Sharing Problem

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

For player , call the number of games left to win  the "quota." For two players, let  and  be the probabilities of winning a single game, and  and  be the number of games needed for each player to win the tournament. Then the stakes should be divided in the ratio , where

			(1)
			(2)

(Kraitchik 1942).

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

(3)

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

(4)

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

(5)

If  and , then  and  reduce to  as they must. Let  be the joint probability that the players would be statistically ranked in the order of the s in the argument list if the contest were completed. For ,

(6)

For  with quota vector  and ,

(7)

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

REFERENCES:

Kraitchik, M. "The Unfinished Game." §6.1 in Mathematical Recreations. New York: W. W. Norton, pp. 117-118, 1942.

Sobel, M. and Frankowski, K. "The 500th Anniversary of the Sharing Problem (The Oldest Problem in the Theory of Probability)." Amer. Math. Monthly 101, 833-847, 1994.