Eco514|Game Theory
Problem Set 2,Due Thursday,October 14
Recall the following de nitions,in any model M = ( ;(Ti;ai;pi)i2N),Ri is the event
\Player i is rational"; R = Ti2N Ri,Also,Bi(E) is the event \Player i is certain that E is
true" and B(E) = Ti2N Bi(E),This is as in Lecture 7.
Let me introduce the following notation for iterated mutual certainty,B(0)(E) = E,
B(k)(E) = B(B(k 1)(E)),Then the de nition of Bk in Lecture 7 can be rewritten as Bk =
B(k)(R) for k 0.
1,A characterization of Correlated Rationalizability
Consider a game G = (N;(Ai;ui)i2N),Construct a ( nite) model M = ( ;(Ti;ai;pi)i2N)
for G such that,for any k 0 and for any pro le of actions a2A,a2Ak+1 if and only if
there exists a state !2 such that (i) ai(!) = ai for all i2N,and (ii) !2Sk‘=0Bk.
A comment,the \if" part follows from Part (1) of Proposition 5 in Lecture 7; however,
notice that Part (2) therein asserts the existence of a model that \works" for a single ratio-
nalizable action pro le,The \only if" part of the claim I am asking you to prove instead
applies to all such pro les,Also,[hint! hint!] an elegant proof of the claim does not rely on
Proposition 5 in Lecture 7 for the \if" part.
Also,I specify that I am looking for a nite model because obviously the complicated
\universal" model constructed in the notes for Lecture 6 will \work".
2,Mutual and Common Certainty
Consider the normal-form game with payo uncertainty in Fig,1:
A N
A,1,0
N 0,1 0,0
Figure 1,2f12; 32g; Pr( = 32) =,
1
(i) Begin by assuming that the probability that = 32 is commonly known among
the players,Formalize this assumption by representing the situation as a game with payo
uncertainty G = (N; ;(Ai;ui;Ti)i2N) as de ned in Lecture 4,and compute all its Bayesian
Nash equilibria (again as de ned in Lectures 4 and 5).
(ii) Now x in the region where (N,N) and (A,A) are both Bayesian Nash equilibria of
the game in which is common knowledge,Consider the following (k + 1) statements:
E = B(0)(E) = Every player believes that = 32 with probability ;
B(1)(E) = Every player is certain that E is true;
:::
B(k)(E) = Every player is certain that B(k 1)(E) is true.
Show that,for any k = 0;1;:::,you can provide an alternative formulation of the game
Gk = (N; k;(Ai;uki;Tki )i2N) (with the same action sets) and a pair of priors pki 2 ( k),
i = 1;2,such that:
(i) there exists some state !2 k for which !2Tk‘=0B(‘)(E) (where of course E and
certainty operators are de ned in k and with respect to the priors pki );
(ii) yet,there is a unique Bayesian Nash equilibrium of the game in which the players’
priors are given by pki,i = 1;2,and in this equilibrium both players choose A at every state,
i.e,regardless of their respective types (in the notation of Lecture 4,i(ti) = A for i = 1;2
and all ti2Tki ).
In words,to support (N,N) as an equilibrium,the value of must be common certainty;
mutual certainty of arbitrarily high (but nite) order is not enough.
4,From OR,19.1,35.2,42.1 (important!)
2
Problem Set 2,Due Thursday,October 14
Recall the following de nitions,in any model M = ( ;(Ti;ai;pi)i2N),Ri is the event
\Player i is rational"; R = Ti2N Ri,Also,Bi(E) is the event \Player i is certain that E is
true" and B(E) = Ti2N Bi(E),This is as in Lecture 7.
Let me introduce the following notation for iterated mutual certainty,B(0)(E) = E,
B(k)(E) = B(B(k 1)(E)),Then the de nition of Bk in Lecture 7 can be rewritten as Bk =
B(k)(R) for k 0.
1,A characterization of Correlated Rationalizability
Consider a game G = (N;(Ai;ui)i2N),Construct a ( nite) model M = ( ;(Ti;ai;pi)i2N)
for G such that,for any k 0 and for any pro le of actions a2A,a2Ak+1 if and only if
there exists a state !2 such that (i) ai(!) = ai for all i2N,and (ii) !2Sk‘=0Bk.
A comment,the \if" part follows from Part (1) of Proposition 5 in Lecture 7; however,
notice that Part (2) therein asserts the existence of a model that \works" for a single ratio-
nalizable action pro le,The \only if" part of the claim I am asking you to prove instead
applies to all such pro les,Also,[hint! hint!] an elegant proof of the claim does not rely on
Proposition 5 in Lecture 7 for the \if" part.
Also,I specify that I am looking for a nite model because obviously the complicated
\universal" model constructed in the notes for Lecture 6 will \work".
2,Mutual and Common Certainty
Consider the normal-form game with payo uncertainty in Fig,1:
A N
A,1,0
N 0,1 0,0
Figure 1,2f12; 32g; Pr( = 32) =,
1
(i) Begin by assuming that the probability that = 32 is commonly known among
the players,Formalize this assumption by representing the situation as a game with payo
uncertainty G = (N; ;(Ai;ui;Ti)i2N) as de ned in Lecture 4,and compute all its Bayesian
Nash equilibria (again as de ned in Lectures 4 and 5).
(ii) Now x in the region where (N,N) and (A,A) are both Bayesian Nash equilibria of
the game in which is common knowledge,Consider the following (k + 1) statements:
E = B(0)(E) = Every player believes that = 32 with probability ;
B(1)(E) = Every player is certain that E is true;
:::
B(k)(E) = Every player is certain that B(k 1)(E) is true.
Show that,for any k = 0;1;:::,you can provide an alternative formulation of the game
Gk = (N; k;(Ai;uki;Tki )i2N) (with the same action sets) and a pair of priors pki 2 ( k),
i = 1;2,such that:
(i) there exists some state !2 k for which !2Tk‘=0B(‘)(E) (where of course E and
certainty operators are de ned in k and with respect to the priors pki );
(ii) yet,there is a unique Bayesian Nash equilibrium of the game in which the players’
priors are given by pki,i = 1;2,and in this equilibrium both players choose A at every state,
i.e,regardless of their respective types (in the notation of Lecture 4,i(ti) = A for i = 1;2
and all ti2Tki ).
In words,to support (N,N) as an equilibrium,the value of must be common certainty;
mutual certainty of arbitrarily high (but nite) order is not enough.
4,From OR,19.1,35.2,42.1 (important!)
2
