Game Theory: An Introduction, 2nd Edition

CHAPTER TWO

Solution Methods for Matrix Games

I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong,  . . . ; but time and chance happeneth to them all.

—Ecclesiastes 9:11

2.1 Solution of Some Special Games

Graphical methods reveal a lot about exactly how a player reasons her way to a solution, but it is not a very practical method. Now we will consider some special types of games for which we actually have a formula giving the value and the mixed strategy saddle points. Let’s start with the easiest possible class of games that can always be solved explicitly and without using a graphical method.

2.1.1 2 × 2 GAMES REVISITED

We have seen that any 2 × 2 matrix game can be solved graphically, and many times that is the fastest and best way to do it. But there are also explicit formulas giving the value and optimal strategies with the advantage that they can be run on a calculator or computer. Also the method we use to get the formulas is instructive because it uses calculus.

Each player has exactly two strategies, so the matrix and strategies look like

For any mixed strategies, we have E(X, Y) = X AYT, which, written out, is

Now here is the theorem giving the solution of this game.

Theorem 2.1.1 In the 2 × 2 game with matrix A, assume that ...