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Equilibrium Selection in Signaling Games
Author(s): Jeffrey S. Banks and Joel Sobel
Source:
Econometrica,
Vol. 55, No. 3 (May, 1987), pp. 647-661
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1913604
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This content downloaded from 131.215.70.231 on Thu, 17 Mar 2016 23:13:09 UTC
All use subject to JSTOR Terms and Conditions
Econometrica, Vol. 55, No. 3 (May, 1987), 647-661
EQUILIBRIUM
SELECTION
IN
SIGNALING
GAMES
BY
JEFFREY
S.
BANKS
AND
JOEL
SoBEL
1
This
paper
studies
the
sequential equilibria
of
signaling games.
It
introduces a new
solution concept, divine equilibrium, that refines
the
set
of
sequential equilibria
by
requiring
that
off-the-equilibrium-path beliefs satisfy
an
additional restriction. This restriction rules
out
implausible sequential equilibria in many examples. We show
that
divine equilibria
exist by demonstrating
that
a sequential equilibrium
that
fails
to
be
divine
cannot
be
in a
stable component. However, the stable
component
of
signaling games is typically smaller
than
the
set
of
divine equilibria.
We
demonstrate this fact through examples. We also
present a characterization
of
the stable equilibria in generic signaling games.
KEYWORDS: Strategic stability, equilibrium selection, signaling, game theory.
1.
INTRODUCTION
Tms
PAPER
INVESTIGATES
the
relationship between Kreps
and
Wilson's (1982)
concept
of
sequential equilibria
and
Kohlberg
and
Mertens's (1986) concept
of
stability.
It
introduces a restriction
on
off-the-equilibrium-path beliefs
that
refines
the set
of
sequential equilibria in signaling games. We call all sequential equilibria
that satisfy
our
restriction
on
beliefs divine.
For
generic signaling games, every
equilibrium contained in a stable
component
is
divine. Moreover,
the
solution
concept
is
restrictive
enough
to rule
out
all
of
the equilibria
that
Kreps (1985)
2
and
others dismiss
on
intuitive grounds. Thus, divinity provides
an
independent
theoretical foundation for discarding nonintuitive equilibria in signaling games.
We provide a generic example to show
that
divine equilibria may
not
be
contained in any stable component. However, the
paper
presents an explicit
characterization
of
stability in terms
of
off-the-equilibrium-path beliefs.
That
is,
an
equilibrium
of
a generic signaling game
is
in a stable
component
if
and
only
if
it
can
be supported as a sequential equilibrium with restricted off-the-equili-
brium-path beliefs. Just as Kreps
and
Wilson (1982) characterize perfect equilibria
for generic extensive-form games in terms
of
sequential equilibrium strategies
and
beliefs,
our
result characterizes stable outcomes for generic signaling games
in terms
of
sequential equilibrium ·strategies
and
restriction
on
beliefs. The
characterization may be a useful way
to
compute
stable equilibrium outcomes
and
to evaluate the consequences
of
using stability to select equilibria in extensive-
forni games.
Independent
of
our
work,
Cho
and
Kreps (1987) analyze the
power
of
stability
to
select equilibria in signaling games. Their results closely parallel
our
own.
They identify restrictions
on
equilibria similar
to
those embodied by divinity.
In
1
The original version
of
this
paper
was written while Banks was a
graduate
student
and
Sobel
was a visitor
at
Caltech. We
thank
participants
of
Caltech,
UCSD,
and
Rand
Corporation
Theory
Workshops, Drew Fudenberg, David Kreps,
and
two referees for valuable comments. Sobel thanks
Joe Farrell
and
Chris Harris for many conversations
on
related topics
and
the National Science
Foundation for partial
support
under
Grant
SES 84-08655.
2
Kreps (1985) stimulated
our
interest in this area.
Cho
and
Kreps (1987) contains some
of
the
results
of
this paper,
647
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648
JEFFREY
S.
BANKS
AND
JOEL
SOBEL
addition,
they
also
state
our
characterization
result
(Theorem
3).
Cho
(1987)
extends a restriction identified in
Cho
and
Kreps
to
obtain
a
solution
concept
that
refines
the
set
of
sequential equilibria
in
general extensive-form games.
Our
debt
to the existing literature
on
solution
concepts
for
noncooperative
games is
obvious
. Recent
work
on
this
topic
includes papers by Selten (1975),
Kreps
and
Wilson ( 1982),
and
McLennan
( 1985),
who
present
refinement concepts
for
extensive-form games;
and
Myerson (1978), Kalai
and
Samet
(1984),
and
Kohlberg
and
Mertens (1986),
who
present
refinement
concepts
for
normal-form
games.
2.
THE
MODEL
In this
paper
we analyze
the
equilibria
of
signaling games with finite
action
sets. There
are
two players, a
Sender
(S)
and
a Receiver
(R).
The
Sender
has
private
information,
summarized
by
his type, t,
an
element
of
a finite
set
T.
There
is a strictly positive probability distribution
p(t)
on
T;
p(t),
which is
common
knowledge, is
the
ex
ante
probability
that
S's
type
is
t.
After S learns his type
he
sends a message,
m,
to
R;
m is
an
element
of
a finite
set
M.
In
response to
m, R selects
an
action, a,
from
a finite set
A(m);
k(m)
is
the
cardinality
of
A(m).
S
and
R have von
Neumann-Morgenstern
utility functions
u(t,
m,
a)
and
v(t,
m,
a),
respectively.
For
fixed
T,
M,
and
A(m)
for
m
EM,
the
utility functions
u(t,
m,
a)
and
v(t,m,a)
completely
determine
the
game.
Therefore,
if
L=[fxI;'°!
1
k(i)]2,
where f is
the
cardinality
of
T
and
M is
the
cardinality
of
M,
then
every element
of
IRL
determines
a signaling game. We call a
property
of
a signaling
game
generic
if
there exists D c
IRL
such
that
the
property
holds
for
all signaling games
determined
by d
ED
and
a
closed
set
of
Lebesgue measure zero
contains
IRL\D.
If
a
property
of
a signaling
game
is generic,
then
we say it
holds
for
generic
signaling games.
For
any
positive integer
k,
let ..::lk={8=(8(1), . . .
,8(k)):
8(i)
:;o;:
OVi
and
I~
~
1
8(i)=l}
be
the
(k-1)-dimensional
simplex. We refer
to
the
(f-1)-
dimensional simplex most
often;
to simplify
notation,
we write
..:1
instead
of
..:1
"f·
A signaling rule
for
Sis
a function
q:
T'..::l,w;
q(mlt)
is
the
probability
that
S
sends the message m, given
that
his type is
t.
An
action rule
for
R is
an
element
of
IlmEM..::lk
<m>;
r(alm)
is
the
probability
that
Ruses
the
pure
strategy a
when
he
receives
the
message
m.
We
extend
the
utility
functions
u
and
v
to
the
strategy spaces
Llk
<m>
by
taking
expected
values;
for
all t E
T,
let
u(t,
m,
r)
=
L:
u(t,
m,
a)r(a
Im),
a c
A(m
)
v(t,m,r)=
L:
v(t,m,a)r(alm).
a E
A(m)
Also, for
each
A E
..:1
and
m E M let
BR(A,
m)
=
~~~
E
~~~
.
,~T
v(t,
m,
r(m))A
(t)
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EQUILIBRIUM
SELECTION
649
be
the best-response
correspondence
for
R
and
for
Ac
Llk(mh
let
BR(A,
m)
=
u.\
EA
BR(A,
m).
DEFINITION:
A
sequential
equilibrium
for
a signaling
game
consists
of
signal-
ing rules
q(
t) for
S,
action
rules
r(
m)
for R,
and
beliefs µ (
·Im)
E
Ll
for
R,
such
that
(i)
VtE
T,
q(m*I
t)>O
only
if
u(t,
m*,
r(m*))
=max
u(t,
m,
r(m));
m E M
(ii)
VmEM,
r(a*Jm)>O
only
if
I
v(t,m,a*)µ(tJm)=
max
I
v(t,m,a)µ(tJm);
I E T a E
A(m)
I E T
(iii)
ifI,ETq(mJt)p(t)>O,
then
µ(t*I
m)
=
q(m
I
t*)p(t*)
.
LET
q(m
I
t)p(t)
In
words, (i) states
that
q(
·)maximizes
S's
expected
utility, given
R's
strategy;
(ii) states
that
r(
·)
maximizes
R's
expected
utility, given beliefs
µ(
· );
and
(iii)
states
that
R's
beliefs given
S's
strategy
are
rational
in
the
sense
that
Bayes' Rule
determines
µ ( t
Im)
whenever
the
probability
that
S
sends
m
in
equilibrium
is
positive.
If
q(m
It)=
0,
for
all t
ET,
then
sequential
rationality
does
not
determine
µ(ti
m). However,
the
refinement
concept
introduced
in
Section
3 restricts
the
values
that
these beliefs
may
take
.
Next,
we describe
stable
equilibria.
Our
introduction
follows
Cho
and
Kreps
(1987). Fix a signaling
game;
let
p=(pR,Ps)
satisfy
O<p;<l,
i=R,
S,
and
let
ij
and
;
be
strategies
for
Sand
R respectively
that
satisfy
ij(m
It)>
0, Vm
EM,
Vt
E
Tand
r(a
Im)>
0,
Va
E
A(m),
Vm
EM.
A (p,
ij,
i}-perturbation
of
the
original
game
is
the
signaling
game
in which,
if
the
players
choose
strategies q
and
r
from
the
original
game,
then
the
outcome
is
the
outcome
of
the
original
game
if
the
strategy
chosen
by
S is
(1-
Ps) q + p
5
ij
and
the
strategy
chosen
by
R is
(1-pR)r+
PR;.
We refer
to
(p,
ij,
r)
as trembles. Let (q,
r)
be
Nash
equilibrium
strategies for a
perturbed
game.
If
q(m
It)>
0, we
say
that
a
type
t
Sender
voluntarily
sends
m
and
we say
that
R voluntarily uses
the
mixed
strategy
r(
m ).
For
a given signaling
game,
we call a
subset
C
of
the
set
of
Nash
equilibria
stable if,
for
every e > 0,
there
exists 8 > 0
such
that
every (p,
ij,
r)-perturbation
of
the original
game
with 0 <
p;
<
8,
i = R, S
has
an
equilibrium
no
more
than
e
from
the
set
C.
DEFINITION
: A stable component is a
minimal
(by
set
inclusion)
stable
set
of
equilibria.
Our
analysis
depends
on
several facts
about
extensive-form
games
and
stable
sets.
3
To
state these facts, we need
one
more
definition.
Given
an
extensive-form
3
Kreps
and
Wilson (1982) prove Proposition
I.
Kohlberg and Mertens (1986) prove Propositions
1-3.
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650
JEFFREY
S.
BANKS
AND
JOEL
SOBEL
game, the strategy choices for the players induce a probability distribution over
the
endpoints
of
the game. We call this probability distribution
the
outcome
of
the game (associated with particular strategies).
PROPOSITION
1:
For generic extensive-form games, the set
of
Nash equilibrium
outcomes
is
finite and all Nash equilibria within a given connected component induce
the same outcome.
PROPOSITION
2:
Every game has at least one stable component.
PROPOSITION
3: A stable set
of
equilibria for a given game remains a stable set
for the game obtained by deleting a strategy that
is
not a weak best response against
any equilibrium
in
the set.
Therefore, in generic signaling games, there exists a stable set
of
equilibria
with the property that every equilibrium in the set agrees along the equilibrium
path; the equilibria may vary off the equilibrium path. A variety
of
off-the-
equilibrium-path responses may be needed to guarantee that
any
perturbation
of
the game has an equilibrium
path
close to a particular equilibrium path.
Therefore, a single equilibrium need not
be
a stable set. However, we use
Proposition 1
to
justify
an
abuse
of
terminology. We call
an
equilibrium stable
if
it agrees with an element
of
a stable
component
along the equilibrium path.
In particular, in generic signaling games,
if
an
equilibrium
is
stable, then every
perturbation has an equilibrium with payoffs close to the original equilibrium
payoffs.
3.
DIVINE
EQUILIBRIA
Previous refinements
of
the
Nash
equilibrium concept place rationality restric-
tions
on
zero-probability events. In particular, sequential rationality requires that
players respond optimally to some consistent assessment
of
how the game has
been
played. These equilibrium concepts do not require a player to draw any
conclusion when a zero-probability event takes place. That is, although the
refinements concepts
embodied
in sequential rationality
and
perfectness require
that equilibria
of
games induce equilibria on any continuation
of
the game, these
concepts
do
not require that a player systematically draw
an
inference from
an
opponent's
unexpected move. Nevertheless, in
order
to decide how to respond
to
an
unexpected signal, R should evaluate the willingness
of
S-types to deviate
from equilibrium,
and
then incorporate into his beliefs the information that
deviations from equilibrium might reveal.
This section presents
an
equilibrium concept
that
refines
the
set
of
sequential
equilibria in signaling games by placing restrictions
on
off-the-equilibrium-path
beliefs. We begin
by
describing two restrictions
on
beliefs along with the intuition
behind
them,
and
then
proceed
to define an equilibrium concept that incorporates
these restrictions.