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\begin{LARGE}

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  19 Sept. 2006  Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org. Overheads for Chapter 4 of  {\it Games and Information}

  
 \newpage

Latex\\

Finance--- risk aversion, conglomerates, multinationals\\

BEPP Seminar Friday at 3:30--Raphael Rob, U. of Penn. . CG 2069. Repeated Games.   \\

TUESDAY: 


 1. Perfectness

 2. Repeated Games

3. Reputation

THURSDAY

1. Rob's paper

2. Perfect bayesian equilibrium

\newpage

\includegraphics[width=150mm]{fig04-01.jpg}

\begin{center}
 {\bf Figure 1:    {\it   Follow the Leader I} } \end{center}

  

We say that equilibria $E_1$ and $E_3$ are Nash equilibria but not ``perfect''
Nash equilibria. A strategy profile is a perfect equilibrium if it remains an
equilibrium on all possible paths, including not only the equilibrium path but
all the other paths, which branch off into different ``subgames.''

 

\noindent
{\it A strategy profile is a {\bf subgame perfect Nash equilibrium} if (a) it is
a Nash equilibrium for the entire game; and (b) its relevant action rules are a
Nash equilibrium for every subgame.}

  
 
\newpage

TREMBLES


   A second reason is that a weak Nash equilibrium is not robust to
small changes in the game.  So long as he is certain that Smith will not choose
{\it Large}, Jones is indifferent between the never-to-be-used responses ({\it
Small } if $Large$) and ({\it Large} if $Large$).  Equilibria $E_1$, $E_2$, and
$E_3$ are all weak Nash equilibria because of this. But if there is even a small
probability that Smith will choose {\it Large}--- perhaps by mistake--- then
Jones would prefer the response ({\it Large} if {\it Large}), and equilibria
$E_1$ and $E_3$ are no longer valid. Perfectness is a way to eliminate some of
these less robust weak equilibria. The   small probability of a mistake is
called a {\bf tremble}, and  Section 6.1 returns to this  {\bf trembling hand}
approach   as one way to extend the notion of perfectness to games of asymmetric
information.

\newpage

The tremble approach is
distinct from   sequential rationality.

 Consider  Figure 2's   { Tremble Game}.
This game has three Nash equilibria, all weak: {\it (Out, Down)},  {\it (Out,
Up)},  and {\it (In, Up)}. Only  {\it (Out,  Up)} and {\it (In, Up)} are subgame
perfect, because  although $Down$ is weakly Jones's   best response to Smith's
$Out$, it is inferior if Smith chooses $In$. In  the subgame starting with
Jones's   move, the only subgame perfect equilibrium is for Jones to choose
$Up$.  The possibility of trembles, however, rules out {\it (In, Up)} as an
equilibrium.   If Jones has even an infinitesimal chance of trembling and
choosing $Down$,   Smith will choose $Out$ instead of $In$.  Also,  Jones will
choose $Up$, not $Down$, because if Smith trembles and chooses $In$, Jones
prefers $Up$ to $Down$.  This leaves only   {\it (Out, Up)} as an equilibrium,
despite the fact that it is weakly pareto dominated by {\it (In, Up)}.

\includegraphics[width=150mm]{fig04-02.jpg}

\begin{center} {\bf Figure 2:   The     Tremble Game:     Trembling Hand Versus
Subgame Perfectness  } \end{center}


\newpage

 {\bf   Entry Deterrence I}

 
\begin{center}
{\bf  Entry Deterrence I }\\
 \end{center}
 {\bf Players}\\ Two firms, the entrant and the incumbent.

 \noindent
 {\bf The Order of Play} \vspace{-18pt} \begin{enumerate} \item[1] The entrant
decides whether to {\it Enter} or {\it Stay Out.} \item[2] If the entrant
enters, the incumbent can $Collude$ with him, or $Fight$ by cutting the price
drastically. \end{enumerate}

\noindent {\bf Payoffs}\\ Market profits are 300 at the monopoly price and 0 at
the fighting price.  Entry costs are 10. Duopoly competition reduces market
revenue to 100, which is split evenly.

\begin{center} {\bf Table   1: Entry Deterrence I}

 \begin{tabular}{lllccc} &       &             &\multicolumn{3}{c}{\bf
Incumbent}\\ &       &             &  {\it   Collude}    &   &  {\it Fight}
\\ &   & {\it Enter}      &     {\bf 40,50} & $\leftarrow$  &  $-10,0$ \\ & {\bf
Entrant:} &&$\uparrow$& & $\downarrow$ \\ &  &    {\it Stay Out }     &
$0,300$  & $\leftrightarrow$  & {\bf 0,300} \\
 \end{tabular} \end{center}
\vspace{-24pt}
 {\it Payoffs  to:  (Entrant, Incumbent). Arrows show how a player can increase
his payoff.   }

\bigskip

  

\includegraphics[width=150mm]{fig04-03.jpg}

\begin{center}
{\bf Figure 3:  Entry Deterrence I}\end{center}

  

\end{LARGE}
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