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\begin{document}
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\blfootnote{Omer Tamuz. Email: tamuz@caltech.edu.}
\section*{PS/Ec 172, Set 4\\Due Friday, May 5\textsuperscript{th} at
11:59pm \\ Resubmission due Friday, May 19\textsuperscript{th} at
11:59pm}
Collaboration on homework is encouraged, but individually written
solutions are required. Also, please name all collaborators and
sources of information on each assignment; any such named source may
be used.
\mbox{}
\begin{myenumerate}
\item Consider the following game played by $n$ players who are
sitting in a circle. Each player chooses one of two actions: $X$ or
$Y$. The players make this choice simultaneously. The payoff to a
player is 0 if she chooses the same action as the person on her
right, and 1 otherwise.
\begin{myenumerate}
\item {\em 15 points.} Let $n$ be even. Find a pure Nash equilibrium
or explain why none exist.
\item {\em 15 points.} Let $n$ be odd. Find a pure Nash equilibrium
or explain why none exist.
\item {\em 15 points.} Find a completely mixed Nash equilibrium for
those values of $n$ for which no pure one exists. What is the
expected utility to each player?
\item {\em 15 points.} For those values of $n$ for which no pure
Nash equilibria exist, find a correlated equilibrium in which the
expected utility to every player is $1-1/n$.
\end{myenumerate}
\item {\em 40 points.} Construct an example of a knowledge space with
two players, a finite set of states of the world, an event $A$ and a
state of the world $\omega$ such that $\omega \in K_1A$, $\omega \in
K_2A$, $\omega \in K_1K_2A$, $\omega \in K_2K_1A$, but $\omega \not
\in K_1K_2K_1A$. That is, at $\omega$ both players know that $A$ has
occured, both know that the other knows, but player 1 does not know
that player 2 knows that player 1 knows that $A$ has occured.
\end{myenumerate}
\end{document}