Signaling Games and Stable Equilibria

TL;DR: In this paper, the authors present a number of formal restrictions of this sort, investigate their behavior in specific examples, and relate these restrictions to Kohlberg and Mertens' notion of stability.

Abstract: Games in which one party conveys private information to a second through messages typically admit large numbers of sequential equilibria, as the second party may entertain a wealth of beliefs in response to out-of-equilibrium messages. By restricting those out-of equilibrium beliefs, one can sometimes eliminate many unintuitive equilibria. We present a number of formal restrictions of this sort, investigate their behavior in specific examples, and relate these restrictions to Kohlberg and Mertens` notion of stability.

Summary

I. INTRODUCTION

• Much of information economics has been concerned with situations in which the following simple signaling game is embedded: one party, hereafter called party A, possesses private information.
• This is only a partial list (and apologies are tendered to those left off), and in most cases the models are variations on the general theme outlined above.
• Suppose that there is a second possible message m' with the following properties: if A knows t, then A would strictly prefer (in comparison with the equilibrium outcome) not to send m', no matter how B interprets this.
• This particular criterion, and minor extensions to it, have appeared in several of the applications described above.

1. Signaling Games

• To require that the Nash equilibrium is sequential is to add the requirement that, for every message m that is sent with zero probability by A (for all m such that Xt7r(t)p(m;t) = 0), there must be some probability distribution over types T(m), which the authors shall write pi(-I m), such that (1) holds.
• The general program the authors shall follow is to restrict the set of sequential equilibria by posing restrictions on these out-ofequilibrium beliefs.

2. Three Facts about the Equilibria of Games

• Note that genericity for their signaling games is defined in the space of all payoff assignments to all endpoints.
• There may be a finite number of equilibrium outcomes for such games-witness the beer-quiche game.
• Thus, their justification is not valid for the games analyzed by Crawford and Sobel [1982] or Farrell [1985] .
• (In any event, the criteria that the authors subsequently develop would not have force in games in which signals are costless to the sender.).

3. Stability-A Review

• Suppose that an equilibrium outcome is stable, in the sense that the set of all the equilibria that give rise to the outcome is a stable set.
• Then the set of all sequential equilibria that give rise to the outcome is also a stable set.
• Also, every stable set of equilibria contains at least one sequential equilibrium. ).
• And it is likewise easy to show that, for a given stable set of equilibria, the subset of all normal form perfect equilibria in that set will be stable.

2. Dominance

• This seems rather counterintuitive, but it is not hard to see why this is happening.
• The authors shall try to avoid intertypal comparisons of utility for the rest of the paper, which means, among other things, that they abandon properness.
• One further word on this: while the properness of the m,r1 equilibrium depends on the prior, there is another equilibrium with the m outcome, namely where B randomizes evenly between the two responses, which is not proper for any prior.
• The power of iterated dominance in signaling games has long been noted.

3. Equilibrium Dominance and the Intuitive Criterion

• For a generic class of signaling games, a stable equilibrium outcome will pass the equilibrium domination test.
• Every signaling game from this generic class has at least one equilibrium outcome that will pass this test.

5. Never a Weak Best Response

• Hence only screening equilibria can survive this test.
• And among those, only the Riley outcome will do so.
• Finally, the authors observe that the Intuitive Criterion would suffice if they modified the game form.
• Suppose that the authors modified things slightly, so that the worker obtains education, and then the worker proposes the wage that he wishes; the firms then (simultaneously and independently) signify whether they are willing to hire the worker at that wage.
• (The worker cannot ask for more because this would change beliefs.).

VI. CONCLUDING REMARKS-THE FULL IMPLICATIONS OF STABILITY

• The arguments just given are not meant to justify restriction to the Riley outcome in the Spence signaling model.
• The authors thank Anat Admati for this observation.
• That is, at beliefs gt, B has an equilibrium response that makes m a weak best response for all the types in S simultaneously.
• The characterization given in Proposition 4 shows that stability (for generic signaling games) entails two considerations that their earlier criteria did not.
• It is also worth noting that the example of subsection IV.5 is an unstable equilibrium that Grossman and Perry accept.

Figures

TABLE I GAME OF FIGURE III IN NORMAL FORM

FIGURE IXB

FIGURE II

FIGURE I

FIGURE IV

FIGURE V

TABLE II PAYOFFS TO A AND B IF MESSAGE m IS SENT

FIGURE VI

FIGURE III

FIGURE IXB

FIGURE IXB

FIGURE VII

FIGURE VIII

Full text

Signaling Games and Stable Equilibria
Author(s): In-Koo Cho and David M. Kreps
Source:
The Quarterly Journal of Economics,
Vol. 102, No. 2 (May, 1987), pp. 179-222
Published by: The MIT Press
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THE
QUARTERLY
JOURNAL
OF
ECONOMICS
Vol.
CII
May
1987
Issue
2
SIGNALING GAMES
AND STABLE
EQUILIBRIA*
IN-KOo CHO
AND
DAVID
M. KREPS
Games
in which
one
party
conveys private
information to
a second
through
messages
typically
admit large
numbers
of
sequential
equilibria,
as the second
party
may
entertain
a
wealth
of
beliefs
in
response
to out-of-equilibrium
messages.
By
restricting
those
out-of-equilibrium
beliefs,
one can sometimes
eliminate
many
unintuitive
equilibria.
We
present
a
number of formal restrictions of
this
sort,
investigate
their
behavior
in
specific
examples,
and
relate these
restrictions
to
Kohlberg
and
Mertens' notion
of
stability.
I.
INTRODUCTION
Much
of
information
economics has been
concerned
with
situations
in
which
the following
simple signaling
game
is embed-
ded:
one
party,
hereafter called
party
A, possesses
private
informa-
tion. On the basis of
this information,
A
sends
a signal
to a second
party
B,
who
thereupon
takes an
action.
Examples abound:
Spence's
[1974]
model
of
job
market signaling
is one example,
if
we
modify things
slightly
so there
are
two
or
more
parties
B.
(We
shall
develop
a
simple
case
of
the
Spence
model
in
this
format
in
Section
V.)
Grossman
[1981]
examines
the role
of
warranties
and product
quality
using
this sort of
model. Models of
bargaining
with incom-
plete
information (see,
for example,
Grossman
and Perry
[1986a]
or
*We are
grateful
to Anat
Admati,
Drew Fudenberg,
Elon Kohlberg, Paul
Milgrom,
Richard
McKelvey,
Jean-Francois
Mertens,
Motty
Perry, John
Roberts,
Joel
Sobel, Gyu
Ho
Wang,
and
especially
Hugo
Sonnenschein
for
helpful
discussion,
and to three
referees and
an editor for helpful
suggestions.
The
financial support
of
Harvard
University,
the
Korea
Foundation for Advanced
Studies,
the National
Science Foundation
(Grants
SES80-06407
and
SES84-05865),
the Sloan
Foundation,
and the
Institute for
Advanced
Studies
at the Hebrew
University,
are
all
gratefully
acknowledged.
The
material
in
this
paper
originally
appeared
in
two separate
papers,
one with the above
title,
and a second entitled
"More
Signaling
Games
and
Stable
Equilibria."
We
hope
that anachronistic references
to the
earlier
incarnations
of these
ideas
will not
prove
too troublesome
to the
reader.
?
1987
by
the
President and Fellows
of Harvard
College
and the
Massachusetts
Institute
of
Technology.
The
Quarterly
Journal
of
Economics,
May
1987

180 QUARTERLY JOURNAL OF ECONOMICS
Rubinstein
[1985])
constitute another class of
examples.
In
the
literature of industrial organization, there is the
entry-deterrence
limit pricing model of Milgrom and Roberts [1982a],
the
analyses
of
the chain-store
game
of
Kreps
and
Wilson
[1982b]
and
Milgrom
and
Roberts
[1982b],
and recent
work on the role of
advertising by
Milgrom
and
Roberts
[1986].
Theoretical accountants often
employ
this sort of
model (see,
for
example,
Demski and
Sappington
[1986]). And, on a slightly higher plane, there is the general
analysis
of a
game
of this sort
due
to Crawford and Sobel
[1982],
and related
work on mechanism
design by
an informed
principal
[Myerson,
1983]. This is only a partial list (and apologies are tendered
to
those
left off), and in most cases the models are variations on
the general
theme
outlined above. But this theme, together with
variations, has
been
played
a lot
recently.
In
most of these recitals, one finds a plethora of
equilibria. This
paper takes
a
noncooperative game-theoretic approach,
and we
mean here a plethora of Nash equilibria. One can cut
back on the
number of equilibria by invoking notions of perfection
(or sequen-
tiality),
but this is
only
of minor
help-in many games the
wealth of
off-the-equilibrium path beliefs that can be imposed gives
rise to a
wealth of
equilibria. That is, what constitutes an
equilibrium is
powerfully
affected
by
the
"interpretations" that would be
given by
B
to
messages that
A
might have sent, but
in
equilibrium
does not
send.
In
a
sequential equilibrium,
B
is
required
to frame some
hypothesis (probability assessment) over what is A's
private infor-
mation
and respond accordingly.
As one varies those
hypotheses,
one varies the
optimal responses
of
B,
and hence the incentives of
A
to
send the various
messages.
At this point,
in
many of these contextually based
analyses,
the
analyst(s)
resorts
to
various
intuitive
criteria based
on the conclu-
sions that
B
"ought"
to
draw
from
sundry out-of-equilibrium
messages.
If
one can restrict the
out-of-equilibrium
beliefs
(or
hypotheses)
of
B,
one can sometimes eliminate
many
of the
equilib-
ria.
An
example of this that is particularly prevalent runs as follows:
suppose,
for
simplicity,
that
A
's
private
information
must be
one of
two
things,
called
t
and
t'.
Suppose
that
in
equilibrium
A
sends
message
m
with
probability
one.
Suppose
that there is a second
possible message
m'
with the
following properties:
if A
knows
t,
then
A
would strictly prefer (in comparison with the equilibrium out-
come)
not to send
m',
no matter how
B
interprets
this. And
if A
knows
t',
then
A
would
prefer
to
send m'
to
what
A
gets
in
the
equilibrium if by sending
m'
A could
convince
B
that
A knew t'.

SIGNALING GAMES
AND STABLE
EQUILIBRIA
181
The former
condition, it
is argued,
implies that
B
should
not
entertain the hypothesis
that
the
message
did come from
an A who
knows t. B
should infer
from the message
that
A knows
t'.
And,
therefore,
if
A
knows t', he
should send
the message (thus upsetting
the
given equilibrium).
It
is as
if
A,
if he
knows t',
is
(by
sending m')
implicitly
making the speech:
I
am
sending
the
message
m',
which
ought
to convince
you
that I know
t'.
For I
would never
wish
to send
m'
if
I
know t,
while
if
I
know
t',
and
if
sending
this
message
so convinces you,
then, as you
can
see,
it is
in
my
interest
to send it.
This particular
criterion,
and
minor extensions
to it, have
appeared
in
several of the
applications described
above.
It has been
applied directly
in
Grossman
[1981], Milgrom
and Roberts
[1982a],
and
Kreps
and Wilson [1982b],
and
it is applied indirectly
in
Rubinstein
[1985]. Its powers
can be considerable:
in a
simple (two
type) Spence
signaling
model, there is a single
equilibrium
outcome
that
survives
this criterion
(see Section V).
While analyses of particular
examples
have been
based on
intuitive criteria
for out-of-equilibrium
beliefs such as the
one just
given, there
have been, at
the same time,
further attempts
to refine
generally the
notion
of
a
Nash equilibrium.
An important
recent
example
of
this is the Kohlberg-Mertens
[1986] theory
of stability
and
stable
equilibrium
outcomes.
Because the Kohlberg-Mertens
development
takes
place
in
a
very abstract
context, it
is hard to see
what stability
entails for
concrete
examples. One point
of this paper
is to see what
stability
does entail for
(generic)
signaling games.
Roughly put,
we
find
that
stability
implies a number
of (progressively
stronger)
restrictions
on
out-of-equilibrium
beliefs
in this simple
class of games.
Some of
the
restrictions
we
find
quite intuitive;
for example,
stability
implies
the intuitive restriction
given
above.
As the
restrictions
mount,
however,
our intuition
becomes progressively
weaker;
until
we come to
implications
of
stability
that
we
(at least)
are unable to
motivate
so nicely.
In
the
end,
we
have mixed
feelings
about
stability,
at
least
insofar as
it
applies
to
signaling games:
it
captures
quite
beautifully some
restrictions
that we find
very
satisfactory;
but
in
other
cases
it seems
very strong.
We have two
objectives
in
this
study.
Our
first concern is with
signaling
games
alone.
These
games,
and
elaborations
of these
games,
have
proved
to be
very important
to recent
work
in
theoreti-
cal microeconomics.
By
developing
a
sequence
of
(progressively
stronger) general
criteria
for
the
equilibria
in
these
games,
we
hope

182
QUARTERLY
JOURNAL
OF ECONOMICS
to provide
analysts with
a general language
for the
discussion of
what level
of restrictions
they must
impose
in
order
to obtain a
particular
equilibrium
outcome. Second,
the
theory
of
stability,
either as
it
stands
or as it develops,
will
certainly prove
to be an
important
idea in noncooperative
game
theory. We
hope that our
examples
and characterization
of stability
for
signaling
games
will
help
in
the
further general
development
of these ideas.
The paper
is organized
as follows.
We begin,
in
Section
II, with
an
example
that illustrates
the basic
program
that
we
are following.
Then, in
Section III we
introduce a
general framework
for our
analysis.
In
subsection
111.1
we
define
the general signaling
game,
and we recall
in
subsection
111.2
some propositions
concerning
equilibria
of extensive games
from
the
literature.
In
subsection
III.3, we recall
basic concepts
and definitions
from
Kohlberg and
Mertens
[1986],
with
emphasis
on a
particular
result that
they
give.
Section IV is
the
heart
of the
paper.
The
general
program
that
we follow
for restricting
beliefs
in
testing
equilibrium
outcomes of
signaling games
and the connection
of
this
program
to stability are
given
in
subsection
IV.1.
The rest of
Section IV develops
some
specifications
of the
general
program:
subsection
IV.2 concerns
the
well-known
and
much
used
criterion
of
domination.
Subsection
IV.3 takes
up what we call
equilibrium domination;
included
here is
the criterion
that follows
from the "speech"
given above,
which we
refer to as
the
Intuitive
Criterion (the
uppercase letters
signifying
this
particular
criterion).
Subsection
IV.4
briefly discusses
(varia-
tions
on)
the Banks and Sobel
[forthcoming]
criteria
of
divinity
and
universal
divinity.
And subsection IV.5
discusses
the "never a weak
best
response"
criterion of
Kohlberg
and Mertens
[1986].
In
Section
V we
apply
these various
criteria to a
simple version
of
Spence's
signaling
model, showing
how they work to
rule out all
but
a
single
equilibrium
outcome
in
the
game (namely
the
separat-
ing equilibrium
identified
by Riley [1979]).
We conclude
in
Section
VI
with a discussion
of the
full implications
of stability
for
signaling
games,
and with a
summary
of what
(we
think)
we
have
learned.
McLennan
[1985]
pioneers
the
approach
of
refining
Nash
equilibria
by
formal restrictions
on
out-of-equilibrium
beliefs.
His
approach
differs
from
our own
in
one important respect,
and
we
shall offer a few remarks on
this in
subsection
IV.3.
He should be
credited,
however,
with
initiating
the general program
we follow
here.
Contemporaneously,
Banks and Sobel
[forthcoming]
have ana-