Let's say there are two friends A and B and they want to regularly play a non-drawable (one party always wins) competitive game (e.g. US 8-balls) with each other with bets. Every time a player wins, the losing player has to pay some non-zero amount X to the other. In order to simplify things, assume that player A has a winning probability of Pa
Find a system to calculate X (it can depend upon the past win/loss history of the player) for each game, such that, in the long run, net gain of both players converge to zero (i.e. the total money they have lost and won are close to equal).
What doesn't work: When Pa = 0.5, and X = $1 (always), the net money won is equivalent to the position of a symmetric random walk, and it doesn't converge to zero.
Can such a betting system exist? (Except for a trivial case of X=0).
The motivation here is that it should be possible to win money in the short term, but in long term, it has to converge to zero.