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Prof.~David Draper \\
Department of Statistics \\
University of California, Santa Cruz

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\textbf{\large STAT 131: Quiz 8} \textit{\fbox{85 total points}}

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Name: \underline{\hspace*{5.85in}}

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Someone offers you the possibility to play a gambling game with the following rules. First, you decide how much money you're willing to put at risk in this game: this amount --- let's call it $A > 0$ --- is referred to as your \textit{stake} (all the monetary quantities are in dollars in this problem); think of $A$ as a fixed positive real number in what follows. Having chosen your stake, you're allowed to bet any amount $0 \le B \le A$ (thus, as a decision problem, your possible actions in this situation correspond to values of $B$). If you win the bet, which occurs with probability $0 < p < 1$, your stake becomes $( A + B )$; if you lose, it becomes $( A - B )$, and this (of course) occurs with probability $( 1 - p )$; and (crucially) \textit{$p$ is known to you}. Let $X$ denote the value of your stake after the gamble has occurred. The point of this problem is to explore your optimal betting strategy, \textit{assuming the Principle of Maximization of Expected Utility (MEU)}, for two different reasonable-looking utility functions, to see which one leads to more sensible behavior.

\begin{itemize}

\item[(a)]

Write out the probability mass function (PMF) for $X$. \textit{\fbox{5 points}}

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\item[(b)]

First let's suppose that for you, utility coincides with money, i.e., $U_1 ( x ) = x$. Work out your expected utility $E [ U_1 ( X ) ]$ as a function of $A, B$ and $p$. Holding $A$ and $p$ constant, what type of function of $B$ is this? Explain briefly, and sketch this function for $0 \le B \le A$ in the three separate cases $( p < \frac{ 1 }{ 2 } ), ( p = \frac{ 1 }{ 2 } ), ( p > \frac{ 1 }{ 2 } )$. \textit{\fbox{15 points}}

\vspace*{1.1in}

\item[(c)]

Now let's maximize $E [ U_1 ( X ) ]$ as a function of $B$.

\begin{itemize}

\item[(i)]

By considering the three different cases for $p$ mentioned above, and with particular reference to your sketches in (b), briefly explain why $B_1^*$, the optimal $B$ under $U_1$, is as follows:
\begin{equation} \label{e:utility-2}
B_1^* = \left\{ \begin{array}{ccc} 0 \ \textrm{(don't bet)} & \textrm{for} & p < \frac{ 1 }{ 2 } \\ \left( \begin{array}{c} \textrm{bet any number} \\ \textrm{between 0 and } A \end{array} \right) & & p = \frac{ 1 }{ 2 } \\ A \ \textrm{(bet it all)} & & p > \frac{ 1 }{ 2 } \end{array} \right\}
\end{equation}
\textit{\fbox{10 points}}

\newpage

\item[(ii) ]

Compute the first partial derivative of $E [ U_1 ( X ) ]$ as a function of $B$, and try setting it equal to 0 and solving for $B$. With reference to your sketches in (b), briefly explain why this standard calculus approach to maximizing a function won't work in this problem. \textit{\fbox{10 points}}

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\item[(iii)]

Identify one feature of this betting strategy that seems reasonable, and two features that seem unreasonable; explain briefly. \textit{\fbox{10 points}}

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\item[(d)]

Now let's suppose instead that you use Daniel Bernoulli's utility function $U_2 ( x ) = 1 + \log ( x )$. 

\begin{itemize}

\item[(i)]

Work out your expected utility $E [ U_2 ( X ) ]$ as a function of $A, B$ and $p$. 
Take $A = 10$ for illustration and get \texttt{Wolfram Alpha} (or some other equivalent environment) to plot this as a function of $B$ from $-5$ to $A$ for two different values of $p$: 0.4 and 0.7; reproduce these plots in your solutions, either electronically or by hand-sketching what your software environment showed you. Holding $A$ and $p$ constant, what type of function of $B$ is this (increasing, decreasing, concave, convex)? Explain briefly. \textit{\fbox{10 points}}

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\item[(ii)]

Compute the first partial derivative of $E [ U_2 ( X ) ]$ with respect to $B$, set this expression to 0, and solve for $B$. In many problems this would be the optimal $B^*$ (the value of $B$ that maximizes the expected utility), but the situation is a bit more subtle here. Identify the range of values of $p$ for which the $B^*$ you obtained by differentiation cannot be the solution to this optimality problem, and briefly explain why. \textit{\fbox{10 points}}

\end{itemize}

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\item[(e)]

By examining the two different cases $( p \le \frac{ 1 }{ 2 } ), ( p > \frac{ 1 }{ 2 } )$, briefly explain why $B_2^*$, the optimal $B$ under $U_2$, is as follows:
\begin{equation} \label{e:utility-2}
B_2^* = \left\{ \begin{array}{ccc} 0 \ \textrm{(don't bet)} & \textrm{for} & p \le \frac{ 1 }{ 2 } \\ \left( \begin{array}{c} 2 \, \left( p - \frac{ 1 }{ 2 } \right) \, A \\ \textrm{(bet an amount} \\ \textrm{between 0 and $A$} \\ \textrm{in proportion to} \\ \textrm{how much bigger} \\ p \textrm{ is than } \frac{ 1 }{ 2 } ) \end{array} \right) & & p > \frac{ 1 }{ 2 } \end{array} \right\} \, ,
\end{equation}
Identify two features of this betting strategy that seem eminently reasonable; explain briefly. \textit{\fbox{10 points}}

\vspace*{1.1in}

\item[(f)]

Is it fair to describe one of the two betting strategies above, based on the two different utility functions, as more risk-seeking than the other one? If so, which is which? If not, why not? Explain briefly. \textit{\fbox{5 points}}

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