Eco514|Game Theory
Lecture 7,Interactive Epistemology (2)
Marciano Siniscalchi
October 7,1999
Introduction
This lecture presents the two main contributions of \interactive epistemology" to the the-
ory of normal-form games,a characterization of Nash equilibrium beliefs,and a full (i.e.
behavioral) characterization of rationalizability.
A review of the basic de nitions
For your convenience,I summarize the essential de nitions pertaining to models of interactive
beliefs; please consult the notes for Lecture 6 for details.
We x a simultaneous game G = (N;(Ai;ui)i2N,A frame for G is a tuple F =
( ;(Ti;ai)i2N) such that,for every player i 2 N,Ti is a partition of,and ai is a Ti-
mesurable map ai,!Ai.
We denote by ti(!) the cell of the partition Ti containing !.
A model for G is a tuple M = (F;(pi)i2N),where F is a frame for G and each pi is a
probability distribution on ( ).
Given any belief i 2 (A i) for Player i,ri( i) is the set of best replies for i given
i,The rst-order beliefs function i,! (A i) extracts Player i’s beliefs about A i
from her beliefs about,
Armed with this notation,we can de ne the event,\Player i is rational" by
Ri =f!2,ai(!)2ri( i(!))g
and the event,\Every player is rational" by R = Ti2N Ri.
De ne the belief operator Bi,2 !2 by
8E ; Bi(E) =f!2,pi(Ejti(!)) = 1g:
1
We read Bi(E) \Player i is certain that E is true." Also de ne the event,\Everybody is
certain that E is true" by B(E) = Ti2N Bi(E).
Beliefs operators satisfy the following properties:
(1) ti = Bi(ti);
(2) E F implies Bi(E) Bi(F);
(3) Bi(E\F) = Bi(E)\Bi(F);
(4) Bi(E) Bi(Bi(E)) and nBi(E) Bi( nBi(E));
(5) Ri Bi(Ri).
Finally,a few shorthand de nitions:
8i2N; q2 (A i),[ i = q] =f!,i(!) = qg
and
8i2N;ai2Ai,[ai = ai] =f!,ai(!) = aig
Nash Equilibrium
I will provide two distinct epistemic characterizations of Nash equilibrium,The rst is
behavioral,nonstandard,and rather trivial,The second is actually a characterization of
equilibrium beliefs,but it is the standard one,and requires a modicum of work.
Simple-minded characterization of Nash equilibrium
You will recall from our past informal discussions that Nash equilibrium incorporates two
key assumptions,(1) Players are Bauesian rational; (2) Their beliefs are correct,what they
believe their opponents will do is exactly what they in fact do.
We now have the machinery we need to formalize this statement,The key idea is that
correctness of rst-order beliefs is easy to de ne in our model.
De nition 1 Fix a game G = (N;(Ai;ui)i2N) and a model M = ( ;(Ti;ai;pi)i2N) for G.
For every i 2 N and ! 2,Player i has correct rst-order beliefs at ! i there exists
a i = (aj)j6=i2A i such that:
(1) 8j6= i,aj(!) = aj;
(2) i(!)(fa ig) = 1.
Let CFBi denote the set of states where Player i’s rst-order beliefs are correct,That
is,concisely,
CFBi =f!,i(!) = (aj(!))j6=ig
At every state in a model,players choose a single action|they do not randomize,If
randomization is an actual strategic option,it must be modelled explicitly,The present
approach only deals with pure Nash equilibria.
2
Proposition 0.1 Fix a game G = (N;(Ai;ui)i2N) and a pro le of actions a = (ai)i2N.
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such
that !2R\Ti2N CFBi and ai(!) = ai for all i2N,then a is a Nash equilibrium of G.
(2) If a2A is a Nash equilibrium of G,there exists a model M= ( ;(Ti;ai;pi)i2N) for
G and a state ! in the model such that !2R\Ti2N CFBi and ai(!) = ai for all i2N.
Proof,(1) At !,ai = ai(!) 2ri( i(!)) = ri( a i) for all i2N; hence,a is a Nash
equilibrium,(2) Consider a model with a single state ! (so that = f!g = ti(!) for
all i2N) such that ai(!) = ai,Since a is a Nash equilibrium,! 2R; since necessarily
pi(!) = 1 for all i2N,!2CSBi for all i2N.
You are authorized to feel cheated,This characterization does not add much to the
de nition of Nash equilibrium,However,it does emphasize what is entailed by correctness
of beliefs.
I mentioned this in the informal language we had to make do with a while ago,but I can
be more precise now,Player i’s beliefs at ! are correct if
i(!) = (ai(!))i2N
which is a condition relating one player’s beliefs with her opponents’ choices at a state,If at
! this is true for all players,and if players are rational,then the pro le (ai(!))i2N they play
at ! must be a Nash equilibrium.
On the other hand,if a pro le a is a Nash equilibrium,the assumption that there exists
a state ! where players are rational and
i(!) = (aj)j6=i
for all i2N,but still it may be the case that ai(!) 6= ai for some i2N,and the pro le
(ai(!))i2N that is actually played at ! may fail to be a Nash equilibrium.
In some sense,the most interesting part of Proposition 0.1 is (2),because it states that
the strong notion of correctness of rst-order beliefs is necessary if we are to interpret Nash
equilibrium as yielding behavioral predictions.
Equilibrium beliefs
The \standard" approach,usually attributed to Aumann and Brandenburger,is intellectually
more satisfying,Instead of asking what conditions imply that players’ behavior will be
consistent with Nash equilibrium,Aumann and Brandenburger ask what conditions imply
that their beliefs will be.
The result is most transparent in two-player games,The idea is to consider a pair ( 1; 2)
of (possibly degenerate) mixed actions as a pair of beliefs,Thus,1,which is conventionally
3
interpreted as Player 1’s mixed action,is actually viewed as Player 2’s beliefs about Player
1’s actions; and similarly for 2.
Then,instead of assuming that players are rational,we assume that they are certain that
their opponent is rational,And,instead of assuming that their rst-order beliefs are correct,
we assume that they are certain of their opponent’s rst-order beliefs.
Roughly speaking,if Player 1 believes that Player 2 will choose an action a2 with positive
probability (i.e,if 2(a2) > 0); if she is certain that he is rational; and if she is certain that
his belief about her actions is given by 1,then it must be the case that a2 is a best reply
to 1,A similar argument holds for Player 2’s beliefs about Player 1,and we can conclude
that ( 1; 2) must be a Nash equilibrium.
Let us state and prove the result; additional comments will follow.
Proposition 0.2 Fix a two-player game G = (N;(Ai;ui)i2N),with N =f1;2g,and a pro le
of mixed actions = ( i)i2N (A1) (A2).
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such
that
!2B(R)\B([ 1 = 1]\[ 2 = 2])
then is a Nash equilibrium of (the mixed extension of) G.
(2) If is a Nash equilibrium of (the mixed extension of) G,then there exists a model
M = ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such that ! 2B(R)\B([ 1 =
1]\[ 2 = 2]).
We begin with two preliminary results.
Lemma 0.3 For any event E,every Player i2N,and every state !2,!2Bi(E)
i supppi( jti(!)) E.
Proof of Lemma 0.3,suppose !2Bi(E),Then pi(Ejti(!)) = 1,This requires that
pi(!0jti(!)) > 0)!02E,so supppi( jti(!)) E,The converse is obvious.
Lemma 0.4 For every player i2N,every ti 2Ti and every state ! 2,! 2Bi(ti) i
ti = ti(!); hence,Bi([ai = ai]) = [ai = ai] and Bi([ i = q]) = [ i = q],and Bi(Ri) = Ri.
Proof,For every !,pi(tijti(!)) = 1 if ti = ti(!),and pi(tijti(!)) = 0 otherwise,This
proves the rst claim,Now ai( ) is Ti-measurable; hence,if !2[ai = ai],then ti(!) [ai =
ai),and if !2Bi([ai = ai]),then necessarily !2ti(!) [ai = ai] (because otherwise we
could not have pi([ai = ai]jti(!)) = 1); and similarly for [ i = q],The second claim follows
immediately,Finally,note that Ri =f!,ai(!)2ri( i(!))gmust clearly be Ti-measurable,
and the third claim follows by the same argument as above.
4
Proof of Proposition 0.2,(1) Fix a1 2A1 such that 1(!)(a1) > 0; that is,consider
an action which,at !,Player 2 expects Player 1 to choose with positive probability,This
implies that there exists !0 such that a1(!0) = a1 and p2(!0jt2(!)) > 0.
Our assumptions imply that ! 2B2(R1)\B2([ 2 = 2]) = B2(R1 \[ 2 = 2]) (here
we are using a key property of B2,B2(E\F) = B2(E)\B2(F)),Hence,by Lemma
0.3,suppp2( jt2(!)) R1 \[ 2 = 2],But then !0 2R1 \[ 2 = 2],which implies that
a1 = a1(!0)2r1( 2(!0)) = r1( 2).
Thus,every a1 such that 1(!)(a1) > 0 is a best reply to 2,To complete the proof,
we must show that 1(!) = 1,Note that this is not part of our assumptions; however,
our assumptions do imply that !2B2([ 1 = 1]),and by Lemma 0.4,the required equality
follows,The proof of Part (1) is complete.
(2) Construct a model as follows,let = Qi2N supp i and de ne via the possibility
correspondence ti( ) by letting
8(a1;a2)2 ; ti(a1;a2) =faig supp i
for i = 1;2,This is similar to (but simpler than) the construction we used to prove the
\revelation principle" in the last lecture notes,States are pro les that have positive proba-
bility in equilibrium,and each player is informed of her action at any state,This makes it
is possible to de ne ai(a1;a2) = ai.
Finally,we can use a common prior on to complete the de nition of a model,p1(a1;a2) =
p2(a1;a2) = 1(a1) 2(a2),There is nothing special about this; the key point is that,for ev-
ery (a1;a2) 2,pi(a1;a2jti(a1;a2)) = i(a i) and therefore i(a1;a2) = i with these
de nitions,Note that this implies in particular that [ i = i] = for i = 1;2; hence,
clearly (a1;a2)2B([ 1 = 1]\[ 2 = 2]) at any state (a1;a2)2,
Now consider any state ! = (a1;a2),Then ai(!) = ai2ri( i) = ri( i(!)),so !2Ri;
that is,R1 = R2 = (which is obvious,if you think about it!) and therefore,at any state
!2,!2B(R),We are done.
Observe that,to prove (1),we do not actually need ! 2Bi(Ri),However,this comes
almost for free in part (2),Also,the assumptions imply (via Lemma 0.4) that players are
indeed rational at the state under consideration; this is not used in the proof of Part (1),
but again it comes for free in part (2).
A key observation is that Aumann and Brandenburger’s construction solves another key
interpretational problem almost automatically,Mixed actions are viewed as beliefs,thus,
one does not even need to invoke randomizations to justify them.
Rationalizability
We now turn to Tan and Werlang’s characterization of correlated rationalizability,Actually,
I will again present the argument for two-player games,but here everything generalizes
5
readily,and it is easy to add stochastic independence as an explicit restriction on beliefs.
Indeed,any restriction on rst-order beliefs can be easily added to the characterization.
Let me remind you of the de nition rst,Let A0i = Ai for i = 1;2,Next,for k 1,say
that ai2Aki i there exists i2 (A i) such that i(Ak 1 i ) = 1 and ai2ri( i).
We now de ne a sequence of events:
B0 = R; 8k 1;Bk = B(Bk 1)
Thus,B1 = \Everybody is certain that everybody is rational"; B2 = \Everybody is certain
that everybody is certain that everybody is rational"; and so on.
Proposition 0.5 Fix a two-player game G = (N;(Ai;ui)i2N),with N =f1;2g,and a pro le
of actions a = (a1;a2).
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model with
ai(!) = ai for i = 1;2 such that
!2
k\
‘=0
B‘
then a2Ak+1,Hence,if there is a model M and a state ! in that model with ai(!) = ai
for i = 1;2 such that !2T1‘=0 B‘,then a is rationalizable.
(2) If a2Ak (k 1),then there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state
! in the model such that ai(!) = ai,i = 1;2 and !2Tk 1‘=0 B‘,If a is rationalizable,there
exists a model M and a state ! such that ai(!) = ai,i = 1;2 and !2T1‘=0 B‘.
Proof,Note rst that,for k 1,
k\
‘=0
B‘ = R\
k\
‘=1
B(B‘ 1) = R\B
k 1\
‘=0
B‘
!
(1,k = 0) Fix !2B0 = R,Then trivially ai = ai(!)2ri( i(!)),so ai2A1i.
(1,k> 0) By induction,suppose the claim is true for‘ = 0;:::;k 1,Fix!2Tk‘=0Bk; by
the above decomposition,!2Ri and !2Bi
T
k 1
‘=0 B
‘
,By Lemma 0.3,supppi( jti(!))
Tk 1
‘=0 B
‘; but for all !0 2 Tk 1
‘=0 B
‘,by the induction hypothesis a i(!0) 2 Ak 1
i,Hence,
i(!)(Ak 1 i ) = 1,and we are done,The claim concerning rationalizable pro les follows
from the fact that,since the game is nite,there exists K such that k K implies Aki = AKi
for all i2N.
(2),left as an exercise for the interested reader.
The characterization result is quite straightforward,I do point out that the assumption
that ! 2Sk‘=0B‘ has behavioral implications,what players actually do at ! is consistent
with k + 1 steps of the iterative procedure de ning rationalizability,As I have remarked
many times,this is not the case with Nash equilibrium.
6
Lecture 7,Interactive Epistemology (2)
Marciano Siniscalchi
October 7,1999
Introduction
This lecture presents the two main contributions of \interactive epistemology" to the the-
ory of normal-form games,a characterization of Nash equilibrium beliefs,and a full (i.e.
behavioral) characterization of rationalizability.
A review of the basic de nitions
For your convenience,I summarize the essential de nitions pertaining to models of interactive
beliefs; please consult the notes for Lecture 6 for details.
We x a simultaneous game G = (N;(Ai;ui)i2N,A frame for G is a tuple F =
( ;(Ti;ai)i2N) such that,for every player i 2 N,Ti is a partition of,and ai is a Ti-
mesurable map ai,!Ai.
We denote by ti(!) the cell of the partition Ti containing !.
A model for G is a tuple M = (F;(pi)i2N),where F is a frame for G and each pi is a
probability distribution on ( ).
Given any belief i 2 (A i) for Player i,ri( i) is the set of best replies for i given
i,The rst-order beliefs function i,! (A i) extracts Player i’s beliefs about A i
from her beliefs about,
Armed with this notation,we can de ne the event,\Player i is rational" by
Ri =f!2,ai(!)2ri( i(!))g
and the event,\Every player is rational" by R = Ti2N Ri.
De ne the belief operator Bi,2 !2 by
8E ; Bi(E) =f!2,pi(Ejti(!)) = 1g:
1
We read Bi(E) \Player i is certain that E is true." Also de ne the event,\Everybody is
certain that E is true" by B(E) = Ti2N Bi(E).
Beliefs operators satisfy the following properties:
(1) ti = Bi(ti);
(2) E F implies Bi(E) Bi(F);
(3) Bi(E\F) = Bi(E)\Bi(F);
(4) Bi(E) Bi(Bi(E)) and nBi(E) Bi( nBi(E));
(5) Ri Bi(Ri).
Finally,a few shorthand de nitions:
8i2N; q2 (A i),[ i = q] =f!,i(!) = qg
and
8i2N;ai2Ai,[ai = ai] =f!,ai(!) = aig
Nash Equilibrium
I will provide two distinct epistemic characterizations of Nash equilibrium,The rst is
behavioral,nonstandard,and rather trivial,The second is actually a characterization of
equilibrium beliefs,but it is the standard one,and requires a modicum of work.
Simple-minded characterization of Nash equilibrium
You will recall from our past informal discussions that Nash equilibrium incorporates two
key assumptions,(1) Players are Bauesian rational; (2) Their beliefs are correct,what they
believe their opponents will do is exactly what they in fact do.
We now have the machinery we need to formalize this statement,The key idea is that
correctness of rst-order beliefs is easy to de ne in our model.
De nition 1 Fix a game G = (N;(Ai;ui)i2N) and a model M = ( ;(Ti;ai;pi)i2N) for G.
For every i 2 N and ! 2,Player i has correct rst-order beliefs at ! i there exists
a i = (aj)j6=i2A i such that:
(1) 8j6= i,aj(!) = aj;
(2) i(!)(fa ig) = 1.
Let CFBi denote the set of states where Player i’s rst-order beliefs are correct,That
is,concisely,
CFBi =f!,i(!) = (aj(!))j6=ig
At every state in a model,players choose a single action|they do not randomize,If
randomization is an actual strategic option,it must be modelled explicitly,The present
approach only deals with pure Nash equilibria.
2
Proposition 0.1 Fix a game G = (N;(Ai;ui)i2N) and a pro le of actions a = (ai)i2N.
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such
that !2R\Ti2N CFBi and ai(!) = ai for all i2N,then a is a Nash equilibrium of G.
(2) If a2A is a Nash equilibrium of G,there exists a model M= ( ;(Ti;ai;pi)i2N) for
G and a state ! in the model such that !2R\Ti2N CFBi and ai(!) = ai for all i2N.
Proof,(1) At !,ai = ai(!) 2ri( i(!)) = ri( a i) for all i2N; hence,a is a Nash
equilibrium,(2) Consider a model with a single state ! (so that = f!g = ti(!) for
all i2N) such that ai(!) = ai,Since a is a Nash equilibrium,! 2R; since necessarily
pi(!) = 1 for all i2N,!2CSBi for all i2N.
You are authorized to feel cheated,This characterization does not add much to the
de nition of Nash equilibrium,However,it does emphasize what is entailed by correctness
of beliefs.
I mentioned this in the informal language we had to make do with a while ago,but I can
be more precise now,Player i’s beliefs at ! are correct if
i(!) = (ai(!))i2N
which is a condition relating one player’s beliefs with her opponents’ choices at a state,If at
! this is true for all players,and if players are rational,then the pro le (ai(!))i2N they play
at ! must be a Nash equilibrium.
On the other hand,if a pro le a is a Nash equilibrium,the assumption that there exists
a state ! where players are rational and
i(!) = (aj)j6=i
for all i2N,but still it may be the case that ai(!) 6= ai for some i2N,and the pro le
(ai(!))i2N that is actually played at ! may fail to be a Nash equilibrium.
In some sense,the most interesting part of Proposition 0.1 is (2),because it states that
the strong notion of correctness of rst-order beliefs is necessary if we are to interpret Nash
equilibrium as yielding behavioral predictions.
Equilibrium beliefs
The \standard" approach,usually attributed to Aumann and Brandenburger,is intellectually
more satisfying,Instead of asking what conditions imply that players’ behavior will be
consistent with Nash equilibrium,Aumann and Brandenburger ask what conditions imply
that their beliefs will be.
The result is most transparent in two-player games,The idea is to consider a pair ( 1; 2)
of (possibly degenerate) mixed actions as a pair of beliefs,Thus,1,which is conventionally
3
interpreted as Player 1’s mixed action,is actually viewed as Player 2’s beliefs about Player
1’s actions; and similarly for 2.
Then,instead of assuming that players are rational,we assume that they are certain that
their opponent is rational,And,instead of assuming that their rst-order beliefs are correct,
we assume that they are certain of their opponent’s rst-order beliefs.
Roughly speaking,if Player 1 believes that Player 2 will choose an action a2 with positive
probability (i.e,if 2(a2) > 0); if she is certain that he is rational; and if she is certain that
his belief about her actions is given by 1,then it must be the case that a2 is a best reply
to 1,A similar argument holds for Player 2’s beliefs about Player 1,and we can conclude
that ( 1; 2) must be a Nash equilibrium.
Let us state and prove the result; additional comments will follow.
Proposition 0.2 Fix a two-player game G = (N;(Ai;ui)i2N),with N =f1;2g,and a pro le
of mixed actions = ( i)i2N (A1) (A2).
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such
that
!2B(R)\B([ 1 = 1]\[ 2 = 2])
then is a Nash equilibrium of (the mixed extension of) G.
(2) If is a Nash equilibrium of (the mixed extension of) G,then there exists a model
M = ( ;(Ti;ai;pi)i2N) for G and a state ! in the model such that ! 2B(R)\B([ 1 =
1]\[ 2 = 2]).
We begin with two preliminary results.
Lemma 0.3 For any event E,every Player i2N,and every state !2,!2Bi(E)
i supppi( jti(!)) E.
Proof of Lemma 0.3,suppose !2Bi(E),Then pi(Ejti(!)) = 1,This requires that
pi(!0jti(!)) > 0)!02E,so supppi( jti(!)) E,The converse is obvious.
Lemma 0.4 For every player i2N,every ti 2Ti and every state ! 2,! 2Bi(ti) i
ti = ti(!); hence,Bi([ai = ai]) = [ai = ai] and Bi([ i = q]) = [ i = q],and Bi(Ri) = Ri.
Proof,For every !,pi(tijti(!)) = 1 if ti = ti(!),and pi(tijti(!)) = 0 otherwise,This
proves the rst claim,Now ai( ) is Ti-measurable; hence,if !2[ai = ai],then ti(!) [ai =
ai),and if !2Bi([ai = ai]),then necessarily !2ti(!) [ai = ai] (because otherwise we
could not have pi([ai = ai]jti(!)) = 1); and similarly for [ i = q],The second claim follows
immediately,Finally,note that Ri =f!,ai(!)2ri( i(!))gmust clearly be Ti-measurable,
and the third claim follows by the same argument as above.
4
Proof of Proposition 0.2,(1) Fix a1 2A1 such that 1(!)(a1) > 0; that is,consider
an action which,at !,Player 2 expects Player 1 to choose with positive probability,This
implies that there exists !0 such that a1(!0) = a1 and p2(!0jt2(!)) > 0.
Our assumptions imply that ! 2B2(R1)\B2([ 2 = 2]) = B2(R1 \[ 2 = 2]) (here
we are using a key property of B2,B2(E\F) = B2(E)\B2(F)),Hence,by Lemma
0.3,suppp2( jt2(!)) R1 \[ 2 = 2],But then !0 2R1 \[ 2 = 2],which implies that
a1 = a1(!0)2r1( 2(!0)) = r1( 2).
Thus,every a1 such that 1(!)(a1) > 0 is a best reply to 2,To complete the proof,
we must show that 1(!) = 1,Note that this is not part of our assumptions; however,
our assumptions do imply that !2B2([ 1 = 1]),and by Lemma 0.4,the required equality
follows,The proof of Part (1) is complete.
(2) Construct a model as follows,let = Qi2N supp i and de ne via the possibility
correspondence ti( ) by letting
8(a1;a2)2 ; ti(a1;a2) =faig supp i
for i = 1;2,This is similar to (but simpler than) the construction we used to prove the
\revelation principle" in the last lecture notes,States are pro les that have positive proba-
bility in equilibrium,and each player is informed of her action at any state,This makes it
is possible to de ne ai(a1;a2) = ai.
Finally,we can use a common prior on to complete the de nition of a model,p1(a1;a2) =
p2(a1;a2) = 1(a1) 2(a2),There is nothing special about this; the key point is that,for ev-
ery (a1;a2) 2,pi(a1;a2jti(a1;a2)) = i(a i) and therefore i(a1;a2) = i with these
de nitions,Note that this implies in particular that [ i = i] = for i = 1;2; hence,
clearly (a1;a2)2B([ 1 = 1]\[ 2 = 2]) at any state (a1;a2)2,
Now consider any state ! = (a1;a2),Then ai(!) = ai2ri( i) = ri( i(!)),so !2Ri;
that is,R1 = R2 = (which is obvious,if you think about it!) and therefore,at any state
!2,!2B(R),We are done.
Observe that,to prove (1),we do not actually need ! 2Bi(Ri),However,this comes
almost for free in part (2),Also,the assumptions imply (via Lemma 0.4) that players are
indeed rational at the state under consideration; this is not used in the proof of Part (1),
but again it comes for free in part (2).
A key observation is that Aumann and Brandenburger’s construction solves another key
interpretational problem almost automatically,Mixed actions are viewed as beliefs,thus,
one does not even need to invoke randomizations to justify them.
Rationalizability
We now turn to Tan and Werlang’s characterization of correlated rationalizability,Actually,
I will again present the argument for two-player games,but here everything generalizes
5
readily,and it is easy to add stochastic independence as an explicit restriction on beliefs.
Indeed,any restriction on rst-order beliefs can be easily added to the characterization.
Let me remind you of the de nition rst,Let A0i = Ai for i = 1;2,Next,for k 1,say
that ai2Aki i there exists i2 (A i) such that i(Ak 1 i ) = 1 and ai2ri( i).
We now de ne a sequence of events:
B0 = R; 8k 1;Bk = B(Bk 1)
Thus,B1 = \Everybody is certain that everybody is rational"; B2 = \Everybody is certain
that everybody is certain that everybody is rational"; and so on.
Proposition 0.5 Fix a two-player game G = (N;(Ai;ui)i2N),with N =f1;2g,and a pro le
of actions a = (a1;a2).
(1) If there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state ! in the model with
ai(!) = ai for i = 1;2 such that
!2
k\
‘=0
B‘
then a2Ak+1,Hence,if there is a model M and a state ! in that model with ai(!) = ai
for i = 1;2 such that !2T1‘=0 B‘,then a is rationalizable.
(2) If a2Ak (k 1),then there exists a modelM= ( ;(Ti;ai;pi)i2N) for G and a state
! in the model such that ai(!) = ai,i = 1;2 and !2Tk 1‘=0 B‘,If a is rationalizable,there
exists a model M and a state ! such that ai(!) = ai,i = 1;2 and !2T1‘=0 B‘.
Proof,Note rst that,for k 1,
k\
‘=0
B‘ = R\
k\
‘=1
B(B‘ 1) = R\B
k 1\
‘=0
B‘
!
(1,k = 0) Fix !2B0 = R,Then trivially ai = ai(!)2ri( i(!)),so ai2A1i.
(1,k> 0) By induction,suppose the claim is true for‘ = 0;:::;k 1,Fix!2Tk‘=0Bk; by
the above decomposition,!2Ri and !2Bi
T
k 1
‘=0 B
‘
,By Lemma 0.3,supppi( jti(!))
Tk 1
‘=0 B
‘; but for all !0 2 Tk 1
‘=0 B
‘,by the induction hypothesis a i(!0) 2 Ak 1
i,Hence,
i(!)(Ak 1 i ) = 1,and we are done,The claim concerning rationalizable pro les follows
from the fact that,since the game is nite,there exists K such that k K implies Aki = AKi
for all i2N.
(2),left as an exercise for the interested reader.
The characterization result is quite straightforward,I do point out that the assumption
that ! 2Sk‘=0B‘ has behavioral implications,what players actually do at ! is consistent
with k + 1 steps of the iterative procedure de ning rationalizability,As I have remarked
many times,this is not the case with Nash equilibrium.
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