Sequential Games

Dr. Kyle Siegrist

Department of Mathematical SciencesUniversity of Alabama in Huntsville

February 1, 2002, 3:00 PM (coffee and cookies at 2:30)

Abstract

Suppose that k players play a sequence of random points (each point is won by exactly one player). A sequential game has two components:

1. stopping time that specifies when the game ends.
2. decision rule that specifies the winner of the game.

The games, sets, and match in tennis are examples of sequential games, as are the standard "best 3 of 5" and "best 4 of 7" tournaments used in baseball and basketball.  Sequential games also have important applications in statistics.  Usually we will assume that the points are independent and identically distributed (IID).

We will give a general construction of sequential games and discuss some interesting properties such as coherence (the sequential game really makes sense), symmetry (the players are treated the same way), and independence (between the number of points and the winner).  We will also give a general construction of composition, the most common way that sequential games are combined to form new games.  We will note a number of analogies between sequential games and structural reliability.

In the case of two players playing IID points, we will reveal the the class of sequential games that are optimal, in a certain statistical sense, and discuss the efficiency of sequential games (a method of comparing a sequential game with an optimal one).  Finally, we will show that the class of optimal games has a number of very nice mathematical properties, including the independence property and closure under composition.

This is joint work with John Steele.