91Ó°ÊÓ

There are three gambles X, Y, and Z

Their probability distributions of the gains are,

\(\begin{align}\Pr \left( {X = 5} \right) = \Pr \left( {X = 25} \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ } \!\lower0.7ex\hbox{$2$}},\\\Pr \left( {Y = 10} \right) = \Pr \left( {Y = 20} \right) = {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\ }\!\lower0.7ex\hbox{$2$}},\\\Pr \left( {Z = 15} \right) = 1.\end{align}\)

The utility function is given by,

\(U\left( x \right) = {x^2}\,for\,x > 0\)

The expectation of gamble X is,

\(\begin{align}E\left) {U\left( X \right)} \right) &= \left( {\frac{1}{2} \times {5^2}} \right) + \left( {\frac{1}{2} \times {{25}^2}} \right)\\ &= 12.5 + 312.5\\ &= 325\end{align}\)

The expectation of gamble Y is,

\(\begin{align}E\left) {U\left( Y \right)} \right) &= \left( {\frac{1}{2} \times {{10}^2}} \right) + \left( {\frac{1}{2} \times {{20}^2}} \right)\\ &= 50 + 200\\ &= 250\end{align}\)

The expectation of gamble Z is,

\(\begin{align}E\left) {U\left( Z \right)} \right) &= \left( {1 \times {{15}^2}} \right)\\ &= \left( {1 \times 225} \right)\\ &= 225\end{align}\)

Hence, gamble X is preferred.