SECTION $11\mathrm{.}4$ PROBLEM SET: CHAPTER REVIEW

Determine whether the games are strietly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game.

a$)\left[\begin{array}{cc}2& 3\\ 3& 4\end{array}\right]$

b$)\left[\begin{array}{ccc}0& 3& -1\\ 1& 3& -2\\ -1& 2& -5\end{array}\right]$

c$)\left[\begin{array}{ccc}3& 2& -1\\ 5& 3& 4\end{array}\right]$

d$)\left[\begin{array}{cc}4& 2\\ -1& 3\\ 4& 3\\ 1& -3\end{array}\right]$

Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than $10$ cents, Player I gets both the coins; otherwise, Player II gets both the coins.

a$)$ Write a payoff matrix for Player I.

b$)$ Find the optimal strategies for each player and the value of the game.

Lacy's department store is thinking of having a major sale in the month of February, but does not know if its competitor store Hordstrom's is also planning one. If Lacy's has a sale and Hordstrom's does not, Lacy's sales go up by $30\%,$ but if both stores have a sale simultaneously, Lacy's sales go up by only $5\%.$ On the other hand, if Lacy's does not have a sale and Hordstrom's does, Lacy's loses $5\%$ of its sales to Hordstrom's, and if neither of the stores has a sale, Lacy's experiences no gain in sales.

a$)$ Write a payoff matrix for Lacy's.

b$)$ Find the optimal strategies for both stores.

Mr$.$ Halsey has a choice of three investments: Investment $A,$ Investment $B,$ and Investment $C.$ If the economy booms, then Investment A yields $14\%$ return, Investment B returns $8\%,$ and Investment $C11\%.$ If the economy grows moderately, then Investment A yields $12\%$ return, Investment B returns $11\%,$ and Investment $C11\%.$ If the economy experiences a recession, then Investment A yields a $6\%$ return, Investment B returns $9\%,$ and Investment C $10\%.$

a$)$ Write a payoff matrix for Mr$.$ Halsey.

b$)$ What would you advise him?

Mr$.$ Thaggert is trying to decide whether to invest in stocks or in CD$\text{'}$s$($Certificate of deposit$).$ If he invests in stocks and the interest rates go up$,$ his stock investments go down by $2\%,$ but he gains $1\%$ in his CD$\text{'}$s$.$ On the other hand if the interest rates go down, he gains $3\%$ in his stock investments, but he loses $1\%$ in his CD$\text{'}$s$.$

a$)$ Write a payoff matrix for Mr$.$ Thaggert.

b$)$ If you were his investment advisor, what strategy would you advise?