Let's tackle this problem step$-$by$-$step:

Part $($a$)$: Constructing the Decision Tree

The decision tree structure will follow this sequence:

Hale decides whether to produce the pilot $($d$1)$ or sell to a competitor $($d$2).$

If Hale chooses to produce the pilot, the network's response will be one of three outcomes: reject, purchase for $1$ year, or purchase for $2$ years.

If Hale chooses to use the agency's review, we branch according to the favorable $($F$)$ or unfavorable $($U$)$ outcome and update the probabilities for each state of nature $($S$1,$ S$2,$ S$3)$ accordingly.

I can create the decision tree visualization for you if you'd like. Let me know if you'd prefer that.

Part $($b$)$: Recommended Decision without the Agency Opinion

To find the recommended decision without consulting the agency, calculate the expected monetary value $($EMV$)$ for each decision based on the given probabilities:

Expected Value for Producing the Pilot $($d$1)$:

EMV$($d$1)=(1000\mathrm{.}20)+(500\mathrm{.}30)+(1500\mathrm{.}50)=20+15+75=70\backslash$text$\{$EMV$($d$1)\}=(-100\backslash$times $0\mathrm{.}20)+(50\backslash$times $0\mathrm{.}30)+(150\backslash$times $0\mathrm{.}50)=-20+15+75=70$EMV$($d$1)=(1000\mathrm{.}20)+(500\mathrm{.}30)+(1500\mathrm{.}50)=20+15+75=70$

Expected Value for Selling to Competitor $($d$2)$:

EMV$($d$2)=100(0\mathrm{.}20+0\mathrm{.}30+0\mathrm{.}50)=100\backslash$text$\{$EMV$($d$2)\}=100\backslash$times $(0\mathrm{.}20+0\mathrm{.}30+0\mathrm{.}50)=100$EMV$($d$2)=100(0\mathrm{.}20+0\mathrm{.}30+0\mathrm{.}50)=100$

Recommended Decision without Agency Opinion: Since EMV$($d$2)=100\backslash$text$\{$EMV$($d$2)\}=100$EMV$($d$2)=100$ is greater than EMV$($d$1)=70\backslash$text$\{$EMV$($d$1)\}=70$EMV$($d$1)=70,$ the optimal decision is to sell to the competitor, with an expected value of $$100,000.$

Part $($c$)$: Expected Value of Perfect Information $($EVPI$)$

To calculate EVPI, find the expected value with perfect information and then subtract the highest EMV without perfect information.

EV $($with Perfect Information$)$:

EV$($with$$PI$)=(1500\mathrm{.}50)+(500\mathrm{.}30)+(1000\mathrm{.}20)=75+1520=70\backslash$text$\{$EV$($with PI$)\}=(150\backslash$times $0\mathrm{.}50)+(50\backslash$times $0\mathrm{.}30)+(-100\backslash$times $0\mathrm{.}20)=75+15-20=70$EV$($with$$PI$)=(1500\mathrm{.}50)+(500\mathrm{.}30)+(1000\mathrm{.}20)=75+1520=70$

Since selling to a competitor yields a certain value of $$100,000$ in the absence of perfect information, the EVPI here would be zero, indicating that perfect information does not increase Hale's potential profit under these circumstances.

Part $($d$)$: Optimal Decision Strategy Using Agency Information

Using the agency's probabilities and Bayes' theorem, calculate the EMV for each decision after accounting for the agency's favorable or unfavorable review.

Expected Values After Favorable Review $($F$)$:

Using the probabilities with a favorable review:

EMV$($d$1|$F$)=(1000\mathrm{.}09)+(500\mathrm{.}26)+(1500\mathrm{.}65)=9+13+97\mathrm{.}5=101\mathrm{.}5\backslash$text$\{$EMV$($d$1|$ F$)\}=(-100\backslash$times $0\mathrm{.}09)+(50\backslash$times $0\mathrm{.}26)+(150\backslash$times $0\mathrm{.}65)=-9+13+97\mathrm{.}5=101\mathrm{.}5$EMV$($d$1|$F$)=(1000\mathrm{.}09)+(500\mathrm{.}26)+(1500\mathrm{.}65)=9+13+97\mathrm{.}5=101\mathrm{.}5$ EMV$($d$2|$F$)=100\backslash$text$\{$EMV$($d$2|$ F$)\}=100$EMV$($d$2|$F$)=100$

Since EMV$($d$1|$F$)=101\mathrm{.}5\backslash$text$\{$EMV$($d$1|$ F$)\}=101\mathrm{.}5$EMV$($d$1|$F$)=101\mathrm{.}5$ is higher than EMV$($d$2|$F$)=100\backslash$text$\{$EMV$($d$2|$ F$)\}=100$EMV$($d$2|$F$)=100,$ the optimal decision after a favorable review is to produce the pilot.

Expected Values After Unfavorable Review $($U$)$:

Using the probabilities with an unfavorable review:

EMV$($d$1|$U$)=(1000\mathrm{.}04)+(500\mathrm{.}3)+(1500\mathrm{.}01)=4+15+1\mathrm{.}5=12\mathrm{.}5\backslash$text$\{$EMV$($d$1|$ U$)\}=(-100\backslash$times $0\mathrm{.}04)+(50\backslash$times $0\mathrm{.}3)+(150\backslash$times $0\mathrm{.}01)=-4+15+1\mathrm{.}5=12\mathrm{.}5$EMV$($d$1|$U$)=(1000\mathrm{.}04)+(500\mathrm{.}3)+(1500\mathrm{.}01)=4+15+1\mathrm{.}5=12\mathrm{.}5$ EMV$($d$2|$U$)=100\backslash$text$\{$EMV$($d$2|$ U$)\}=100$EMV$($d$2|$U$)=100$

Since EMV$($d$2|$U$)=100\backslash$text$\{$EMV$($d$2|$ U$)\}=100$EMV$($d$2|$U$)=100$ is higher than EMV$($d$1|$U$)=12\mathrm{.}5\backslash$text$\{$EMV$($d$1|$ U$)\}=12\mathrm{.}5$EMV$($d$1|$U$)=12\mathrm{.}5,$ the optimal decision after an unfavorable review is to sell to the competitor.

Part $($e$)$: Expected Value of the Agency$$s Information $($EVSI$)$

The EVSI is calculated by determining the overall expected value of using the agency's recommendation and subtracting the best EMV without using the agency's information.

Expected Value with Agency$$s Information:

EV$($F$)=(0\mathrm{.}69101\mathrm{.}5)+(0\mathrm{.}31100)=70\mathrm{.}035+31=101\mathrm{.}035\backslash$text$\{$EV$($F$)\}=(0\mathrm{.}69\backslash$times $101\mathrm{.}5)+(0\mathrm{.}31\backslash$times $100)=70\mathrm{.}035+31=101\mathrm{.}035$EV$($F$)=(0\mathrm{.}69101\mathrm{.}5)+(0\mathrm{.}31100)=70\mathrm{.}035+31=101\mathrm{.}035$

The EVSI is:

EVSI$=101\mathrm{.}035100=1\mathrm{.}035\backslash$text$\{$EVSI$\}=101\mathrm{.}035-100=1\mathrm{.}035$EVSI$=101\mathrm{.}035100=1\mathrm{.}035$

Part $($f$)$: Is the Agency$$s Information Worth the $$5000$ Fee?

Since the EVSI $(1\mathrm{.}035)$ is less than $$5,000,$ the agency's information is not worth the $$5,000$ fee. The maximum that Hale should be willing to pay for this information is $$1,035.$

Part $($g$)$: Recommended Decision

Given that the agency

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