[AaronBrown]

I apologize, I thought you were asking a simpler question.

There still is an equilibrium however. The difference between head's-up one-bet hold'em and zero/one games is the range of probabilities in hold'em go from something like 0.3 to 0.85 instead of 0 to 1.

A more subtle difference is the probabilities are not fixed, but depend on the hands your opponent will play. A pair of 7's has a small edge over suited Jack 10 if one of the sevens matches the suit. Suited Jack 10 has a small advantage over unsuited Ace 2. But unsuited Ace 2 beats 7's 70% of the time.

Despite the complications, there has to be a fixed point equilibrium. We can represent a strategy for the first player by the probability of checking for each of the 178 hold'em hands, the strategy for the second player is the probability of folding for the same set of hands. For every vector for the first player, there is a vector for the second player that maximizes expectation. Therefore, under the usual game theory assumptions, the first player knows the expected outcome for every vector, and can choose the maximum.

Of course, in real life, the second player does not know the first player's strategy and so the first player has to guess what he will do. With both players guessing, you can't predict the outcome.