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I am interested in the following simplified version of poker. Each player gets a card (for example, either A or B). Then they bet knowing their own cards (for example, the pot initially has 1 euro, then they go in order and decide to fold or add $\alpha$ euros to the pot, then after the last player the betting is over). After the betting, either those distribute the money who have A or those who have B (for example, with 60% chance A wins and with 40% chance B wins). What are the optimal strategies?

Note that I am most interested if the game has at least 3 players and it is crucial that the outcome can depend on the flop, unlike most theoretically studied versions, like Kuhn poker or when the inputs are from the [0,1] interval. Has this ever been studied?

I wonder if paradoxical situations can happen, like the last player should bet with a higher chance when his card is B than with A (given in the above example).

I am interested in the following simplified version of poker. Each player gets a card (for example, either A or B). Then they bet knowing their own cards (for example, the pot initially has 1 euro, then they go in order and decide to fold or add 10 euros to the pot, then after the last player the betting is over). After the betting, either those distribute the money who have A or those who have B (for example, with 60% chance A wins and with 40% chance B wins). What are the optimal strategies if the players want to maximize their expected income?

Note that I am most interested if the game has at least 3 players and it is crucial that the outcome can depend on the flop, unlike most theoretically studied versions, like Kuhn poker or when the inputs are from the [0,1] interval. Has this ever been studied?

I wonder if paradoxical situations can happen, like the last player should bet with a higher probability when his card is B than with A (when A has a higher chance of winning).