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DEWEV
HB31
.M415
'0^
working paper
department
of
economics
RA TIONAL DEBA TE AND
ONE-DIMENSIONAL CONFLICT
David Spector
No. 99-09
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
RATIONAL DEBATE AND
ONE-DIMENSIONAL CONFLICT
David Spector
No. 99-09
March 1999
,
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 02142
Rational debate and one-dimensional conflict*
David Spector (MIT)
Revised version: February 1999
"It's
like
G.H. Hardy's old crack:
says he believes in God, that's
he doesn't, one can take
C.P.
This paper owes
the
Archbishop of Canterbury
way of business,
but
if
he says
he means what he says."
Snow, The Affair
to Thomas Piketty, who took an active part in an earlier version. I am also grateful
Michael Kremer, Augustin Landier, and two anonymous referees for their helpful
much
to Abhijit Banerjee,
it
all
If the
comments. All errors are mine.
Abstract
This
paper
studies
repeated
communication
collective decision in a large population.
regarding
When
a
multidimensional
preferences coincide but beliefs
it is shown, under some
communication causes the disagreement between
about the consequences of the various decisions diverge,
specific assumptions, that public
beliefs
either
to
Multidimensional
vanish
or
become
to
disagreement
allows
one-dimensional
indeed
communication, including some which are orthogonal
agents can
communicate
credibly.
The
for
many
the
limit.
directions
to the conflict,
of
along which
possible convergence towards a one-
may be related
many countries.
dimensional conflict where no further communication takes place
to the empirically
at
observed geometry of the
political conflict in
Introduction
1.
This paper develops a model of repeated communication between agents facing
We
a collective decision.
consider a situation where agents' interests coincide, while
of the collective decision diverge, and
their beliefs about the uncertain payoffs
we
analyze the dynamics of beliefs induced by strategic communication regarding this
The
uncertainty.
main
result
multidimensional forever, as beliefs either converge
some one-dimensional disagreement.
We
disagreement
the
that
states
all
towards the
model communication
randomly selected agent observes a signal about the
and
publicly,
agent
is
it
which causes
randomly selected
to
make
all
them
state
game^:
of the world
agents to update their beliefs. Then, another
the collective decision.
allows to ascribe the possible persistence of
failure, as the infinite
towards
truth, or
as a cheap-talk
in every period, a
talks about
remain
cannot
The information
some disagreement
sequence of signals would be enough,
to a
known
if
structure
communication
to all agents, to let
learn the truth.
The
intuition of the
main
simple geometric expression: a one-
result has a
dimensional conflict can be a steady-state, because there are only two "directions" of
communication, so that a "left-wing" agent always wants to report a "left-wing" signal,
and conversely, making communication impossible. The same logic implies
long as the disagreement
that, as
multidimensional, agents manage to communicate credibly
is
along a more "neutral" direction, orthogonal to their conflict, and beliefs keep changing.
These
two
arguments
together
imply
starting
that,
from
multidimensional
a
disagreement, repeated communication causes beliefs to change and converge toward a
steady-state, characterized
However
by
full
agreement or one-dimensional disagreement.
general the intuition seems,
its
formal modeling requires very specific
assumptions: agents are infinitely impatient, communication
is
bound
to
be public, there
a continuum of agents, the payoffs of the collective decision are not observed, and the
model only considers a particular
assumptions,
we show
We
is
common knowledge
function and signal structure. Under these
measure
set
of initial
beliefs,
convergence
to a
a positive probability event.
believe this result
theoretical viewpoint,
utility
that for a positive
one-dimensional conflict
in
is
we know
may be
of interest for two main reasons.
relatively little about
communication
of diverging priors, of which this model
is
a case.
in
From
a
games with
More important
our view, our results might be related to an empirical fact generally considered a
See Crawford and Sobel (1982).
puzzle; contrary to the predictions of the
most natural economic reasoning^,
conflicts tend be organized along a one-dimensional axis^.
model explains
this,
we
the idea, explored in a
Without claiming
political
that
our
believe that this coincidence pleads in favor of taking seriously
growing
literature"*, that political
beliefs as about interests, so that studying
much about
conflicts are as
communication about collective decisions
is
probably an important research direction for formal political science.
The paper
is
organized as follows: section 2 introduces what
simplest possible model allowing us to
states;
section 4 describes
make our
point; section 3 analyzes
convergence and
its
we view
stability
its
as the
steady-
and section 5
properties,
concludes.
2.
The model
2.1.
Players and preferences
We
consider a continuum of agents partitioned into two groups
into repeated
communication over a discrete
collective decision
Mj = (x,
,
^y, )
e 91
infinite
contingent utility function U(M,s), depending on the decision
state
of the world
s.
M
^
= {x^,y^)
of the world
is
2),
and engaged
all
M
have the same
t
a
state-
and some uncertain
U is given by
U{M,s) =
where
and
horizon t=0,l,... In every period
be taken. Agents
is to
(1
is
unknown
-\M-M,f
the optimal decision
to the agents,
when
the state of the world
and remains the same
is s.
The
state
in all periods.
In addition, agents have an infinite rate of time preference, so that in period
t
they
maximize the expectation of U(Mt).
Along each of the two dimensions, there
each be equal
to either
or
1.
are
two possible
ideal decisions: Xs
and ys can
There are therefore four possible optimal decisions
^
See for example Arrow (1963).
3
Poole and Rosenthal (1991), Snyder (1996).
'*
For example, Banerjee and Somanathan (1998), Piketty (1995).
2
(0,0),
and
(0,1), (1,0)
we
decision,
(1,1).
will identify
optimal decision
A
belief
them and write
is
relevant only through the optimal
for the state of the
(i,j)
world where the
is (i,j).
about the
fx
Since the state of the world
state
of the world
an element (/iQo.MiO'A'oi'/^ll) °f
is
•^he
three-
dimensional simplex A.
We
assume
denoted
that all agents belonging to
T
The
/x'^.
fi
.
group
have identical beliefs in period zero,
i
difference between beliefs across groups
is
the only heterogeneity
between agents.
We
characterize the indirect preferences induced
first
by some belief
|i.
The
quadratic
state-contingent utility function yields the following simple relationship between beliefs
and induced preferences: indifference curves corresponding
circles
around the most preferred decision, which
is
to indirect preferences are
simply the expected most preferred
decision.
Lemma
An
1:
are represented
agent's indirect preferences given a belief
by the
|i
about the state of the world
indirect utility function
ViM) = -\\M-M*(n)f
with M*(/i)
= £(Mj/z) = (/iio+/iii,//oi+Mii)
Proof: Appendix.
Remark:
as
A belief
can be any element of the three-dimensional simplex A, but as far
induced preferences are concerned, only M*(|i), which varies in a two-
the
dimensional
to that
|J.
set, matters.
The information which
conveyed by M*(fx)
Xs and about ys: infinitely
additional
information
preferences,
the
way
it is
is
is
new
conveyed by the belief p.
related to the correlation
many
beliefs
irrelevant
\i
for
information.
between the agent's
in addition
beliefs about
correspond the same M*(|i). Although
the
relevant for the analysis of the
agents update
is
this
induced preferences given quadratic
communication game, since
One might argue
that since
we view
it
affects
the various
dimensions of the collective choice as "independent" (as mentioned in the introduction),
3
we
should
restrict
our attention
to beliefs that are
such that fi{(x^,y^)-(i,j))=fi{x^-i)/j.{y^
=
independent across dimensions, that
This
j).
is
wrong, because with the
is
dynamics generated by our communication game (described below),
beliefs
would
generally cease to be independent after communication took place.
are going to write
Q
most preferred decisions belong
to
we
Notation: Throughout the paper
Lemma
1
implies that
all
Q.
The communication game
2.2.
The following sequence of events
A signal
(i)
for the square [0,l]x[0,l].
is
is
infinitely repeated:
drawn randomly according
to a probability distribution
depending on the
true state of the world,
(ii)
A randomly
is
selected agent observes the signal and everyone learns "what the signal
about" (see section 2.3 below),
(iii)
The agent who observed
(iv)
A randomly selected agent belonging to the group other than the speaker's takes the
the signal speaks,
collective decision.
To keep
the repeated
game simple enough, we assume
that in every
1
(resp.
group 2) according
to the
(resp.
randomly chosen
in
uniform distribution. The randomization
is
even) period, prior to decision-making, an agent (the speaker)
group
odd
is
independent across periods. The speaker then observes a signal at providing some
information about the state of the world, updates his beliefs, and can attempt to
communicate
to other agents
receiver) is then
decision.
to
We
by sending a message.
randomly chosen
assume
that
to
r(t)
agent in the other group (the
be a dictator in period
communication
is
public:
everybody, and not only to the agents of his
and
An
when an
own
group.
t
and take the collective
agent talks, he has to talk
We
are going to write
respectively for the speaker's and the receiver's group in period
(s(t),r(t)) is (1,2) if
t
is
t,
s(t)
so that
odd, and (2,1) otherwise.
The assumptions of a continuum of agents and of public communication imply
that,
except for the finite number of agents
received the
who
directly observed a signal, all the other
same information, coming exclusively from publicly
sent messages.
Therefore in period
probability
1,
t
almost
all
neither the agent
observed a signal
agents in group
who
i
have the same belief
observes a signal in period
t
a
receiver's decision following a
,
the speaker chooses
mt
B(|j.,a)
from the belief
message mt sent by the speaker
Given the form of indirect preferences and the
is 5(jU'^^'^' ,CT, )
starting
)i,
.
With
denoting
that the
M *(B(/i'^^'^' ,tni)).
is
fact that the speaker's belief
to
ji"
nor the dictator directly
an earlier period. This implies, with the expression
in
the belief held after updating the information
speaks
(i=l,2)
when he
minimize
^ A\ x^ *rr>r,i'^Wt
\M*(B(fl'"",(7,))-M*(B(fi'"",m,
j(')'
In other words, the speaker sends the
new most preferred
his
We
assume
expected
utility.
decision and the future dictator's.
that although agents act,
utility
message which minimizes the distance between
whenever they can,
in order to
maximize the
derived from the collective decision, they do not observe their
Therefore, agents learn nothing on their
own and
by the messages they hear. This strong assumption
on learning imposed by communication.
If
is
their beliefs are affected
needed
if
we
are to focus
on
own
only
limits
agents could learn from their utility level,
then they would learn the true state of the world at once: except for a zero-measure set
of decisions, the four possible states of the world yield four different utility levels, so
observing
2.3.
The
utility
would amount
to observing the state
of the world^.
signal structure
To make
the
too informative.
On
problem
interesting, signals
the one hand,
we
must be neither too uninformative, nor
are interested in the extent to
which
strategic
communication may leave room for some disagreement. Therefore, a desirable feature
of the model should be that making the infinite sequence of signals public would cause
all
agents to learn the true state of the world (under the mild assumption that initial
beliefs assign a
non-zero probability to the
learn the truth results
strategic
truth): this will
from a communication
communication non
trivial, signals
failure.
A similar assumption
is
made by Hart
(1985).
5
the other hand, to
who
to
make
should not be completely informative:
they were, then given identical preferences, the agent
^
On
imply that any failure
if
observed the signal would
.
simply report
We
assumed
make some very
to
about Xs
and would be believed.
it,
specific assumptions about the signal
be rich enough to allow not only
only
or
only)
ys
but
also
It
is
for pure signals (providing information
"mixed"
for
structure.
signals
providing
information
simultaneously along both dimensions, with various "ratios of informativeness" along
both directions.
We
signal) along the
two dimensions
assume
that this relative informativeness (the "direction" of the
is
drawn
but that this piece of information alone
precisely, the signal
First,
tells
g
[0,
k)
drawn
is
at
world,
with
A<
[O,
—
z=
.
A]
is
The
also
drawn
direction
at
More
steps.
random, from a probability distribution
independent of the state of the world with
"signal intensity" a e
to all agents,
nothing about the state of the world.
can be decomposed in two
direction
a
random and becomes known
at
strictly positive
density everywhere.
random independently of the
state
and the signal intensity become
knowledge, and they obviously provide no information about the
state
A
of the
common
of world.
Then, the agent randomly selected to be the speaker observe a signal which can take two
values Safi or Sa^e+nstate
of the world
We
write fijiS^fi) for the probabihty of observing
)
=
I
if the
is (i,j).
We assume that the functions
fij iSa,e
S^q
+ a(A- COS0 + pj
f^ have the following form:
sin 6)
with^o=-l.A=lThis implies that /^ (S^q )
The assumption
A<
+ /^ (S^^e+j^ ) =
—z= implies
that
1
fij(S^Q)>0 foralli,j, a and
6.
2v2
This functional form has the following simple property:
weights to
if |i is
fl
all
four states of the world (so that
the belief assigning equal
O
M*{^.)= —,— )then
V
2 2
M*(B{iJ.,Sae)) =
1
^2
so that
the
1
^
2
J
\- + acose,- + asm9
change of the agent's most preferred decision, and a
the direction of the
is
magnitude of this change.
This example allows to
signal direction
(mod
direction 9
more
precise.
information
provides
fio{SaQ) =
make
It
only about
— + aPi=fii(SaQ).
,
x^
and not
all
results: as will
assumed above
particular,
occurs and
it
(mod
about
jt)
y^,
is
is
the signal
is
is
since
for
i=0,l,
then the signal provides
more and more
to the vertical.
one of many which would yield the same
bounded and
common knowledge
may seem more
is
that the signal
appear throughout the proofs of the various propositions, what
only that informativeness
direction
all
other directions are mixed, and are
informative about y^ (relative to x^ ) the closer
signal structure
at
Similarly if Q=k/2
and
of the
should be interpreted in the following way: the
example spelled out above, 6=0 (mod k) means
information only about y^
The
common knowledge
the assumption of
some commonly known information about "what
it) is
about". For the particular
is
is
that in infinitely
with a
many
is
needed
periods, each
strictly positive probability. In
natural to consider a signal structure
which would allow
for all directions of updating in the three-dimensional simplex, since there are four
states
of the world. The results would
and the updating direction
form above
2.4.
is
just
common
is
chosen for
still
hold
-
as long as informativeness is
knowledge. Beyond
bounded
that, the specific functional
its clarity.
Role of the various assumptions
2.4.1.
Continuum of agents
This assumption
observe infinitely
the truth
2.4.2.
many
is
necessary: with finitely
many
agents, each of
signals directly with probability one, and
even without any communication.
Public communication
them would
would therefore
learn
If
communication were not bound
communicate the signal
at least to his
to the truth with probability
1.
The
learn infinitely
fact that this paper
about the political debate (involving
2.4.3. Infinite
group, since there
many
signals,
and converge
communication
may be more
relevant to think
agents and characterized by a "public"
situations.
is
made
for simplicity.
Given
necessary for the results. At the very
the complexity of
we do
forward-looking behavior in this stochastic context,
it is
many
no prior divergence
impatience
This assumption
extent
is
necessity of this assumption, as well as that of the
continuum assumption, highlights the
character) than about other
own
would
within a group. Therefore each group
be public, then the speaker would
to
not really
least, the results
modeling
know
to
what
should carry over, by
continuity, to the case of very impatient agents.
2.4.4.
Knowledge of the signal direction
This assumption
is
made
two signals along each direction
of the communication
games
for simplicity. Together with the fact that there are only
(for a given intensity),
it
either completely
are
implies that the pure equilibria
uninformative or completely
informative, which facilitates the analysis.
3.
Steady states
As we
stressed in section 2.2 above, the existence of a
implies that in the beginning of any period
beliefs
fi"
starting
from
,
and receivers
resulting
/i'°.
is
from updating
all
t,
almost
all
continuum of agents
agents in group
i
have the same
the messages sent between periods
This, and the assumption that the
and
t
random determination of speakers
independent across periods, ensures that in the beginning of period t+1,
the speaker's and the receiver's respective beliefs are
/i^^'''
and
/i''*'"
with probability
one. This allows us to keep track of the entire dynamics by using (/^'',/z^')
as an
exhaustive state variable.
As always
arises,
since
in
there
pure communication games, the question of equilibrium selection
always
exists,
alongside any other equilibrium,
8
a
"babbling"
equilibrium where no communication takes place. However, since there are only two
signals (once the uncertainty about a and
belief distribution
has been resolved), there exists for each
and signal direction either the babbling equilibrium only, or the
babbling equilibrium and the "communicative" equilibrium where the speaker reveals
Whenever such
the signal^.
it.
a communicative equilibrium exists,
This defines a stochastic process
We
condition (/z'°,^^°).
describe
its
dynamics
first
(fi" ,fi^'),^Q starting
characterize
such
i.e.
probability
from some
arbitrary initial
steady states, which will allow us to
its
that
if
(fi" ,fi^')
(/i\/i^) such that no further communication
=
(ii\fi^)
then
(^'"',/i"^')
=
is
(//',/i')with
1.
Notations: For
i,j
in
{0,1 }2 let
denote the vertex of
Sjj
probability distribution assigning probability
other three events.
the edges of A.
two
are going to select
in Section 4.
A steady-state is a pair of beliefs
possible,
we
We
corresponding to the
and zero
to the event {s=(i,j)},
1
are going to call the segments [5ij,5i'j'] (where
They correspond
states of the
A
world
(i,j)
and
(/,/') ?i
to beliefs assigning a positive probability
to the
(j,j'))
only to the
There are six such edges, since there are four
(i',j')-
vertices.
We
first
show why
many
there are
dimensional conflict of beliefs, that
is,
steady-states characterized
such that
all beliefs
by a one-
belong to the same edge of
A.
The proposition below
states that there exist steady state belief distributions
with
support in the interior of any given edge.
Proposition
{fl\^^)G
1:
Consider an edge
l} which has
L
of
A
a non-empty interior
There exists a set of steady states
in L?,
and
therefore,
a
strictly positive
measure.
Proof:
^
We omit
mixed equilibrium, since one can easily show that for any belief distribution, they occur
zero-measure
set
of signal directions and intensities.
9
for a
^
If L=[5ij,5i'j'] (with (/,/')
number p
(j,j')) then each belief belonging to
by a
single
(i,j),
the probability assigned to (i'J') being then equal to 1-p.
L
can be summarized
denoting the probability assigned to the state of the world
in [0,1]
We consider a belief distribution such that the respective beliefs of groups and 2 are p
and p'. We assume that there exists a communicative equilibrium when the signal
1
direction
is 9,
the intensity
is
a and the speaker belongs to group
1
.
Given the
signal
structure, for all signal directions 9'
< r with r =
1
+ 2aV2
"
1-2aV2
Therefore,
if in
equilibrium the speaker reports the signal, the posterior beliefs of
members of groups
and 2 belong, respectively,
1
P'r
pr
and
p + (\-p)r pr + (i-p)
If
pr
pT + l-p
if
p'Hl-p')r
than the receiver would after observing any signal.
(i,j)
the receiver expects the speaker to report the signal truthfully, then the best
response for the speaker
receiver to
p'Hi-p')r'pT+i-p-
then whatever the signal was, the speaker's posterior belief
c
assigns a lower probability to
Therefore,
(i,j),
that
is,
to report the signal increasing the probability assigned
is
S^fi if
^'j''^"'^^
>
frf(^a,e)
Since this best response
report of the signal
By
is
group 2
is
pT + \-p
This inequality
by the
s^fi+ji otherwise.
fi'j'^^ajd+n)
independent of which signal the speaker observed, truthful
there exists
<
if
fij^^^fi+-^^
not an equilibrium and there
same argument,
the
to the intervals
is
no communicative equilibrium.
no communicative equilibrium with the speaker
in
.
p'+(l-p')r
is satisfied
by a
positive
measure subset of L-^, which yields the
result.
q.e.d.
Proposition
disagreement
is
1
is
very intuitive:
if all beliefs
one-dimensional. Therefore any
10
new
are
on the same edge of A,
the
information (signals or messages)
changes beliefs in a way
one dimensional
signals
is
makes new most preferred
that
belong to the same
still
are initially far apart, then, since the informativeness of
set. If beliefs
limited,
policies
extreme beliefs remain extreme whatever the signals (and, therefore,
by prior
the messages) they observe: posterior beliefs are determined mostly
rather than
by new information. This implies
the receiver a "right-winger", the speaker
that if the
speaker
is
beliefs
a "left-winger" and
would always prefer the receiver
to believe he
observed a "left-wing" signal, irrespective of the signal he truly observed. This means
he cannot be credible, and agents are stuck
communication can take
further
The
in this polarized belief distribution
place.
distance between priors in our model
preferences, and Proposition
greater the distance of preferences, the lesser
Notice that these steady
between x and
correlation
for i=0 or
then
1,
distributions
edge containing them
may
may
or
not display
all is [5oi,5ii] (resp. [5io,6ii])
agree that ys=i (resp. Xs=i) so that the disagreement
about the dimension x (resp. y) and beliefs are not correlated. But
[5oo>Sll] (resp. [5oi,5io]), there
and
between
the equivalent of a distance
communication takes place^.
state beliefs
y: if the
all beliefs
is
the equivalent, in our context, of results stating that the
is
1
where no
is
if
a positive (resp. negative) correlation
is
only
the edge
is
between xs
ys.
What
by showing
steady-state.
are the other steady-states? Proposition 2
that a belief distribution such that
It is
the
most important
below provides a
no belief belongs to an edge of A
result of the paper,
argument mentioned in the introduction:
partial
and
it
relies
answer
is
not a
on the orthogonal
as long as agents assign a positive probability
to at least three states of the world, there exists a signal direction
moving the
receiver's
induced most preferred decision "orthogonally" to the disagreement, and information
can
be
credibly
communication
transmitted
along
this
direction.
Given
will take place in the future with probability one,
distribution is not a steady-state. In other words. Proposition
converse of the result in Proposition
1
:
we do
infinite
all
horizon,
and such a belief
2 establishes a
the steady states of Proposition
only ones (there also exist steady states such that
edges of A but
the
1
partial
are almost the
beliefs are located
on
different
not mention them because, as will appear below, the probability of
converging toward them
is
zero
Before proving Proposition
2,
if
tne initial beliefs are not degenerate).
we need
to state a
^ For example, Crawford and Sobel (1982).
11
few general
results about
how
a signal
most preferred decision of an agent whose behef does not belong
affects the
Lemma
of A.
to an
edge
2 below states a few simple properties about the change of the most
preferred decision induced by a signal.
formal statement and
Its
its
proof are in the
appendix.
Lemma 2:
If the initial belief assigns a positive probability to at least three states
of the world,
then
(i)
any signal with a positive
(ii) the direction
intensity induces the
most preferred decision
to change,
of this change depends only on the signal direction 6 and not on the
signal intensity
a. It
moves clockwise when 6 does, which implies
that there exists a
one-to-one correspondence between signal directions and directions of the induced
change of the most preferred decision.
(Hi) as the signal intensity converges toward zero, the changes in the most preferred
same magnitude.
decision corresponding to the two possible signals tend to have the
Remark:
Part
(iii)
reflects
the fact that as a converges to zero, signals
become
uninformative and therefore equally likely a priori. The agent's expectation of the
change of
change
his
most preferred decision being zero
in belief), this implies that the
(it is
a linear function of his expected
two changes have the same magnitude and cancel
each other in expectation.
The next lemma addresses a more
summarized
in the introduction relies
beliefs in similar
ways, so that
if
on the idea
agent
1,
is
not true in general.
different beliefs
may
As
interpret the
telling the truth
argument
changes different
without any information,
the following
same
orthogonality
that a given signal
between saying black or white, he should then prefer
This
the
point:
difficult
is
indifferent
when he knows
example shows, agents
it.
starting with
signal in very different ways. If agent I's belief
assigns a positive probability only to the states of the world (0,0) and (0,1), and agent
2's only to (0,0)
and
preferred decision
"downward",
decision
is
- — while
,
(1,1), then
it
observing a signal
that
is,
moves agent
S
^^
moves
agent's
the direction of the change of his
2's
most
most preferred
most preferred decision upward
12
1
to the right
.
r
K
(the direction of this
change
—
is
since for
),
a >
all
/j
,
S
j
V
Lemma
work
3
below qualifies
remark, and
this
proof of Proposition
in the
2.
>fQO
a,
6 J
allow the orthogonal argument to
this will
formal statement and
Its
—
jj^
a,
its
proof are in the
appendix.
Lemma 3:
given a direction
there exists a signal direction
(p,
decision of one of the groups along the direction
and
(p,
moving the most preferred
that of the other
group along a
K
direction forming
an angle with
Lemma
Notice that
be possible to
(p
smaller than
—
3 introduces a possible asymmetry between agents: given
move group
along the direction
1
forming an acute angle with 9, while the converse
cp, it
may
and group 2 along a direction
cp,
may be
false, as the
above example
shows.
We can now state the central result of the paper.
Proposition 2: Assume
probability to
to
{jl
more than
,^
)
is
such that
jl
and
three states of the world (that
an edge of A). Then (ll\/l
)
is
is,
assign a strictly positive
jl
none of these
beliefs belongs
not a steady-state.
Sketch of the proof:
The exact proof
Figure
1.
is
in the appendix.
Let us write
M^
for
M
The argument summarized here
*{iLi')
(the
M*{B(n' ,Sa^e)) ^nd
Mf
illustrated
most preferred decision of group
Lemma
before any communication takes place).
is
3 implies that, writing
for M*(B(/i',5q0+;j-))
,
i
on
agents
M^
for
there exists a signal direction
6
such that
•
M2M2
•
Mj and M2
•
(for
example)
is
are located
orthogonal to
on the same
M1M2
side of the Af j
Lemma
2 implies in addition that the distances
equal
a
if
is
small.
13
M2
•
MiM^
and
MiMf
are roughly
One can check
•
•
that this implies that
M1M2 <MxM2
M{ M2 < Ml M2
Therefore a group
and
agent wants to truthfully report
1
expects to be believed: his most preferred decision
is
of the indirect
function stated in
utility
going
M2
the receiver in group 2
to
be
if
Mj
is
he reports
Lemma
1
S^q
he observed S^ q and
if
and the decision chosen by
S^q
,
MJ
above implies that
inequality implies that a
group
this is
1
done by reporting S^q
intensity
is
small and the sender
neighborhood around
that with probability
beliefs are
9,
1,
.
receiver's.
is
in
group
so they occur with a
some communication
going to change. Therefore
1.
if
continuity there exist
will take place in
first
he observed
the signal direction
strictly positive probability.
(/x' ,fl^) is
14
By
when
The
Similarly, the second
agent wants to truthfully report S^ q^^^
Therefore there exists a communicative equilibrium
The form
implies that the sender wants to
minimize the distance between his most preferred decision and the
inequality
otherwise.
some
is 6,
many
in a
This means
future period,
not a steady-state. Q.e.d.
the
and
(1,1)
(0,1)
(1.0)
(0,0)
Figure 1
4.
Dynamics,
stability
For any
initial
and convergence
condition
bounded martingale,
just as
member
belief
of group
(//'°,/i^°),
any bayesian learning process. That
time
t,
convergence theorem^ applies, and as
t
i's
the stochastic process
//'^
at
Eyfi"'^^ /x"
)
=
fi"
15
..Its
is
a
conditional on a
Therefore the martingale
goes to infinity {}l'
°See for example Neveu 0975).
is,
.h
(^l^',^'^')
,jl
')
converges toward
'
some
steady-state
{jl
The
on
with probability
)
we examine
states in Section 3,
beliefs
,ii
in this section
that set varies as a function
description of the belief
1^.
of
Having characterized the
how
set
of steady
the probabihty distribution of limit
initial beliefs.
dynamics
starts
with the following restriction about
possible limits:
Proposition 3: Assume that
2
1
distribution
(I) either
(II)
or
Proof:
jl
A
{fl
,jl
is
)
fl
and
and
/i
fl
and
assign probability 1 to the true state of the world
jl
both belong to some edge [5^,5^] of A containing the true
general result about bayesian learning
does as welll^.
>
.
If
limit belief
such that
strictly positive probability to the true state
jUj^
have full support. Then the
fl
case
all
agents in both groups
(all
belief.
that if a belief assigns initially a
of the world, then with probability
does not hold, there exists a
(i)
Since almost
is
i.e.
s,
state
of the world
who
except those
a signal) received the same information between periods
and
w^ s
its
1
limit
such that
directly observed
Bayes' rule implies
°°,
that
li\,li]^
fl
}°/i J
_
_
jJ-wfij^
t
fl }^fi1
2
Therefore ji^^,>0
,
.
Pr(messages up to t|w)
.
^~
Pr(messagesuptot|5)
We know
12
interior of A, for {fi ,fL
)
from Proposition 2
would not be a
2
I
that neither
nor
/x
/x
12
can be
steady-state then. Therefore both
fi
in the
and
fi
belong to [5y,5n,]. Q.e.d.
"strictly speaking, the martingale
somewhere with
only
if
probability
1.
The
convergence theorem only implies
limit has to
be a steady-state
the transition correspondence has adequate continuity properties,
most informative equilibrium
at
that the
(in the sense
system will converge
defined in section 3 above)
which
is
the case
if
we
select the
each stage (selecting the truth-telling equilibrium whenever
defines a continuous transition correspondence).
^0Aghionetal.(1991).
16
it
exists
.
The second case corresponds
Proposition
only steady
1,
convergence toward the steady
to
states described in
characterized by a one-dimensional conflict. Although these are not the
states.
Proposition 3 rules out convergence to any other (for example belief
distributions such that beliefs belong to different edges).
There remains to prove
one-dimensional conflict indeed happens
strictly positive probability.
Although we do not
cases: the one
state
initial belief distribution is
and the one where
(Proposition 4),
with a
we
dynamics,
fully describe the
where the
at the limit
that
focus below on the following two
close to a one-dimensional steadyare
beliefs
initial
close
to
each other
(Proposition 5).
Proposition 4 shows that if the
belief distribution
initial
steady-state, then there is positive probability of
that this probability
converges
to
1
is
close to a one-dimensional
converging toward a nearby one, and
as the initial belief distribution converges to such a
steady-state. This result is important:
it
means
that the steady-states of Proposition
locally stable (as long as they assign a positive probability to the truth): if
state is locally perturbed, the
probability. This
is
equal to
1
is
a
system converges back
weak notion of stability,
for small perturbations
since
to a similar
we do
- we only prove
that
1
are
such a steady-
one with positive
not prove that this probability
it
converges to
1
as the size of
the perturbation converges toward zero. This proposition implies that the steady-states
of Proposition
1
are meaningful: convergence
toward a one-dimensional conflict occurs
with a positive probability for a positive-measure set of
Proposition 4: Consider an edge
{fl*^,H*^)
1.
in the interior
L
of
A
containing the true belief 5^, a steady state
of L^ and a neighborhood
,
There exists a neighborhood
V
initial beliefs.
of (fl*^ ,fl*^)
then with a strictly positive probability (fl
W of (fl*^ ,fl*^)
in
It
\t
,/l
A^ such
in
l}
that if (fl^^ ,il^^)
eV
)f>o converges toward a limit
belonging to W.
2.
This probability converges to I as
(/z
,fl
is
in the
Sketch of the proof: The exact proof
(see figure 2):
"
y^
=
1"
(the
assume
proof
)
converge toward (fl* ,fl*
Appendix.
Its
main
).
lines are as follows
that steady state beliefs assign a zero probability to the event
is
dimensions). If (/i* ,/i*
less
)
is
intuitive
if
steady state beliefs are correlated across
a steady state corresponding to most preferred decisions
17
(*
Mj
*
,
M2
N
I,
Mi,M2j
then one can check that beliefs inducing most preferred decisions close to
must be close
to
(fi* ,ji*
).
One shows
such beliefs are
that
first
characterized by "almost vertical" communication (Result 2 of the proof): horizontal
1
communication
that as long as
is
infeasible at (/i* ,/i*
2
and
),
this
implies by a continuity argument
most preferred decisions are respectively
in the interior
A^BiCiDy and A2B2C2D2, communication can cause them
directions very close to the vertical.
initial beliefs
this probability
move
to
only along
standard property of martingales implies that
assign a very small probability to the event
enough probability
initially the
A
of the squares
"
y^
=
remains small forever (Result
most preferred decisions are
in the small squares
1"
4).
around
if
then with a large
,
This means that
Mj
and
M2
,
if
then
with a large enough probability, the most preferred decisions cannot leave the squares
Ai^iCjDi and A2B2C2D2 by moving across
the lines
cannot either cross the vertical lines Aj fij or A2 B2
:
fijCj
and 52C2. But they
the fact that
communication
almost along the vertical direction implies indeed that a given
movement corresponds
total vertical oscillation
to a total vertical "oscillation"
total
probability assigned to the event
y^
=
i"
is
horizontal
of comparable magnitude, but the
must remain very small (with a large probability)
"
is
if the initial
small, and in particular if initially the most
preferred decisions are in the small squares around
Mj
and
M2
(this is
shown
in
Result 4, building on the theorem about the quadratic variation of positive martingales).
This implies that
if initially
the
most preferred decisions are
in these small squares, then
with a positive probability, the most preferred decisions never leave the "large" squares
AiB^CiDi and A2B2C2D2- Belief convergence and Proposition 2 then imply
limit distribution
I'sbelief lying
that the
must be characterized by a one-dimensional disagreement, with group
between Aj and Dj and group
18
2's belief
between A2 and D2-
(1,1)
(0,1)
(0,0)
Ai
(1,0)
Figure 2
19
—
The next
.
then convergence toward the truth - and total agreement
Proposition
enough
enough
result qualifies Proposition 4: if initial beliefs are close
5: If the ratios
to 1, then all beliefs
^^
.„
-
to
each other,
occurs with probability
(for all pairs (p,q) of states
1
of the world) are close
converge towards the truth with probability one.
Proof: Appendix
This result
stronger than a continuity property around identical initial beliefs:
is
one could have expected the probability of convergence
instead of being equal to
The proof relies on
one around a given
to the truth to
with identical
initial distribution
the following idea: since in period
t
almost
converge to one,
all
agents in both groups
observe the same information (the sequence of messages up to period
—
-
implies that
be close to
This
is
1
at
y-
is
independent of
t,
so that
any steady-state, meaning
if it is
close to
1
beliefs.
t),
Bayes' rule
initially, it
that the limit beliefs are close to
must
also
each other.
impossible, as one can show, with arguments similar to those of the proof of
Proposition
1,
that one-dimensional conflict requires
To summarize, we have shown
some
distance between beliefs.
in this section that
convergence toward one-
dimensional conflict occurs with a positive probability for a positive measure-set
(though not the entire
5.
set)
of
initial belief distributions.
Concluding remarks
One may wonder how
sensitive the results are to the various assumptions.
explained in Section 2.4, they would
number of agents, or
if
fail
for obvious reasons if there
an agent could speak to his
own group
only.
were a
As
finite
We have a less clear
idea about the possibility of relaxing the assumption of infinite impatience, as the
forward-looking behavior this would induce makes the analysis of the communication
game
difficult (at the
very
least,
a continuity argument should allow our results to carry
over to the case of very impatient agents, with the set of possible steady-states
converging toward the one found here as the rate of time preference goes to
20
infinity).
we do
Similarly,
know
not
The reason of
extended.
to
which
class of utility
uncertainty
this
geometry of bayesian updating
in several
functions our results can be
the very counterintuitive nature of the
is
dimensions, exemplified by the fact that the
changes of the most preferred decisions of two agents having observed the same signal
can go
in
very different directions (Section
These problems may weaken the
3).
orthogonality argument. Extending the results to
particularly
welcome:
would make
it
more meaningful. The
the link with the evidence
here
difficulty,
again,
establishing that updating directions can be
be possible,
at least, to
extend our
close to
more than two groups would be
results,
made
is
small enough, or
if all initial beliefs
-
Political Science
lemma
with generalizing
close enough to each other.
by continuity,
two main groups
belief distribution
lies
from
for
to the case
example
if all
It
where the
3
-
should
initial
groups except two are
can be partitioned into two sets of neighboring
beliefs.
Generalizing
the
model
to
more than two dimensions seems
straightforward: after extending the signal structure in a natural way,
it
relatively
should be
sufficient to consider a plane containing both beliefs in the corresponding simplex,
to apply Proposition
in that plane.
and
2 to that plane by restricting the attention to signals leaving beliefs
This would show that steady-state beliefs must belong to a
common
edge
of the simplex.
Further developments in bayesian theory might allow to assess
argument
is,
and
how
relevant
it
may be
in terms of political
the subject of future research.
21
how
general our
economy. This should be
.
Appendix
Proof of Lemma
1:
Given a belief n about the
M,
equal to (writing
is
V(M,fi) =
state of the world, the expected utility
M*(n)
induced by the decision
for E[M^\fij)
= -2/^J MM*{jl)+M*{jl)M^
-J^nAMM,
J
V
c
2\
+ 2MM*{ii).M*(fl)M^ + M*{pL)M^
s
2
X/^. M*(fl)M^
which yields the
result. Q.e.d.
Lemma 2:
Iffi assigns
(i)
for
a
all
strictly positive probability to at least three states
a>Q
and
M *{jJ.)M *{B{p.,S^Q))
denote
in)
(ill)
it
M *(B(fi,Safi))* M *(^).
6,
is
of the world then
The direction
of the
vector
independent of a and moves clockwise as 6 does.
(pu(0) hereafter, (p^
is
a one-to-one function from [0,2k] into
(MHB(^,S^^e)))^0.
Y
da
a=0
— M*ifi)M*(B(fl,S^^e))+M*{^)M*iB(fl,S^^0+^))
We
itself
= 0.
aa a=OV
Proof of Lemma
We consider a
states
2:
belief /i
= (//qO' Ao'j"oi Ai)
'
assigning a positive weight to at least three
of the world.
22
.
M
* (//)
= £(mJm) =
The coordinates of M*(^)
U//) =
\y(fi)
=
+ /^i
(/^lo
1
'/^i
+ Ml
I
)
are
/iio+//ii
Moi
+ flu
Bayes' rule implies that
observed the signal S^q
if
,
an agent's
then his
new
initial belief
belief //'=
2^y"'oo('^a,0)
= /^o(l + 2a(-cos0-sin0))
^^'oi(Sa,e)
=
MQi{^
+ 2(^(-cosO + sme))
Z/i'lO (5^
=
/iio(l
+ 2a(cos0 - sin0))
=
/Xii(l
+ 2a(-cos0-sin0))
was
//
B{ii,S^q)
=
(/iQO'/^iO'/^i'/^ii) ^^i^
is
he
given by
(1)
)
I//'ll(5a,e)
with
E
(a function
of a and 6)
is
given by
Z - jUqo(i + 2(3(-cos0 - sin0)) + //qi(i + 2<3(-cos0 + sin0))
+ /iio(l + 2a(cos0 - sin0)) + jUj i(l + 2a(-cos0 - sin0))
Notice that the assumption
The change
in
A < —=r
ensures that
E>
M*(|l) induced by observing ^a,6
,
(2)
.
denoted
AM* = (Ax,A3')
Substituting (1) and (2) in this expression yields
-^=
(cos0
+ sin0)[(/iii+//oo)-(/^ll-/^o)(j"ll-/%)+/^l-/^lo)]
+ (COS0 - sin 6l)[(/iio +^0-(llio-lJQi )(/fio - /ioi - /^O + /^l
-1^=
(cos0
I
)]
+ sin0)[(/Xii+/A)o)-()"ll-i%)Xj"ll-AADO-/^l+/^lo)]
- (COS0 - sin 0)[(/Xio + /ioi ) - (A^IO - ;^l X/ilO - /^l + /%) - /^l 1 )]
23
is
equal to
n
Result
We assume
1:
&^
that
Then Ajc>0_ Ax + A3'>0. This implies that
0,-
TT
(Ax, Ay)
^
(0,0)
for any 6 e
0,.
In addition the direction of (Ax, Ay)
independent of
is
Proof:
n
For^e
L.(Ajc
0,-
(3) implies
+ Ay)
indeed that
(cos0-sin0)/
oi
r
^[(/^ii+/^o)-(/^ii+/^o) J-
cos6>
+ sin0
\/
\
i^n"^oA^io~^w
(cos0-sin0)
(/^ll+/A)o)UlO+/A)l)If
three
at
least
of the
cosO + s\n6
(mi
probabilities
1-/^0 )(/^io-/^i )
/^0'/^lO'i"oi'^ll
(/^11+/%))(Mio+A^i)>(/^11-/%))(/^10-/^i)- Also,
1
>
are
strictly
(cos
0- sin 0)
positive,
if
cos0 + sin0
0e
then
K
Therefore
(cos0-sin0)
(/^ll+/^)(/^l0+P0l)>
and Ax + Aj >
(/^ii-/%))(mio-/^i)
.
Similarly, for
•
cos0 + sin0
6e
"I
^
(3) implies that
Z.(Ajc)
=
(cos
+ sin 0)[(/Xi +/%))- (/^l - /^ )()"l - /^O - /^lO + ^1 ))]
1
1
1
+ (cos0-sin0)[(/iio+/^l)-(/XlO-/^l)(/ilO-/^l+Mll-/^o))]-
If
1^1
three
1
at
least
of the
~ /A)0 ~ A'lO + Ak)i <
I
1
probabilities
^"d
|/^10
/^O'A^lO'/^l'/^ll
~ A^l + A^l ~ A)0 <
1
24
^
are
strictly
positive,
then
^^ ^^at the factors in brackets
The
are strictly positive.
that
Ax >
Any
•
cos0 + sin0>O and
inequalities
is
of these two
For every 6
2;
different from (0,0)
kK
e
[k
.
In addition (3)
independent of a. Q.e.d.
is
[0,2k] the change (AK,Ay) ofM*(fl) induced by observing
in
and
its
depend on
direction does not
(k
implies that
a. If
one writes
<P
•S^.e
(0) for
k
+ l)K
T'
^
[Ax, Ay) ^(0,0)
inequalities implies that
strict
this direction, then for every integer
(i)
imply then
.
implies that the direction of [Ax, Ay)
Result
cos0-sin0>O
(jo^
4
- l)7t
{6)
4
(k
+ 2)7r
4
'
^
^
and
^\k-l)K
(kK\
{k
+ l)K^
e
(ii) (Pn
V
^
y
V
Proof:
7t
By
Result
1
we know
7t
respectively (p^iO)
if
^^
that if
0,-
then
integers
and
We proved
7t
By symmetry
two
n
—
..
<
(i)
for k=0,
and
it
that
remains true for
all
k because of the symmetry of the model.
>-—
..
These two inclusions imply
(p^ (6)
^^ {Q)
In particular (p^(0)
(p^ (Oj
A(x + y)>0 and Ax>0, implying
n K
57t
4'4
n K
0e
then
tnese
(p^
A)
inequalities
<
<jO^
—
>
,
n
have the following counterparts:
^;^^
..
,
and
and
so that
^^
(kn\
—
—
k=l.
25
({k-\)n {k^\)K\
e
,
is
true for
k=0 and
.
Again, the symmetry argument implies that
it is
K
Result
3:
We assume
that
^^
true for every integer k. Q.e.d.
Ay
0,-
Then
is
.
a increasing function of 6.
Proof:
K
^y
Notice
first that
by Result
1
is
.
always defined
if
^
0,.
+ sin 0) - C(cos - sin 0)
ns
^ + sm 0)
^r
o - sm
+ E(cos
D(cos 6
6
9)
5(cos
derivative
BE+CD
of the function
7;;
,
implies that the derivative of
~^
,
Ay
—
Ac
(3)
^^^
and the
fact that the
^^™^
^^^
^^
^^g"^
has the same sign as
[(i"ii+/^oo)-(Mii-/^o)(j"ii-/^oo+i^i-i"io)I(/^io+/^i)-(/^io-/^i)(/^io-/^i+Moo-i"ii)]
+ [(/^io+i"oi)-(Mio-/^oi)(i"io--"oi
But the term
each bracket
in
--"oo+/^ii)][(/^ii +/^oo)-(/^ii -i"oo)(/^ii -i"oo-/^oi +/^io)]
is strictly
%
positive if three at least of the
are strictly
Ay
positive, so the
whole expression
Result 4; The function
and
(P
(0)
(P
(d)
is
is
positive
^
"•I
,
^„
(^)
belongs to
direction of the vector {Ax,Ay)^
—=
increasing with 6, and that^^ (^)
a increasing function of 6. Q-e.d.
itself
;r^
7t
by Result
4'2
tan((i()^
(0)J
.
moves clockwise
Result
as
2.
Since
4 implies then
does.
(6)
(p
that
By symmetry,
9^
is
(^)
the
is
this is true
[0,2;r]
This implies that the function
to the
is
a one-to-one correspondence from [0,2k] into
7t
9e
T~
moves clockwise as 6 does.
Proof: If^e
for
and
same value
Result 5: For any
as
^ W)
moves from
9 -— (m
[$
a one-to-oae correspondence, since
to 27i. Q-e.d.
^0.
*(B(/i,5q g))]
a=0
26
it
comes back
.
Proof:
K
e
Again, consider
Z(a,0)(Ax)
^
(3) implies
(cos0
+ sin0)[(//i,+/A3o)-(Mll-/^o)(/^ll-/%) + /^l-Mio)]
+ (COS0 - sin 0)[(/iio + /ioi ) - (/^lO - /^i )(j"iO - /^i - /A)0 + /^i
The right-hand
side of this equality
in Result 1), let us
denote
it
is
)]
independent of a and different from zero (as proved
C{0)
Differentiating the above equality at
0=0
yields
=^C(e)^O.Q.e.d.
(Ax)
da
1
a=Q
^(0'^)
Result 6: For any
e,
=
—[Mi*{B{^i,Sa^e)) + m*{B{^,S^Q^j,)))
(0,0)
a-0
Proof:
We proved above that
—-(Ac(5^ q)]
da
Therefore
—
(ax(5^^0)
+ Ax(5^
= 0(6)
a=0
= C(d) + C(e + K) =
6+^^))
0.
a=0
Similarly
=
—[^y{S^^Q) + ^y{S^^Q^J,)]
,
implying
a=Q
Y(^HB(^,S^^e)) + ^*(B(^,S^,e+n)))
a=0
-(Ax{S^^Q) + Ax(S^^e+n))
_ '-^(^yiSa,e)
27
+ ^y(Sa,e+n))
(0,0)
a=Oj
.
Q.e.d.
Results
Result
Lemma 2:
6 prove
to
1
part
(i) is
just Results 2
and
(ii) is
4,
Result
5,
and
(iii) is
6.
Lemma
3: If ^^
and
the world, then for
function of (pn
[12
assign a strictly positive probability to at least three states of
any direction
(p
in [0,2 n],
we have
(writing 6^
((p)
for the inverse
):
Min\^ (p^^ (0^2 (<?)) -(P^(Pti2
^^^^
^^/^l
~^)^2
In other words, one at least of the scalar products
-^
M*(/ii)M*(5(//,,5^.(^))).M*(/i2)M*(5(//2,^^.(,p)))
is strictly
positive (i=I or i=2).
Proof of Lemma
3:
We use the same notations as in the proof of Lemma 2.
—
IcTt
Result
1: If
-^
9=
(k integer)
and
(Ax, Ay) is the
2
observation of S^g, then the product Ax. Ay has the
across dimensions, that
- /ioi/Zio
Hi iHqq
is
Proof:
It is
enough
the
same
sign as
same
/^n/^oo
positive (resp. negative), then (p^ (O)
^n^
(negative) while (p^
is,
change of M*(il) induced by the
K
-—
\^J
to
2
~ A'oiA'lO-
-
and
(p^^
•^^'^
{k)
implies that if
- Tt are positive
3k
(2>K
and
"••"
sign as the correlation of il
are negative.
^^IT
prove the result for k=0.
We
know
already (from Result 2 in the
proof of Lemma 2) that
^
(P^(0)
V
prove
is
K K^
4'4y
that if
.
Since
Ay and
(p„ (O)
-
have the same sign, the only thing
0=0, then Ay has the same sign as
28
/ij
j/iQo
- //qi/^iq
-
we need
to
.
Substituting
A}'
9=0
in (3) (in the
proof of Lemma 2) implies that
= //i + /ioo - (/^l - /^o)^ - (A^IO + Poi ) + (^10 - /^l )^
1
1
-Uo + /^i) + (/^io+/^if +4(/illA^0-j"l0/^l)
= (/^ll+/^o)-{Mll+/^o)
= 4(/^iiA^o-i"io/^i)
(the third equality uses the fact that the probabilities
which yields the
This result
is
result.
add up
to 1),
Q-e.d.
intuitive:
a signal providing information about only one dimension
if
changes the belief regarding the other dimension, then the changes along the two
dimensions must have the same (opposite) sign
if
the initial belief
is
characterized by a
positive (negative) correlation across dimensions.
n
Result 2:
We know
Lemma 3
holds with
from Result 2
s
(p
(in the
proof of
Lemma
2) that ^i(^)
We distinguish three cases, according to the correlation of
•
If
Ml
and
(Pi\^j<—
/^2
If
A'l
and
(i=l,2) so that if for
/^2
<
=
(p
-><p =
<pj
example
(^j (cp)]
<
(p^
6i((p)
[62
<
and M2 across dimensions.
by Result
0,-
((p)
>
e^y
(O)
>
that
(i=l,2). If for
(p2fe(<p))><P2(^l('P))^<P2(0)>-T so
29
1
then by Result 4 (in the
Ojicp)
((p))<— so
belong to
\(pi
(Oj ((p))
- <p| < —
display negative correlation across dimensions, then by Result
<p,(0)<0 (i=l,2) so that
then
Oiicp)
display positive correlation across dimensions, then
proof of Lemma 2)
•
A'l
and
example
e^icp)
that \(p2{d^{(p))-(p\<
<
—
1
&2.{(p)
.
>>
•
If j"i
and ^2 display respectively positive and negative correlation then, by the same
n
| > 02
argument,
<
that
<pi
(<?>)
<
(02 {(p))
By symmetry, Lemma
Proof of Proposition
^
-
^2 (O)
and
^
\(p^
3 holds for
and [<p,(0),(Pi(f)
(Oj (<p))
all
- <p| <
y
values of
(p.
•
(Result 1) implying
Q-e-d.
Q.e.d.
2:
->
Let
^e[0,;r)be
M*{}1
1
)
^
the
M *(ll
2
)
,
direction
orthogonal
to
the
vector
M *{ll
1
)M
9
*{il
)
if
and any direction otherwise. Let us define for every 6 the vector
—>
V(d) = (cos 0, sin 0)
that,
Using the notations of lemmas 2 and
.
with 62 = 62{(p)
,
3,
we can assume
for
example
the inequality
V{(p^{e2)).v{(p2{e2))-vi(pi(e2)).v{(p)>0 (i)
holds (this
Parts
(i)
is
and
an application of Lemma
(ii)
of
Lemma
3).
2 implies the existence of X,X'>0 such that for a
enough
""
2
2
—
->
M*(r)M*(5(//^5^.
))
= a(A + (9(a))V(<?))
and
^2
—
M*(^^)M*(BmKs^.)) = aa'+0(a))V((p2e2))
30
small
>
We
now
claim
signal
is
after
^a,ei+K
By symmetry,
M^
M *(/i^)
is
,
a
group
show
1
and
enough,
small
suffices to
it
M * {B{[i\S^fi J)
for
the difference
it is
commonly known
there
a
exists
that the
communicative
that truth-telling is optimal for the speaker
M^M2
-
identity
2
M1M2
-
V,
-
.
M,"
M *{B{ii\SaQ^^^))
for
between the speaker's expected
when he
believed) and
M1M2
The
with
he observed Sa,Qi
Writing
(and
o"^
>
>
that if the speaker belongs to
S^^Oi
equilibrium.
)
lied
utility
when he
and M,tells
for
the truth
is
(2).
= v,-V2
V2
2
M1M2
.
=M^M2.
i'i+V2
implies that
—
-^
Mi"^M2 + Mi"^M2
7
r
-2a{X +
—
->
—
->
2MiM2 + 2Mj^Mi-l-M2M2 + M2M2
0(a))V((p).
V
—
y
/
-2a(A-l-C>(a))V((p).
2^
\
->
->
A
2M1M2 - 2a{X'+0(a)) V((pi (62 ))+ aO(a)
^
= 4AA'a2 V((p).V((Pl(02))+O(a)
which
positive
is strictly
definition of V((p)
The expression
observed 5^
,
implying
in (2)
gu+^j-
by
(1) if
a
is
small enough (the last equality results from the
Mj M2 .V((p) =
).
and the corresponding expression
open (and therefore positive probability) subset of
odd period the speaker
signal's intensity
in the case
where the speaker
are continuous functions of 9 and a so they are strictly positive for an
is
in
group
1
and there
and direction are going
to
be in
31
is
(0,A]x[0,7t). In the
beginning of any
a strictly positive probability that the
this subset,
so there
is
a strictly positive
probability that an informative equilibrium will occur. Therefore
i/i
,ii
)
is
not a
steady-state. Q.e.d.
Proof of Proposition 4:
Let
L
be an edge of
(ji* ,11*
A
containing the true belief
in the iriterior
)
direction orthogonal to L).
[5y,5n,]
L
Note
.
first that
We
consider a steady state
there exists a unique direction
along the direction 6i does not affect beliefs belonging to
that a signal
L-
of
<5y.
We
write hereafter
and p,q for the two other
,
states
w
for the state of the
L
0^ such
(it
is
the
world such that
of the world.
Notations:
It
be convenient to denote the coordinates of a most preferred policy (an element of
will
the square
sides of
Q
Q) with respect
(in
diagonal of Q.
case
s
and
to the axis
w
formed by
[s,w]
and
[p,q].
These axis are either
agree on one dimension and disagree on the other) or a
One can check that
the coordinates u(fx) and v((x) defined by
^s-^w
uW
2
if [s,w]
if
and
[p,q] are sides
«(/!)
= 2 '(ji,-nj
v(/x)
= 2-^(/i^-^^)
[s,w]
and
[p,q] are
are coordinates
this
of Q, and by
diagonals of
Q
(modulo a constant) of
M *(//)
according to these
system of coordinates to describe elements of
example
M*
(ji)
=
(m(^), v(jU)) where u(ft) and
32
Q
uew
hereafter, so that
v(fx) are
axis.
We will
use
one can write for
defined as above.
Result
1;
12
As
(fl
,fl
1
converges towards {jl* ,/i*
)
7
), the
change
in beliefs
induced by
communication converges toward 0.
Proof: This
sequence {}i'\li ")
converging
converging towards {ji* ,/!*
some 0'^dj^,di+n:
to
would
a simple continuity result. If this were not true, there
is
such that communication
is
,
),
exist a
a sequence of directions
6^
and a sequence «„ converging toward some a>0
possible with a speaker in group
1
,
or equivalently
M*(Bifi^\S^^^e))M*(B(ii^\S^^^e))
< M*(B(H^'\S^ ^e))M*(B(n^\s^ ^e+^))
for
e =
e,^
and e = 0„+7t.
This implies
at the limit that
M*{B{fi*KSa^e))MHBifi*^,Sa,e)) ^ M*(B{fl*\Sa,e))M*(Bifi*^,Sa,e+K))
for
6-6' and 6 - 6'+n
distribution
is
12
{/i* ,fi*
so that there exists a communicative equilibrium
),
which
is
12
impossible since (jl* ,fl*
)
is
when
the belief
a steady-state.
Result 2: Consider the edge L'=[s,w] of [0,l]x[0,l] containing the most preferred
policies associated to beliefs in L.
12
neighborhood Vq of (fi* ,fl*
equilibrium
and
,
]
that u(/l*
and some K>0 such
2
)
,
Av,-
)
,
the inequality
< K{Max[Avi Av2 ])^
,
holds.
33
— u{fl*
that if {jl
realization of uncertainty, denoting the
preferred decision by (Au^
MojcfAwj Aw2
)
Assume
1
)
= B >0.
12
,fl
)
There exist a
G Vq then for any
change of a group
,
i
agent's most
Remark:
this
of the changes of the most preferred
result implies that the direction
decisions tend to
become orthogonal to L. It is
Max[Aui,Au2\
would only require
in fact stronger
than that (orthogonality
converge to zero).
that the ratio
M<3x[Avj Av2
,
J
Proof:
Assume
for
intensity
is
change
example
that there
an equilibrium where the speaker
a and the signal direction
in a
group
i
agent's
is 6.
We denote
is in
(Ax/",A>'/") (resp.
most preferred decision induced by the
5q e+n: ) Since the most preferred decision
group
1,
(Axf ,Ayf)
signal
S^ q
a linear function of the beliefs
is
the signal
)
the
(resp.
(it is
the
expectation) and bayesian updating implies that the expectation of next period's belief
the current belief, the expected change in the
Pr5'a.e ll^\Aut ,Avtyv\s^^e+KWVui
The
signal structure
is
such that the ratios
most preferred decision
is
is
zero, or
)=0.
'Av,-
—
jy are
-,
bounded away from zero
"^^Sa^e+nV']
and
infinity
uniformly (that
is,
ri-2AV2 H-2AV2
,-
jU
and
independently of a,
)
as they belong to the interval
j=
f=,
Ll+2Av2 l-2Av2
Therefore there exists some uniform K^ such that
_l_
Aui < Au'l < K^Aui
and
—
Av,-
<
Av,^
< KiAv}
K,
Let
us
suppose
without
loss
of
communicative equilibrium implies
generality
that a
group
most preferred decision moved by (Am/',Av/"))
observed
S^q
than by falsely reporting S^
q+jj-
The existence of a
AmJ >
agent who observed S^q (and whose
that
,
1
is
better off
.
by
truthfully revealing
or
II
19
The
left-hand side
is
>0.
equal to
34
he
.
[b
- 1(B + Au2 -Auy, Av2 - Avi
+ Am2 - ^ui Av2 - Avj*"
J
,
J
+ Au2 - Aui f + (av2 - Avi"^ f -\B + Am2 - AuJ^ f + (avj - Av^ f
= (25 + Am2 + Am2 - 2Ami'*" J(Am2 - Aa2 j+ iAv2 - Av2 J(Av2 + Av2 - Avf - AvJ^ j
=
[b
If
(/i\fi^)
is
close
enough
Au2, Am2 have opposite
SAwJ
absolute value than
2(1
+ Ki )v^Max(Av^
5Am2 ^
2(1
)< 2(1
+ Ki)[Max[Av^
close enough to
,
Av^
)j
,/i
*
)
then
>0
and
negative and greater in
smaller in absolute value than
Av'^ )f
second term
is
greater in absolute value than
sum
is
positive), or
.
L = [S^,5^^,],
are interior to
(ju *
that the
,
is
is
negative and the condition states that their
first in
+ Ki )[Max(Av2
Since fi* ,/i*
(and the facts that Am2
1
Similarly, the second term
•
The equilibrium condition implies
the first (since the
Result
),
signs) implies that the first term
Avj*"
,
(jU*^/i*
to
Am2 <
there exists
AmJ*"
< K2AU2
K2 such
,
that if (//
,/i
)
is
so that the inequality above
implies
BMax[Aui Am2 )<
,
2(1
+ K^ )(1 + K2)[Max[Av^
,
AvJ^ )f
.
This implies in turn
BMax\Au^ ,Au2]< 2{Ki f (1 + K^ )(1 + K2)\Max[Av2 Avf )f
,
so that the result holds with
Result
3:
Let
us fix
ifl*\ld*^)such that
K = 2(/i:i)^(l+ A:j)(1+ /^2)-
some
77,0 0. There
if (fi^^,ll^^)
exists
a
neighborhood
Vi(ri,£)
belongs to Vi(r7,e) then, writing [Au'' ,Av'')
35
of
for
the
change
in the
most preferred decision of a group
i
agent between periods
t
and
t+1,
the following inequality holds:
<£.
Pr
Vt>0
Proof: Note
first that if
L
consists of beliefs agreeing along one dimension (so that the
corresponding set of most preferred decisions
while
if
[p,q]
is
a diagonal then then
Av'
is
a side of Q) then Av
= 1 -X
Afi^i-Afiij
In
A/i^'+A/ij
both cases, the
following inequality holds:
r>l
r>l
The following
any (real-valued) positive martingale
results holds for
ji^
(see
Neveu
(1975),p.l86):
Pr
l(M'-/-lf>; <3a-y.
\t>l
This inequality applies to {ip and
beliefs.
/I'J
which are martingales conditional on agent
Therefore
2
2
>a
Pr
(0
fi
<3a
ftp
Kt>i
f
But Pr
2
1
/O
/i
>a
>/.fp.
so that
\t>\
iO
Pr
<3a-l-^andPr
l^s
iO
XlA/^i')
U>1
36
>«
<3a -l^P
d'
i's
^
I(av"F >4a'
implying Pr
3a-'{^f+H'^^)
Mf
t>\
This provides the result:
JO
fip
.
77
= 4a
then defining the set Vy(ri,£) by
./^re^-^
..io
+^q <
12
1
•
/*
/"O
if
implies that Pr
,*
I
<£
l(^v^'f>ri
if (jd^^,ii^^)
belongs to
V'i(t7,£)
f>i
Result
(//
*
I
Let
4:
,/i *
?
some
us fix
)such that
90
10
if {{i
ri,£>0.
,ji
)
There
exists
L (z=p or z=q),
Pr
..\
I
\
k)>n
t>Q
implies that:
/
Max k)>n M'
t>Q
r
f
and the inequality Pr Max\ii^l]>r]
,/0
>MfPr
r>0
leads to Pr
Max\^^[]>r]
t>Q
\j^
fTn
Therefore defining the set V2(,ri,e) by the inequalities
37
k)>n
Max^
t>0
J
of
J
<£
Proof: the "maximal inequality" for positive martingales (see
Pr
V2{'T],£)
belongs to V2(rj,£)
thenfo,
for any of the two states of
¥2(7], £) then
f
the world z not belonging to
a neighborhood
Neveu
(1975), p. 23)
(0
.iO
f^s
(0
,,('
^^
I*
1
-^s <2^s
(
implies that Pr
<e
Max k)>^
if (/i^^,/z^°)
belongs to V2(77,e).
f>0
Result 5: Let us
fix
some neighborhood
dimensional" belief pairs) and some
(lJ.*^,^i*^)
A^
in
such
that
W of
(jJ.*
12
,jl*
£>0. There
)
in
exists
(that
is,
a set of "one-
a neighborhood
belongs
(/i^^,/i^°)
if
L2
to
V(£)
V(£)
of
then
\
Pr L^m(//^^JU^')
eW
>l-3e
Proof:
Notice
Am'
first
=
that
Am'
A/if-A/i''
IV
2" A/if-A/i^
if
s
and
w
agree on one dimension
otherwise so that in both cases Am'
<
and
1
A/i^'-A/ilt
Therefore the set of belief pairs
<(2K + l)rifor =
1
converges towards
Let
us
j(//
* ,// *
choose
V^) c {(/i\/i2|^l^
such
r\
>
)j as
and
r\
1,2]
converges to zero.
V'(ri)(zVQ,
that
/4 > o|.
Consider a pair of beliefs (/i^^,//"^^) belonging to
Vi(r],£)nV2{T],e)nV'(-n)n-\\ jii,ii2 m' -M(/i'
38
)
<77
V'{ri)nL (zW
,
and
,
(//^°,//^^)
e V2(T],£) implies by Result 4
that for i=l,2
f
MaxWL ]<
Pr
rj
and Maxui'' 1<
r>0
>l-2e.
77
^
f>0
(/i^°,/i-^°)eV2(^>e) implies by Result 3 that for i= 1,2
<£
Pr
t>]
Therefore the event E:
Maxi/ip
]
<
T]
and Maxi/i^^'
j
<
77
occurs with a probability greater than
Let us assume that
E
=
The
Mm
{l
u''-u(fl'*)
fact that
E occurs
(1
.
j
>
T]
— 3e)
occurs, and let us
u" -u(j2'*) >{2K + l)T]
tQ
^ (Av"
and
assume
that for
some period
t,
and i=l or i=2,
Consider then
>(2K + l)riOT
implies than for
Therefore Result 2, applied to
all
u^'
all t
-u(^^*) >(2K +
<tQ, (^
If
pairs of beliefs (/x
,,2r.
,^
)
,//
e
)
1)77}.
V'(??)
for
t
c Vq
<tQ, implies
i=l and i=2
''+l-«''
0<t<t„
(
<K
5;(vl'-^l_,l'f
+ (,2r+l_,2r]2 +
.'0
«"^
-«(/:'
0<r<f„
39
)
<2/ir77
+ 77 =
(2/:
+ l)77
that for
,
Therefore
it is
impossible that
This implies that
if (//^^,/x^^)
u''o
-U{fl'*)
>(2K + l)T]
for i= lor 1=2.
belonging to
{li^^,^^^)GVi(T],e)nV2(ri,£)nV'(Ti)n\(iii,H2\u'-u(^'*)
then Pr((/x'',/i^')G\/'(n) for
But (fi',/1
°°
')
some {n°° ,ii
same
the
to
edge
V(U) )> l-3e.
Pr((/i^°°,/i^~)e
V'(ri)
> 1 - 3e
has to converge to
belong
ji
all t)
e |(/i \ //^ ) /xi
(i=l,2) so that iji
°°
°°
,fi
j
g
L
.
of
°°),
and by Proposition 2
We
A.
Therefore
and /i^ > o| implies
>
<n
just
the
that if (/i '" /i^~
,
)
/i
and
showed
that
assumption
that
e
>
V'(ri) then //£"
Therefore with a probability greater than
1
— 3£
fi^°°,fi^'-)e(v'(T])nL^)czW.Q. e.d.
Proof of Proposition
We
5:
consider a pair of
initial beliefs ifi
,fi
j
such that for
all
pairs of states of
the world (p,q)
«
<£.
^'?4''
We
want
to
show
that
,
if
e
is
small
foo
Let us assume that with a positive probability
40
enough,
then
with
probability
1.
By
we know
proposition 2
assume
for
//
and
°°
/i
belong to the same edge of A. Let us
example they both assign weight only
Let us define for i=l,2,
(1,0).
°°
that
= {Iiq and
p,-
,
let
to the states
of the world (0,0) and
us assume that pj
< p2 (group
1
is
"to
the left" of group 2).
Step
1:
Assume
(a
the signal direction
is
horizontal (6=0) and the signal intensity
is
maximal
-A).
The
signal structure implies that for i=l,2:
5(/i'~,S^,o)(l'0)
+ 1A)
Cpi
+ 2A) + (l-pi)(\-2A)
Cpi+I-Pi
Pl{\
=
Plil
and
Pi(l-2A)
B{fi'"',s.^)im
With
C=
Since
(/z
group
1,
1
Pl(l-2A) + (l-pi)(l + 2A)
Pi+C(l-pi)
+ 2A
1-2A
~ ,/i
>
I
°°
1
is
a steady
state,
it
must be the case
he wants to report the "left-wing signal"
wing signal" 5^
q,
which
is
5^
;j
that
whenever the speaker
even
after observing the "right-
true if
P\C
Pi
P\C
PlC
P\C+\-p\
P2 + C(l-P2)
PlC+\-Pi
P2C+I-P2
or
Pl{\-p2)C -P2(l-Pi)
P2+C(l-P2)
which implies |(pi-p2)c|>
(P\-P2)C
<
P2C+I-P2
^^fX!
+ C(l—
Pt
^\ pi(^-P2)C^-P2(^-pO
P-)
)
41
is in
C>
1
K^l
- P2 )c\ >
a
1-
and pi < P2 imply that
/^2 (1
P2 (-/'.)
(l-Z'l)
i^-Pl)
P\ (I-P2)
P\
Step
— PiC^-Pi )C^ -
-
Pi
)
.
or equivalently
2:
For every period
Bayes' rule implies that
t,
/^iVio ^ /^fp/^^ ^ Pr(messages up
/zj>;0
to t|(l,0))
Pr(messagesuptot|(0,0))
/iJ'o/iJO
^10^00 ^ ^10^00
so that
^00'"lO
Taking the
^00^10
limit of this identity as
t
tends to infinity yields
P2(^-Pl) ^ ^10^00
Step
3:
Steps 2 and 3 imply that
if for all pairs (p,q)
of states of the world,
-1 <e
then C"
1-
P2
Pl
C^
Pi (I-P2)
so that
eC^ > C^
P2
Pi (1-/^2)
and
/'I
<e
Pi (1-P2)
(1-P2)
P2
>C^-l-£
C^
/'I
(l-/'2)
42
.
which cannot be
true if e
This proves that
if
characterized
by
e
is
a
is
close
enough
to zero,
given that
close enough to zero, then i^
one-dimensional
conflict,
43
,/i
so
C>
j
that
1
cannot be a steady-state
(by
Proposition
2),
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Aghion,
P.
P.,
C.
Bolton,
Harris,
and
P.
JuIIien,
1991,
"Optimal
learning
by
experimentation", Review of Economic Studies, vol. 58, 621-654.
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K., 1963, Social choice
and
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New York
:
Wiley.
Banerjee, A. and R. Somanathan, 1999, "A simple model of voice", mimeo.
Crawford, V. and
vol. 50,
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Hart,
S.,
6,
J. Sobel, 1982, "Strategic Information Transmission",
1431-1451.
Two-Person
"Nonzero-Sum
1985,
Repeated
Information", Mathematics of Operations Research, vol.10.
Neveu,
Piketty, T.,
Economics,
1, 1
with
Incomplete
17-153.
1995, "Social Mobility and Redistributive Politics", Quarterly journal of
vol. 110,
Poole, K.T.
No.
551-584.
3,
and H. Rosenthal,
Journal of Political Science, vol. 35,
J.,
No
Games
1975, Discrete-Parameter Martingales, Oxford: North-Holland..
J.,
Snyder,
Econometrica,
1991, "Patterns of congressional voting", American
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44
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