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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}
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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
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\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.}
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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$E_1$ and $E_3$ are no longer valid.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.
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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}
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{\bf Entry Deterrence I}
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{\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. }
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{\bf Figure 3: Entry Deterrence I}\end{center}
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