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 (x1, 1-x1), 0 ≤ x1 ≤ 1, and Player II uses
mixed strategy (y1, 1-y1), 0 ≤ y1 ≤ 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 y1 is being fixed to y1=0.3, find the value(s)
of x1 that maximizes E(x,y) for fixed y1. 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.