Thanks, it's clearer now. So we could say that every pair of players in $A$ flip a fair coin and the winner gets $\$2$. Every pair of players with one from $A$ and one from $B$ flip a fair coin and the winner gets $\$1$. Interesting problem!

Wouldn't any two players in A have the same number of transfers, and the same for B? I guess I don't understand the game.

This is very helpful!

Perhaps I'm missing something here. The game ends at a countable stage so pR's plays from $R\setminus C$ won't produce an injection from $\omega_1$ to $R$. If we need a more precise strategy, what about this? We may assume that the injective copy of pN's set is the set of positive integers $P$. For each turn $n<\omega$, pR plays -n. At every turn after that, pR plays the continued fraction defined by all previous plays of pN.