“Sharing Problem”的概念、定义、习题解题方法及翻译-数学辞典

单词

Sharing Problem

释义

Sharing Problem

A problem also known as the Points Problem or Unfinished Game. Consider a tournament involving playersplaying the same game repetitively. Each game has a single winner, and denote the number of games won by player atsome juncture . The games are independent, and the probability of the th player winning a game is . Thetournament is specified to continue until one player has won games. If the tournament is discontinued before anyplayer has won games so that for , ..., , how should the prize money be shared in order todistribute 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 bethe 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, thegeneral Dirichlet integral

must be used, where

(5)

and

, then

and

reduce to

as they must. Let

be the jointprobability that the players would be Ranked in the order of the

s in the argument list if the contest werecompleted. For

(6)

For

with quota vector

and


			(7)

An expression for

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

See also Dirichlet Integrals

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.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。