Sean Pegado

Link to article: http://arxiv.org/pdf/0911.3100

$\\$ In his paper "The Independent Chip Model and Risk Aversion," George Gilbert at Texas Christian University looks into expected values of poker tournaments. Specifically, he looks at chip-based tournaments, in which every player begins with an equal number of chips, and a player is eliminated when he runs out of chips. The last remaining player wins first place, the second-to-last wins second place, and so on. The game is complicated, however, by the fact that although when the game ends the winning player has all chips, he only retains a portion of those chips: the second place player gets a portion of the money, the third place gets a portion, smaller than second place, and so on. Essentially, the winning player must relinquish a portion of his chips once the game is over. 

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The paper uses the Independent Chip Model (ICM) to model poker tournaments. Under the ICM, a player's probability of winning is equal to the fraction of total chips in play he is holding. Two technical lemmas are introduced, which follow through induction, the product rule of derivatives, and a fair amount of algebra. 

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The first main theorem proves a surprising result: in a tournament as described above, any fair bet between exactly two players in which only one can gain or lose chips in the current hand will lower both players' expected prize money. This is shown by breaking down the "winner take all" tournament into $n$ sub-tournaments (for number of players $n$): one a winner-take-all paying a sum to first place, one paying a sum to the first and second place finishers, one paying a sum to the first, second and third place finishers, and so on. It is shown that there will be no expected gain in any of these sub-tournaments, and that at least one will have an expected loss. Thus the overall expected value will fall.

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The second main theorem of the paper mirrors the first, and focuses on the non-betting players in the game. The theorem states that the expected prize money of any player not involved in a fair bet between two players will increase. The proof too is similar to the first theorem, breaking down a non-betting player's expected winnings into a sum of expected winnings of smaller sub-tournaments. If $A$ and $B$ are betting against each other and $C$ is a non-betting player, it is shown that the expected winnings of $C$ is unaffected by a fair bet, or when neither $A$ nor $B$ finishes ahead of $C$. Curiously, $C$'s expected winnings increase when $B$ finishes ahead of both $A$ and $C$. This is shown by breaking down this final scenario into further sub-scenarios.

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This is not a perfect theorem: it does not hold for every possible game: "Theorem 2 holds for tournaments paying only two places for fair bets among three or more players, but is false in general."

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This is a very interesting paper nonetheless! The introduction provides some intuition for understanding the results: since the winning player must give back some of his chips (for the second place prize, the third place prize, etc.), his best strategy is to play in a somewhat risk averse manner. Not participating in bets thus increases his expected winnings.