## Bluffing Against Two or More Opponents

It is rarely correct to try to bluff out two or more people when all the cards are out; your chances of success decrease geometrically with each additional player in the pot. Paradoxically you might have a profitable bluffing opportunity against each of two opponents individually, but not against both of them as a group. Suppose, for example, you are heads-up on the end in a \$10-\$20 game. There is \$80 in the pot, and you think you can get away with a bluff one out of three times. Clearly this is an extremely profitable bluffing situation. Once you will win \$80, and twice you will lose \$20 for a net profit of \$40 or an average profit of \$ 13.3 3 per bet.

Now suppose you are in the identical situation except that you are up against two players instead of one. We'll assume each player has put \$40 in the pot to expand it to \$120, and you think, as in the former case, that each opponent will fold one time out of three. You are now getting 6-to-l instead of 4-to-l from the pot. Nevertheless, an attempt at a bluff is no longer profitable because the probability that both of your opponents will fold is V3 X V3, which equals 1/9. In other words, eight times out of nine one or the other of your opponents will call on average. So you stand to lose \$20 eight times for a total of \$ 160 and to win \$ 120 once. Your net loss is \$40 or \$4.44 per bet. Thus, opposing each individual player by himself results in a profitable bluffing situation, but if they're both in against you, you have gone from a profitable situation to an unprofitable one.

(It should be pointed out that in most bluffing situations against more than one player the probabilities that each player will fold are not independent. The player in the middle will frequently fold a hand that he would call with if he was last, and sometimes the player who is last will call with a hand he would have folded without hesitation had he been in the middle, expecting the player behind him to call. Nevertheless, the general principle still holds that it is usually more profitable to try to bluff one player out of a pot containing 2X dollars than to bluff two players out of a pot containing 3X dollars.)