Problem : Consider a constant-sum game with payoff matrix $$\left( \begin{array}{ccc} 0 & 2 & -1 \\ 1 & -2 & 0 \\ -2 & 0 & 2 \end{array} \right)$$.

• Find $v^-$ and $v^+$ for this game.
• Does this game have an equilibrium in pure strategies?
• Compute the value of $E(X,Y)$ for $X=(0 {1 \over 2} {1 \over 2})$ and $Y=({2 \over 3} 0 {1 \over 3})$.
• Using Theorem 1.3.8c, verify that $X^*=({2 \over 5} {2 \over 5} {1 \over 5})$,$Y^*=({2 \over 5} {1 \over 5} {2 \over 5})$ is an equilibrium for this game.
• Find player I's best response to $Y=({1 \over 3} {1 \over 3} {1 \over 3})$.

Problem : Find an equilibrium and its value for the constant-sum game with payoff matrix $$ \left( \begin{array}{cc} 3 & 2 \\ 0 & 4 \\ \end{array} \right) $$

Problem : Set up the linear program to find an equilibrium for the constant-sum game with payoff matrix $$ \left( \begin{array}{ccc} 4 & 7 & 3 \\ 0 & 5 & 4 \\ 10 & 6 & 2 \\ \end{array} \right) $$