2021-58.md

course	course_year	question_number	tags	title	year
Optimization	IB	58	IB 2021 Optimization	Paper 3, Section II, H	2021

Explain what is meant by a two-person zero-sum game with $m \times n$ payoff matrix $A$, and define what is meant by an optimal strategy for each player. What are the relationships between the optimal strategies and the value of the game?

Suppose now that

$$A=\left(\begin{array}{cccc} 0 & 1 & 1 & -4 \\ -1 & 0 & 2 & 2 \\ -1 & -2 & 0 & 3 \\ 4 & -2 & -3 & 0 \end{array}\right)$$

Show that if strategy $p=\left(p_{1}, p_{2}, p_{3}, p_{4}\right)^{T}$ is optimal for player I, it must also be optimal for player II. What is the value of the game in this case? Justify your answer.

Explain why we must have $(A p)_{i} \leqslant 0$ for all $i$. Hence or otherwise, find the optimal strategy $p$ and prove that it is unique.