I'm interested in zero-sum symmetric games which have the following form. Each player has a counter which starts at 0. Each turn, a player may choose from a fixed set of actions. A player's counter is raised or lowered according to the cost of the chosen action. (If this drops the player's counter below 0, they lose.) Otherwise, look up the respective actions on a payoff matrix to determine if one or the other wins. If so, the game ends with the selected winner. If neither wins, the game is played again, but retaining the counter.
It is like a repeated game, except that the counter is not reset between rounds and the number of games is not fixed but depends on the players' actions. In that way it's more like a Markov chain with a countably-infinite state space.
Here's a toy example with three actions:
- A: Add 1 to your counter.
- B: Subtract 1 from your counter. You win unless your opponent chooses B or C.
- C: Subtract 2 from your counter. You win unless your opponent chooses C.
I'm interested in the existence of mixed strategies for this type of game, and how to find them if they do exist. Also, if there's a name for this type of game I'd like to know (which should make finding information much easier).
