Turnout equilibria with two supporters and one opponent

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