Chapter 22.2, Problem 35E

To determine

Calculate the EVSI.

Explanation of Solution

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	0.15	0.5	0.075	0.25
s2	0.55	0.3	0.165	0.55
s3	0.3	0.2	0.06	0.2
Total			0.3

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	0.15	0.3	0.045	0.103
s2	0.55	0.6	0.33	0.759
s3	0.3	0.2	0.06	0.138
Total			0.435

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	0.15	0.2	0.03	0.113
s2	0.55	0.1	0.055	0.208
s3	0.3	0.6	0.18	0.679
Total			0.265

The EMV value of a₁ with I₁ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_1}=\left({\text{Payoff}}_{{\text{a}}_1{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_1}\right)\right)\\ =\left(\left(-220\times 0.25\right)+\left(-330\times 0.55\right)+\left(-440\times 0.2\right)\right)\\ =-324.5\end{array}$

The value of EMV of a₁ is -324.5.

The EMV value of a₂ with I₁ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_1}\right)\right)\\ \left(-300\times 0.25\right)+\left(-320\times 0.55\right)+\left(-390\times 0.2\right)\\ =-329\end{array}$

The value of EMV of a₂ is -329.

 The EMV value of a₃ with I₁ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_1}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_1}\right)\right)\\ \left(-350\times 0.25\right)+\left(-350\times 0.55\right)+\left(-350\times 0.2\right)\\ =-350\end{array}$

The value of EMV of a₃ is -350. Since the EMV value of a₁ is greater, select the option a₁.

The EMV value of a₁ with I₂ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_1}=\left({\text{Payoff}}_{{\text{a}}_1{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_2}\right)\right)\\ =\left(\left(-220\times 0.103\right)+\left(-330\times 0.759\right)+\left(-440\times 0.138\right)\right)\\ =-333.85\end{array}$

The value of EMV of a₁ is -333.85.

The EMV value of a₂ with I₂ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_2}\right)\right)\\ \left(-300\times 0.103\right)+\left(-320\times 0.759\right)+\left(-390\times 0.138\right)\\ =-327.59\end{array}$

The value of EMV of a₂ is -327.59.

 The EMV value of a₃ with I₂ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_2}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_2}\right)\right)\\ \left(-350\times 0.103\right)+\left(-350\times 0.759\right)+\left(-350\times 0.138\right)\\ =-350\end{array}$

The value of EMV of a₃ is -350. Since the EMV value of a₂ is greater, select the option a₂.

The EMV value of a₁ with I₃ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_1}=\left({\text{Payoff}}_{{\text{a}}_1{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_1{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_3}\right)\right)\\ =\left(\left(-220\times \mathrm{0.0.113}\right)+\left(-330\times 0.208\right)+\left(-440\times 0.679\right)\right)\\ =-392.26\end{array}$

The value of EMV of a₁ is -392.26.

The EMV value of a₂ with I₃ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left({\text{Payoff}}_{{\text{a}}_2{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_3}\right)\right)\\ \left(-300\times \mathrm{0.0.113}\right)+\left(-320\times 0.208\right)+\left(-390\times 0.679\right)\\ =-365.28\end{array}$

The value of EMV of a₂ is -365.28.

 The EMV value of a₃ with I₃ can be calculated as follows.

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left({\text{Payoff}}_{{\text{a}}_3{\text{, s}}_1}\times \text{P}\left(\frac{{\text{s}}_1}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_2}\times \text{P}\left(\frac{{\text{s}}_2}{{\text{I}}_3}\right)+{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left(\frac{{\text{s}}_3}{{\text{I}}_3}\right)\right)\\ \left(-350\times 0.113\right)+\left(-350\times 0.208\right)+\left(-350\times 0.679\right)\\ =-350\end{array}$

The value of EMV of a₃ is -350. Since the EMV value of a₃ is greater, select the option a₃.

The EMV value can be calculated as follows.

$\begin{array}{c}\text{EMV}=\left(\begin{array}{l}{\text{EMV highest}}_{{\text{I}}_1}\times \text{P}\left({\text{I}}_1\right)+{\text{EMV highest}}_{{\text{I}}_2}\times \text{P}\left({\text{I}}_2\right)\\ +{\text{EMV highest}}_{{\text{I}}_3}\times \text{P}\left({\text{I}}_3\right)\end{array}\right)\\ =\left(\left(-324.5\times 0.3\right)+\left(-\text{327}\text{.59}\times 0.435\right)+\left(-\text{350}\times 0.265\right)\right)\\ =-332.6\end{array}$

The value of EMV is -332.6.

The EVSI value can be calculated as follows.

$\begin{array}{c}\text{EVSI}=\text{EMV}-{\text{EMV}}_{\text{Prior probability}}\\ =-332.6-\left(-338\right)\\ =5.4\end{array}$

The value of EVSI is 5.4.