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
If players have equal probability of winning (``cell probability''), then the chance of player winning for quotas ,
See also Dirichlet Integrals
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.
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.
© 1996-9 Eric W. Weisstein