Rock paper scissors against a player who never throws rock

You play repeated rounds of rock-paper-scissors against someone who is not allowed to play rock. A win pays you $1, a loss costs you $1, and a tie pays nothing. If both sides play optimally, what is your expected gain per round?

$\tfrac13$ dollar. Paper is dominated for you, so mix rock (prob. $a$) and scissors; the opponent mixes paper (prob. $b$) and scissors. Your payoff is $a+b-3ab$. Choosing $a=\tfrac13$ makes it $\tfrac13$ whatever $b$ is, and the opponent's $b=\tfrac13$ caps it at $\tfrac13$, so the game's value is $\tfrac13$.