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Credible Neologisms in
Games of Communication
Joseph Farrell
No.
386
June 1985
massachusetts
institute of
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50 memorial drive
Cambridge, mass. 02139
\m
Credible Neologisms in
Games of Communication
Joseph Farrell
No.
386
June 1985
Credible Neologisms in Games of Communication
by
Joseph Farrell
GTE Laboratories
40 Sylvan Road
Waltham, MA 02254
claA
December 1983,
M.\.T.
revised June 1985
Thanks are due to many people for the much-delayed revision of
this paper. Vincent Crawford, Jerry Green, Eric Maskin, Andrew
McLennan, and especially Roger Myerson and Joel Sobel have
contributed ideas and encouragement. A special debt is due to
Suzanne Scotchmer and Joel Sobel, both of whom kept encouraging
me. I also wish to thank Kathleen Stevens for programming
assistance
Introduction
1.
Little attention has been paid to communication in games:
the most part,
for
game theorists think as if no player would ever
trust another (as is the case with two-person zero-sum games).
But an important role of language is coordination of actions,
and
game theory is the natural discipline in which to study language
from that point of view,
In
[3],
showed that
as
Lewis
[9]
showed.
we defined communication equilibrium in games,
it
is
a
strict and useful extension of Bayesian
like that used by Crawford
equilibrium.
However,
and Sobel
and Green and Stokey [4],
[2]
that notion,
the way language is interpreted,
In
and
requires only that given
there is no incentive to lie
.
the present paper we extend the analysis by examining
incentives to introduce credible neologisms.
Our principal interpretation
This can be viewed in two ways.
is
that it is an equilibrium condition that nobody should have an
incentive to introduce
has,
credible neologism:
for,
if someone
then the purported equilibrium does not describe how the
game will be played.
a
a
We elaborate on this below,
neologism-proof equilibrium.
when we define
Another related approach, which
we only hint at here in Section 5,
is
to ask when an evolving
language will develop neologisms that will not immediately be
discredited by lies:
truthfully,
at
a
self-signaling neologism will be used
least at first.
More technically,
the paper can be viewed as examining the
implications of
in
a
certain restriction on disequilibrium beliefs
a
The restriction is that,
sequential equilibrium.
set
X
to what
if there
is
of types that strictly prefer being treated as a group
they get in equilibrium,
to treated as
X
and if no other types would wish
then it must be possible for the message-sender
X,
to induce the belief X.
This restriction is similar to that independently proposed by
Grossman and Perry [5]. However,
they,
like the literature on
stability of signaling equilibrium (Banks and Sobel [1], Kohlberg
and Mertens
payoffs.
Kreps
[6],
[7])
consider signals that directly affect
Their criteria, being based on dominance arguments, have
no force if,
as
here,
signals do not directly affect payoffs.
The paper is organized as follows.
simple communication game,
(an X with the properties
there is such
is
a
Section
2,
we define a
and a sequential equilibrium in such a
We introduce the notion of
game.
In
a
self-signaling set of types
just described),
and argue that if
set then the equilibrium is unpersuas ive
vulnerable to the introduction of
a
,
as
it
credible neologism. An
equilibrium not subject to such an objection we call neologismproof
.
In Section 3,
concept,
we give four examples to illustrate our
and relate it to Myerson's notion
mechanism.
[10]
of core
We show by example that neologism-proof equilibrium
need not exist
In Section 4,
proof equilibrium.
we prove some existence results for neologismIn
Section
5,
we consider what happens if
agents repeatedly play (adjusting towards best responses)
a
that has no neologism-proof equilibrium.
the
our example,
In
game
average result depends on relative speeds of adjustment, and we
suggest that the usual game- theoret ic paradigm (in which all
those speeds are assumed infinite) may be misleading.
Section
6
concludes
2
Assumptions and Definitions
.
2.1 Simple Communication Games
First,
we define our object of analysis:
communication game
G.
and the Receiver (R).
finite,
(S)
has payoff-relevant private information
which we sometimes call his type
t€T,
imple
the Sender
players:
There are two
S
a s
and that R's prior
-n
on T,
.
We assume that
which
is
T
is
common knowledge,
assigns strictly positive probability to each type
t.
The only
choice that enters directly into payoffs is R's choice of an
action a
on t.
u
e
A,
where
is
a
payoffs also depend
fixed finite set;
Players maximize their expected payoffs
(a, t)
S,
possible messages;
as
u
(a,t)
and
.
The Sender,
of M
A
chooses
R
a
message m from
a
class M
then learns m before choosing
infinite but discrete
-
for instance,
(arbitrarily long) utterances in English.
a.
of
We think
the set of all
2.2 Sequential Equilibrium
sequential equilibrium
A
G
=
*
R
S
(T,A,M ,u (•,«),u («,•),
function
A
(i)
(Kreps and Wilson,
T
s:
-»
A(M
ti )
)
A
function
p
:
which assigns to each teT
probability distribution p(m) on
(iii)
A
function
r
:
probability distribution r(m) on
The functions
s
and
to each meM
a
to each meM
a
T.
A.
are S's and R's Bayesian
r
strategiesrespectively.
a
.
A(A) which assigns
-
M
M
A(T) which assigns
-
M
of the game
consists of the following:
probability distribution s(t) on
(ii)
[8])
The distribution p(m)
R's posterior beliefs about t€T,
on T represents
after hearing the message
m.
These functions must satisfy three conditions:
First,
for each teT and each meM
n(t)s(t)(m)
=
p(m)(t)
t
|: T
,
-n(t)s(t) (m)
(2.1)
This condition says that the probability distributions on
T*M
described by
the same.
-n (
•
)
and s(»),
In other words,
and by p(»)
and s(»), must be
given s(»), R's posteriors must be
consistent with his priors on
We can re-write (2.1)
t.
more
in
familiar Bayes' rule form by dividing by the prior probability of
m,
q(m)
=
.
£_"n
(
t
)
s
(
t
)
(m)
if q(m)
,
0,
>
but so as to allow for
zero-probability messages we use the form (2.1), which can be rewritten as
:
T,(t)s(t)(m)
=
q(m)p(m)(t)
(2.2)
The second condition is that R's choice of action be optimal
Write
given his beliefs.
v
R
(m)
sup
=
[
,
E.p m)
(
(
t
)
optimized expected payoff when he hears message
u
R
m.
(a,t)]
for R's
Then we
require that:
r(m)(a)
>
t
Z p(m)
T
R
(
t
)
(
a, t
Our third requirement is that,
r(#),
)
=
v
R
given R's response function
S's choices of messages should be optimal.
syn^t Z.r(m)(a)u (a,t)]
he observes t.
s(t)(m)
>
Having defined
a
a
S r(m)(a)u
A
Define
v
g
(t)
=
to be S's optimized expected payoff when
Then we require
-
(2.3)
(m)
2
S
(a,t)
=
v
S
(t)
(2.4)
sequential equilibrium, we prove
sequential equilibrium always exists:
that a
Propos i t i on
1
Every simple communication game has
:
a
sequential
equil ibr ium.
Proof
f(»)
:
Let a
A be
€
optimal for R given his prior beliefs
be any probability distribution on M
s(t)
=
f(»)
for all t€T
p(m)
=
n
for all meM
(
•
)
"1
(m)(a)
=
(J
10
if
L
?.
a
a
=
Define:
.
In
this equilibrium,
communicated:
we call
of
it
the
we define a notion of equivalence of equilibria.
We
t
Let
3
otherwise
no information about
.
for all meM
.
This is a sequential equilibrium.
course,
*
-n
is
uncommunicative equilibrium.
2.3
Outcomes and Equivalence of Equilibria
Next,
are not really concerned with the choice of messages;
care about how communication affects the outcome of G.
the outcome w of an equilibrium (s,p,r)
from
T
to A( A
)
given by:
of
G
to be the
rather, we
We define
function
vv
w(t)(a)
probability that
=
Z
meM
t
is
a
chosen if
t
is
the state;
s(t) (m)r(m) (a)
(2.5)
Two equilibria are equivalent
if they have the same outcome.
While there are always infinitely many sequential equilibria
there may be many fewer equivalence classes,
language slightly by saying (e.g.)
that
(s,p,r)
unique, when
is
(s,p,r).
Further Notation
2.4
Let
X
be a non-empty subset of T.
probability distribution on
T
"(t |X)
f
=
t
X
We write
n
(
•
|
X
for the
)
given by
(t)
if teX
Tt(t')
otherwise
We write a (X)
(
•
,
and we often abuse
we mean that every equilibrium is equivalent to
n
4
|
X
Since
)
,
T
for R's optimal action if his beliefs are
and assume that a*(X)
and A are finite,
is
unique for each
this is generically true.
holds if R's preferences are strictly concave in
Crawford and Sobel
s
a,
also
It
as
in
[2].
We write v(X,t)
v(X,t)
X.
for t's payoff if R has beliefs
u
S
(a*(X),t).
tt (
•
|
X
)
:
Given
a
response rule
(r,t)
=
sup^
meM
and recall that,
v(r,t)
is
a £A
if
achieved.
on R's part,
r
r(m)(a)u (a,t)
part of
is
r
we define t's payoff:
a
sequential equilibrium, then
We define the set P(X,r)
of types who
strictly prefer to be thought of as unknown members of
be treated according to
P(X,r)
=
{teT
|
than to
r:
v(X,t)
v(r,t)}
>
(2.6)
When there is no danger of confusion, we write P(X)
2.5
X
for P(X,r)
Self-Signaling Subsets and Neologism-Proof Equilibrium
We say that the subset
P(X)
=
is
X
self-signaling if
(2.7)
X
Now suppose that in a sequential equilibrium (s,p,r),
confidently believes that
and suppose that,
S
expects to be treated according to
instead of sending
probability in equilibrium,
S
"t
to expect
that R should believe that
in
X."
If
a
r;
message sent with positive
sends an unexpected message or
neologism
is
R
(2.7)
holds,
t
then it seems reasonable
is
in
X.
For if teX,
then
(by (2.7))
S
would not try to persuade
every type
t
with teX would try to persuade
submit,
a
self-signaling message
Moreover,
that t€X.
R
R
that teX.
a
we
highly compelling.
is
Based on this, we give the following definition:
equilibrium (s,p,r) of
Thus,
A
sequential
simple communication game is neo logism-
proof if there is no non-empty subset
X
of T that
is
self-
signaling relative to the payoffs v(r,t).
sequential equilibrium is
A
in
equilibrium,
lying.
language (set of messages used
a
and their interpretations p)
that will not induce
Our concept of neologism-proof equilibrium not only
requires that (we start with
a
sequential equilibrium), but also
requires that no credible neologisms (new messages,
literally
"new words") will be used.
There is always more language available than is used in
non-equilibrium messages exist
given game:
5
Moreover,
.
non-equilibrium messages have focal meanings:
meanings. Thus it is unreasonable to assume,
sequential equilibrium does,
there is
that
t
is
in
a
these
their literal
as
the concept of
that R will not understand
neologisms. We will assume that,
T,
a
for every non-empty subset
X
of
neologism n(X) that (it is common knowledge) claims
X.
We will focus on whether n(X)
While we certainly should not assume that
is
R
credib le
is
.
credulous
enough to believe arbitrary statements, we also should not assume
that he will not believe a convincingly credible neologism.
Because
a
self-signaling neologism
is
knowledge focal meaning is such that
a
S
message whose commonwould like R to believe
it
10
if and only if it
is
we assert that only
true,
Receiver would refuse to believe
Accordingly, we assume that,
is
available,
that
t
is
in
then
X.
S
S
paranoid
self-signaling neologism.
a
if a self-signaling neologism n(X)
can (and,
if
t
is
in
X,
will) make R believe
since all and only types
Moreover,
t
would
X
€
R's beliefs ought to be simply
use the neologism n(X),
Therefore,
a
ti
|
knows that by using n(X) he can induce beliefs
and action a*(X).
So
it
is
a
Myerson [10] has discussed
"core mechanism")
the present paper.
|
X
a
in which fewer
X.
related equilibrium concept (the
neologisms are credible than in
He requires that a neologism (in his
terms,
proposed alternative mechanism) be preferred by some set
and be incentive-compatible (have R making
to his beliefs)
-n
compelling restriction on sequential
equilibrium that there be no self-signaling subset
types,
X
whatever R's beliefs "between"
tt
a
and
X
a
of
best response
n
|
X
.
This
stronger requirement on neologisms makes fewer neologisms
credible,
proof.
and thus makes more sequential equilibria neologism-
Myerson obtains an existence theorem (his Theorem 4),
while we show below in
a
very simple example that neologism-proof
equilibrium need not exist.
11
3
.
Some Simple Examples
we analyse the sequential and the neologism-
In this section,
proof equilibria of some simple communication games.
We show how
the requirement of neologism-proof ness eliminates many
"unsatisfactory" sequential equilibria.
even in non-pathological games,
fail to exist
Example
We will also see that,
neologism-proof equilibrium may
.
Eliminating an Uncommunicative Equilibrium
1:
there
As we saw above,
is
always
a
sequential equilibrium in
which different types are treated alike, and any attempt to
communicate
1
ignored,
is
even if it is entirely credible.
Example
shows how our condition of neologism-proof ness may rule out
this equilibrium.
We define a simple communication game as follows:
T
(t
=
TT(t
1
A
=
)
{
1
,t
=
a,
2}
T,(t
,
2
a„ a„
,
1/2
)
}
Payoffs can be concisely described as follows
12
S
R
'
'
s
s
payoff
1
2
1
-2
-2
-1
payoff
:
2
3
2
R's best responses in terms of his posterior probability p that
is
t
,
are
S
13
"
a*(p)
=
if p
<
2
a,
1/3
if p
>
2/3
a
if 1/3
a
3
p
<
2/3
<
the uncommunicative one,
There are two sequential equilibria:
and one in which
equilibrium
is
S
reveals his type to
better (ex-ante) for
S.
R.
The uncommunicative
However,
even if
S
can
propose behavior ex-ante, he cannot get that allocation, because
if he turns
out to be of type
1
neologism.
That is,
is
payoff
S
X
=
{t,}
he will
use a self-signaling
self-signaling relative to the
gets in the uncommunicative equilibrium.
equilibrium
is
not neologism-proof.
Therefore that
The revealing equilibrium is
neologism-proof.
Example
2:
Eliminating an Informative Equilibrium
This game also has an informative and an un inf ormat i ve
sequential equilibrium.
However,
in this
case it is the
informative equilibrium that fails to be neologism-proof.
The game is the same as in Example
is
changed from
1
to -1.
1,
except that u (a.,t.)
Now S's payoffs are:
14
'2
a
-1
-2
-2
1
l
a
a
2
°
3
If R responds
others with
a9
However,
is
it
,
you the state;
to
some equilibrium messages with
a,
and to all
then truth-telling is a sequential equilibrium.
vulnerable to the neologism n(T):
since this
better
is
(0
won't tell
and for
-1)
for
t
That
is,
X
>
you shouldn't infer anything about t."
"I
-,
=
T
therefore the revealing equilibrium
Example
In
3:
is
9
,
self-
is
signaling relative to payoffs resulting from revelation,
t
and
not neologism-proof.
No neologism-proof equilibrium
this example,
we produce a game that has no neologism-
proof equilibrium.
As
A
=
in
examples
{a, ,a2,a„}.
and
{t,,t ? },
1/2
=
-n(t 9 ),
R's payoffs are also as in examples
1
and
1
S's payoffs are given by:
2,
T
=
n
(
t
,
)
=
2.
and
15
a
a
1
2
2
-1
-1
-2
2
°
3
There is now just one sequential equilibrium outcome:
a„
is
chosen with probability one in both states.
self-signaling relative to equilibrium payoffs.
is no
But
action
{t,}
Therefore,
is
there
neologism-proof equilibrium.
Example
4:
The
C
rawf ord-Sobe 1 Game
We now show that the example discussed by Crawford and Sobel
([2],
Section 4) often fails to have
equilibrium,
a
neologism-proof
and never has a neologism-proof equilibrium with
communication. The game has infinite type and action spaces, but
concavity ensures that it is well-behaved.
prior
ti (
•
)
Payoffs are
is
uniform.
T
=
[0,1],
The action space A is also
and the
[0,1].
16
u
u
R
S
2
(a,t)
-
~(a-t)
(a,t)
=
-(a-(t+b))
2
(b
Crawford and Sobel show that,
N
satisfying 2N(N-l)b
<
1,
0)
>
if N(b)
integer
the largest
is
then there is one sequential
equilibrium for each integer
N
£
)
u...u [X
i
The N-equi 1 ibr ium is
N(b).
1
described by the partition
[0,1]
[X
=
Q
,X
,X
[X
u
)
1
1
2
N_
,X
1
N
where
X.
i/N
=
2bi(i-N)
+
If te[X. ,,X.),
l-l
i
(0
<
i
IN).
then R chooses the action
a
=
(X.
,+X.)/2 in
l-l
l
equi librium.
Since
always wants a to be higher than R wants
S
might expect that,
if
signaling neologism
For such an
t
=
is
very close to
1,
S
S
y,
is
X
=
(y,l],
where
y
to be self-signaling,
X
>
a
self-
X.,,.
it
must be the case that
and using his equilibrium message (thus
a
=
[l+y]/2)
inducing
a
=
[1+X„ ,]/2).
This implies that y+b
,
just half way between those two possible actions:
-
one
0),
indifferent between using the neologism (thus
inducing
(l+y)/2
>
will try to reveal
We therefore investigate whether there is
the fact.
when
t
(b
[y+b]
-
y+b - (l+X _ )/2
N 1
S's
ideal,
is
17
This implies that
+ y - 4y - 4b
1
3y
+
=
2
-
4b +
X
=
2
-
4b +
[N-l]/N
=
3- 1/N - 2b(N+l)
X
N-1
1
'
N-l
N_ 1
To check whether this
(
+
1
-
2b(N-l)
describes
value of
a
y
the range
in
we write
')
'
y =
[l/N+2b(N+l) ]/3
-
1
1
<
and
3(y-X _
N 1
Thus,
if N
>
and only if
3/N
=
)
6b(N-l)
+
=
2/N
+
4bN
=
2/N
+
4b(N-2)
y
€
(X
2,
b
<
1/2.
-
-
1/N - 2b(N+l)
8b
,,1).
Hence,
then y
If N = 1,
if b
proof equilibrium in this example;
<
1/2,
if b
>
€
there is
1/2,
(X
N _ 1>
n_o
1)
neologism-
the
uncommunicative equilibrium is the unique neologism-proof
equilibrium.
(It
is
if
also the unique sequential equilibrium.)
18
Discussion of the Examples
Example
1
:
For the uncommunicative outcome to be
equilibrium requires beliefs p(m)
in M
it
(i)
is
implausible that all the messages
will be used for meaningless babble;
i.e.
S
will be something like "Really,
since such
and (iii)
message has
a
would want believed by
that R will believe it.
R
a
focal
we assume that
(ii)
some message not used in equilibrium will have
meaning,
,
We find such beliefs
whether or not used in equilibrium.
implausible because
sequential
for all meM
[1/3,2/3]
€
a
t,
t,
as
focal
has occurred";
(literal) meaning that
if and only if it
is
true,
we assume
Thus the possibility of "unauthorized"
communication rules out the outcome in which R always takes
action a„
:
it
is
a
sequential equilibrium but
is
not neologism-
proof.
Example
2
:
The notion of neologism-proof equilibrium does not
simply mean selecting
a
most-communicative sequential
equilibrium, as one might suspect.
19
Example
3
:
We do not propose as a solution concept that type-l's can
induce action
and type-2's can induce action a„
a,
is
The fact that the sequential
plainly not an equilibrium.
equilibrium (in which all get a„)
type-l's to get
This
.
is
blocked by the ability of
does not imply that there is an equilibrium in
a,
which they do so.
This language suggests
Type
t,
can get
payoff of
a
core-like approach to the problem.
a
2
can separate himself.
if he
The
usual core concept would automatically allow him to do so,
that the types will be treated separately.
context,
our
in
and being so treated is
if pretending to be t,
attractive to t~,
However,
so
relative to his alternative,
then he cannot be
"kept out" by t,.
In Myerson's
in
[10]
the uncommunicative equilibrium
framework,
this game is a core mechanism.
According to the concept of
core mechanism,
type-1 cannot say "I am of type
you take action
a,
if
I
;
notice that
were in fact of type 2":
credibility of
a
I
1;
I
propose that
would not wish to propose this
Myerson's requirement for
neologism insists that R's proposed action be
a
best response to his beliefs even if his beliefs do not shift as
a
result of the neologism.
Another way of seeing the contrast is as follows:
Myerson's framework,
In
the neologism consists of proposing
allocation that is an equilibrium if
R
makes inferences
appropriate direction) from the proposal,
and is also an
a
(in
new
the
20
our framework,
equilibrium if he does not.
In
not worry about what happens
if R fails
the proposer need
to make
the appropriate
inference, because we assume that the message is so credible that
will make the inference.
R
What will happen in this game, given that there is no
Clearly, we cannot give
convincing equilibrium?
equilibrium argument for any outcome.
4
.
it
is
interesting
happens if agents adjust at finite speed towards best
to see what
See Section
responses.
However,
convincing
a
below.
5
Existence Results
Examples
and
3
above show that we cannot generally expect
4
neologism-proof equilibrium to exist.
a
this section we provide
In
some sufficient conditions for existence.
Our first result concerns games in which S's preferences over
R's beliefs are unrelated to S's true type.
for instance,
case,
in
This is often the
labor-market models
in simple
which private information concerns "ability."
say that S's preferences are uniform if,
all a,,a„
u
S
(a
for all
t,
,
t~
e
T
A,
e
,t
1
Formally, we
)
1
>
u
S
(a
2
,t
)
1
- u
S
(a
,t
:
2
)
>
u
S
(a
2
,t
2
)
(4.1)
and
21
Since we assume that
is
a
*
"
R's best response to beliefs
(X),
unique for each non-empty subset
X
£
T,
(4.1)
-n
implies that
|
X
T
itself is the only possible self-signaling subset (since in
equilibrium each type must be indifferent among all equilibrium
messages).
It
Propos it ion
2
follows that:
If S's
:
preferences are uniform,
uncommunicative equilibrium
then the
neologism-proof.
is
Our second result concerns games of pure conflict:
that
is,
the case where S's and R's preferences over actions are always
diametrically opposed:
prefers
to a.
b
if,
given
t,
S
prefers
a
to b
includes the zero-sum case.)
(This
In
then R
this case,
there can be no self-signaling subset relative to the no-
communication equilibrium.
signaling subset.
v(X, t)
By assumption,
u
R
>
To see this,
Then we know that
v(T,t)
this
(a*(X),t)
if teX
implies:
<
u
R
(a*(T),t)
and hence R's expected payoff,
action a (X) than from action
definition of
a
(X).
Hence we get:
let X be a self-
if teX;
given teX,
a
(T).
is
strictly lower from
But this contradicts the
22
Propos i t i on
3
equilibrium
is
(In
[3],
In
:
a
game of pure conflict,
the uncommunicative
neologism-proof.
we also showed that every sequential equilibrium in
a
two-person zero-sum game has the same payoffs as the
uncommunicative equilibrium.)
These two results concern cases in which there can be no
important communication (although
it
is
possible to have
equilibria with "inessential" communication
).
We now turn to
the harder task of providing partial existence results when
communication matters.
in
First,
we note that,
in
the case of a game
which S's and R's interests coincide completely,
revealing equilibrium (which always exists)
neologism-proof.
is
the fully-
necessarily
(We would like to be able to state that this
the unique neologism-proof equilibrium in such a game.
is
T
true when
T
has only one,
has four elements:
two or three elements,
see the Appendix.)
it
is
While this
fails when
23
We now give some less intuitive conditions for
existence.
In
the process, we see why equilibrium fails to exist
Example
3:
this,
in
and similar insight,
our motivation for
is
inquiring into some intrinsically unappealing conditions.
We say that S's preferences are quas i-linear if,
and all non-empty A,B
min[v(A,
v(B,b)
t)
]
c^
T,
<
v(AuB,t)
Notice that Example
t„,
A
=
{t,},
B
=
3
{t„}).
<
:
If S's
[
v
(
A
,
t
)
,
v
(
B
,
t
)
(4.2)
]
violates this condition (look at
t
=
The condition is a combination of
quas i-concavi ty and quas i-convexi ty
Propos it ion 4
max
for all teT
,
hence the name.
preferences are quas i- 1 inear
,
then either
the uncommunicative sequential equilibrium is neologism-proof,
there is
Proof
is
a
:
a
In
or
communicative sequential equilibrium.
uncommunicative equilibrium,
self-signaling neologism.
t
gets v(T,t).
Then by definition
Suppose
X
24
v(X,t)
v(X,t)
When
t
e
v(T\X,
and when
v(X'
,
But
>
v(T, t)
i
v(T,t)
X,
(4.3)
t)
<
if t€X
v(T, t)
€
X'
t)
>
v(T,t)
equilibrium
in
v(X, t)
<
1
(4.4)
they imply that
T\X,
t
(4.4)
(4.3)
imply that
and (4.2)
=
"I
otherwise.]
v(X, t)
and (4.5)
(4.5)
imply that there is
a
sequential
which some equilibrium messages mean "t
all others mean "t
€
X'."
neologism-proof,
X"
and
This proves Proposition 4.
We would like to be able to extend Proposition
conditions under which,
6
4
by finding
if a sequential equilibrium is not
then there exists a more-communicative
sequential equilibrium.
That would imply the existence of a
neologism-proof equilibrium, given that
T
is
finite.
(Evidently,
the conditions would have to be violated in Example 2.)
However,
I
have been unable to find natural sufficient conditions
for that, extension.
One result along these lines, but with very
stronTT assumptions,
is:
25
Propos it ion
5
Consider
:
sequential equilibrium in pure
a
strategies which partitions
T
into
u...u
T,
T,
.
Let
signaling subset wholly contained within one of the
Suppose that
X'
=
T,\X is also self-signaling.
T.
Proof
(a)
:
u
X
u
'
Tn
2
u.
.
.
u
a
say i=l.
:
T
into
T.
k
We need to show three things:
then v(X,t)
If teX,
v(X,t)
1
v(X',t) and
1
v(T.,t), where j>l.
>
v(T,,t) since teX and
We have
v(X,t)
vCT^t)
>
v(X'
v(T,,t)
2
v(T.,t) since teT,,
t)
since t^X'
self-
Then there is
sequential equilibrium in pure strategies partitioning
X
be
X
X
is
self-signaling;
and
X'
is
self-signaling;
and we began at
a
sequential equilibrium.
a
26
(b)
Similarly, we prove that if tcX',
and v(X'
(c)
,
v(T
1
t)
.,
v(X,t) and v(T ,t)
>
>
v(X',t) follow
J
J
J
v(X,t)
1
t)
then v(T.,t)
If teT.,
v(X',t)
immediately from the fact that
This proves Proposition
X
and
X'
are self-signaling.
5
We can also prove another partial result to the effect that a
self-signaling
a
contained in
X
partition set can be made part of
a
finer sequential equilibrium.
we assume quasi-linear
For this,
and an extra condition that ensures that,
preferences,
if a vague
then so is a purportedly more precise lie.
lie is unprofitable,
We defer the precise condition until the reader has seen what
is
needed.
Propos it ion
6
:
Consider
strategies, so that
each
i,
all types
T
in
a
partitioned into
is
T.
l
sequential equilibrium in pure
T,
induce action a (T.)-
u...u
T,
,
If there
is
X
which lies wholly within (say)
S's preferences are quasi-linear,
X
u
(T, \X)
u
1
u.
.
.
u
T,
,
then the following partition
defines another sequential equilibrium in pure strategies:
=
a
l
sel'f-sjgna 1 ing neologism
T
for
and,
T,
.
and
27
provided that an extra condition
Proof
We need to check that none of three classes of types will
:
want to lie:
some
T
satisfied (see below).
is
.
j
,
those in
those in
X,
but not in
T
X,
and those in
*1
J
(a)
If teX,
then v(X,t)
v(T,,t)
>
1
will not claim falsely to be in T.,
are quasi-linear and v(X,t)
<
v(X,t),
(b)
£
so that
If teX',
vCTj.t),
it
t
v(X',t)
be in
X
(c)
If teT.,
Also,
Hence
t
since preferences
v(T,,t), we have v(X',t)
£
v(T,,t)
then since preferences are quasi-linear and v(X,t)
follows that v(X',t)
for j*l.
Also,
j*l.
for j*l.
will not falsely claim to be in X'.
original partition was
v(T.,t)
>
v(T.,t)
1
1
vCT-.t).
But since the
sequential equilibrium, v(T,,t)
a
>
Hence
t
will not falsely claim to be in T..
v(T,,t)
>
v(X,t);
then we know that
v(T ,t)
>
v(T lf t)
and that
v(X,t)
We need to show that
<
v(T
t)
so
t
will not falsely claim to
28
v(X'
t)
i
v(T,t)
Unfortunately, our assumptions so far do not imply this:
is
possible that teT. would like to lie and claim to be
it
X'.
in
J
although
This would be a somewhat "surprising" case:
to
tell the truth than to tell the lie T,,
If
proof of Proposition
complete.
6
is
A
prefers
he strictly prefers a
this can be ruled out,
more precise lie X*£ T,.
T
then the
sufficient assumption
is
the following:
If teT.
v(T ,t)
and A £
or v(B,t)
Next,
>
B,
and
v(T
,
t)
B
T.
n
=
#,
then either
v(/>,,t)
<
.
we prove a result that guarantees,
assumption than quas i- 1 inear i ty
,
under
a
weaker
the full-revelation
that
sequential equilibrium (if it exists) will be neologism-proof.
Propos i t ion
all teT,
u
S
7
If S's preferences are quasi-convex,
:
if
for
A.BcT,
(a*(A
u
B),
t)
S
<
max[u (a*(A)
then no self-signaling set
In particular,
is
i.e.
if full
X
t),
u
S
(a*(B),t)]
can be a union of partition sets T..
revelation
also neologism-proof.
,
is
a
sequential equilibrium,
it
29
Proof
Let
:
=
X
u...u T.
T,
£
T
=
u...u
Tj
T
.
k
Then by quasi-
convexity applied (j-1) times
u
so that
S
X
(a*(X),t)
partitions
Note
)
.
.
.
,u
S
(a*(T ),t)]
j
cannot be self-signaling.
But now if
sets
S
max[u (a*(T 1 ,t),
i
T
sequential equilibrium in pure strategies
a
then there can be no self-signaling
into singletons,
X
:
Observe how quas i-convexi ty is violated in Example
2
above
A
further existence result has been obtained by Joel Sobel
(personal communication).
flO]
concept of
a
This is phrased in terms of Myerson's
neutral optimum for
class of games which
a
includes our simple communication games.
optimum is
a
Recall that a neutral
sequential equilibrium (Myerson uses more general
terminology) which maximizes
a
weighted sum X\(t)v
sequential equilibria, where the weights x(t)
2
among
(t)
must satisfy
additional conditions which generically determine them.
Sometimes, in a neutral optimum some of the weights x(t) will be
zero.
Sobel defines
optimum that has x(t)
existence of
shows that,
a
strict neutral optimum to be
a
for all
>
teT.
neutral optimum, but
if there
is
a
neologism-proof. However,
it
is
neutral
Myerson proves the
may not be strict.
strict neutral optimum,
there
a
Sobel
then it is
no reason to believe that a
30
strict neutral optimum will "usually" exist.
These partial existence results do not imply that neologism-
proof equilibrium will "usually" exist.
What is one to make of
that fact?
There is
equilibrium,
a
core-like quality to the notion of neologism-proof
as Myerson
"core mechanisms."
recognised
Types,
in
naming his related concept
or groups of types,
can block an
allocation if they can do better by distinguishing themselves,
and if their distinguishing claim is credible.
While coalitions
in
the usual core concept can keep out undesirable members,
is
not possible here
said),
so that
this
(the undesirable members can mimic anything
the set of neologism-proof equilibria is not just
the core of some related game.
However,
problems have the same flavor,
the non-existence
and in the same way they are not
to be dismissed as exceptional.
What can we predict for games without equilibrium?
possibility is to look at how
behaves over time.
a
In Section 5,
One
reasonable dynamic process
we explore that approach.
31
5
.
Evolution of Language and Lying
In this section we re-examine Example
3
above under the
assumption that agents do not instantly optimize against their
strategies, but rather move in that direction.
opponents'
assume
a
finite speed of introduction of neologisms,
and
We
a
finite
rate of imitation by type-2's when enough type-l's are using
a
neologism that using the old strategy means being recognised as
type-2
(which,
gives the lowest payoff).
recall,
consists of the positive integers
We suppose that M
Initially,
{1,2,3,...}.
time,
a
types of
all
S
send message
and R always responds with action a„.
1
all
the
This is the (unique)
sequential equilibrium.
Now,
few
(f
<<
every
k
the next neologism is discovered by a
periods,
of the type l's,
1)
who then use it effectively to
distinguish themselves from the other S's.
simplicity that R's learning
to how fast
S
can change).
is
We assume for
instantaneous (very fast compared
Thus,
type-l's using
discovered neologism induce action a,.
a
newly
Those using an older
message induce R's best response to the mix of types using that
message
In each period,
both type-l's and type-2's change their
message-sending behavior
Thus,
if old messages
in
response to differences in payoffs.
are inducing action a„,
then type-2's
continue to send old messages, but type-l's move (at
rate)
into sending a new message,
if one
is
a
finite
available.
After
32
enough type-l's have Moved away from the old messages,
those
messages begin to induce action a, (since most of the S's sending
so now it pays
them are now type-2's),
to
for the type-2's
Thus the old messages get abandoned,
imitate the type-l's.
and the new ones become "discredited" by type-2's
l's
(in preference to being
perhaps,
while,
there
is
to begin
imitating type-
identified as type-2's).
For some
no effective communication.
next neologism is discovered,
Then the
communication by type-l's
and
begins again.
He wrote
a
simple computer program (available on request) to
simulate this process.
The proportions of type-l's and of type-
2's that get treated with each of R's three actions
on -Figure
1,
for different values of
k
are plotted
(the number of periods
change to
(the rate at which type-l's
between neologisms),
r
better moves),
(the corresponding rate for type-2's).
Table
1,
and
In
summary statistics are given for various parameter
In Table 2,
itaiues.
s
we show the results of regressing the average
proportions of the two types treated with the three actions on
S*e parameter values.
As
can be seen,
increasing the
......
and
s
r
speed" parameters
does net lead to a predominance of action a„
"
j
.
f,
1/k,
The fuliy-
optimizing model of game theory corresponds to ail parameters
being very large:
compare,
but without more knowledge of how they
we cannot properl}' say that the uncommunicative
equilibrium correctly describes limiting behavior as the
parameters grow large.
r
33
6
.
Conclus ion
In a sequential
introduced
equilibrium, no agent can profitably lie.
We
plausible condition expressing the further
a
equilibrium requirement that no agent can profitably deviate by
coining
a
credible neologism.
We showed that
implausible sequential equilibria;
out all sequential equilibria.
behavior in
a
this rules out some
and that sometimes it rules
We then simulated adaptive
game with no neologism-proof equilibrium,
that the predictions of sequential equilibrium were not
fulfilled.
and found
Appendix: The Case
full revelation of
In this case,
u
R
S
=
u
is a neologism-proof equilibrium,
t
Pareto dominates every other sequential equilibrium (generically
is
the
|T|
=
or
1
but if
we will show,
For
=
|T|
easy to see
is
|T|
=4
suppose
3,
the full-revelation equilibrium
that
If
|T|
=
3,
tins
still true,
is
|T|
equally
S
u (a*(T),
then it need not be true.
equilibrium was
sequential
other
some
neologism-
types (otherwise some two types would find themselves in the
=2).
By scaling payoffs,
likely
and
=
t)
= u
that
R
(a*(T),
can suppose
we
equilibrium
the
is
that all
three
types
uncommunicative.
S
Normalize
t)
for all
>
t:
otherwise, some
never appear in P(X) for any X, and we would be reduced to the case
S
Normalize so that u (a*({t}), t) =
Choose
t
E
T.
Write
a
= u
S
1
(a*(t
is not self-signaling, we know max
(a,
definition of a*(T), we know
c
= u
d = u
e
= u
(a*({t
S
S
(a*({t
(a*({t
,
t
r
t
r
t
2
2
}),
t
}),
t
>),
t
)
2
3
)
).
t
|T|
would
=
2.
for all t.
1
Now write
are
for all t.
t)
We can assume that u (a*({t}),
"By
as
Each message used in equilibrium must be used with positive probabil-
ity by all three
case
it
2,
only neologism-proof equilibrium.
proof.
up to equiv-
But it need not be the only neologism-proof equilibrium.
alence).
If
,
also
1
S
),
t
),
b = u (a*(t
b)
>
0:
say a
+ a + b < 0.
>
0.
),
t
).
Since {t
}
Now since {t
d)
< 0.
t
,
}
is not self-signaling,
If min (c,
definition of a*({t
{t
so
t
},
,
than a*({t
that a*({t
,
t
})
proves the result for
then
<
d)
t
}),
t
}).
c+d<l<l+a;
since
If
says
it
mm
=3.
d)
(c,
does better than
|T|
we know that either
that
a*({t
and
>
but
e
>
given T:
a*(T)
or min
(c,
this contradicts
the
e
})
does
0,
then
a
>
given
better,
+
c
d + e
contradiction.
>
0,
This
Now we show by example that the uncommunica-
tive equilibrium can be neologism-proof when
|T|
=4.
(This also shows,
inci-
dentally, that neologism-proof equilibrium need not be unique, since full-rev-
elation is also neologism-proof.)
There are
four equally likely types,
labelled
1,
3,
2,
4.
There are
fifteen
actions, each of which is the optimal action a*(X) for just one non-empty subset X of T.
Payoffs can be described as follows:
Type
Action:
a = a*({
})
=
1
1
.1
10
2
1
.1
-10
-10
3
-10
-10
•10
1
.1
4
.1
•10
10
1
12
23
34
41
13
24
.5
.6
.1
-10
-10
.6
.6
.1
.1
10
6
.6
.6
.1
10
.6
.6
10
.6
.1
10
.6
-
1
.
123.
- .1
.9
.9
-10
234
-10
.9
.9
341
412
.9
.1
10
.1
.9
.9
.9
10
-.1
1234
easy to
It is
For example,
proof.
P({t
2
-
(t
3
.
check
=
))
t
4
(One
{t
t
,
}.
that the
uncommunicative
equilibrium is
neologism-
the neologism {t_} fails because
The neologism {t 2<
t
,
3
t
4
)
fails because P({t
,
t^,
t^})
2
>.
could
(even though not
including not
imagine
in
fact,
a
neologism
{t
}
would
be
believed
self-signaling) because R v:ould think of S's alternative as
only the
This line of argument
low in qeneral.
that,
equilibrium but
also some
other possible
neologisms.
seems persuasive in this case, but is difficult
to fol-
Figure
1:
Two
graphs of the proportion
of
type-l's
inducing action
a
.
In
The relatively small disconthe first, we see discontinuities of two types.
tinuity at T = 250 is due to the introduction of a neologism (on the part of
The other discontinuities, including the large one, re8% of the type-l's).
The
sult from a critical change in the mixture of types using a message.
downward discontinuity results from a neologism becoming discredited by the
influx of type-2's.
In the second graph, we applied a smoothing algorithm and used
value of k. We see the oscillatory behavior clearly.
a
much smaller
TABLE
.15
(35.9)
XM
(-4.7)
.44
(50.1)
YL
(8.7)
-.05
(-2.6)
-.13
(-6.1)
(5.3)
YM
.48
YH
(5.1)
.18
.54
.29
-.28
(-13.6)
.50
.70
(11.3)
-.36
(-20.1)
-.25
(-13.2)
.38
.70
(20.0)
.05
.04
(22.9)
-.02
(-5.2)
.05
(11.0)
(10.9)
.51
Explanation
The variable named e.g., XH denotes the fraction of type-l's
(X's) treated as being of high (H) probability of being type 1 (and thus getwho induce action a
ting action a ).
Thus YM is the fraction of type 2's
:
(corresponding to middling probability of type 1).
Parameters result from regression using 256 data points corresponding to
10, 20, 40, 80; f = .01,
Numbers under
.04, .08.
.02,
.04,
08,
r
=
.01,
.02,
parameters are t-statistics
2
(14.6)
.20
.10
(59.5)
R
-.01
(-1.1)
.11
-.14
(-8.1)
.48
(63.1)
Log(s/.02)
(-5.8)
.17
.41
(47.6)
XH
Log(r/.02)
-.06
Log(K/40)
-.05
Intercept
Variable
XL
1
.04,
.08;
s
=
.01,
k
=
.02,
FXL
rxM
FXH
FYL
FYM
TYH
15
.25
.60
.60
.25
.15
.30
.20
.61
.
19
.41
.55
.04
.30
.60
'.09
.40
.51
.57
.43
.00
.10
.30
.
15
.
13
.24
.63
.60
.25
.15
7
.10
.30
.30
.
18
.60
.22
.41
.55
.04
7
.
10
.30
.60
.08
.42
.50
.56
.44
.00
10
.08
.01
.01
.20
.38
.42
.50
.50
.00
10
.08
.01
.02
.22
.40
.39
.46
.54
.00
10
.08
.01
.04
12
.50
.38
.27
.73
.00
10
.08
.02
.01
.21
.28
.51
.64
.35
.00
10
.08
.02
.02
.25
.31
.44
.60
.38
.02
10
.08
.02
.04
.17
.43
.40
.39
.60
.01
10
.08
.04
.01
.16
.22
.63
.74
.26
.00
10
.08
.04
.02
.
19
.25
.56
.69
.30
.01
10
.08
.04
.04
.25
.32
.44
.61
.36
.03
15
.01
.30
.15
.07
.70
.23
.37
.58
.05
15
.01
.30
.30
.10
.81
.09
.20
.78
.02
15
.01
.30
.60
.08
.59
.33
.39
.59
.02
15
.
10
.30
.15
.09
.71
.20
.37
.58
.05
15
.
10
.30
.30
.09
.81
.
10
.20
.78
.02
15
.10
.30
.60
.08
.58
.34
.39
.58
.02
no
.08
.01
.01
.26
.31
.43
.60
.37
.03
40
.08
.01
.02
.15
.48
.37
.33
.65
.02
40
.08
.01
.04
.15
.45
.39
.35
.63
.03
40
.08
.02
.01
.17
.28
.55
.64
.31
.05
40
.08
.02
.02
.26
.34
.41
.58
.38
.03
40
.08
.02
.04
.16
.46
.38
.34
.63
.03
40
.08
.04
.01
.10
.23
.67
.71
.24
.05
40
.08
.04
.02
.15
.30
.54
.61
.32
.08
40
.08
.04
.04
.21
.53
.26
.45
.51
.04
50
.08
.01
.01
.25
.36
.40
.57
.39
.04
80
.08
.01
.01
.25
.34
.40
.58
.39
.03
80
708
.01
.02
.15
.48
.37
.32
.65
.03
80
.08
.01
.04
.15
.48
.37
.32
.66
.02
80
.08
.02
.01
.15
.31
.54
.60
.32
.08
80
.08
.02
.02
.21
.53
.26
.44
.52
.04
80
.08
.02
.04
.
10
.62
.28
.22
.73
.05
80
.08
.04
.01
.08
.24
.67
.61
.25
.
80
.08
.04
.02
.
1
1
.57
.32
.45
.48
.08
80
.08
.04
.04
.
13
.74
.13
.27
.71
.02
K
F
R
S
7
.01
.30
.15
.
7
.01
.30
7
.01
7
I
.
TAR
I
F
?
14
FOOTNOTES
1.
With more than two players, communication becomes
complicated affair.
to each other,
For example,
a
much more
two players could whisper
keeping information secret from
a
third.
Or
they could merely pretend to whisper to each other, without
actually passing information.
We assume that there are two
players in order to avoid these problems:
thus,
whatever
is
announced is publicly heard.
2.
(2.4)
implies that the supremum in the definition of
attained, which is not otherwise guaranteed since M
infinite.
attained,
If r(»)
then
(t)
is
is
were such that the supremum could not be
would have no best response,
S
v
and r(») could
not be part of a sequential equilibrium.
3.
Since the strategy spaces are infinite,
and Wilson
4.
Let
p
[8]
permute M
do not apply directly.
.
equilibrium, so is
5.
the theorems of Kreps
Then if (s,p,r)
(p
o
s,p
o
o
,r
o
p
sequential
is
a
),
in obvious
notation
This is why we take M* to be infinite. With A and T finite,
it
is
possible to show that every sequential equilibrium
is
equivalent to one in which only finitely many messages are
used.
For example,
let u
R
(a,t)
=
=
u
S
Then any pattern of revelation is
equi 1 ib r ium
(a,t)
a
for all
(a, t)eAxT.
neologism-proof
References
Banks, J., and J. Sobel, "Equilibrium Selection in Signaling
Games," mimeo, San Diego, 1985.
[2] Crawford, V., and J. Sobel, "Strategic Information
Transmission," Economet r ica 50 (1982), pp. 1431-1451.
[1]
[3]
Farrell, J., "Communication Equilibrium in Games," mimeo, GTE
Labs, 1985.
[4]
Green,
[5]
Grossman, S., and M. Perry,
mimeo, Chicago, 1985.
[6]
Kohlberg, E., and J.-F. Mertens, "On the Strategic Stability
of Equilibria," mimeo, Harvard, 1984.
[7]
J., and N. Stokey, "A Two-Person Game of Information
Transmission," mimeo, Harvard University, 1980.
"Perfect Sequential Equilibrium,"
Kreps, D., "Signaling Games and Stable Equilibria," mimeo,
Stanford, 1984.
[8]
Kreps,
[9]
Lewis,
[10]
D., and R. Wilson, "Sequential Equilibria,"
Econometrica 50 (1982), pp. 863-894.
D., Convention
1969.
,
Cambridge:
Harvard University Press,
Myerson, R., "Mechanism Design by an Informed Principal,"
Econometrica 51 (1983), pp. 1767-1797.
^353" 002
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