The optimal strategy (as many posters have said) is R = 1/2, P = 1/6, S = 1/3 for the constrained player (A) and r = 1/3, p = 2/3 for the unconstrained player (B), and the value of the game is 1/6 in favour of B (pretty straightforward game theory). If a fee is charged on every round of the game, then $16.66 is the fair price. If a fee is only charged on rounds with a definite result, then the fair price is $23.07 since 5/18 of the time the result will be a draw. Fair odds for each player, if draws are counted as 'push' bets, are 8/5 for A and 5/8 for B. If B can choose the stake for each round, and is paid off at evens, he should bet 3/13 (23.08%) of his current bankroll to maximise his long-term expectation.