Game Theory Part 3: Observations

Mathematics, Strategy

April 13, 2008

In the last game theory column, I presented a real-world poker problem, and started through the process of figuring out the equilibrium solution. As you probably noticed, I simply told you what the solution was without explaining how I got that solution. Now I owe it to you to explain how I did it.

The traditional way of solving for the equilibrium solution in game theory problems is by means of matrix arithmetic. This is obviously far more complicated math than you would want to apply at the poker table. In other words it’s bad poker math. To avoid that, I used a nifty little arithmetical trick. In the first game theory article, we noticed a interesting fact in the roshambo example: when player 1 chooses an equilibrium strategy, all of player 2’s strategies become equally good (or bad). This isn’t always true, but there’s a very similar statement we can make that is:

In a two player game, when player 1 chooses an equilibrium solution strategy, all of player 2’s simple strategies in the head of the rho (or any combination thereof) are equally good in response.

That’s pretty much what I just said, but notice that it only works in the roshambo case because all of the simple strategies are in the head of the rho – there is no tail on the rho. Notice also that our poker problem is the same way – there are only two simple strategies for player 2 (call, fold) and both are in the head of the rho. This means the same logic can be applied here: player 1’s equilibrium strategy will make player 2’s decision not matter. This lets us solve for player 1’s strategy very simply as follows:

  • Player 2 is being offered 11:1 on this call
  • In order for player 2’s decision to NOT matter, he has to be facing odds of exactly 11:1 against his opponent bluffing.
  • value bets amount to 70% of player 1’s range. To give player 2 those odds, bluffs must account for 70%/11 or 6.36% of his range. Hands where player 1 has the option to bluff account for 30%, so if he bluffs just over 1 time in 5 when he has the opportunity, he’ll have the requisite number of bluffs in his range. 1 in 5 looks a lot like the 20%ish number I got last time.

This is a very powerful result: when bluffing mixing bluffs and value bets on the end, the equilibrium strategy is to set your bluff rate such that the odds against you bluffing are the same as the pot odds being offered to call. The point here is that there are certain problems in poker where it’s not terribly difficult to come up with an equilibrium strategy.

Next article, I’ll talk about how all this fits in with practical poker play. What I say may surprise you.

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