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\centerline{\bf \Large Math 315 Homework 4}
\smallskip
\centerline{\bf Due Friday, March 3}
\bigskip
Solutions must be written in \LaTeX. You are encouraged to work with others on the assignment,
but you should write up your own solutions independently.
You should reference all of your sources, including your
collaborators.
\smallskip
\begin{enumerate}
\item Suppose that a person whose preferences satisfy the von Neumann-Morgenstern axioms, and who always prefers more money to less money
says that:
\begin{itemize}
\item He is indifferent between receiving \$500 and participating in a lottery in which he receives \$1000 with probability 2/3 and
receives \$0 with probability 1/3.
\item He is indifferent between receiving \$100 and participating in a lottery in which he receives \$500 with probability 3/8 and
and receives \$0 with probability 5/8.
\end{itemize}
\begin{enumerate}
\item Find a linear utility function $u$ representing this person's preferences and also satisfying $u(\$0) = 0$ and $u(\$1000) = 1$. What is $u(\$500)$ and $u(\$100)$?
\smallskip
\item Find a linear utility function $v$ representing this person's preferences and also satisfying $v(\$0) = 3$ and $v(\$1000) = 8$. What is $v(\$500)$ and $v(\$100)$?
\smallskip
\item Determine which of the following two lotteries would be preferred by this person:
\begin{align*}
L_1 &= \left[\tfrac{3}{10} (\$1000), \tfrac{1}{10} (\$500), \tfrac{1}{2} (\$100), \tfrac{1}{10}(\$0)\right] \\[10pt]
L_2 &= \left[\tfrac{2}{10} (\$1000), \tfrac{3}{10} (\$500), \tfrac{2}{10} (\$100), \tfrac{3}{10}(\$0)\right]
\end{align*}
\smallskip
\item Does your answer to part (c) depend on whether you use the utilities from part (a) or part (b)?
\smallskip
\item Is it possible to determine whether the person would prefer to receive \$400 or participate in lottery $L_1$ (where $L_1$ is the lottery from part (c))? Justify your answer.
\smallskip
\item Is it possible to determine whether the person would prefer to receive \$600 or participate in lottery $L_1$ (where $L_1$ is the lottery from part (c))? Justify your answer.
\end{enumerate}
\bigskip
\item \begin{enumerate}
\item Prove that if $v$ is a positive affine transformation of $u$, then $u$ is a positive affine transformation of $v$.
\smallskip
\item Prove that if $v$ is a positive affine transformation of $u$, and if $w$ is a positive affine transformation of $v$, then $w$ is a positive affine transformation of $u$.
\end{enumerate}
\bigskip
\item A farmer wishes to dig a well in a square field whose vertices have coordinates $(0, 0)$, $(0, 1000)$, $(1000, 0)$ and $(1000, 1000)$. The well must be located at
a point whose coordinates $(x, y)$ are integers. The farmer's preferences are lexicographic: if $x_1 > x_2$, he prefers that the well be dug at the point $(x_1, y_1)$ to the point
$(x_2, y_2)$ for all $y_1, y_2$. If $x_1 = x_2$, he prefers the first point only if $y_1 > y_2$. Give an example of a utility function representing such a preference relation.
(Since no information is provided about the preference relation on lotteries, there are many possible utility functions.)
\bigskip
\item In this exercise, we will consider the situation in problem 3, but we will now allow the well to be located at any point $(x,y)$ in the square (with $x$ and $y$ being real numbers).
We will show that in this case, it is not possible to have a preference relation that satisfies the von Neumann-Morgenstern axioms (and thus, we are showing that when the set of possible
outcomes is uncountable, it is not necessarily possible to have a preference relation that satisfies the axioms).
The proof will use contradiction, so first, we assume that there does exist a preference relation over lotteries of points in the square that satisfies the von Neumann-Morgenstern axioms.
\begin{enumerate}
\item First, prove the following statement (assuming that there is a preference relation satisfying the axioms):
For all $(x,y)$ in the square there exists a unique number $p_{x,y}$ with $0 \le p_{x,y} \le 1$ such that the farmer is indifferent between locating the well at the point $(x,y)$
and a lottery in which the well is located at point $(0,0)$ with probability $1-p_{x,y}$ and located at point $(1000,1000)$ with probability $p_{x,y}$.
\smallskip
\item Prove that the function $f:(x,y) \rightarrow p_{x,y}$ is one-to-one. (That is, prove that $f(x_1, y_1) = f(x_2, y_2)$ implies that $(x_1, y_1) = (x_2, y_2)$.)
\smallskip
\item For each $x$ with $0 \le x \le 1000$, define $A_x = \{ p_{x,y}: 0 \le y\le 1000\}$. Prove that for each $x$ the set $A_x$ contains at least two elements.
\smallskip
\item Prove that the sets $A_x$ are pairwise disjoint. (That is, for all $x_1, x_2$, we have that $A_{x_1} \cap A_{x_2} = \emptyset$.)
\smallskip
\item Prove that if $x_1 < x_2$ then for all $p_1 \in A_{x_1}$ and $p_2 \in A_{x_2}$, we have that $p_1 < p_2$.
(There do not exist sets $A_x$ satisfying (c), (d), (e); thus, we have a contradiction. Proving that such sets do not exist is a little complicated, so we will not prove it
in this exercise, but if you think about it, it should make intuitive sense that such sets cannot exist.)
\end{enumerate}
\end{enumerate}
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