In this page 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 page 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.

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.

However, if I only bluff some of the time, the situation is much different. Suppose I were to bluff only when I caught the king of spades. In other words, I would bet whenever I caught any of my 18 good cards and also when I caught the king of spades. If I bluffed this infrequently, your proper play would still be to fold when I bet because the odds against my bluffing are 18-to-1. But notice how this improves my position. Bluffing when I catch the king of spades still doesn't give me a profit, but it allows me to win 19 times instead of 18 and lose only 23 times instead of 24. That single bluff once out of 19 times has begun to close the gap between your status as a favorite and mine as an underdog. Notice too that you have no way of knowing when I am bluffing since I am randomizing my bluffs by using a card, an object as inanimate as the coin in the odds-evens game, to make my bluffing decision for me.

If bluffing with one card makes me less of an underdog than never bluffing at all, suppose I choose two - say, the king of spades and the jack of spades. Once again your correct play is to fold when I bet. But now you win only 22 times when I don't bet, and I win 20 times when I do. Assuming you have no way of knowing when I'm bluffing and when I'm not, my using just two key cards to bluff, in addition to my 18 good cards, has reduced you from 24-to-18 favorite to a 22-to-20 favorite - that is, from a 4-to-3 favorite to an 11-to-10 favorite. This bluffing seems to have possibilities. Suppose instead of two cards, I picked five key cards - the king of spades and all four jacks. That means I would be betting 23 times - 18 times with the best hand and five times on a bluff. Now all of a sudden you are in a bad situation with your pat 9,8 because you have to guess whether I'm bluffing when I bet. I could even tell you precisely the strategy I am using, but you would still have to lose your money.

What would happen? You know there are 18 cards that will make me my hand and five other cards I will bluff with. Thus, the odds are 18-to-5 or 3.6-to-1 against my bluffing. With the $200 in antes and my $100 bet, the pot is $300. So you are getting 3-to-1 odds from the pot. You cannot profitably call a 3.6-to-1 shot when you stand to win only 3-to-1 for your money. Lo and behold, by using five cards to bluff with, I win that pot from you 23 out of 42 times, and you win it only 19 times. I make a profit of $400. Thus, my occasional random bluffing has swung a hand that is a 24-to-18 underdog into a 23-to-19 favorite. To assure yourself there is no arithmetical sleight of hand here, you can work out what happens if you call every time I bet. You will win $200 from me the five times I am bluffing and $100 from me the 19 times I don't bet, for a total of $2,900. But you will lose $200 to me the 18 times I have the best hand for a total of $3,600. Your net loss when you call is $700, which is $300 more than you lose if you simply fold when I bet.

Had I picked seven cards to bluff with instead of five, the odds would then be 18-to-7 against my bluffing, and since the pot odds you're getting are 3-to-1, you would be forced to call when I bet. However, you would still end up losing! Seven times, when I'm bluffing, you would win $200 from me for a total of $1,400 and the 17 times I don't bet at all you would win $100 from me for a total of $1,700. Your wins after 42 hands would total $3,100. But I would win $200 from you the 18 times I bet with my good cards for a total of $3,600, giving me a net profit and you a net loss of $500 after 42 hands. It should be pointed out - once again to make it clear there are no tricks to this arithmetic - that you would lose even more money if you folded every time I bet with my 18 good cards and seven bluffing cards. You would win $100 from me the 17 times I don't bet, while I would win $100 from you the 25 times I do. Your net loss would now be $800 instead of $500.