Tough, Important, General Case, Game Theory Problem

[David Sklansky]

I'm not sure how hard this question is, nor whether it has ever been solved in print. Probably it has. But if it hasn't, the first one to do it would get a feather in his cap.

It involves an obvious variation of regular poker. One round of betting. Except the bets are simultaneous rather than sequential. Everybody antes a certain amount. Then after looking at their cards (or to make it more precise, a real number from zero to one) they SECRETLY press a button indicating whether they are in for a second bet. Those who indicated that they stayed in now compare hands (numbers) and the winner takes the whole pot. If no one has made the second bet there is no action.

In real life this game, while interesting, could never be dealt seriously because of the collusion risk. Partners would never play more than one hand.

Notice that the correct strategy is the same for everybody. Given a certain number of players, a certain ante per player, and a certain size bet per player, the optimum strategy would be to bet some amount of your best hands.
For example if three players each antied one dollar and then had to bet two dollars secretly to reach a showdown, the optimum strategy might be to bet something like the top 30% of your hands (in other words-.7 or higher). If so that would mean about 35% of hands would be no decision.

So here is the question. "n" players are dealt a real number between zero and one. Each player antes "a" dollars. Each player secretly and independently, chooses whether to enter a showdown with a bet of "b" dollars. No action if no one bets. What proportion of your hands should you bet?

The answer will be in terms of n, a, and b. (It would be helpful if you translated it into the answer for my three player example.) This problem might require calculus.

Remember this answer assumes that you have no "read" on the other players. You are assuming that they are also playing their optimimum strategy. If so you will break even. I would expect that if they deviated from optimum (without colluding) your playing the optimum strategy would now probably win and certainly not lose. But that may be wrong. I have not thought about or attempted to find a solution myself. That's your job.