Using Game Theory to Bluff
In this site we are mainly concerned with how game theory can be applied to the art of bluffing and calling possible bluffs in poker. For this purpose we will talk about mixed strategy, a strategy in which you make a certain play - specifically a bluff or a call of a possible bluff - a predetermined percentage of the time, but you introduce a random element so that your opponent cannot know when you are making the play and when you are not.

You will recall from the last site that, everything else being equal, the player who never bluffs and the player who bluffs too much are at a decided disadvantage against a player who bluffs correctly. To illustrate this point and to show how game theory can be used to decide correctly when to bluff, we'll set up a proposition.

We are playing draw lowball with no joker, and we give you a pat :

We take a :

You stand pat, and I must draw one card. If I catch a five, a six, a seven, an eight, or a nine, I beat you with a better low than yours. If I catch any other card, you win. That means that of the 42 cards remaining in the deck, I have 18 winners (4 fives, 4 sixes, 4 sevens, 3 eights, and 3 nines) and 24 losers, which makes me a 24-to-18 or 4-to-3 underdog. We each ante \$100, but after the draw - which you do not see -- I can bet \$100.

Suppose I said I'm going to bet \$100 every time. Clearly you would call every time because you would stand to win \$200 the 24 times I'm bluffing and lose \$200 the 18 times I have the best hand for a net profit of \$1,200. On the other hand, suppose I said I will never bluff; I will only bet when I have your 9,8 low beat. Then you would fold every time I bet, and once again you would win 24 times (when I don't bet) and lose 18 times (when I do) for a net profit of \$600 since you win or lose \$100 in each of these hands. So with either of these variations of the proposition, you definitely have the best of it.
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I'll Show Me

Your spouse comes home one day from shopping. She or he has bought the most hideous, impractical, overpriced sofa/motorcycle/carpet/computer you can possibly imagine. You think that the purchase was a huge, ponderous waste of money, and you say so. "Oh, yeah?" your spouse snaps back. "Well, what about all the money you spend on poker?" You don't bother pointing out that you're investing, not spending; you've been down this road be fore. You break off the argument before it goes ballistic, but it's on your mind later that day when you head down to the club. You carry it with you into battle. Maybe somewhere in a dark corner of your mind there's even a tiny motivation to lose-or at least a higher tolerance for losing than usual. After all, your spouse has shown a fabulous capacity to waste money, why should you be any different?

Those prone to this "I'll show me!" state of mind willfully commit self-destructive acts in the misguided hope or expectation that the damage they're doing to themselves is actually afflicting someone else. Other stimuli can trigger this sort of revenge-against-the-world response. Maybe you're being treated unfairly at work. Maybe your parents/children/ siblings are driving you crazy. Maybe you think you're only seeking a little peace and quiet, or alone time, or a distraction when you race off to play poker. That may seem benign enough-we all can use a little distraction from time to time. But it's hardly a reason for playing poker, and it hardly augers well for the success of your session. Worse, it may mask a deeper resentment, which will manifest itself in repressed anger, poor judgment, flawed decision-making and, ultimately, failure and loss.