A Little Bit About Games

Abstract

In this chapter, we discuss several games; more along this line will follow in the next chapter. The first problem, solved by both Pascal and Fermat, goes back to the earliest days of probability as a formal theory. Suppose two people are playing a game with the winner receiving prize money at the end. If the game is forced to end before either player wins, how should the prize money be divided between the players? Pascal introduced the principle that the prize money should be divided in proportion to each player’s conditional probability of winning if the game were to be continued, given the score when the game is forced to end. Suppose, for example, that the plays of the game constitute a sequence of Bernoulli trials where A wins a point with probability p (success) and B wins a point with probability 1— p (failure), and n points are needed to win. We will not derive the general formula but will give the solution for the case where A has n — 1 points and B has n — 2 points. Then A needs one point to win and B needs two points. A can win in two ways if the game were to be continued at this moment: (1) A can win the next point, and (2) B can win the next point and A can win the succeeding point. This gives the value p + p(1 — p) for the conditional probability of A winning. If p = 1 — p =.5 and the purse is $100, then, according to Pascal’s principle, A should receive $75 and B $25.