Let's say I choose specifically 6 key cards to bluff with. That means I will bet 24 times. 18 of those times I have the best hand, and 6 of those times I am bluffing. Therefore, the odds against my bluffing are exactly 3-to-1. The pot is $200, and when I bet, there is $300 in the pot. Thus, your pot odds are also 3-to-1. You are calling $100 to win $300. Now when the odds against my bluffing are identical to the odds you are getting from the pot, it makes absolutely no difference whether you call or fold. Furthermore, whatever you do, you will still lose exactly $600 after 42 hands. If you were to fold every time I bet, I would beat you out of $100 24 times when I bet and lose $100 to you 18 times, when I don't bet, for a profit of $600. If you were to call me every time, you would beat me out of $200 six times when I'm bluffing and $100 18 times, when I don't bet, for a total of $3,000; but I would beat you out of $200 18 times when I bet with my good hands for a total of $3,600. Once again my profit is $600. So other than being a psychic, there is no way in the world you can prevent me from winning that $600 per 42 hands, giving me a positive expectation of $14.29 per hand. Bluffing exactly 6 times out of 24 has turned a hand that was a 4-to-3 underdog when I didn't bluff at all into a 4-to-3 favorite - no matter what strategy you use against me.

We can now move to the heart of game theory and bluffing. Notice first that the percentage of bluffing I did was predetermined- one time every 19 bets or 5 times every 23 bets or 7 times every 25 bets. Notice secondly that my bluffing was completely random; it was based on certain key cards I caught, which my opponent could never see. He could never know whether the card I drew was one of my 18 good cards or a bluff card. Finally, notice what happened when I bluffed with precisely six cards - which made the odds against my bluffing in this particular instance identical to the pot odds my opponent was getting. In this unique case my opponent stood to lose exactly the same amount by calling or folding.

This is optimum bluffing strategy - it makes no difference how your opponent plays. We can say, then, that if you come up with a bluffing strategy that makes your opponent do equally badly no matter how he plays, then you have an optimum strategy. And this optimum strategy is to bluff in such a way that the odds against your bluffing are identical to the odds your opponent is getting from the pot. In the situation we have been discussing, I had 18 good cards, and when I bet my $100, creating a $300 pot, my opponent was getting 3-to-1 odds from the pot. Therefore, my optimum strategy was to bluff with six additional cards, making the odds against my bluffing 3-to-1, identical to the pot odds my opponent was getting.

Let's say the pot was $500 instead of $200 before I bet. Once again I had 18 winning cards, and my opponent could only beat a bluff. The bet is $100, and so my opponent would be getting $600-to-$100 pot odds when he called. Now my optimum strategy would be to bluff with 3 cards. With 18 good cards and 3 bluffing cards, the odds against my bluffing would be 6-to-l, identical to the pot odds my opponent would be getting to call my bet. If the pot were $100 and I bet $100, I'd have to bluff with 9 cards when I had 18 good cards, making the odds against my bluffing identical to the 2-to-1 odds my opponent would be getting from the pot.

It is important to realize that when the results are the same whether your opponent calls or folds, you will still average the same no matter how that opponent mixes up his calls and folds. Returning to the initial optimum strategy example, where I make a $100 bluff with 6 cards and bet 18 good cards into a $200 pot, I will still average $600 in profits per 42 hands in the long run whether my opponent calls 12 times and folds 12 times or calls 6 times and folds 18 times, or whatever. The inability of a player to find any response to offset his disadvantage is the key to game theory problems, though most game theory books don't put it in this form.
Bluffing on the basis of game theory can also be described in terms of percentages. Suppose you have a 25 percent chance of making your hand, the pot is $100, and the bet is $100. Thus, if you bet, your opponent is getting 2-to-1 odds from the pot. Since there is a 25 percent chance of making your hand, there should be a 121/2 percent chance you are bluffing to create the 2-to-1 odds against your bluffing, which is the optimum strategy. For example, in draw lowball there are 48 cards you do not see when you draw one card, and we'll assume 12 of them (25 percent) will make your hand. So you should pick 6 other cards (12'/z percent) out of the 48 to use for a bluff.

You pick cards, of course, to randomize your bets. Without the random factor, the good opponents against whom you use game theory to bluff would quickly pick up your pattern and destroy you. The beautiful thing about game theory is that even if your opponent knows you are using it, there is nothing he can