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Getting all-in while behind It is not just when you are ahead that you might want to get all-in against someone who has an information advantage. Suppose the pot is $1$ and the effective stack depth ...

The Clairvoyant Game Here is a well-known toy problem (the Clairvoyant Game) that doesn't converge: Suppose your hand is face-up. You have no hidden information. You don't know whether your opponent'...

Here is an answer to the updated question: Suppose that there are two betting rounds. Darth Vader has three types of hands. Type 1 wins with probability 1. Type 2 is a draw that hits (becomes a ...

There is one failure of literal convergence. Consider a situation in the final round of betting where both players have the nuts, i.e. a hand that could not possibly lose, could only win or chop. Then ...

I just encountered this, and you're right: Y prefers bluffing to checking for all bluffs below the threshold. It's pretty simple to prove this: for y_a and y_b below the bluffing threshold with y_a &...