Thursday, September 20, 2007

When It Is Correct To Play Optimally

So I went to the trouble to do a blind steal EV calculation using Game theory. It took several hours, and was not easy to do at all. Then Hoy drops the comment that he hears tons of people talking about game theory and optimal play, but he has yet to meet someone who plays optimally. So the conclusions that you draw from game theory (which are based on optimal play from everyone in the hand), are called into question. Hoy has a valid point here, but I would like to dig a bit deeper before surrendering. First of all when people say they use game theory at the poker table, most are pretty much full of shit. It took me hours to calculate a very simplified blind steal problem for even and small stacks. There is no way the Chris Ferguson or any of the "Math" players, can make a calculation like that on the fly for a simple hand, let alone a real life situation that is much more complicated. What they are really saying is that they understand game theory, have taken the time to analyze certain situations, and take advantage when those situations arise at the table. Since those situations are pretty rare, most of the "game theory" poker players, rarely actually use it at the table. But it can be used, and it can be right to use it.

When you are the one who is forcing the action it will often be most correct to use the strategy game theory suggests. If your opponent plays optimally he will limit what your EV will be for the hand. If they don't play optimally, all the more EV for you. It is to your benefit when they don't play optimally. Now when the action is forced to you, you need to understand if your opponent is playing optimally. If you know that they are not, then you must make an adjustment to what game theory suggests. You just can't call down a huge% of your hands, if your opponent is not raising a high enough % of theirs. A simple example may help here. For this example we are playing a 1/2 NL cash game, and it folds to the SB. Either the SB or the BB only have $20 in front of them. It really is not important which one, as the shorter stack will limit the pot size. So lets just say the BB has $20 to start the hand and the SB has $200. I typical preflop raise from the SB would lead to awkward stack sizes post flop, so the SB either puts the BB all in or folds. With what range should the SB push, and with what range should the BB call if both are playing optimally? The optimal solution is shown below.

The SB pushes in with his top 50% range and the BB calls with about a 32.5% range. You may ask how this is optimal if the SB's EV is negative for the hand. There are no solutions to this problem with a positive expectation value for the SB. He is out of position, and there are no dead antes to fight over this time. However, folding 100% of the SB hands will yield an expectation value of -$1. Any solution that is better than -$1 is more optimal. So now you are the SB, and have done the calculation so you push your top 50% of hands in a cash game when you are the SB and it folds to you, and you or the BB has a stack of about 10x the big blind. If the BB is playing optimally he will call with his top 32.5% of hands. But lets say that the BB is not playing optimally. If he calls down a higher percentage, or a lower percentage, the SBs expectation value increases. The BB must play optimally to limit the SBs EV. Any deviation hurts the BB and helps the SB. So in this case, you use the result from game theory even if you know the BB will not play optimally.

Now lets turn the tables, and you are the BB, but the SB is not playing optimally. You figure out that he really only pushes about 30% of his hands in this situation. It is now incorrect to call down with the top 32.5%. This will result in a lower EV, although your EV will already be higher due to the expanded number of walks the SB is giving you. You want to be calling with about 1/3 less hands than the SB is raising with (a result from game theory as well), so you now call down with your top 20% range, even though that is not the optimal game theory range. You have made an adjustment based on your opponent not playing optimally. This is not strange, as it is done all the time at the poker tables. You sense a blind is weak and will not call down optimally so you attack. Edges are found in this way at the poker table, and game theory can identify where some of those edges are.

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