Costly turnout game

Each voter chooses whether to incur a voting cost. The collective result changes each voter's payoff, and abstention can be optimal when other voters are likely to determine the outcome. The payoff difference between voting and abstaining is the benefit of being pivotal, weighted by the pivotal probability, minus the turnout cost.

// Target: mathematical-optimization.bigb

Suppose turnout costs $3c>0$, each supporter gains $4c$ on passage, and the opponent loses $8c$. The symmetric-supporter Nash equilibria $(\alpha ,\beta ,\gamma )$ are $(0,1/4,1/4)$ and $(1,3/4,3/4)$. Allowing different supporter probabilities adds $(1/4,0,3/8)$ and $(1/4,3/8,0)$. These follow by comparing the pivotal voting probabilities with the turnout cost; no player benefits by changing their mixed strategy.

// Target: mathematical-optimization.bigb

A voter is pivotal when changing only that voter's participation changes the election result. For an opponent facing two independent supporters with participation probabilities $\beta ,\gamma$, under a rule requiring strictly more votes in favour, the opponent is pivotal when exactly one supporter votes. This probability is $\beta (1-\gamma )+(1-\beta )\gamma$.

// Target: mathematical-optimization.bigb