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A Tournament AnomalyThis section concerns a mathematical paradox that arises in freeze-out poker tournaments that play down to one player but pay prize money to the last few players or winner of the poker game eliminated. To illustrate this, let’s say there is a tournament that pays sixty percent for first place, thirty percent for second place, and ten percent for third winning poker player. Now let’s say there are only three players left in the tournament and each one has exactly $ 1,000 in front of him. With $ 3,000 in the tournament this means first place is worth $ 1,800, second is worth $ 900, and the third finisher receivers $ 300. 1,800 = (.60) (3,000) However, if they are all equal players, then each player at this point should expect to win an average of $ 1,000. 1,000 = (1) (1,800) + (1) (900) + (1) (300) Suppose at this point that players A and B get all in against each other while player C sits out. Suppose also that before the last card it is determined that it is exactly 50-50 as far as which of the two hands will win. Let us now determine player A’s expectation for the tournament. One half of the time he will lose the pot odds and thus get the $ 300 third prize. The other half of the time he will win the pot and thus be a 2-to-1 favorite (since he now has $ 2,000) to beat player C heads –up. He will therefore come in third one-half of the time, second one-sixth of the time, and first one-third of the time. His expectation is now $ 900. 900 = (1) (300) + (1) (900) + (1) (1,800) The same calculation would hold true for player B. But, how can this be? Both players A and B started their hand with an expectation of $ 1,000. They are now gambling on a dead–even proposition to play . Yet somehow this even gamble seems to be costing both of them money in the long run. Where has this extra $ 200 gone? Has player C somehow gotten it just by sitting out the pot? Well, let’s figure it out. Regardless of the outcome of the contest between A and B, player C will find himself a 2-1 underdog to the winner. His (1) (1,800) + (2) (900) This comes out to $ 1,200. He does indeed pick up $ 200 “equity” by just watching. It is up to the reader to decide how to best use this insight when considering tournament play.
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