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\begin{LARGE}
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\noindent 25 January 2006. Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org.
Overheads for parts of Sections 6.1 and 6.3 of {\it Games and Information}
\begin{LARGE}
\begin{center} { \bf 6 Dynamic Games with Incomplete Information }
\end{center}
\end{LARGE}
\noindent
{\bf Entry Deterrence II: Fighting Is Never Profitable}
In { Entry Deterrence II}, $X=1$.
The sensible Nash equilibrium is
({\it Enter$|$Strong},{\it Enter$|$Weak}, {\it Collude}).
A subgame cannot start at nodes $E_1$ or $E_2$.
Thus, subgame perfectness does not rule out the
implausible Nash equilibrium, ({\it Stay Out$|$Strong},{\it Stay Out$|$Weak}, {\it Fight}).
Also, note that Bayesian updating fails. Suppose in the implausible equilibrium that the entrant enters, deviating from equilibrium. What is the incumbent's belief, $Pr(Strong|Enter)$?
$$
Pr(Strong|Enter) = \frac{ Pr(Enter|Strong) Pr (Strong)}{Pr(Enter)}.
$$
But $Pr(Enter) = 0$, and we can't divide by zero, even if we have a zero in the numerator too.
\newpage
\noindent
{\bf Perfect Bayesian Equilibrium }
\noindent {\it A {\bf perfect bayesian equilibrium} is a strategy profile s and
a set of beliefs $\mu$ such that at each node of the game: }
\noindent { (1) The strategies for the remainder of the game are Nash given
the beliefs and strategies of the other players.}\\
{ 2) The beliefs at each information set are rational given the evidence
appearing thus far in the game (meaning that they are based, if possible, on
priors updated by Bayes' s Rule, given the observed actions of the other players
under the hypothesis that they are in equilibrium). }
In { Entry Deterrence II}, a PBE is
\noindent Entrant: $ Enter|Weak$, $Enter|Strong$\\
Incumbent: {\it Collude} \\
Beliefs: $Prob$( $Strong|$ {\it Stay Out}) = 0.4
There is no equilibrium with ({\it Stay Out$|$Strong},{\it Stay Out$|$Weak}, {\it Fight}), because no out-of-equilibrium beliefs can be found to support those strategies.
\newpage
\noindent {\bf Entry Deterrence III: Fighting Is Sometimes
Profitable }
\noindent
In { Entry Deterrence III}, $X=60$.
Fighting is more profitable for the incumbent than collusion if
the entrant is $Weak$.
\begin{quotation}
\noindent {\bf A plausible pooling equilibrium for Entry Deterrence III} \\
Entrant: $Enter|Weak$, $Enter|Strong$\\
Incumbent: {\it Collude},
Out-of-equilibrium beliefs: $Prob ( Strong|$ {\it Stay Out}) = 0.5
\end{quotation}
The expected
payoff to the incumbent from choosing {\it Fight} is 30 ($=0.5[0] +0.5[60]$),
which is less than the payoff of 50 from $Collude$.
The incumbent will
collude, so the entrant enters. The entrant may know that the incumbent's payoff
is actually 60, but that is irrelevant to the incumbent's behavior.
The out-of-equilibrium belief does not matter to this first equilibrium.
\noindent
{\bf An implausible equilibrium for Entry Deterrence III}\\
Entrant: {\it
Stay Out}$| Weak$, {\it Stay Out}$| Strong$\\
Incumbent: {\it Fight}, \\
Out-of-equilibrium beliefs: $Prob(Strong | Enter) = 0.1$
If the entrant were to deviate and enter, the
incumbent would calculate his payoff from fighting to be 54 ($=0.1[0] +0.9[60]
$), more than the {\it Collude} payoff of 50. The entrant would
therefore stay out.
A {\bf separating} equilibrium would
look like this:
\begin{quotation}
\noindent
{\bf A conjectured separating equilibrium for Entry Deterrence III}
\\
Entrant: {\it Stay Out}$| Weak$, {\it Enter}$| Strong$\\
Incumbent: $
Collude $
\end{quotation}
No out-of-equilibrium beliefs are specified for the conjectures in the
separating equilibrium because there is no out-of-equilibrium behavior
about which to specify them.
\newpage
\noindent
{\bf Entry Deterrence IV: The Incumbent Benefits from Ignorance}
\noindent
Let $X = 300$, so fighting
is even more profitable than in { Entry Deterrence III} but the game is
otherwise the same.
The unique (in its strategies) perfect bayesian equilibrium in pure
strategies is
\noindent {\bf Equilibrium for Entry Deterrence IV}\\
Entrant: {\it Stay Out} $|Weak$, {\it Stay Out} $|Strong$ \\
Incumbent: {\it Fight},\\
Out-of-equilibrium beliefs:
$Prob$({\it Strong}$|${\it Enter}) = 0.5 (passive conjectures)
\newpage
\noindent
{\bf Entry Deterrence V: Lack of Common Knowledge of Ignorance}
\noindent
In { Entry Deterrence V}, it may happen that both the entrant and the
incumbent know the payoff from ({\it Enter, Fight}), but the entrant does not
know whether the incumbent knows. The information is known to both players,
but is not common knowledge.
The game begins with
Nature assigning the entrant a type, {\it Strong} or {\it Weak},
observed by the entrant but not by the incumbent.
Next, Nature moves again
and either tells the incumbent the entrant's type or remains silent,
observed by the incumbent, but not by the entrant.
\includegraphics[width=150mm]{fig06-03.jpg}
\begin{quotation}
\noindent
{\bf Equilibrium for Entry Deterrence V}\\
Entrant: {\it Stay Out}$|Weak$, {\it Stay Out}$|Strong$ \\
Incumbent: {\it Fight}$|${\it Nature said ``Weak''}, {\it Collude} $|${\it
Nature said ``Strong''}, {\it Fight} $|${\it Nature said nothing}, Out-of-
equilibrium
beliefs: $Prob$( $Strong|${\it Enter, Nature said nothing)} = 0.5 (passive
conjectures)
\end{quotation}
\newpage
\includegraphics[width=150mm]{fig06-03.jpg}
The incumbent will fight for either of two
reasons.
With probability 0.9, Nature has said nothing and the incumbent
calculates his expected payoff from {\it Fight} to be 150.
With probability
0.05 $(=0.1[0.5]$) Nature has told the incumbent that the entrant is weak and
the payoff from {\it Fight} is 300.
\newpage
\includegraphics[width=150mm]{fig06-03.jpg}
Even if the entrant is strong and Nature
tells this to the incumbent, the entrant would choose {\it Stay Out}, because
he does not know that the incumbent knows, and his expected payoff from {\it
Enter} would be $-5$ ($= [0.9][-10] + 0.1[40]$).
If it were common knowledge that the entrant was strong, the entrant would enter
and the incumbent would collude. If it is known by both players, but not common
knowledge, the entrant stays out.
\end{LARGE}
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