Question

A television network earns an average of $18 million each season from a hit program and loses an average of $8 million each season on a program that turns out to be a flop. Of all programs picked up by this network in recent years, 25% turn out to be hits and 75% turn out to be flops. At a cost of C dollars, a market research firm will analyze a pilot episode of a prospective program and issue a report predicting whether the given program will end up being a hit. If the program is actually going to be a hit, there is a 75% chance that the market researchers will predict the program to be a hit. If the program is actually going to be a flop, there is only a 30% chance that the market researchers will predict the program to be a hit. a. What is the maximum value of C that the network should be willing to pay the market research firm? Enter your answer in dollars, not in million dollars. b. Calculate EVPI for this decision problem. Enter your answer in dollars, not in million dollars.

Answer 1

Of all the programs picked 25% turn out to be hits. This is same as the probability of any given program picked turns out to be a hit is 0.25

P(hit)=0.25 with a payoff of $18 millions

Of all the programs picked 75% turn out to be flops. This is same as the probability of any given program picked turns out to be a flop is 0.75

P(flop)=0.75 with a payoff of -$8 millions

If the program is actually going to be a hit, there is a 75% chance that the market researchers will predict hit. This is same as the conditional probability of market researchers predict hit given that the  program is actually going to be a hit is 0.75

$P(\text{predict hit}\mid \text{hit})=0.75$

If the program is actually going to be a flop, there is only a 30% chance that the market researchers will predict hit. This is same as the conditional probability of market researchers predict hit given that the  program is actually going to be a flop is 0.30

$P(\text{predict hit}\mid \text{flop})=0.30$

Using these, we calculate the following

$\begin{align*} P(\text{predict hit})&=P(\text{predict hit}\cap \text{hit})+P(\text{predict hit}\cap \text{flop})\\ &=P(\text{predict hit}\mid \text{hit})P(\text{hit})+P(\text{predict hit}\mid \text{flop})P(\text{flop})\\ &=0.75\times 0.25+0.30\times 0.75\\ &=0.4125\\ P(\text{predict flop})&=1-P(\text{predict hit})=1-0.4125=0.5875\\ \end{align*}$

We calculate the following conditional probabilities

$\begin{align*} P(\text{hit}\mid \text{predict hit} )&=\frac{P(\text{predict hit}\mid \text{hit} )P(\text{hit})}{P(\text{predict hit})}\quad\text{using the Bayes' Rule}\\ &=\frac{0.75\times 0.25}{0.4125}\\ &=0.4545\\ P(\text{flop}\mid \text{predict hit} )&=1-P(\text{hit}\mid \text{predict hit} )=1-0.4545=0.5455\\ P(\text{hit}\mid \text{predict flop} )&=\frac{P(\text{predict flop}\mid \text{hit} )P(\text{hit})}{P(\text{predict flop})}\quad\text{using the Bayes' Rule}\\ &=\frac{(1-P(\text{predict hit}\mid \text{hit} ))P(\text{hit})}{P(\text{predict flop})}\\ &=\frac{(1-0.75\times 0.25)}{0.5875}\\ &=0.1064\\ P(\text{flop}\mid \text{predict flop} )&=1-P(\text{hit}\mid \text{predict flop} )=1-0.1064=0.8936 \end{align*}$

We prepare the following decision tree, ignoring the cost C of market research

hit P(Hit predict hit)=0.4545 payoff=$18 million Pick the program 6 Get a Positive review P(predict hit)=0.4125 payoffs-$8 million 3 flop P(flop predict hit) =0.5455 payoff=$0 Do not Pick the program 2 Hire Market research payoff=$18 million Pick the program hit P(hit|Predict flop)=0.1064 7 flop P(flop predict flop)=0.8936 payoff=$0 payoff=-$8 million get a negative review P(Predict Flop)=0.5875 1 Do not Pick the program hit P(hit)=0.25 payoff=$18 million Pick the program 8 do not hire Market research flop P(flop)=0.75 payoff=-$8 million 5 Do not Pick the program payoff=$0

Node 6: chance node

The expected value at node 6 is

$\begin{align*} EV(6)&=P(\text{hit}\mid \text{predict hit})\text{(payoff when hit)}+\\ &\qquad P(\text{flop}\mid \text{predict hit})\text{(payoff when flop)}\\ &=0.4545\times 18+0.5455\times (-8)\\ &=3.818 \end{align*}$

Node 7: chance node

The expected value at node 7 is

$\begin{align*} EV(7)&=P(\text{hit}\mid \text{predict flop})\text{(payoff when hit)}+\\ &\qquad P(\text{flop}\mid \text{predict flop})\text{(payoff when flop)}\\ &=0.1064\times 18+0.8936\times (-8)\\ &=-5.2336 \end{align*}$

Node 8: chance node

The expected value at node 8 is

$\begin{align*} EV(8)&=P(\text{hit})\text{(payoff when hit)}+\\ &\qquad P(\text{flop})\text{(payoff when flop)}\\ &=0.25\times 18+0.75\times (-8)\\ &=-1.5 \end{align*}$

Decision node 3:

Chose between 2 alternatives

1. pick the program at an expected value $3.817 million
2. do not pick the program at an expected value $0 million

The optimum decision is to pick the program

EV(3)=$3.817

Decision node 4:

Chose between 2 alternatives

1. pick the program at an expected value -$5.2336 million
2. do not pick the program at an expected value $0 million

The optimum decision is to not pick the program

EV(4)=$0

Decision node 5:

Chose between 2 alternatives

1. pick the program at an expected value -$1.5 million
2. do not pick the program at an expected value $0 million

The optimum decision is to not pick the program

EV(5)=$0

chance node 2:

The expected value is

$\begin{align*} EV(2)&=P(\text{predict hit})\text{(payoff when predicted hit)}+\\ &\qquad P(\text{predict flop})\text{(payoff when predicted flop)}\\ &=0.4125\times 3.818+0.5875\times 0\\ &=1.575 \end{align*}$

a) Using the above, we get the following

The expected value with sample(the market research) information is

$\begin{align*} \text{EVwSI}=EV(2)=1.575 \end{align*}$

The expected value without sample(the market research) information is

$\begin{align*} \text{EVwoSI}=EV(5)=0 \end{align*}$

The expected value of sample(the market research) information is

$\begin{align*}\text{EVSI}=\text{EVwSI}- \text{EVwoSI}=EV(5)=1.575-0 =1.575 \end{align*}$

This is the maximum value C that the network should be willing to pay the market research firm

ans: $1,575,000 (when using unrounded probabilities)

b) If the network knows exactly (has the perfect information) if the program being picked is going to be a hit, it will pick the program at a payoff of $18 million. If it knows that if the program being picked is going to be a flop, it will not pick the program at a payoff of $0 million

The expected value with perfect information is

$\begin{align*}\text{EVwPI}&=P(hit)\times \text{payoff when knowing it is a hit}+\\ &\qquad P(flop)\times \text{payoff when knowing it is a flop}\\ &=0.25\times 18+0.75\times 0\\ &=4.5 \end{align*}$

The expected value without perfect information is the expected value at node 5

$\begin{align*}\text{EVwoPI}&=EV(5)=0 \end{align*}$

The expected value of perfect information is

$\begin{align*}\text{EVPI}=\text{EVwPI}-\text{EVwoPI}=4.5-0=4.5 \end{align*}$

ans: EVPI=$4,500,000