Chapter 22.2, Problem 42E

To determine

Calculate the EVSI.

Explanation of Solution

The EMV of 25 calls is 50,000, EMV of 50 calls is 45,000$\left(\left(0.5\times 30,000\right)+\left(0.25\times 60,000\right)+\left(0.25\times 60,000\right)\right)$, EMV of 100 calls is 40,000$\left(\left(0.5\times 20,000\right)+\left(0.25\times 40,000\right)+\left(0.25\times 80,000\right)\right)$, Since the EMV of 25 calls is greater, select 25 calls.

The terms I₁ small number of calls, I₁ indicates the medium number of calls and I₂ indicates the large number of calls.

Table 1 shows that the posterior probabilities for I₁.

Table 1

sj	P(sj)	$\text{P}\left({\text{I}}_1,{\text{s}}_{\text{j}}\right)$	$\text{P}\left({\text{s}}_{\text{j}}\times {\text{I}}_1\right)$	$\text{P}\left(\frac{{\text{s}}_{\text{j}}}{{\text{I}}_1}\right)$
s1	0.5	0.8667	0.4333	0.8792
s2	0.25	0.2202	0.0551	0.1117
s3	0.25	0.018	0.0045	0.0091
Total			0.4929

Table 2 shows that the posterior probabilities for I₂.

Table 2

sj	P(sj)	$\text{P}\left({\text{I}}_2,{\text{s}}_{\text{j}}\right)$	$\text{P}\left({\text{s}}_{\text{j}}\times {\text{I}}_2\right)$	$\text{P}\left(\frac{{\text{s}}_{\text{j}}}{{\text{I}}_2}\right)$
s1	0.5	0.1334	0.0667	0.1601
s2	0.25	0.7527	0.1882	0.4519
s3	0.25	0.6461	0.1615	0.3879
Total			0.4164

Table 3 shows that the posterior probabilities for I₃.

Table 2

sj	P(sj)	$\text{P}\left({\text{I}}_3,{\text{s}}_{\text{j}}\right)$	$\text{P}\left({\text{s}}_{\text{j}}\times {\text{I}}_3\right)$	$\text{P}\left(\frac{{\text{s}}_{\text{j}}}{{\text{I}}_3}\right)$
s1	0.5	0	0	0
s2	0.25	0.027	0.0068	0.0745
s3	0.25	0.3359	0.084	0.9254
Total

The EMV value of 25 calls with I₁ is 50,000.

The EMV value of 50 calls (a₂) with I₁ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_1}\right)\right)\\ =\left(30,000\times 0.8792\right)+\left(60,000\times 0.1117\right)+\left(60,000\times 0.0091\right)\\ =33,624\end{array}$

The value of EMV of a₂ is 33,624.

 The EMV value of 100 calls (a₃) with I₁ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_1}\right)\right)\\ =\left(20,000\times 0.8792\right)+\left(40,000\times 0.1117\right)+\left(80,000\times 0.0091\right)\\ =22,780\end{array}$

The value of EMV of a₃ is 22,780. Since the EMV value of 25 calls is greater, select the option 25 calls.

The EMV value of 25 calls with I₂ is 50,000.

The EMV value of 50 calls (a₂) with I₂ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_2}\right)\right)\\ =\left(30,000\times 0.1601\right)+\left(60,000\times 0.4519\right)+\left(60,000\times 0.3879\right)\\ =55,191\end{array}$

The value of EMV of a₂ is 55,191.

 The EMV value of 100 calls (a₃) with I₂ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_2}\right)\right)\\ =\left(20,000\times 0.1601\right)+\left(40,000\times 0.4519\right)+\left(80,000\times 0.38791\right)\\ =52,310\end{array}$

The value of EMV of a₃ is 52,310. Since the EMV value of 50 calls is greater, select the option 50 calls.

The EMV value of 25 calls with I₃ is 50,000.

The EMV value of 50 calls (a₂) with I₃ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_3}\right)\right)\\ =\left(30,000\times 0\right)+\left(60,000\times 0.0745\right)+\left(60,000\times 0.9254\right)\\ =60,000\end{array}$

The value of EMV of a₂ is 60,000.

 The EMV value of 100 calls with I₃ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_3}\right)\right)\\ =\left(20,000\times 0\right)+\left(40,000\times 0.0745\right)+\left(80,000\times 0.9254\right)\\ =77,012\end{array}$

The value of EMV of a₃ is 77,012. Since the EMV value of 100 calls is greater, select the option 100 calls.

The EMV value can be calculated as follows.

$\begin{array}{c}\text{EMV}=\left(\begin{array}{l}{\text{EMV highest}}_{{\text{I}}_1}\times \text{P}\left({\text{I}}_1\right)+{\text{EMV highest}}_{{\text{I}}_2}\times \text{P}\left({\text{I}}_2\right)\\ +{\text{EMV highest}}_{{\text{I}}_3}\times \text{P}\left({\text{I}}_3\right)\end{array}\right)\\ =\left(\left(50,000\times 0.4929\right)+\left(55,191\times 0.4164\right)+\left(77,012\times 0.0907\right)\right)\\ =54,612\end{array}$

The value of EMV is 54,612.

The EVSI value can be calculated as follows.

$\begin{array}{c}\text{EVSI}=\text{EMV}-{\text{EMV}}_{\text{Prior probability}}\\ =54,612-50,000\\ =4,612\end{array}$

The value of EVSI is 4,612.