C. L. Anderson has used Fermat's argument for the problem of points to prove the following result due to J. G. Kingston. You are playing the game of points (see Exercise Exercise) but, at each point, when you serve you win with probability [math]p[/math], and when your opponent serves you win with probability [math]\bar{p}[/math]. You will serve first, but you can choose one of the following two conventions for serving: for the first convention you alternate service (tennis), and for the second the person serving continues to serve until he loses a point and then the other player serves (racquetball). The first player to win [math]N[/math] points wins the game. The problem is to show that the probability of winning the game is the same under either convention.

• Show that, under either convention, you will serve at most [math]N[/math] points and your opponent at most [math]N - 1[/math] points.
• Extend the number of points to [math]2N - 1[/math] so that you serve [math]N[/math] points and your opponent serves [math]N - 1[/math]. For example, you serve any additional points necessary to make [math]N[/math] serves and then your opponent serves any additional points necessary to make him serve [math]N - 1[/math] points. The winner is now the person, in the extended game, who wins the most points. Show that playing these additional points has not changed the winner.
• Show that (a) and (b) prove that you have the same probability of winning the game under either convention.

Notes