\documentclass{amsart}
\usepackage{amsfonts}
\usepackage{amsmath}
\usepackage[T1]{fontenc} % Output font encoding for international characters
\usepackage{fouriernc} % Use the New Century Schoolbook font
\usepackage{hyperref}
\hypersetup{
  colorlinks=true,
  linkcolor=blue,
  citecolor=blue
}
\usepackage{sgame}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{bb}[theorem]{Blackboard}


\newcommand{\vecl}[1]{\overrightarrow{#1}}
\newcommand{\dls}[1]{\overrightarrow{#1}}

\def\vproj{\mbox{proj}}
\def\half{{\textstyle \frac12}}
\def\third{{\textstyle \frac13}}

\def\C{\mathbb{C}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\pleq{\preceq}
\def\pgt{\succ}
\def\Z{\mathbb{Z}}

\newenvironment{myenumerate}
{\begin{enumerate}
  \setlength{\itemsep}{10pt}}
%  \setlength{\parskip}{0pt}
{\end{enumerate}}



\newcommand\blfootnote[1]{%
  \begingroup
  \renewcommand\thefootnote{}\footnote{#1}%
  \addtocounter{footnote}{-1}%
  \endgroup
}


\newcommand{\solution}[1]{}
%\newcommand{\solution}[1]{\vspace{0.15in}{\bf Solution:  {#1}}\vspace{0.15in}}

\begin{document}
%\newcommand{\itemt}[2]{\item {\bf {#1}} \newline{#2}}


\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}