 Solution

Let $(q_R,q_S,q_P)$ be player two's mixed strategy in Rock paper scissors. The respective expected payoffs from choosing rock, scissors and paper are $1-p$, $-p$ and $2p-1$. Thus player two solves the linear programming problemA linear objective on this probability simplex is maximized by a largest-payoff pure response, or by any mixture of tied maximizers. Scissors is never a maximizing response: rock exceeds it by one. Comparing the other two gives $1-p⋛2p-1$ according as $p⋚2/3$. ThereforeThe optimal expected payoff is $1-p$ below the tie and $2p-1$ above it, with value $1/3$ at the tie.