Finding an optimal strategy is easier if you frame it this way.

1) Let's play Rock, Paper, Scissors, n.

2) Both players make a throw, player A cannot throw paper.

3) If player B wins, player B gets +1, player A gets -1, done

4) If player A wins, player A gets +1, player B gets -1, done

5) If a draw happens, and n = 1, player A gets +1, player B gets -1, done

6) Otherwise, play a game of Rock, Paper, Scissors, n-1, and use the payout from that instead.

From here you can try some inductive/recursive definition of the winning strategy based on n. Don't have the time to actually work it out, but there's the framework.