Consider the following betting game: Two players each have 100 cents to bet. If one player bets more than the other then that player gains a point and the other player loses a point. The goal of the game is to get two points. The betted money is lost from the game entirely (In other words, each player can only bet 100 cents in total).
In principle one could make a finite (expect for the zero loop) game tree for this game and there would be a winning strategy. Now suppose we modify the game, players can bet any real number between zero and one. The game tree is no longer finite. But is there still a winning strategy? What is this strategy?
Naively, you could in principle construct game trees for the finite games, but further subdivide the interval $[0,1]$. Is it plausible to expect some kind of convergence of winning strategies to the continuous case?
I have no knowledge of game theory whatsoever, but I find this an interesting question. Maybe this is well-known, but google didn't find a solution immediately.
Thanks in advance for any ideas.