Sharing Problem
المؤلف:
Kraitchik, M
المصدر:
"The Unfinished Game." §6.1 in Mathematical Recreations. New York: W. W. Norton
الجزء والصفحة:
...
9-11-2021
1186
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
(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 
,
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.
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