Sharing Problem

A problem also known as the points problem or unfinished game. Consider a tournament involving k players playing the same game repetitively. Each game has a single winner, and denote the number of games won by player i at some juncture w_i. The games are independent, and the probability of the ith player winning a game is p_i. The tournament is specified to continue until one player has won n games. If the tournament is discontinued before any player has won n games so that w_i<n for i=1, ..., k, how should the prize money be shared in order to distribute it proportionally to the players' chances of winning?

For player i, call the number of games left to win r_i=n-w_i>0 the "quota." For two players, let p=p_1 and q=p_2=1-p be the probabilities of winning a single game, and a=r_1=n-w_1 and b=r_2=n-w_2 be the number of games needed for each player to win the tournament. Then the stakes should be divided in the ratio m:n, where

(Kraitchik 1942).

If i players have equal probability of winning ("cell probability"), then the chance of player i winning for quotas r_1, ..., r_k is

where D is the Dirichlet integral of type 2D. Similarly, the chance of player i losing is

where C is the Dirichlet integral of type 2C. If the cell quotas are not equal, the general Dirichlet integral D_(a) must be used, where

If r_i=r and a_i=1, then W_i and L_i reduce to 1/k as they must. Let P(r_1,...,r_k) be the joint probability that the players would be statistically ranked in the order of the r_is in the argument list if the contest were completed. For k=3,

For k=4 with quota vector r=(r_1,r_2,r_3,r_4) and Delta=p_2+p_3+p_4,

An expression for k=5 is given by Sobel and Frankowski (1994, p. 838).

See also

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References

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Cite this as:

Weisstein, Eric W. "Sharing Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SharingProblem.html

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