Eco514|Game Theory
Signaling Games
Marciano Siniscalchi
January 10,2000
Introduction
Signaling games are used to model the following situation,Player 1,the Sender,
receives some private information 2 and sends a message m2M to Player 2,the
Receiver,The latter,in turn,observes m but not,and chooses a response r2R.
Players’ payo s depend on,m and r.
What could be simpler? Yet,there is a huge number of economically interesting
games that t nicely within this framework,Spence’s job market signaling model
is the leading example,but applications abound in IO (limit pricing,disclosure...),
nance (security design) and political economics.
Also,the analysis of signaling games is not entirely straightforward,The point is,
in general all (reasonable) Nash equilibria are also sequential,trembling-hand perfect,
etc.; that is,they satisfy all sorts of backward-induction criteria.
However,many equilibria appear to be intuitively unreasonable,Moreover,thanks
to the particularly simple dynamic and informational structure,it is not hard to
generalize intuitions concerning speci c models to the whole class of signaling games.
As a result,a sizable literature on re nements for these games has developed,We
shall only look at the most important (and most successful) notions,the intuitive
criterion of Cho and Kreps,and divinity-like ideas a la Banks and Sobel.
Beer-Quiche and the Intuitive Criterion
Consider the game of Figure 1.
I told you the story behind this game in class,so I’m not going to bother you
with it again,Let me just point out the essential features,either type of the Sender
prefers to avoid a ght,even if this requires not enjoying her favorite breakfast (the
incremental payo from avoiding a ght is 2,whereas the preferred breakfast yields an
1
bw
f:1g
BrHH
HYF0,1
N2,0
-Q r
*F 1,1
HH
HjN 3,0
bs
f:9g
BrHH
HYF1,0
N3,1
-Q r
*F 0,0
HH
HjN 2,1
Figure 1,Beer-Quiche.
incremental payo of 1); also,the Receiver strictly prefers ghting the weak Sender,
and not ghting the strong one.
That all Nash equilibria in this game are pooling is not surprising at all,However,
there are two classes of pooling equilibria,one in which both types of the Sender
order Beer for breakfast,under the threat that Quiche will induce a Fight,and one
in which the exact opposite is true|both types order Quiche,under the threat of a
Fight if they deviate to Beer,Let us indicate the pure-strategy equilibria by (B,B,
NF) (the Weak type orders Beer,the Strong type orders Beer,and the receiver plays
N if he observes Beer and F if he observes Quiche) and (Q,Q,FN).
Why exactly do we nd the second type of equilibrium unreasonable? A moment’s
re ection is su cient to conclude that the reasonableness of the equilibrium hinges
on the reasonableness of the Receiver’s response,which in turn depends on his o -
equilibrium beliefs.
Now,for the Receiver to be willing to choose F after observing B,it must be
that he is su ciently con dent that the Sender is Weak,Hence,we might recast
our question as follows,Is it reasonable for the Receiver to believe that the Sender is
Weak after observing Beer?
Suppose we changed the game so u1(B,N,w) = 0,then we could simply invoke a
dominance argument,That is,we could say:
In the modi ed game a Weak sender gets 0 by choosing B; this is strictly
less than she can get by choosing Q,regardless of the Receiver’s strategy.
That is,B is a strictly dominated message for the Weak Sender,Hence,if
the Receiver thinks that both types of the Sender are rational,he cannot
place positive probability on the Weak Sender having chosen B,Thus,the
out-of-equilibrium beliefs required to support the equilibrium (Q,Q,FN)
are unreasonable,and so is the equilibrium itself.
Note that this argument has a forward induction avor,we consider the out-of-
equilibrium choice of B as being intentional,and we attempt to interpret it assuming
2
that the Sender is rational.
However,in this game u1(B,N,w) = 2,so B is not strictly dominated by Q for the
Weak Sender,For instance,if the Sender expects the Receiver to play NF,contrary
to the equilibrium prescription,then it’s perfectly OK for her to choose B.
Still,a slightly modi ed version of the dominance idea works here,One way to
motivate it is to note that,if we want to take the notion of equilibrium at least
a little seriously,we should not allow the Weak Sender to expect the Receiver to
play anything other than what the equilibrium prescribes,as long as no deviation
has occurred,In particular,the Weak Sender should believe that the Receiver will
respond to Q with N.
Note well that we do not need to assume that the Receiver will respond to the
out-of-equilibrium message B with F,as prescribed by the equilibrium,After all,the
whole point is that we are not so sure this response is reasonable! What we do wish
to maintain,though,is the assumption that,on the equilibrium path,players behave
as prescribed by the equilibrium.
But this su ces to eliminate B for the Weak Sender:
Why should the Weak Sender choose a message which,even under the
most optimistic circumstances (i.e,even if the Receiver responds with N
to B),leaves her with a payo of 2,which is strictly less than what she
gets in equilibrium? On the other hand,the Strong Sender is getting 2 in
equilibrium,and if the Receiver was convinced that only a Strong Sender
might choose B,she could potentially get a payo of 3.
To sum up,the Weak Sender can only lose (compared with the equilib-
rium outcome) by sending B,whereas the Strong Sender has a positive
incentive to send B (relative to the equilibrium outcome.) But the out-
of-equilibrium beliefs necessary to support (Q,Q,FN) do not re ect this,
so the equilibrium is unreasonable.
The Intuitive Criterion formalizes these intuitions,with an additional twist,Sup-
pose that,following each message,the Receiver had another response available|to
donate $10,000 out of his own pockets to the Sender,Then the preceding argument
would not work,because even the Weak Sender might have an incentive to deviate
to B|if she somehow expected the Receiver to donate $10,000 to her.
We cannot accept this as a reasonable justi cation for the choice of B,after all,
donating $10,000 is (conditionally) strictly dominated for the Receiver,i.e,it is never
a best response,Thus,we restrict the beliefs of the Weak Sender to best replies of
the Receiver,and the argument goes through as before,The resulting test is called
equilibrium domination,for obvious reasons.
3
Consider the following de nition,let BR2( 0;m) the set of conditional best re-
sponses of the Receiver after the partial history (m),for m2M,and subject to the
constraint that 1(m)( 0) = 1 for 0,1 That is,
BR2( 0;m) =fr2R,9 1(m)2 ( ) s.t,1(m)( 0) = 1 andX
02 0
[u2(m;r; 0) u2(m;r0; 0)] 1(m)( 0) 08r02Rg
Now x a sequential equilibrium (( 1( )) 2 ;( 2(m))m2M); denote by u 1( ) the
equilibrium payo for the Sender2,and,for any out-of-equilibrium message m,let:
R1(m) = BR2( ;m); 1(m) =f 2,u 1( ) > max
r2R1(m)
u1(m;r; )g
The interpretation should be clear,R1(m) is the set of best replies to m,and 1(m)
is the set of types that lose by deviating to m relative to the equilibrium,That is,we
rst delete message-response pairs,then we delete message-type pairs,The reason
for the superscripts will be clear momentarily.
The candidate equilibrium passes the Intuitive Criterion if it can be supported by
out-of-equilibrium beliefs which assign zero probability to types in 1( ),that is,if,
for every out-of-equilibrium m2M such that 1(m) 6= (see below),we can nd
1(m)2 ( ) such that 1(m)( 1(m)) = 0 and 2(m) is sequentially optimal after
history (m) given 1(m).
Two observations are in order,First,if 1(m) = for some out-of-equilibrium
message m,then we need not worry about the Receiver’s beliefs,intuitively,if m is
equally \bad" for all types,there are no inferences the Receiver can make about
after observing m,Second,note that the candidate equilibrium fails the Intuitive
Criterion if at least one type has a positive incentive to deviate to m whenever the
Receiver’s beliefs are constrained as above,Formally,the candidate equilibrium fails
the Intuitive Criterion i
u 1( ) < min
r2BR2( n 1(m);m)
u1(m;r; )
Observe that necessarily 62 1(m),so 1(m)6=,
I conclude with two additional observations,First,Cho and Kreps allow for more
general signaling games in which the messages available to the Sender may be type-
dependent,and the responses available to the Receiver may be message-dependent.
But the analysis is identical.
1Recall the notation for Bayesian extensive games with observable actions,1(m)2 ( ) is the
belief,shared by all players,about Player 1’s type,after history (m).
2u 1( ) = P
m2M
P
r2Ru1(m;r; ) 1( )(m) 2(m)(r).
4
Second,one may think about constructing more stringent tests,according to the
following intuition,After all,if types in 1(m) cannot gain by sending the message
m,then the Receiver should take this into account; this much is incorporated in the
above alternative formalization of the Intuitive Criterion test,but not in the de nition
of R1(m).
Thus,we can iterate our de nitions:
R2(m) = BR2( n 1(m);m); 2(m) =f 2 n 1(m),u 1( ) > max
r2R2(m)
u1(m;r; )g
and so on (what would be the third step?) This leads to the Iterated Intuitive Crite-
rion.
D1,D2,Divinity and Friends
[This material goes beyond what we covered in class,but I thought you might nd it
useful,Check Cho and Kreps’s original paper for further info.]
The Intuitive Criterion is not a panache,however,Consider once again the game
of Figure 1,but change the payo s by letting u1(Q,N,w) = 2,Then the IC does not
eliminate the (Q,Q,FN) equilibrium,because u 1(w) = 2 = maxr2R1 u1(m;r;w).
However,consider the following argument.
The Weak Sender never gains by sending the message B,Also,the only
case in which she is indi erent is when the Receiver responds with N,On
the other hand,if the Receiver responds to B with N,the Strong Sender
gains by deviating.
In other words,for any response that makes the Weak Sender indi er-
ent between deviating and playing as per the candidate equilibrium,the
Strong Sender has a positive incentive to deviate,This suggests that the
latter should be more likely to be the deviator.
This idea leads to a class of re nements known as D1,D2 and Divinity,First,
xing a candidate equilibrium as above,for any type 2 and out-of-equilibrium
message m,de ne the set of mixed responses of the Receiver that provide type with
a strict incentive to deviate to m as
D (m) =f’2MBR2( ;m),u 1( ) <
X
r2R
u1(m;r; )’(r)g
where MBR2( ;M) is de ned analogously to BR2( ;M) (note that the former is not
the convex hull of the latter|think about it!) Also,de ne the set of mixed responses
5
which leave type indi erent between deviating and playing the equilibrium strategy:
D0 (m) =f’2MBR2( ;m),u 1( ) =
X
r2R
u1(m;r; )’(r)g
Criterion D1 says that type can be eliminated for m i there exists another type
0 such that D (m)[D0 (m) D 0(m),whenever (weakly) prefers to deviate to m,
0 strictly prefers to do so.
Criterion D2 says that type can be eliminated for m i D (m)[D0 (m) S
06= D 0(m),whenever (weakly) prefers to deviate to m,there is some type
0
that strictly prefers to do so,Clearly,D2 is stronger than D1 (unless there are only
two types,in which case the criteria coincide.)
One can think of two tests derived from the criteria,rst,one can require that
the candidate equilibrium be supported by out-of-equilibrium beliefs which assign
zero probability to eliminated types,This is rather strong,and corresponds to the
so-called \D1 or D2 re nement."
An alternative test requires that,whenever a pair ; 0 satisfy the condition in the
de nition of D1,the posterior likelihood ratio 1(m)( ) 1(m)( 0) should not shift towards,
that is,we require
1(m)( )
1(m)( 0)
1( )( )
1( )( 0)
This leads to Divinity and related concepts,Note that,however,this class of tests is
rather strong|the intuitive story is somehow..,less intuitive than the one underlying
the Intuitive Criterion (no pun intended.) In general,divinity and friends capture
notions of \monotonicity." Finally,both D1 and D2 imply equilibrium domination.
6
Signaling Games
Marciano Siniscalchi
January 10,2000
Introduction
Signaling games are used to model the following situation,Player 1,the Sender,
receives some private information 2 and sends a message m2M to Player 2,the
Receiver,The latter,in turn,observes m but not,and chooses a response r2R.
Players’ payo s depend on,m and r.
What could be simpler? Yet,there is a huge number of economically interesting
games that t nicely within this framework,Spence’s job market signaling model
is the leading example,but applications abound in IO (limit pricing,disclosure...),
nance (security design) and political economics.
Also,the analysis of signaling games is not entirely straightforward,The point is,
in general all (reasonable) Nash equilibria are also sequential,trembling-hand perfect,
etc.; that is,they satisfy all sorts of backward-induction criteria.
However,many equilibria appear to be intuitively unreasonable,Moreover,thanks
to the particularly simple dynamic and informational structure,it is not hard to
generalize intuitions concerning speci c models to the whole class of signaling games.
As a result,a sizable literature on re nements for these games has developed,We
shall only look at the most important (and most successful) notions,the intuitive
criterion of Cho and Kreps,and divinity-like ideas a la Banks and Sobel.
Beer-Quiche and the Intuitive Criterion
Consider the game of Figure 1.
I told you the story behind this game in class,so I’m not going to bother you
with it again,Let me just point out the essential features,either type of the Sender
prefers to avoid a ght,even if this requires not enjoying her favorite breakfast (the
incremental payo from avoiding a ght is 2,whereas the preferred breakfast yields an
1
bw
f:1g
BrHH
HYF0,1
N2,0
-Q r
*F 1,1
HH
HjN 3,0
bs
f:9g
BrHH
HYF1,0
N3,1
-Q r
*F 0,0
HH
HjN 2,1
Figure 1,Beer-Quiche.
incremental payo of 1); also,the Receiver strictly prefers ghting the weak Sender,
and not ghting the strong one.
That all Nash equilibria in this game are pooling is not surprising at all,However,
there are two classes of pooling equilibria,one in which both types of the Sender
order Beer for breakfast,under the threat that Quiche will induce a Fight,and one
in which the exact opposite is true|both types order Quiche,under the threat of a
Fight if they deviate to Beer,Let us indicate the pure-strategy equilibria by (B,B,
NF) (the Weak type orders Beer,the Strong type orders Beer,and the receiver plays
N if he observes Beer and F if he observes Quiche) and (Q,Q,FN).
Why exactly do we nd the second type of equilibrium unreasonable? A moment’s
re ection is su cient to conclude that the reasonableness of the equilibrium hinges
on the reasonableness of the Receiver’s response,which in turn depends on his o -
equilibrium beliefs.
Now,for the Receiver to be willing to choose F after observing B,it must be
that he is su ciently con dent that the Sender is Weak,Hence,we might recast
our question as follows,Is it reasonable for the Receiver to believe that the Sender is
Weak after observing Beer?
Suppose we changed the game so u1(B,N,w) = 0,then we could simply invoke a
dominance argument,That is,we could say:
In the modi ed game a Weak sender gets 0 by choosing B; this is strictly
less than she can get by choosing Q,regardless of the Receiver’s strategy.
That is,B is a strictly dominated message for the Weak Sender,Hence,if
the Receiver thinks that both types of the Sender are rational,he cannot
place positive probability on the Weak Sender having chosen B,Thus,the
out-of-equilibrium beliefs required to support the equilibrium (Q,Q,FN)
are unreasonable,and so is the equilibrium itself.
Note that this argument has a forward induction avor,we consider the out-of-
equilibrium choice of B as being intentional,and we attempt to interpret it assuming
2
that the Sender is rational.
However,in this game u1(B,N,w) = 2,so B is not strictly dominated by Q for the
Weak Sender,For instance,if the Sender expects the Receiver to play NF,contrary
to the equilibrium prescription,then it’s perfectly OK for her to choose B.
Still,a slightly modi ed version of the dominance idea works here,One way to
motivate it is to note that,if we want to take the notion of equilibrium at least
a little seriously,we should not allow the Weak Sender to expect the Receiver to
play anything other than what the equilibrium prescribes,as long as no deviation
has occurred,In particular,the Weak Sender should believe that the Receiver will
respond to Q with N.
Note well that we do not need to assume that the Receiver will respond to the
out-of-equilibrium message B with F,as prescribed by the equilibrium,After all,the
whole point is that we are not so sure this response is reasonable! What we do wish
to maintain,though,is the assumption that,on the equilibrium path,players behave
as prescribed by the equilibrium.
But this su ces to eliminate B for the Weak Sender:
Why should the Weak Sender choose a message which,even under the
most optimistic circumstances (i.e,even if the Receiver responds with N
to B),leaves her with a payo of 2,which is strictly less than what she
gets in equilibrium? On the other hand,the Strong Sender is getting 2 in
equilibrium,and if the Receiver was convinced that only a Strong Sender
might choose B,she could potentially get a payo of 3.
To sum up,the Weak Sender can only lose (compared with the equilib-
rium outcome) by sending B,whereas the Strong Sender has a positive
incentive to send B (relative to the equilibrium outcome.) But the out-
of-equilibrium beliefs necessary to support (Q,Q,FN) do not re ect this,
so the equilibrium is unreasonable.
The Intuitive Criterion formalizes these intuitions,with an additional twist,Sup-
pose that,following each message,the Receiver had another response available|to
donate $10,000 out of his own pockets to the Sender,Then the preceding argument
would not work,because even the Weak Sender might have an incentive to deviate
to B|if she somehow expected the Receiver to donate $10,000 to her.
We cannot accept this as a reasonable justi cation for the choice of B,after all,
donating $10,000 is (conditionally) strictly dominated for the Receiver,i.e,it is never
a best response,Thus,we restrict the beliefs of the Weak Sender to best replies of
the Receiver,and the argument goes through as before,The resulting test is called
equilibrium domination,for obvious reasons.
3
Consider the following de nition,let BR2( 0;m) the set of conditional best re-
sponses of the Receiver after the partial history (m),for m2M,and subject to the
constraint that 1(m)( 0) = 1 for 0,1 That is,
BR2( 0;m) =fr2R,9 1(m)2 ( ) s.t,1(m)( 0) = 1 andX
02 0
[u2(m;r; 0) u2(m;r0; 0)] 1(m)( 0) 08r02Rg
Now x a sequential equilibrium (( 1( )) 2 ;( 2(m))m2M); denote by u 1( ) the
equilibrium payo for the Sender2,and,for any out-of-equilibrium message m,let:
R1(m) = BR2( ;m); 1(m) =f 2,u 1( ) > max
r2R1(m)
u1(m;r; )g
The interpretation should be clear,R1(m) is the set of best replies to m,and 1(m)
is the set of types that lose by deviating to m relative to the equilibrium,That is,we
rst delete message-response pairs,then we delete message-type pairs,The reason
for the superscripts will be clear momentarily.
The candidate equilibrium passes the Intuitive Criterion if it can be supported by
out-of-equilibrium beliefs which assign zero probability to types in 1( ),that is,if,
for every out-of-equilibrium m2M such that 1(m) 6= (see below),we can nd
1(m)2 ( ) such that 1(m)( 1(m)) = 0 and 2(m) is sequentially optimal after
history (m) given 1(m).
Two observations are in order,First,if 1(m) = for some out-of-equilibrium
message m,then we need not worry about the Receiver’s beliefs,intuitively,if m is
equally \bad" for all types,there are no inferences the Receiver can make about
after observing m,Second,note that the candidate equilibrium fails the Intuitive
Criterion if at least one type has a positive incentive to deviate to m whenever the
Receiver’s beliefs are constrained as above,Formally,the candidate equilibrium fails
the Intuitive Criterion i
u 1( ) < min
r2BR2( n 1(m);m)
u1(m;r; )
Observe that necessarily 62 1(m),so 1(m)6=,
I conclude with two additional observations,First,Cho and Kreps allow for more
general signaling games in which the messages available to the Sender may be type-
dependent,and the responses available to the Receiver may be message-dependent.
But the analysis is identical.
1Recall the notation for Bayesian extensive games with observable actions,1(m)2 ( ) is the
belief,shared by all players,about Player 1’s type,after history (m).
2u 1( ) = P
m2M
P
r2Ru1(m;r; ) 1( )(m) 2(m)(r).
4
Second,one may think about constructing more stringent tests,according to the
following intuition,After all,if types in 1(m) cannot gain by sending the message
m,then the Receiver should take this into account; this much is incorporated in the
above alternative formalization of the Intuitive Criterion test,but not in the de nition
of R1(m).
Thus,we can iterate our de nitions:
R2(m) = BR2( n 1(m);m); 2(m) =f 2 n 1(m),u 1( ) > max
r2R2(m)
u1(m;r; )g
and so on (what would be the third step?) This leads to the Iterated Intuitive Crite-
rion.
D1,D2,Divinity and Friends
[This material goes beyond what we covered in class,but I thought you might nd it
useful,Check Cho and Kreps’s original paper for further info.]
The Intuitive Criterion is not a panache,however,Consider once again the game
of Figure 1,but change the payo s by letting u1(Q,N,w) = 2,Then the IC does not
eliminate the (Q,Q,FN) equilibrium,because u 1(w) = 2 = maxr2R1 u1(m;r;w).
However,consider the following argument.
The Weak Sender never gains by sending the message B,Also,the only
case in which she is indi erent is when the Receiver responds with N,On
the other hand,if the Receiver responds to B with N,the Strong Sender
gains by deviating.
In other words,for any response that makes the Weak Sender indi er-
ent between deviating and playing as per the candidate equilibrium,the
Strong Sender has a positive incentive to deviate,This suggests that the
latter should be more likely to be the deviator.
This idea leads to a class of re nements known as D1,D2 and Divinity,First,
xing a candidate equilibrium as above,for any type 2 and out-of-equilibrium
message m,de ne the set of mixed responses of the Receiver that provide type with
a strict incentive to deviate to m as
D (m) =f’2MBR2( ;m),u 1( ) <
X
r2R
u1(m;r; )’(r)g
where MBR2( ;M) is de ned analogously to BR2( ;M) (note that the former is not
the convex hull of the latter|think about it!) Also,de ne the set of mixed responses
5
which leave type indi erent between deviating and playing the equilibrium strategy:
D0 (m) =f’2MBR2( ;m),u 1( ) =
X
r2R
u1(m;r; )’(r)g
Criterion D1 says that type can be eliminated for m i there exists another type
0 such that D (m)[D0 (m) D 0(m),whenever (weakly) prefers to deviate to m,
0 strictly prefers to do so.
Criterion D2 says that type can be eliminated for m i D (m)[D0 (m) S
06= D 0(m),whenever (weakly) prefers to deviate to m,there is some type
0
that strictly prefers to do so,Clearly,D2 is stronger than D1 (unless there are only
two types,in which case the criteria coincide.)
One can think of two tests derived from the criteria,rst,one can require that
the candidate equilibrium be supported by out-of-equilibrium beliefs which assign
zero probability to eliminated types,This is rather strong,and corresponds to the
so-called \D1 or D2 re nement."
An alternative test requires that,whenever a pair ; 0 satisfy the condition in the
de nition of D1,the posterior likelihood ratio 1(m)( ) 1(m)( 0) should not shift towards,
that is,we require
1(m)( )
1(m)( 0)
1( )( )
1( )( 0)
This leads to Divinity and related concepts,Note that,however,this class of tests is
rather strong|the intuitive story is somehow..,less intuitive than the one underlying
the Intuitive Criterion (no pun intended.) In general,divinity and friends capture
notions of \monotonicity." Finally,both D1 and D2 imply equilibrium domination.
6
