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Coiranunication Eq uilibrium in Games
!
i
Joseph Farrell
Number
38.2
-%v
June 1985
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Communication Equilibrium in Games
Joseph Farrell
Number 382
June 1985
Communication Equilibrium in Games
Joseph Farrell
GTE Laboratories and MIT; June
]
985
1.
INTRODUCTION
simultaneous-move game of incomplete information, the standard solu-
In a
This is the natural
tion concept is Bayesian equil ibrium.
generalization (to
incomplete-information games) of the notion of Nash equilibrium: the modification
instead
that,
is
choosing
of
each player chooses such
a
probability distribution
a
over
actions,
probability distribution for each of his possible
The different players may be thought of as making their choices in
"types."
separate rooms: there is no communication among them.
Myerson (1983) has argued for
the
possibility
of
a
different solution concept, allowing for
centralized communication.
In
correlated equil ibrium,
a
each player secretly reveals his private information to
worthy mediator who
committed in advance to
is
a
disinterested trust-
known behavioral
a
mediator collates this information, and sends each player secretly
dation for
his
requirement for equilibrium is
The
move.
that
rule.
a
no
The
recommen-
player be
tempted to lie and/or to cheat, given that he believes that all others report
honestly and act obediently.
Correlated equilibrium allows for communication among players,
about
Accordingly,
to
it
make
is
a
Often there is no disinterested mediator available.
somewhat artificial way.
Agents
but in
strategic
choices
often
talk
directly to one
another.
important to ask whether communication without a mediator
is the same as communication with one.
In
an
this paper,
outcome that
specified way.
we
introduce the concept of
a
communication equi ibrium:
equilibrium when players can talk to one another
is
an
We
show
that
every
Bayesian
equilibrium
is
a
in
communication
equilibrium, but not vice versa, and that every communication equilibrium is
correlated equilibrium,
but
not
direct communication can count;
tive value.
vice
on
versa.
Thus on
the other hand,
a
the
one
a
hand
a
costless
mediator can have posi-
We then ask what kinds of information can be revealed by unmediated com-
munication.
the standard case of
In
independently distributed types, we show
that there are invariants that are always preserved under unmediated communi-
cation,
sents
but need not be preserved when
the
fact
that,
without
a
a
mediator
mediator,
it
is
available.
This repre-
impossible for
is
each
of
two
players to make his revelation contingent on what the other says.
Possibly because of game theory's beginnings
case
(von Neumann
the Prisoners'
and Morgenstern,
1944)
in
as
strategic
relationships
were
Many theorists informally
essentially
therefore (absent differential costs of sending messages,
etc.)
communication can be
realize
how
to
Dilemma since the subject turned to the more general case, com-
all
if
two-person zero-sum
and the excessive emphasis given
munication has been badly neglected in game theory.
think
the
important
ignored.
It
is
also
payoff-irrelevant moves
adversarial,
as
in
something of
may
be
for
Spence,
1970,
surprise
a
the
and
outcome
to
of
a
game.
Roughly,
payoff-irrelevant
opportunities for coordination
in
two-person zero-sum games,
in
but
effect to delegate some control
"his opponent"
ing even
is
misleading).
moves
matter
the game.
in
over
general
his
to
the
extent
that
there
are
There are no such opportunities
each player may be prepared
move to the other player
(the
in
term
This can be accomplished through message-send-
if messages do not directly affect payoffs.
DEFINITIONS
2.
consider
We
indexed by
which
move
TT.;
X.
E
•)
(*,
can
.-1
We write
X..
is
given by
=
t
(x
u
t
)
for all
t
a
Players
random variable
)
t
T..
e
-1
and
=
x
n
Player
t.
(x,,
...,
also chooses
i
x
n
).
a
Player i's
-^
The priors tt.(*) and the payoff
.
are
functions
u
are common knowledge.
We write T
T,x ...
=
x
and X
T
in
=
XiX
n
i
probability distributions on any set
w
follows:
as
according to the probability distri-
T.
....
(t,,
described
be
observes the value of
i
we assume that n.(t) >
1
1
payoff
Player
n.
...,
2,
which
G
distributed on the finite set
is
bution
1,
game
a
A.
...xX .We write A(A) for the set of
An outcome of the game
is a function
T ^ A(X).
:
An outcome w:
T
A(X)
-*
stratagies, functions s-:
1.
The strategies s.
2.
The strategies
s.
T.
is
a
Bayesian equil ibrium outcome if there exist
^ A(X.),
constitute
a
such that
Bayesian equilibrium:
lead to the outcome w:
weight assigned by w(t) to the point x
e
that
is,
X is equal
for each t
e
the
T,
to
n
S.(t.)(x.)
7T
It
is
a
standard result
that
a
Bayesian
equilibrium will
exist
in
a
finite game such as ours.
An outcome w:
1983)
if,
T
->
A(X)
is a
correlated equil ibrium outcome (see e.g., Myerson,
when a trusted mediator asks for reports of
t,
performs any neces-
sary randomization, and then instructs players secretly as to the value of
they should choose, there are no incentives for
vate
information or to disobey his
x.
a
player to misreport his pri-
instructions,
assuming that other players
are reporting truthfully and acting obediently.
We say that
a
game
G'
communication version of G if
is
a
T.
(just as in G)
G'
consists of
three phases:
1.
Player
2.
There
learns
i
t
opportunities
are
speak.
t-
assume
(We
unrestrictive if
functions,
but
n
that
=
may
2.)
announce
for
players
to
all
players
hear
messages,
i.e.,
to
this
is
announcements;
all
These messages do not directly enter payoff
convey
information
and
influence
others'
final
choices.
The players simultaneously choose x.
3.
We emphasize that payoffs are given by
there
is
payoffs.
no need
to
w'
(x,
u
X.
t)
(just as in G).
just as in
know what messages were announced
However, as we will
Every outcome
z
of G'
forget the information
see below,
G
in
a
is a
communication equil ibrium outcome of
For instance,
if we make
We will
natural
in
G',
G
is
x.
way (simply
If w'
is
the
then we say
G.
the communication opportunities vacuous,
that any Bayesian equilibrium outcome w of
rium outcome.
and x,
order to calculate
concerning the messages announced).
outcome corresponding to some perfect Bayesian equilibrium
that w
t
sometimes the messages will affect
determines an outcome w of
in w'
in
given
G:
also
a
we see
communication equilib-
show below that there can be others.
RELATIONSHIPS OF THE THRtE EQUILIBRIUM CONCEPTS.
3.
Theorem
1:
Every Bayesian equilibrium outcome is
1.
communication equilibrium out-
a
come, but not vice versa.
equilibrium
communication
Every
2.
outcome
is
correlated
a
equilibrium
outcome, but not vice versa.
Proof:
First, we can always set
1.
come is
1:
< a <
dent.
1
This
< b.
Hence every Bayesian equilibrium out-
G.
communication equilibrium outcome.
a
fails, consider Example
Example
=
G'
is
a
1:
symmetric
=
{y,
z}.
t.
each player,
For
game.
The probability that
For each player, X
To see that the converse
=
a
is
and
p,
Player
indepen-
z
t
(-tp
-t^)
-t2)
lj_
If p < a,
0)
(0,
1
is for
(z,
z)
better move if
the unique Bayesian
to be chosen for all
Let q denote the probability that player
player I's choice.
0)
then the unique rational izable (hence,
equilibrium) outcome of Example
Proof:
and t^ are
t,
where
1
(0,
Lemma
tp
b},
2
y
-
{a,
Payoffs are given, by:
Player
(1
T =
If
t-,
=
b,
2
then z dominates y.
will
If
play y,
t,
=
a,
t^
and consider
then y is
the
q(l
+ (1
a)
if q > a.
i.e.,
know
-
<
q
-
q)(-a) >
However, since z dominates y for player
Hence if p <
p.
wise, nor can player 2.
then player
described
b,
we
Like-
1.
more striking example with infinite T,
a
Farrell, 1983.)
in
The outcome which assigns probability
2j_
=
cannot rationally play y.
1
This proves Lemma
This is an adaptation of
(Note:
Lemma
a,
whenever t^
2
and assigns probability
to
1
y)
(y,
when
t
=
(a,
a),
for other values of t^ is a communication
z)
(z,
to
1
equilibrium outcome.
Proof: Define
as follows.
G'
Stage
"A" or "B".
If
t
=
b,
announce
announces.
If t
=
a,
they do so,
If
Lemma
is
not
Theorem
a
1
Next,
2
outcome w,
complete.
in 6'
"B".
for both players to use this strat-
This proves Lemma 2.
a
communication equilibrium outcome that
(in
G);
thus
the proof of part
(1)
of
the positive statement
is
Costless communication can make
we prove part
a
whatever the other player
the outcome is as described above.
displays an example of
easy to prove:
z
if he announced
z
Bayesian equilibrium outcome
is
play
strategy:
announce "A"; then play y if the other player
perfect Bayesian equilibrium
is a
egy.
and then
"B",
announced "A", and play
It
each player simultaneously announces
I,
Consider the following
is G.
II
Stage
In
(2)
of
Theorem
1.
whatever
Again,
the
a
process
difference.
in
G'
that yields
an
mediator can commit himself to imitating that process internally.
The fact that w was supported as an equilibrium in G'
implies that it will be
an equilibrium for players to report truthfully and to obey instructions.
To see
that not every correlated equilibrium outcome is
equilibrium outcome, consider Example
Example 2^
{z,
sible
t
E
T =
{c,
d)
X
{c,
we use four bi-matrices,
d).
Player 2
t
=
(c,c):
z
Player
t
=
w
2
(1,
1)
(0,
0)
W
(0,
0)
(0,
0)
1
(c,d):
w
z
communication
2.
This too is a symmetric two-player game, with T.
To represent the payoffs,
w).
a
(0,
0)
(-2, 0)
(1,
-2)
(-2,
-3)
=
{c,
d)
and X.
=
one for each pos-
t
=
(d,
c)
w
z
t
=
(d,
(0,
0)
(-2,
1)
(0,
-2)
(-3,
-2)
d)
w
z
z
(0,
0)
(0,
0)
w
(0,
0)
(1,
1)
The types are independently distributed, with
Lemma
3:
In
if t
=
Example
2,
the following
> n.(c)
1
outcome is
>
1/2.
correlated equilibrium out-
a
come:
Otherwise,
Proof
ior,
(z,
z,
x
=
(w,
w);
z).
Suppose
3:
player to play
he will
=
x
Lemma
of
then
(d,d),
a
mediator
announces
unless both players report
instruct both to play w.
If
this
that
their types as d,
a
asked to play
play
Thus,
z
if
c.
If player
1
=
d,
1
reports truthfully when
t,
=
2
honest and obe-
is
reports truthfully, he will be
and he knows that player
whatever player 2's report,
Since np (c) > 1/2,
c,
he
then
will
he
2
will
would not.
obey the mediator's
His expected payoff is TTp(c).
Would player
ti
=
Would he then prefer to play w?
z.
instruction.
that
t,
which case
correlated equilibrium.
Consider player I's incentives, assuming that player
Suppose first that
in
each
induces honest and obedient behav-
then we have shown that the outcome given is
dient.
instruct
will
he
1
benefit
from
lying when
then what happens depends on t^:
t,
=
c?
If
he
falsely reports
probability
With
player
(c),
-np
mediator then asks each player to play
that Xp
=
truthfully
2
Player
z.
sees that he should play z, and gets payoff
now knows that tp
The
=
and
c
probability
1
-
player
(c),
iTp
1
1.
truthfully reports
2
mediator then asks each player to play w.
Xp = w.
t^ = c.
Accordingly, consulting the first of our payoff tables, player
2.
With
1
reports
Player
1
tp = d.
now knows that tp
Consulting the second payoff table, second column,
=
The
d and
sees that what-
1
ever he does he gets payoff (-2).
Hence
Tip
(c) + (1
Next,
expected
the
-
payoff
iT2(c))(-2) <
consider
a
to
player
of type d.
1
c
will
who
c
lies
is
not choose to lie.
tells the truth,
he
If
type
of
A player of type
(c).
ITp
player
a
one of two
things may happen.
With probability
player
(c),
iTp
mediator asks both players to play
of
the third payoff table,
2
truthfully reports
Player
z.
=
Then
c.
the
thus consults the first column
1
that he might as well
and sees
tp
obey.
This gives
him payoff 0.
With probability
players to play
[1
-
•np(c)],
Player
w.
reports tp
2
now knows tp
1
=
Thus if
t,
=
d,
player
he tells the truth he will
If
t,
=
d
and
player
1
1
Whatever tp may be,
it gives
1
t,
=
d.
payoff of 0.
1
-
d.
=
w.
The mediator asks both
Consulting the fourth
1.
'np(c)
by telling the truth;
and if
obey instructions.
Xp = z.
a
can get
Xp
d,
table, he sees he should obey, and gets payoff
=
he will
lies,
x,
Since
=
z
<
not
learn
tp,
and will
is at least as good a move as x,
1
-
TTp(c),
This concludes the proof of Lemma
3.
know that
=
and
w;
lying is unprofitable for
1
if
10
We complete our proof that
a
a
correlated equilibrium mechanism need not be
communication equilibrium mechanism with Lemma
Lemma
The outcome described in Lemma 3
4j_
is
4:
not
communication equilibrium
a
outcome.
Proof of Lemma
Lemma
corollary of Theorem
below.
However,
4
is
a
apparatus is required only
in
order to allow for mixed Bayesian strategies in
G'.
It
4j_
5
very straightforward to show that neither player
is
that
Example
in
2
is
willing to reveal his type before knowing the other's.
notice that,
To see this,
he will
set Xp
sirable
otherwise.
=
n2(c)(-2) + (1
(since
of
choose Xp
=
Lemmas
If
=
d,
•iT2(c))(l)
=
1-3
also
1-3
Since
z).
and
4
not
TTp(c),
reveal
tiplc)
if
1
=
t
the fact would
revealing
t,
will
2
3
This is desirable for player
w.
-
knows or believes that
if player 2
while concealing
and
tr,,
< 0, player
then
1
each
will
essarily be properly modeled by assuming
mediator.
a
mark case in which communication cannot matter:
if
of
payoff
a
1
player will
want
to
conceal.
together complete the proof of Theorem
that decentralized costless communication can matter,
gives
it
payoff
a
1
then
d,
but unde-
d),
(d,
give
=
t,
Theorem
1.
1
shows
and that it cannot necWe next examine
there
so
is
a
bench-
much conflict
that players can never trust one another.
Theorem
2j_
In
two-person zero-sum game, every correlated equilibrium outcome
a
gives the same payoffs in each state
affect payoffs
not
tively,
is
the reason is clear.
affect the action rule
bility
in
with
report
a
as
Bayesian equilibrium.
Another way to say this is that only "inessential" communication that
Remark:
does
t
x,,:
possible
that means that,
when
if 2
T^ ^ A(Xp)
1
such
in
means
that
t,
.
he
But
a
game.
each player (say,
used by the other.
his true type is
is
equilibrium
By choosing a message,
reporting for player
t,
in
prefers
since
Intui1)
may
Incentive-compatithe
rule
the game
is
associated
zero-sum,
asked to follow one of the rules used in equilibrium,
11
he would do better to follow any other.
This proves Theorem 2 for the case
where the rule can be inferred from the mediator's message.
order to allow
In
for the more general case, we give the following proof.
Theorem
of
Proof
Although
2j_
general
in
a
player
in
zero-sum game whose
a
opponent pays attention to his claims can profit by lying,
write down
instead
a
a
lying strategy for the general
successful
deviation from
a
it
difficult to
is
so we
case,
consider
proposed correlated-equil ibrium mechanism to ran-
domized uninformative messages.
Consider
a
correlated-equil ibrium,
in
which each player sends messages to
the mediator revealing his private information,
Suppose that player
appropriate moves.
1
and then the mediator suggests
begins randomizing his messages so
and playing
that they convey no information about his type,
Bayesian equi-
a
librium strategy independent of what the mediator suggests for him.
2
knew this was
strategy,
happening,
his
and both would get Bayesian
response,
best
best response would be to play his
his
not know about I's deviation,
and
equilibrium payoffs.
on
would
If player
equilibrium payoffs.
the other hand,
therefore do
Hence player
1
no
better
than
player
If
then player
2
Bayesian
2
did
would not use
Bayesian
his
get
would do at least as well as his Bayesian
payoffs.
Thus,
least
can
his
do
no
in
any correlated equilibrium,
each
type
Bayesian equilibrium expected payoff.
better
than
his.
The
same
of
This
argument,
player
implies
reversing
1
must get at
that player
players
1
and
2
2,
gives the bounds the other way, thus proving Theorem 2.
Remark:
We cannot
quite say that communication
will
not occur.
For example,
consider the game which gives payoffs of zero to both players whatever happens.
Theorem
Many correlated equilibria will exist in which the players communicate.
2
expresses
a
sense
in
which such uninteresting communication
only kind that can occur in equilibrium in
a
two-person zero-sum game.
is
the
12
Corollary to
Every communication equilibrium outcome gives the same
Theoreir. 2j_
payoffs (to each type) as Bayesian equilibrium.
Discussion: The proof of Theorem
when communication will
in
be
2
suggest the following heuristic analysis of
important.
When
there is essential
communication
equilibrium, each player is allowing the other to choose his move to some
since
extent,
rule,
it
is
common knowledge that one message will
another another.
tions
are sufficiently
(u
12
u
A
general
=
mixed;
)
but
is
is
wise precisely when payoff func-
"positively correlated."
A
has
some
aspects of each
type.
some games they are separable.
Often
they
An example
group of risk-lovers wishing to meet to play poker.
12
game of pure coordination
the polar case, opposite to that of the zero-sum game (u
game
in
Such delegation
induce one action
If
is
are
=
-u
).
inextricably
the problem of a
they coordinate suc-
cessfully on the meeting-place, they will play an enjoyable game of pure conflict;
otherwise they will
drink
estly concerning where to meet,
knowledge of odds, etc.
alone.
We
expect
them to communicate hon-
but not concerning their playing
strategies,
13
4.
AIMNOUIMCEABLE INFORMATION
We now turn to a limitation of decentralized communication that has nothing to do with incentives to tell
possible both for I's revelation of
2's revelation of tp to depend on
is possible
with
the value of
We show that a mediator can
absence
of
a
t,
the value of tp,
Such
.
dependence
mutual
this section, we define the
In
structure having the right
5)
a
and for
(if information is verifiable ex-post) gen-
mediator various
apply this result (Theorem
it is not
structure and announceable information struc -
information
erate any public information
the
to depend on
t,
mediator, as in Example 2.
a
concepts of publ ic
ture.
As in Example 2 above,
the truth.
constraints
other
"mean";
must
satisfied.
be
the proof of Theorem
to complete
that
but
in
We
and to dis-
1
cuss bargaining with two-sided uncertainty.
In
this
we are concerned with constraints on what
section,
ally possible to reveal.
tives,
punishment
is
controlled (e.g.,
if
quate
it may still
available
there is
a
information
is the subject of this
for
liars).
lot
of noise
is
in
actions
each
(and
ade-
cannot
be
the observation of actions),
revelation of
Just what revelations can be achieved without
a
mediator
section.
publicly heard by all.
equilibrium,
private
If
We assume (with some loss of generality if
ken
verifiable ex-post
is
be desirable to have some, but less than complete,
private information.
logic-
Accordingly, we suppress for now the role of incen-
that private
by assuming
is
it
player's
Therefore,
information
together with the information inherent
of course, reduplicate some or all
at
n
each
consists
in
> 2)
stage
of
his
of
a
communication
private
information
what has been announced (which may,
of the player's private information).
latter information we call
the publ ic
cation happens,
information changes (in
the public
that every message spo-
information at that stage.
a
This
As communi-
way that depends on the
true state t) until, just before moves in G are actually selected, each player
sees the final publ ic information together with his own private information.
14
We can represent the final public information as a posterior probability
distribution p on T.
Each player i's information is then the conditional dis-
tribution generated by restricting p to T,x ...xT. ,x {t.}
A
information
publ ic
structure
is
function
a
a:
x
T.,,
1
-*
x...x
T
A(a(T)),
.
where
o(t)(p) describes how likely the posterior p on T is to become the final public information, given that t
plicity that each o(t)
T
e
is
support
finite
has
the true state.
in
We will
A(T),
suppose for sim-
that o maps
so
into A.
T
(A(T)), where Ar(*) represents the set of distributions with finite support.
Write
over
That
T.
...,
{p,,
is,
p-,
for the
Pm]
public information.
as final
while if p
Thus, for each
is not one of the p.
A(T)
£
The mean of o is the distribution
=
via)
^(t)
I
UT
i,
o(t)(p.)
as
t
then o(t)(p)
vi(o)
for some
>
for all
=
varies
t
e
t
T,
e
T.
on T given by
o(t)(p) pdp
/
"
supports of o(t)
the
p^ are the distributions on T which can arise under
...,
,
union of
(4.1)
'
A(T)
N
=
I
I
o(t)
TT{t)
(p.)
ttT i=l
so that,
for any
t'
v(o)(t')
I
I
UT
i
-IT
^(t) o (t)(p.) p.(t').
=l
Every public information structure has mean
Theorem
3:
Proof:
Let m,
t
^
T,
z
N
=
(4.2)
p.
^
,
...,
mj,
final public information is
^
I
p(m.iT)
teT
^
"
it.
be all possible message histories.
leads to message-history m.
q(t|m.)
(4.3).
tt(t)
'
q(''
=
with probability p(m.|t).
|
Suppose that state
Then,
after m.
,
the
m.), where by Bayes' rule
p(m.|t)
"
^
TT(t).
(4.4)
15
Now
in
our previous notation
while o(t)(q('
...
p,
=
p(m.|t).
...,
M),
]x{a)
into our present notation gives us
|m. ))
are
p^
the distributions
q
(•im.)(i = l,
Hence translating the definition of
M
y(o)
=
(t)
I
^(t) p(m.|T) q(t|ni.)
I
teT i=l
^
(4.5)
^
M
q(t|m.)
I
'
i=l
p(m. |t)
I
it
(t)
rearranging
(4.6)
by (4.4)
(4.7)
teT
'
M
p(m.|t)
^(t)
"
I
=l
^
-n(t)
since
i
=
p(m.
Z
This proves Theorem 3.
1
1
)
=
1.
Next we show that \i{a)
=
is also
it
sufficient for
centralized announceabil ity.
Theorem
If via)
=
T ^ Aj:
Let o:
4j_
then
IT,
(A(T)) be any function.
mediator who knows
a
t
can
induce public information struc-
ture o by public announcement.
Proof:
As before,
let
weight in some o(t).
...,
|T|;
j
=
1,
p,
,
....
p^ be
The function
...,
2,
o
(Here
N.
the distributions
is thus
we
ignore the Cartesian product structure.)
1
distribution q(*|m.)
on T
T
given positive
described by o(t )(?•)>
simply
the elements
number
i
=
of
2,
1,
T
and
Suppose that our mediator announces
message m. with probability p^m-lt'') when the state
J
on
is
t
.
Then the posterior
J
is given by
w
i
P(^.i!t^)
teT
J
^(t^)
(4.8)
16
Simple algebra shows
this posterior distribution will
that
be p.
if
we
set
p(ni.|t^)
=
7T(t^'Mp-(t^)
J
remains to check that,
It
^(t) o(t)(p.)]
I
^sT
J
(4.9)
J'
with this specification,
Tr(T)
I
o
(t)
(p.)
•
p.(t')
J
teT
j
=
(a)(t')
y
(4.10)
J
and by assumption this is equal to
iT(t
This proves Theorem
).
T ^
Thus we have seen that every function o:
tor can publicly reveal
This means that
exactly as much or as little as is desirable,
only to the condition on the mean.
tt
can be
a
mediasubject
Without a mediator,
controlled revelation
independent of
T
harder:
is
Theorem
Suppose that
5j_
be elements of T.,
t^j
t.
=
i
z
1,
T.
public information
P(t') p(t")
Proof of Theorem
=
(in
because
Ti
t-
in
a
a
player
from
T,
1,
1
'
s
i,
let
t'.
.
,
,
i
j
?*
iV be
Let
.
t'.
,
First,
5:
is
is
t'.
t'.'
Then if the distribution p(t) can occur as
(4.11)
note that (4.11)
independent of t.,
TT(t,)
TT(t,)
i
?!
-"(t^)
is true of the prior distribution
j
,
tt
so that each
side of (4.11)
(tp)
).
preserved by any revelation by
a
...
a
rT(t
player.
Next,
Suppose
is
we
1
makes
point p of public information satisfying (4.11).
possible public posterior after I's revelation.
I
I
Suppose
For each
e
.
p(t') p(t").
=
revelation, beginning at
Let p' be
n.
t
decentralized revelation system, we have
the obvious notation)
show that (4.11)
is
...,
but not necessarily respectively.
a
4.
Ar(A(T)) with mean
realized through (randomized) central announcements.
p
see
To
1.
^
this, we notice that
I
=
p(m.it'')
Z
=
tp
no new
II
T,
II
=
t,.
Then since
t'
and
t_'
^^^
indistinguishable to
information on the likelihood ratio between them can emerge
revelation:
17
P'(t')
P'(t')
P(t')
P(t')
_
"
Likewise, since
P'(t")
1
(4.12)
cannot distinguish between t" and t"
P(t")
_
(4.13)
Since (4.11) can be rewritten as
p(t"
P(t'
equations
(4.14)
p('
(4.12)
and
(4.13)
assure
us
that,
information p, then it also holds for p'.
if
(4.11)
holds
for
the
public
This concludes the proof of Theorem
18
APPLICATION
5.
A gap in the proof of Theorem
mixed revelation
strategies,
information.
Conceivably
slightly that
t,
with mutual
=
then
x;
circumspection
Theorem
information.
2
until
that
have
final
as
point with p(x, x)
p(x,
x)
and Theorem
one
=
p(y, y)
5
0,
i
of
its
(e.g.)
for
they had mutually revealed
t
have,
possible
be
stochastically convey
1
hint
to
very
might respond cautiously, and they could proceed
information is "whether or not
to
which
for messages
might
ensures
5
1.
given above was that we did not allow for
1
i.e.,
it
PROOF OF THEOREM
1:
=
(x,
cannot
x)."
public
and p(x, y) >
p(x, y) p(y,
this
0,
p(y,
x)
guarantees that this cannot be.
happen.
appropriate
the
For
the
required
Thus the revelation system would
information
x)
> 0.
states,
at
least
one
Hence we would have
(4.15)
19
APPLICATION
6.
2:
BARGAINING:
Consider two players bargaining over an object.
the object is private information,
E
{I's value exceeds 2's) and F
=
If
player
whether
2
or
the object,
owns
event
the
not
The events
{2's value exceeds I's) are both possible.
natural
is
for the players to want to know
We
occurred.
has
E
and the values are independent.
=
it
Each player's value for
whether
ask
this
information
could conceivably be revealed by communication between them.
If
we write
for
u
value and
I's
either of two values (u,, u^;
v,
for
v
value,
2's
and each
can
take on
Vp), and if
,
u^ > v^ > U2 > v^
(4.16)
then we need to reveal whether or not the event
course,
might
revelation would tell
full
necessary
be
us whether
subsequent
for
incentive
=
F
or
(up,
not
reasons
F
v,
)
has occurred.
has occurred,
not
to
reveal
Of
but
(in
it
the
course of discovering whether there are gains from trade) just what the values
are.
Hence we want
system that reveals whether
a
without leaking further information if
With
for
his
a
mediator,
and
type,
such
then
a
players will bargain if "E"
"E"
or
F
has
occurred,
The mediator asks each player
or
"F".
If
everyone expects that
announced, and will walk away if "F"
is
but
has occurred.
E
system is simple.
announces
E
is,
the
then
truth-telling is an equilibrium.
Without
a
mediator,
this, apply Theorem
p(Up v^
Equation
(4.17)
5
the
to get,
p(u2, v^)
=
revelation cannot be achieved.
partial
see
for any public information system, p
p(up
v^)
p(u2, v^)
(4.17)
cannot be satisfied with p(u,,
strictly positive but p(up,
To
v,
)
zero.
Hence
v,),
p(u2,
v^)
and p(u,,
Vp)
any system of public communica-
tion that establishes whether or not there are gains from trade must also leak
20
further information.
It
is
not possible,
without
a
mediator, for the players
simply to find out whether or not there are gains from trade.
21
CONCLUSIONS
7.
Costless
communication
Outcomes
matters.
games
in
which would not be equilibria without communication.
two-person zero-sum games, but in general
become
equilibria
This does not happen in
important.
it can be
Since players
our equilibrium notion should reflect the
very often can talk to one another,
fact.
Players may be prepared to reveal
Mediators matter.
tion
to
a
trusted mediator
(who
pass
will
only
on
it
important
if
it
informa-
will
not
be
exploited against the player), when they would keep the information secret in
the absence of a mediator.
There has been very little work on communication
ing
the
set
the
set
of
Bayesian
equilibria.
algorithm
coordination
to
5
aspects)
gives
5
that
game
a
aspects"
enough conflict
has
communication
Theorem
2
is
a
would be exciting to know how to decompose
tion
necessary conditions,
but
will be complicated.
determine when
would also be very desirable.
It
Theorem
The interaction of incentives with the logical
restrictions analyzed in Theorem
its
Characteriz-
games.
of communication equilibria may be no easier than characterizing
they are far from sufficient.
An
in
and "conflict aspects,"
as
happen
cannot
first
a
step
general
in
(relative
to
in
equilibrium
this
direction.
game into "coordina-
hinted on page 12.
However,
I
have
made no progress on this.
Some communication
structures
are more
direction of work on this, see Farrell
focus
on
the
trivial
messages are ignored.
one.
In
my opinion,
munication, which
is
communication
(1985).
structure
plausible
than
others.
For
one
But there is no good reason to
in
which
all
payoff-irrelevant
That is just one among many, and often not
a
plausible
the literature should begin to take more account of com-
one of the salient features of the human world.
22
8.
1.
2.
REFERENCES
Farrell, J.,
"Communication in Games: Mechanism Design Without a Media-
tor," M.I.T.
Economics Working Paper 334,
Farrell,
"Credible Neologisms
J.,
in
1983.
Simple Communication Games," GTE
Laboratories Technical Note 85-407.5, 1985.
3.
Myerson,
"Incentive
R.,
Compatibility
and
Bayesian
Equilibrium:
An
Introduction," Mimeo, Northwestern University, 1983.
4.
von
Neumann,
J.,
and 0.
Morgenstern,
The
Theory of Games and Economic
Behavior, Princeton University Press, 1944.
5.
Spence,
A.
M.,
Market Signal ing. Harvard University Press, 1970.
S85k
-^'^
-^^^^o
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