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\usepackage{fullpage,amsmath,amsfonts,graphicx}

\begin{document}

\title{Perspectives of Mathematics I}
\author{Homework 3}
\date{due November 16, 2011}
\maketitle

\begin{enumerate}

\item Find the pure-strategy Nash-equilibria for the two-player game
with payoff matrix

\begin{center}
\begin{tabular}{ccccc}
&&\multicolumn{3}{c}{Player 2} \\
& & l & c & r \\
\cline{3-5}
\smash{\raisebox{-1.1cm}{\rotatebox{90}{\makebox[0in][l]{Player 1}}}} &
U & \multicolumn{1}{|c}{$1,1$} & \multicolumn{1}{|c}{$2,0$} &
\multicolumn{1}{|c|}{$3,0$} \\\cline{3-5}
& M & \multicolumn{1}{|c}{$0,2$} & \multicolumn{1}{|c}{$3,3$} &
\multicolumn{1}{|c|}{$0,0$} \\\cline{3-5}
& D & \multicolumn{1}{|c}{$0,3$} & \multicolumn{1}{|c}{$0,0$} &
\multicolumn{1}{|c|}{$4,4$} \\
\cline{3-5}
\end{tabular}
\end{center}

\item Consider the two-player game with payoff matrix
\begin{center}
\begin{tabular}{ccccc}
&&\multicolumn{3}{c}{Player 2} \\
& & l & r \\
\cline{3-4}
\smash{\raisebox{-0.7cm}{\rotatebox{90}{\makebox[0in][l]{Player 1}}}} &
U & \multicolumn{1}{|c}{$3,1$} &
\multicolumn{1}{|c|}{$0,0$} \\\cline{3-4}
& D & \multicolumn{1}{|c}{$0,0$} & 
\multicolumn{1}{|c|}{$1,3$} \\
\cline{3-4}
\end{tabular} 
\end{center} 
\begin{enumerate}
\item Are there pure-strategy Nash equilibria?
\item What are the mixed-strategy Nash equilibria?
\item Sketch the best response functions for case (b).
\end{enumerate}


\item Consider the following slight modification of the Cournot game
considered in class.  Two firms produce quantities $q_1$ and $q_2$ of
the same good with respective unit cost per unit $c_1$ and $c_2$ with
no fixed costs.  The market price is assumed to satisfy the simple
inverse linear relation $P=a-q_1-q_2$.  Now, Firm 1's objective is to
maximize its market share without making a loss.  Firm 2's objective
is to maximize profit.

Compute both firms' best response functions and describe the Nash
equilibria. 


\item 
\begin{enumerate}
\item Show that the relation
\[
  f(x) =
  \begin{cases}
    (\tfrac12,1] & \text{if } x<\tfrac12 \\
    [0,\tfrac12) & \text{if } x\geq \tfrac12 
  \end{cases}
\]
on the interval $[0,1]$ does not have a fixed point $x \in f(x)$.

\item Which condition of Kakutani's theorem is violated?
\end{enumerate}

\end{enumerate}
\end{document}

