Problem $13.$ A decision tree $is$ a graphical representation that employs a branching structure $to$ depict all potential

outcomes $of$ a decision. This visual tool helps provide a clear and organized representation $of$ a decision problem.

Consider the decision tree presented below, complete with associated probabilities and payoffs:

$(a)$ Determine the optimal decision strategy that maximizes the expected payoff.

None $of$ the other four

$If{S}_1,$ choose $d_{11}.If{S}_2,$ choose $d_{22}.$

$If{S}_1,$ choose $d_{12}.If{S}_2,$ choose $d_{21}.$

$If{S}_1,$ choose $d_{12}.If{S}_2,$ choose $d_{22}.$

$If{S}_1,$ choose $d_{11}.If{S}_2,$ choose $d_{21}.$

$(b)$ What $is$ the highest expected payoff achievable $by$ following the optimal strategy identified $in$ part $(a)?$

None $of$ the other four

$$5\mathrm{.}7$

$$7\mathrm{.}0$

$$7\mathrm{.}2$

$$5\mathrm{.}4$