s_i^v(w_i) = rho_i w_i + s_i^v(0)), voter choice probabilities are the weighted sums of candidate quality scores by relative salience, and the winner is defined by the candidate(s) with maximal expected votes. Establish whether a Nash equilibrium (pure or mixed) is guaranteed to exist for an arbitrary number of candidates and issues in this setting.

Background

The paper studies campaign spending for voter priming in multi-issue elections where investment increases issue salience linearly and voters’ probabilities of voting for a candidate depend on quality scores weighted by relative salience. It analyzes parliamentary and presidential settings under several utility variants.

In the presidential setting, the variant denoted ind assigns utility solely based on winning (the victory indicator), without any secondary goal tied to vote share. For two candidates under ind, the authors prove existence of dominant strategies and a Nash equilibrium. However, for a general number of candidates they only establish that a best response exists and explicitly state they do not know whether a Nash equilibrium is guaranteed to exist.

This uncertainty contrasts with other variants (plus and max), for which the authors show non-existence examples of best responses or equilibria. The ind case therefore isolates a core open question about equilibrium existence in the victory-indicator-only game with linear priming. The summary table also marks the Nash equilibrium status for ind as “open.”