But, we can determine what is the best new strategy or strategies to add using linear programming.

The button's strategy for each card is something horrific like:

If Player A bets, call. If A stands pat, draw one. Then, if I get a 2 do AAA, if I get a 3 do BBB, if I get a 4 do CCC etc. If A draws one, stand pat. If A bets the last round, do DDD. If A checks the last round, do EEE. If A checks, bet. If A raises, call. If A stands pat, draw one. Then, if I get a 2 do FFF, if I get a 3 do GGG, etc. If A draws one, stand pat. If A bets the last round do HHH. If A checks the last round do III. If A calls, If A stands pat, draw one. Then, if I get a 2 do JJJ, if I get a 3 do KKK, etc. If A draws one, stand pat. If A bets the last round do LLL. If A checks the last round do MMM. If A draws two, stand pat. If A bets the last round do NNN. If A checks the last round to OOO.

Gross, huh? But, the key point is that each of the innermost last-round actions are a problem we already know how to solve. Given a hand range for A, we can calculate the best strategy in the last round for B (the appropriate AAA/BBB/CCC mix) as a linear program. For 1:1 drawing situations, this is just the problem I've already been solving (except for there being a range of dead cards instead of known ones.)

So, the algorithm should look something like this:

1. Start with a resticted set of pure strategies for A and B.

2. Solve the main linear program to arrive at the best mix of strategies for A and B.

3. For each possible fourth-round situation,

3a. Calculate the hand range for A on the fourth round given the action on the third round.

3b. Solve a subsidiary linear program giving the best strategy for B against A's hand range.

3c. If the fourth-round strategy for B includes strategies not found in the restricted set of pure strategies, add a new pure strategy for B to the restricted set.

4. Iterate steps 2-3, alternately adding strategies for A and B, until no new pure strategy can be found to add. Then the solution to the main linear program cannot be improved further.

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