Comment by OkayPhysicist

Your chess betting example would play slightly differently than standard chess. Since the goal becomes not just to win, but to win the most cash, you'd be trying to leverage your hidden information in order to improve your outcome, which in chess is just "things you've foreseen but your opponent hasn't", which, due to the nature of chess, you'd weaken by jacking up your bets to try and capitalize. The result would probably be a worse game of chess, since without the chance component the better player will win the chess game far more often than the worse player, and then the optimal move for the worse player would be to forfeit early constantly.