If your opponent is not playing optimal game theory strategy, you shouldn’t either. If he isn’t bluffing or calling with a frequency that is “unexploitable,” he’s opening himself up to a counter strategy by you that will beat him. Assuming, of course, that you have identified how he is deviating from Game Theory Optimal.
If you have figured out this deviation (i.e. he’s bluffing or calling too much or too little), there is no longer any reason to “mix up” (i.e. randomize) your play. If he bluffs too much, you always call when your hand could beat a bluff. If he folds too much, you always bluff with a hopeless hand. But, there is a problem with this fixed counter strategy. He will notice it and start changing his strategy for the better. In other words, you will push him into playing more optimally. Therefore, it may be better to continue to mix up your play a little bit to prevent this. But, you wouldn’t do this in the game theoretic way. Let’s look at some simplified examples.
It is the last round of betting, the last card is face down, and one player is known to be 40 percent to have the nuts and 60 percent to have a clearly losing hand. The other player’s hand will always beat a bluff, but never a value bet. He will never have a raising hand. A bet on the last round, if it is made, must be the size of the pot. We will say that the pot has $100 in it and that this money came from Santa Claus.
The game theory strategy for the bettor is to bet the 40% of the time he has made his hand plus another 20%, 60% altogether. By doing that, he has a 2/3 chance of having the other player beaten when he bets the pot. Since that player is getting 2:1 odds when he calls, it makes no difference whether he calls or not. Since it makes no difference, we can assume he will always fold for ease of calculations. If he always folds, the bettor will win $100 60% of the time and break even 40%. His EV is therefore $60. The other player has an EV of $40. Hopefully, you understand that the $60 and $40 EVs will stay the same regardless of the calling frequency.
Suppose the bettor is not bluffing according to game theory. Suppose, for instance, he is only bluffing with 5% of his hands. If so, the other hand should always fold to a bet. This will move his EV up to $55, rather than $40. That’s great. Except that, the other guy will probably start noticing it and bluff more. Say that even though he is a timid player, he moves his bluffs up to 15%. You still have to fold every time he bets, but now your EV has gone down to $45. But, maybe you can prevent this by calling every once in a while. Suppose that calling 20% of the time keeps him from getting frisky. Meaning, he goes back to 5% bluffing. That means that you win $100 55% of the time, win $200 1% of the time, lose $100 8%, and break out even 36%. That’s an EV of $49, even though you are making what you know to be very bad calls every once in a while. Those occasional bad calls hurt you less than the alternative of making him play better when you don’t call at all.
A similar technique can be applied if he bluffs too much. Suppose half his bets are bluffs. Calling him every time is the theoretically best counter strategy. You will win $100 the 20% of the time he checks, win $200 40%, and lose $100 40%. That’s an EV of $60, much better than the $40 you “deserve”. But, if you call every time, he will stop bluffing so much. If he bluffs only 25% of his hands, you still call every time. But, now you will win $100 35% of the time, lose $100 40%, and win $200 25% of the time. That’s only a $45 EV. Suppose instead, you fold 20% of your hands even though you theoretically shouldn’t because you realized that those folds will keep him bluffing as often as he has been. Now, you win $100 20%, break out even 16%, win $200 32%, and lose $100 32%. That’s moves your EV back up to $52.
This type of strategy could also be used if you are the bettor. The proper game theory calling probability for pot limit is 50%. That’s because a bluffer is getting even money on his bluff. Calling 50% means that it no longer matters if he bluffs or not. But, what if he runs into someone who never folds? Does that mean he shouldn’t bluff at all? Probably not. It’s probably still a good idea to bluff once in a while. Suppose this guy folds 90% of the time. Bluffing every time means you win $100 90%, win $20 4%, and lose $100 6%. That’s a whopping $92 EV. It would be terrible if you turned him into something like a 30% caller. Now, you win $100 70%, win $200 12%, and lose $100 18% if you bluff every time. Suppose instead, you give up on your bluffs once in a while, say 10% of all your hands, and this keeps him folding with a 90% frequency. Now, you break even 10%, win $100 81%, win $200 4%, and lose $100 5%. That’s $84. Not as good as $92, but better than the $76.
Finally, we have the situation where he calls way more than the 50% game theory strategy. Say he calls 80% of the time. Bluffing is now silly. But, only if never bluffing doesn’t change his calling ways. As an 80% caller, he will give you an EV of 32% times 200 plus 8% times 100. That’s $72. But, if he changes to a 60% caller because he sees you never bluff, you only get 24% time 200 plus 16% times 100 or $64. If instead, you force yourself to bluff with 5% of your hands and that keeps him from changing his 80% calling strategy, you will win 32% times $200 plus 8% times $100 plus 1% times $100 minus 4% times $100 55% of the time you break even. That’s $69.
Obviously, these situations are never as clear cut as they are in these examples. But, the general principle holds in all poker situations. Don’t be afraid to occasionally make a play that is obviously negative EV against a certain player if the failure to do it will drive him towards proper game theory strategy. A mixed strategy on your part that is neither game theory optimum nor pure exploitation is often best.
