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
 |  | ![p^a[1+a/1q+(a(a+1))/(2!)q^2+...+(a(a+1)...(a+b-2))/((b-1)!)q^(b-1)]](/images/equations/SharingProblem/Inline20.svg) |
(1)
|
 |  | ![q^b[1+b/1p+(b(b+1))/(2!)p^2+...+(b(b+1)...(b+a-2))/((a-1)!)p^(a-1)]](/images/equations/SharingProblem/Inline23.svg) |
(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).
