This question illustrates the following theorem.

**If one player of a 2-person zero-sum game employs a fixed strategy, then ****the opponent has an optimal counter strategy that is pure. In other words, ****if Player I knows that Player II is using a particular mixed strategy y, ****then Player I can maximize their expected payoff by using a pure strategy, ****and vice versa.**

a. Work out the following.

i. Find the expected payoff, *E*(*x*,*y*), to Player I if Player I

uses mixed strategy (*x*1, 1-*x*1), 0 ≤ *x*1 ≤ 1, and Player II uses

mixed strategy (*y*1, 1-*y*1), 0 ≤ *y*1 ≤ 1, in the 2-person zerosum

game with the following payoff matrix.

6 , 1

−2 , 3

(this is a 2x2 matrix)

ii. Now, think of *y*1 is being fixed to *y*1=0.3, find the value(s)

of *x*1 that maximizes *E*(*x*,*y*) for fixed *y*1. Make your case

that, in general, if Player I can use a pure strategy to get

the maximum expected payoff, given that Player I knows

Player II’s strategy.

b. Suppose that Player II knew that Player I is to use a mixed

strategy (0.3, 0.1, 0.6) in the game with the following payoff

matrix. Work out the three expected payoffs if Player II uses a

pure strategy and hence deduce the best strategy for Player II.

⎡ 0 1 −2 ⎤

−1 0 1

⎣ 2 −1 0 ⎦

(this is a 3x3 matix, sorry for the formatting)

Using the same game given in Question b:

Write down the linear programming problems induced by the

game.