The interrupted game (problem of points)

Question

Two players, A and B, play a series of fair, independent rounds (each round won by A or B with probability $\tfrac12$ ). The first to a target score wins the whole pot. The game is halted early: at that moment A needs $2$ more round wins and B needs $3$ more to take the pot.

What fraction of the pot should A receive, that is, what is the probability A would have won?

Accepted Answer

This is Pascal and Fermat's "problem of points," and it uses the Banach idea of extending play to a fixed horizon and counting binomially. A needs 2 wins and B needs 3, so the match is certain to be decided within 2 + 3 - 1 = 4 more rounds (after 4 rounds, someone must have reached their target). Imagine playing all 4 phantom rounds regardless. Then: - A wins the match if A takes at least 2 of these 4 rounds (that reaches A's target). - Otherwise A takes 0 or 1, meaning B took at least 3, so B wins. With fair rounds, the number A wins is Binomial(4, 	frac12), so P(A 	ext{ wins}) = P(	ext{B…

The interrupted game (problem of points)

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