The problem of points

[Gavran]

Two players, A and B, are playing a game, and the probability that either player wins a round is 1/2. They contribute equally to a prize pot. The game ends when one of the players wins six rounds. The game is interrupted for some reason when player A wins 5 rounds and player B wins 3 rounds. How should the prize pot be divided?

The first solution
The prize pot should be divided equally because nobody wins the prize pot. The winner of the prize pot is the player who wins 6 rounds, and there is no player who wins 6 rounds.

The second solution
The whole prize pot should be awarded to player A, while nothing should be awarded to player B because player A has a higher score than player B.

The third solution (Fra Luca Bartolomeo de Pacioli (1445-1517))
The prize pot should be divided in proportion to the number of rounds won by each player; therefore, the share of the prize pot awarded to player A should be 5/8 of the prize pot, while the share of the prize pot awarded to player B should be 3/8 of the prize pot.

The fourth solution
This solution relies on the remaining rounds. The prize pot should be divided in opposite proportion to the number of rounds that each player should win to reach the number of 6 winning rounds. Therefore, the share of the prize pot awarded to player A should be 3/4 of the prize pot, while the share of the prize pot awarded to player B should be 1/4 of the prize pot.

The fifth solution (Blaise Pascal (1623-1662), Pierre de Fermat (1601-1665))
This solution relies on what could happen in the remaining rounds. The prize pot should be divided in proportion to the probability of winning the remaining rounds. Player B needs to win 3 consecutive remaining rounds to win the prize pot (the probability equals 1/8), while to win the prize pot, player A needs to win the first remaining round (the probability equals 1/2), or he needs to lose the first remaining round and win the second remaining round (the probability equals 1/4), or he needs to lose the first remaining round, lose the second remaining round, and win the third remaining round (the probability equals 1/8). It is evident that player A should receive 1/2+1/4+1/8=7/8 of the prize pot, while player B should receive 1/8 of the prize pot.

The fifth solution is the most acceptable solution, but what about other solutions? Should they just be ignored, particularly in the case when the information that the probability that either player wins a round is 1/2 is excluded from the problem statement?

 

[pbuk]

The fifth solution is the most acceptable solution

If an unforeseen condition arises in a game between two players then only those two players can determine what is the most acceptable solution to them. Depending on the context, any of these solutions, or even some other solution (see below) may be the most acceptable solution.

in the case when the information that the probability that either player wins a round is 1/2 is excluded from the problem statement?

In this case there is no foundation for the calculation in the fifth solution at all and so it is unlikely to be acceptable.

I give below two solutions you have not considered which in practice may be the most acceptable to the players (and I have seen each of these happen in similar situations in the real world):

Solution A: The conditions for completing the game and therefore determining the distribution of the prize pot have not been met and so the pot will not be distributed but will instead be returned to whoever contributed to it (note that in this case the outcome will be the same as in the OP Solution 1, however the different logic is important and can also be generalized to other situations e.g. where the prize pot is provided by a third party).

Solution B: The conditions for completing the game and therefore determining the distribution of the prize pot have not been met and so the pot will not be distributed but will instead be donated to a third party, e.g. a charity.

 

[DaveC426913]

If an unforeseen condition arises in a game between two players then only those two players can determine what is the most acceptable solution to them. Depending on the context, any of these solutions, or even some other solution (see below) may be the most acceptable solution.

I host games here and there (darts, cards) and I keep running up against discrepancies for "what should happen in scenario X".

My solution is house rules. Specifically: it doesn't matter so much what the solution is, so long as it has been made clear beforehand. In theory, anyone playing is aware of the house rules, and therefore agrees to them implicitly by playing. If they aren't happy with it, the time to challenge it is before playing.

Another way of framing this is: a given solution may be unfair, but it is an equal opportunity unfairness. It is fairly unfair.