MA3662 Exercise Sheet 4

Question 1

Consider the zero-sum game where player 1 has matrix $$A=\left(\begin{array}{rrrr} -2 & 3 & 5 & -2\\ 3 & -4 & 1 & -6\\ -5 & 3 & 2 & -1\\ -1 & -3 & 2 & 2\end{array}\right).$$ It is claimed that the strategies $\underline{x}=(\frac{1}{9},0,\frac{8}{9},0)$ and $\underline{y}=(0,\frac{7}{9},\frac{2}{9},0)$ are optimal.

1. Is this correct? Give reasons for your answer.

2. If $\underline{x}=(\frac{13}{33},\frac{5}{33},0,\frac{15}{33})$ is optimal and $v(A)=-\frac{26}{33}$ then find $\underline{y}$.

Question 2

In a game of baseball, a batter (player 1) expects the pitcher (player 2) to throw a fastball, slider, or curveball. We model this as a zero sum game with payoff matrix for player 1 given by $$A=\left(\begin{array}{rrr} 0.30 & 0.25 & 0.20 \\ 0.26 & 0.33 & 0.28 \\ 0.28 & 0.30 & 0.33 \end{array}\right).$$

1. Determine the gain floor and loss ceiling of this game.

2. It turns out that an optimal strategy for player 1 is given by $\underline{x}=(\frac{2}{7},0,\frac{5}{7})$, and the value of the game is $\frac{2}{7}$. What is the optimal strategy for player 2?

Question 3

Consider the zero-sum game where player 1 has matrix $$A=\left(\begin{array}{rrr} 4 & 3 & 1 \\ 2 & 2 & 2 \\ 1 & 4 & 3 \\ 0 & 3 & 3 \end{array}\right).$$

1. Determine the gain floor and loss ceiling of this game.

2. Use strict dominance with a suitable linear combination of strategies to eliminate one row from this matrix to form a new game.

3. Use the Equality of Payoffs Theorem to determine whether it is possible to have a mixed Nash equilibrium for this new game where player one’s strategy is of the form $(a,b,c)$ with $a,b,c>0$.