Chapter 22.1, Problem 8E

a:

To determine

Calculate the EMV value.

Explanation of Solution

The EMV value of a₀can be calculated as follows:

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_0}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(0\times 0.25+0\times 0.25+0\times 0.25+0\times 0.25\right)\\ =0\end{array}$

The value of EMV of a₀ is 0.

The EMV value of a₁can be calculated as follows:

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_1}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(-3\times 0.25+5\times 0.25+5\times 0.25+5\times 0.25\right)\\ =3\end{array}$

The value of EMV of a₁ is 3.

The EMV value of a₂can be calculated as follows:

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_2}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(-6\times 0.25+2\times 0.25+10\times 0.25+10\times 0.25\right)\\ =4\end{array}$

The value of EMV of a₂ is 4.

The EMV value of a₃can be calculated as follows:

$\begin{array}{c}{\text{EMV}}_{{\text{a}}_3}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(\left(-9\times 0.25\right)+\left(-1\times 0.25\right)+7\times 0.25+15\times 0.25\right)\\ =3\end{array}$

The value of EMV of a₃ is 3. Since the EMV for a₂ is greater, select the a₂ (Bake 2 cakes).

b:

To determine

Calculate the EOL value.

Explanation of Solution

The EOL value of a₀can be calculated as follows:

$\begin{array}{c}{\text{EOL}}_{{\text{a}}_0}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_0{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(0\times 0.25+5\times 0.25+10\times 0.25+15\times 0.25\right)\\ =7.5\end{array}$

The value of EOL of a₀ is 7.5.

The EOL value of a₁can be calculated as follows:

$\begin{array}{c}{\text{EOL}}_{{\text{a}}_1}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{1}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(3\times 0.25+0\times 0.25+5\times 0.25+10\times 0.25\right)\\ =4.5\end{array}$

The value of EOL of a₁ is 4.5.

The EOL value of a₂can be calculated as follows:

$\begin{array}{c}{\text{EOL}}_{{\text{a}}_2}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_2{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{2}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(6\times 0.25+3\times 0.25+0\times 0.25+5\times 0.25\right)\\ =3.5\end{array}$

The value of EOL of a₂ is 3.5.

The EOL value of a₃can be calculated as follows:

$\begin{array}{c}{\text{EOL}}_{{\text{a}}_3}=\left(\begin{array}{l}{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_1}\times \text{P}\left({\text{s}}_1\right)+{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_2}\times \text{P}\left({\text{s}}_2\right)\\ +{\text{Payoff}}_{{\text{a}}_3{\text{, s}}_3}\times \text{P}\left({\text{s}}_3\right)+{\text{Payoff}}_{{\text{a}}_{\text{3}}{\text{, s}}_4}\times \text{P}\left({\text{s}}_4\right)\end{array}\right)\\ =\left(\left(9\times 0.25\right)+\left(6\times 0.25\right)+3\times 0.25+0\times 0.25\right)\\ =4.5\end{array}$

The value of EMV of a₃ is 4.5. Since the EMV for a₂ is greater, select the a₂ (Bake 2 cakes).