$16.$Consider the following two$-$person zero$-$sum game. Assume the two players have the same two strategy options. The payoff table shows the gains for Player A$.$ Player B Player A Strategy$$b$1$ Strategy$$b$2$ Strategy$$a$148$ Strategy$$a$2115$ Determine the optimal strategy for each player. What is the value of the game?

$17.$Consider the following two$-$person zero$-$sum game. Assume the two players have the same three strategy options. The payoff table shows the gains for Player A$.$ Player B Player A Strategy b$1$ Strategy b$2$ Strategy b$3$ Strategy$$a$1652$ Strategy$$a$2103$ Strategy$$a$3343$ Is there an optimal pure strategy for this game? If so$,$ what is it$?$ If not, can the mixed$-$strategy probabilities be found algebraically?

$18.$Consider the following two$-$person zero$-$sum game. Assume the two players have the same three strategy options. The payoff table below shows the gains for Player A$.$ Player B Player A Strategy b$1$ Strategy b$2$ Strategy b$3$ Strategy$$a$1324$ Strategy$$a$2102$ Strategy$$a$3453$ Is there an optimal pure strategy for this game? If so$,$ what is it$?$ If not, can the mixed$-$strategy probabilities be found algebraically? What is the value of the game?

$19.$Two banks $($Franklin and Lincoln$)$ compete for customers in the growing city of Logantown. Both banks are considering opening a branch office in one of three new neighborhoods: Hillsboro, Fremont, or Oakdale. The strategies, assumed to be the same for both banks, are: Strategy $1$: Open a branch office in the Hillsboro neighborhood. Strategy $2$: Open a branch office in the Fremont neighborhood. Strategy $3$: Open a branch office in the Oakdale neighborhood. Values in the payoff table below indicate the gain $($or loss$)$ of customers $($in thousands$)$ for Franklin Bank based on the strategies selected by the two banks. Lincoln Bank $$ Hillsboro Fremont Oakdale Franklin Bank b$1$ b$2$ b$3$ Hillsboro$$ a$1423$ Fremont$$a$2623$ Oakdale$$ a$3105$ Identify the neighborhood in which each bank should locate a new branch office. What is the value of the game?

$20.$Consider the following problem with four states of nature, three decision alternatives, and the following payoff table $($in $$\text{'}$s$)$: $$ s$1$ s$2$ s$3$ s$4$ d$12002600-1400200$ d$20200-200200$ d$3-2004000200$ The indifference probabilities for three individuals are: Payoff Person $1$ Person $2$ Person $3$ $ $26001\mathrm{.}001\mathrm{.}001\mathrm{.}00$ $$400\mathrm{.}40\mathrm{.}45\mathrm{.}55$ $$200\mathrm{.}35\mathrm{.}40\mathrm{.}50$ $$0\mathrm{.}30\mathrm{.}35\mathrm{.}45-$$$200\mathrm{.}25\mathrm{.}30\mathrm{.}40-$$$1400000$ a$.$Classify each person as a risk avoider, risk taker, or risk neutral. b$.$For the payoff of $$400,$ what is the premium the risk avoider will pay to avoid risk?$$What is the premium the risk taker will pay to have the opportunity of the high payoff? c$.$Suppose each state is equally likely.$$What are the optimal decisions for each of these three people?