[David Sklansky]

By now some of you must realize that my "unusual questions" (that I don't profess to have an exact answer for) are meant to help you see why I believe excellent players should limp fairly often in very early position. Even when your opponents are not fish (but do play worse than you).

To complete this process I now ask two very precise game theory questions (again I have not calculated the solutions but I believe I will know the right answer when I see it.)

Two players, A and B receive one card with a real number between zero and one. High card wins. Player A bets one dollar in the blind. Player B can fold, call, or raise one dollar. If he raises, A can call or fold only. End of game.
What is the optimal strategy for each player and the value of the game to B?

To get you started notice that if there were no raises allowed, B would simply call whenever he had above .5. When he called he would win three quarters of the time (half of the time they were both above .5 and all of the time A was below.) Thus in eight tries he would fold four, win a dollar three times and lose a dollar once. The game is worth 25 cents to him. Since the blind is not live, the existence of an option to raise must help player B because if it didn't he could revert to the never raise strategy.

The second version of this question makes A's blind live. So if B just calls, A can raise if he chooses. Game over. Logic says that the value of this second game to B, if both players play optimally, is less than it would be if the blind wasn't live. Solving this second game for optimal strategy is probaly a lot harder than the first one. But I know we have some two plus twoers who can do it.

For the majority of you who know they are not familiar enough with game theory to tackle these proplems precisely, I ask a more general question. Should B raise more often in the first game or in the second game where's A's blind is live?