Eco514|Game Theory
Lecture 16,Applications of Sequential and Perfect
Bayesian Equilibrium
Marciano Siniscalchi
November 16,1999
Introduction
The purpose of this lecture is to help you familiarize with the workings of sequential equi-
librium and \sequential equilibrium lite",i.e,perfect Bayesian equilibrium.
The main focus is the \reputation" result of Kreps and Wilson (1982),You should refer
to OR for details and de nitions,I am following the textbook quite closely.
We have already mentioned the Entry Deterrence game,Now consider aK-fold repetition
of the game in which an incumbent faces a sequence of K (potential) entrants,It is clear
that,in the subgame in which the last entrant gets a chance to play,the incumbent will
Concede,But then,by backward induction,he will also Concede at any previous round.
This conclusion seems unpalatable,One would expect the chain-store to ght a few times
early on so as to deter further entry; yet,this simply cannot occur in equilibrium.
The Chain-Store Game with Imperfect Information
Kreps and Wilson suggest the following way out of the Chain Store Paradox,Assume that,
with some small probability > 0,the incumbent is \Tough",that is,she really enjoys
beating up entrants.
Consider the behavior of a \Regular" incumbent; clearly,when she faces the last entrant,
she will Concede if the latter moves In,However,in previous rounds,she may threaten
to Fight entry; in this case,if the current entrant moves In and is met with a ght,the
following entrant does not learn whether the incumbent is Regular or Tough,because both
Fight entry.
Hence,the posterior probability that the incumbent is Tough remains constant at,and
if this value is such that the following Entrant prefers not to Enter,the incumbent’s threat
is credible.
1
Things are not quite so simple in actuality,but Kreps and Wilson prove that something
very close to this can be supported in a sequential equilibrium; indeed,while there exist other
sequential equilibria in which no \reputation" is maintained,there exists a simple restriction
on the entrants’ beliefs which ensures that the reputation outcome obtains.
OR analyze the situation using the notion of PBE,and do not investigate outcome
uniqueness,I will follow their treatment,emphasizing certain key steps.
Model
Payo s in the stage game are as follows:
b;c q1
0,a
Out
Inq2
F
C
b 1;f
Figure 1,= 1 =fR;Tg,a> 1 >b> 0,cR = fT = 0,fR = cT = 1.
To understand the equilibrium strategies,observe rst that,if the entrant expect the
incumbent to Fight with probability b,her expected payo from entry equals b(1 b) + (b
1)b = 0; hence,she is indi erent between staying Out and going In.
Now,the Tough incumbent will always Fight in the PBE under consideration,This
implies,among other things,that as soon as an entrant observes C in any previous round,he
will conclude that the incumbent is Regular,and therefore will enter; so will every subsequent
entrant.
Consider the problem faced by the Regular incumbent who observes entry at some stage
k < K,If she Concedes,she gets 0 in the current period,but faces entry thereafter|and
hence,by best-responding,she can secure a payo of no more than 0,Now suppose that
she Fights,and that this leads the next entrant to revise her beliefs in such a way that he is
indi erent between In and Out; suppose further that,this being the case,the next entrant
chooses Out with probability 1a,Then,by Fighting the current entrant (and best-responding
thereafter),the incumbent also gets a continuation payo of 0.
Thus,following entry,the incumbent is indi erent between F and C,Hence,he can
randomize,and the probability of C can be chosen so that the posterior beliefs of the next
entrant will be such that he is indeed indi erent between going In and staying Out.
[Notice that this argument relies heavily on the equilibrium assumption!]
2
Equilibrium
[Refer to OR for de nitions]
The basic idea is as follows.
The rst k 1 entrants will stay Out,where k = minfk,bK k + 1 g,Hence,
nothing will be learned about the incumbent’s type in equilibrium during the rst
k 1 rounds,t(h) <k implies (h)(T) =,
If,however,one of the rst k 1 entrants does enter,the Regular incumbent Fights.
Note that this implies that,even in this case,nothing will be learned about the in-
cumbent’s type; hence,the continuation play following such a deviation resembles
equilibrium play.
Entrant k goes In if = (h)(T) <bK k +1,and randomizes between In and Out in
case of equality,In either case,if there is entry,the Incumbent reacts by randomizing
between C and F,If there is no entry,beliefs are unchanged,so the next entrant
certainly enters.
Beliefs are thus updated via Bayes’ rule,Thus,as soon as the Incumbent plays C,the
following Entrants all choose In,Also,whenever an entrant goes In and the incumbent
Fights,the next entrant is indi erent between In and Out,and hence randomizes.
In the last round,the Incumbent Concedes.
You should convince yourself that the equilibrium described in OR achieves precisely
this,In particular,you should see that,whenever (h)(T) = bK k+1,the probability of a
Fight in the following round is exactly b (so the next entrant is indeed willing to randomize).
Also,the posterior probability of T given F is bK k,as per the equilibrium beliefs,Finally,
as claimed in the previous section,it is easy to see that,following entry at or after Stage k,
the continuation payo to the incumbent if he plays F is indeed 0,as it must be to justify
randomization.
Plausible Beliefs
The preceding arguments go a long way (but not all the way) towards proving that reputation
can be maintained in a PBE,However,the original result due to Kreps and Wilson is stronger:
rst,they use the notion of Sequential Equilibrium; second,they show that any SE which
satis es a certain restriction on beliefs induces the reputation outcome.
The restriction on beliefs has the following form,consider two histories h and h0 such
that h0 di ers from h only in that,at some subhistories following entry,the incumbent
3
Concedes instead of Fighting,Then it must be the case that (h)(R) (h0)(R),This
makes sense|and Kreps and Wilson deem beliefs which satisfy this condition plausible.
The key point is that one could construct \strange" equilibria in which reacting to o -
equilibrium entry with F induces subsequent entrants to believe that the incumbent is Reg-
ular,this could engender further entry out of equilibrium,If this is the case,the incumbent
might Concede early in the game,in order to avoid (!) being identi ed as a regular type
(!!!).
You can convince yourself that the beliefs used in the construction of our \reputational
equilibrium" are indeed plausible.
Building vs,Maintaining a Reputation
Note that,in the equilibrium we have constructed,essentially nothing happens in the rst
k 1 stages,Thus,there is no sense in which the incumbent \builds a reputation" for
being tough,This can only be said of the last few stages|but note that the length of the
randomization phase only depends on,and not on the overall length of the game,Thus,one
may have a very protracted play phase in which no reputation is built,and a comparatively
tiny one in which actual reputation building occurs.
Rather,this story has to do with maintaining a reputation for toughness,Also,it should
be apparent that the result depends crucially on the equilibrium assumption,consider the
delicate interlocking of randomizations.
4
Lecture 16,Applications of Sequential and Perfect
Bayesian Equilibrium
Marciano Siniscalchi
November 16,1999
Introduction
The purpose of this lecture is to help you familiarize with the workings of sequential equi-
librium and \sequential equilibrium lite",i.e,perfect Bayesian equilibrium.
The main focus is the \reputation" result of Kreps and Wilson (1982),You should refer
to OR for details and de nitions,I am following the textbook quite closely.
We have already mentioned the Entry Deterrence game,Now consider aK-fold repetition
of the game in which an incumbent faces a sequence of K (potential) entrants,It is clear
that,in the subgame in which the last entrant gets a chance to play,the incumbent will
Concede,But then,by backward induction,he will also Concede at any previous round.
This conclusion seems unpalatable,One would expect the chain-store to ght a few times
early on so as to deter further entry; yet,this simply cannot occur in equilibrium.
The Chain-Store Game with Imperfect Information
Kreps and Wilson suggest the following way out of the Chain Store Paradox,Assume that,
with some small probability > 0,the incumbent is \Tough",that is,she really enjoys
beating up entrants.
Consider the behavior of a \Regular" incumbent; clearly,when she faces the last entrant,
she will Concede if the latter moves In,However,in previous rounds,she may threaten
to Fight entry; in this case,if the current entrant moves In and is met with a ght,the
following entrant does not learn whether the incumbent is Regular or Tough,because both
Fight entry.
Hence,the posterior probability that the incumbent is Tough remains constant at,and
if this value is such that the following Entrant prefers not to Enter,the incumbent’s threat
is credible.
1
Things are not quite so simple in actuality,but Kreps and Wilson prove that something
very close to this can be supported in a sequential equilibrium; indeed,while there exist other
sequential equilibria in which no \reputation" is maintained,there exists a simple restriction
on the entrants’ beliefs which ensures that the reputation outcome obtains.
OR analyze the situation using the notion of PBE,and do not investigate outcome
uniqueness,I will follow their treatment,emphasizing certain key steps.
Model
Payo s in the stage game are as follows:
b;c q1
0,a
Out
Inq2
F
C
b 1;f
Figure 1,= 1 =fR;Tg,a> 1 >b> 0,cR = fT = 0,fR = cT = 1.
To understand the equilibrium strategies,observe rst that,if the entrant expect the
incumbent to Fight with probability b,her expected payo from entry equals b(1 b) + (b
1)b = 0; hence,she is indi erent between staying Out and going In.
Now,the Tough incumbent will always Fight in the PBE under consideration,This
implies,among other things,that as soon as an entrant observes C in any previous round,he
will conclude that the incumbent is Regular,and therefore will enter; so will every subsequent
entrant.
Consider the problem faced by the Regular incumbent who observes entry at some stage
k < K,If she Concedes,she gets 0 in the current period,but faces entry thereafter|and
hence,by best-responding,she can secure a payo of no more than 0,Now suppose that
she Fights,and that this leads the next entrant to revise her beliefs in such a way that he is
indi erent between In and Out; suppose further that,this being the case,the next entrant
chooses Out with probability 1a,Then,by Fighting the current entrant (and best-responding
thereafter),the incumbent also gets a continuation payo of 0.
Thus,following entry,the incumbent is indi erent between F and C,Hence,he can
randomize,and the probability of C can be chosen so that the posterior beliefs of the next
entrant will be such that he is indeed indi erent between going In and staying Out.
[Notice that this argument relies heavily on the equilibrium assumption!]
2
Equilibrium
[Refer to OR for de nitions]
The basic idea is as follows.
The rst k 1 entrants will stay Out,where k = minfk,bK k + 1 g,Hence,
nothing will be learned about the incumbent’s type in equilibrium during the rst
k 1 rounds,t(h) <k implies (h)(T) =,
If,however,one of the rst k 1 entrants does enter,the Regular incumbent Fights.
Note that this implies that,even in this case,nothing will be learned about the in-
cumbent’s type; hence,the continuation play following such a deviation resembles
equilibrium play.
Entrant k goes In if = (h)(T) <bK k +1,and randomizes between In and Out in
case of equality,In either case,if there is entry,the Incumbent reacts by randomizing
between C and F,If there is no entry,beliefs are unchanged,so the next entrant
certainly enters.
Beliefs are thus updated via Bayes’ rule,Thus,as soon as the Incumbent plays C,the
following Entrants all choose In,Also,whenever an entrant goes In and the incumbent
Fights,the next entrant is indi erent between In and Out,and hence randomizes.
In the last round,the Incumbent Concedes.
You should convince yourself that the equilibrium described in OR achieves precisely
this,In particular,you should see that,whenever (h)(T) = bK k+1,the probability of a
Fight in the following round is exactly b (so the next entrant is indeed willing to randomize).
Also,the posterior probability of T given F is bK k,as per the equilibrium beliefs,Finally,
as claimed in the previous section,it is easy to see that,following entry at or after Stage k,
the continuation payo to the incumbent if he plays F is indeed 0,as it must be to justify
randomization.
Plausible Beliefs
The preceding arguments go a long way (but not all the way) towards proving that reputation
can be maintained in a PBE,However,the original result due to Kreps and Wilson is stronger:
rst,they use the notion of Sequential Equilibrium; second,they show that any SE which
satis es a certain restriction on beliefs induces the reputation outcome.
The restriction on beliefs has the following form,consider two histories h and h0 such
that h0 di ers from h only in that,at some subhistories following entry,the incumbent
3
Concedes instead of Fighting,Then it must be the case that (h)(R) (h0)(R),This
makes sense|and Kreps and Wilson deem beliefs which satisfy this condition plausible.
The key point is that one could construct \strange" equilibria in which reacting to o -
equilibrium entry with F induces subsequent entrants to believe that the incumbent is Reg-
ular,this could engender further entry out of equilibrium,If this is the case,the incumbent
might Concede early in the game,in order to avoid (!) being identi ed as a regular type
(!!!).
You can convince yourself that the beliefs used in the construction of our \reputational
equilibrium" are indeed plausible.
Building vs,Maintaining a Reputation
Note that,in the equilibrium we have constructed,essentially nothing happens in the rst
k 1 stages,Thus,there is no sense in which the incumbent \builds a reputation" for
being tough,This can only be said of the last few stages|but note that the length of the
randomization phase only depends on,and not on the overall length of the game,Thus,one
may have a very protracted play phase in which no reputation is built,and a comparatively
tiny one in which actual reputation building occurs.
Rather,this story has to do with maintaining a reputation for toughness,Also,it should
be apparent that the result depends crucially on the equilibrium assumption,consider the
delicate interlocking of randomizations.
4
