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Pure communication between agents with close preferences
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David Spector
Economics Department, Massachusetts Institute of Technology, Cambridge, MA 02142-1347, USA
Received 13 May 1999; accepted 16 September 1999
Abstract
This paper studies cheap-talk games when the speaker’s and the receiver’s preferences are close. It is shown that, as they tend to coincide, the most informative equilibrium converges toward full information transmission. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Cheap-talk games
JEL classification: C72; D82
1. Introduction
In this paper, we prove a result extending the characterization of the equilibria of pure communication games, or ‘‘cheap-talk’’ games. The framework is the one introduced by Crawford and Sobel (1982) (referred to as C–S hereafter): ‘‘there are two agents, one of whom has private information relevant to both. The better-informed agent, henceforth called the Sender (S), sends a possibly noisy message, based on his private information, to the other agent, henceforth called the Receiver (R). R then makes a decision that affects the welfare of both, based on the information contained in the signal.’’
C–S proved a monotonicity result: the precision of S’s message in the most informative equilibrium
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increases when the distance between S’s and R’s preferences decreases . This paper describes more precisely communication in the case of close preferences: we show that as S’s and R’s preferences tend to coincide, the noise in S’s message tends to zero in the most informative equilibrium. This establishes a continuity property at the point where preferences are identical: S’s private information is almost perfectly transmitted to R if preferences differ very little.
*Tel.:11-617-2589-268; fax: 11-617-2531-330. E-mail address: spector@mit.edu (D. Spector) 1
This statement results from their Lemma 6 and Theorems 4 and 5.
2. The model
2.1. Agents and preferences
There are two agents, a sender (S) and a receiver (R). S observes the value of a random variable s (the ‘‘state of the world’’), which has a probability distribution given by a density function m over
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[s,s]] 5[0,1]. m is assumed to be strictly positive everywhere and infinitely differentiable.
The game proceeds as follows: S observes the value of s, and then sends a message to R. R then
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chooses some action x in [x,x] which affects his utility as well as S’s.]
S’s preferences are characterized by a infinitely differentiable utility function U(x,s), (where x is the decision made by R and s is the state of the world) satisfying
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Similarly, R’s preferences are characterized by a utility function (U1eV )(x,s), where V satisfies
Conditions (1) and (2) above. e.0 is a measure of the distance between S’s and R’s preferences. Condition (1) implies that the optimal x is unique given any belief about the probability distribution of s, and Condition (2) implies that a greater s causes an increase in each agent’s most preferred x. To make the problem smooth enough, we assume that the optimal choice for S is always interior, whatever the state of the world. By continuity, this will also be true of R’s most preferred choice if e
is small enough. Formally,
≠U ] ] ≠U
]≠ s dx,s ,0,]s dx,s] ] (3)
x ≠x
2.2. Messages and equilibria
For each p we consider the game where the message is constrained to belong to some p-element set
m , . . . ,m . The equilibrium concept is the Nash Bayesian equilibrium: given the state of the world
h
1 pj
he observed and the impact of various messages on R’s behavior, S sends the message maximizing his utility. Conversely, R chooses a value of x based upon the information about s provided by the message, given the equilibrium behavior of S.
C–S show that an equilibrium where q#p messages are sent with a positive probability is
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characterized by a partition of [s,s] into q intervals I ,.., I , such that the message m is sent with] 1 q i probability one if s belongs to the interior of I . Therefore an equilibrium can be summarized by suchi
2 ]
a partition of [s,s] .]
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3. Informativeness of equilibria when preferences are close
We do not characterize all equilibria. In general, as C–S show, there are many: if an equilibrium exists with p intervals, then for each q#p there exists one with q intervals as well – the equilibrium
with one interval being the obvious ‘‘babbling’’ equilibrium where no information is transmitted at all. The proposition below implies that if S’s and R’s preferences are very close, the most informative of all these equilibria is very informative.
Proposition. For any integer p there exists e0.0 such that if e,e0, there exists an equilibrium in
which the message set has p elements m , . . . , m , and S sends the message m after observing a1 p h
Proof of the proposition. See Appendix A.
Corollary. For everyh.0 there existse0.0 such that ife,e0, there exists an equilibrium such that
after communication takes place, the variance of R’s belief about the state of the world is smaller than h.
Proof of the corollary. If R knows that s belongs to an interval I of lengthl, then the variance of his
2
belief is smaller than l . Therefore, applying the proposition above to an integer p such that
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A( p)#œh proves the result. h
4. Conclusion
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In their discussion of the equilibrium selection problem, C–S argue that one should select the
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equilibrium with the largest number of messages, because it maximizes R’s and S’s utility ex ante . If we accept this argument, our result implies that private information tends to be perfectly transmitted as preferences tend to coincide.
Proof of the proposition
The proof proceeds as follows: Lemma 1 establishes the existence of an equilibrium with p intervals in the case of identical preferences (e50). In Lemmas 2–4 we show that this equilibrium is characterized by an amount of noise converging to zero as p tends to infinity. Finally, Lemma 5 shows that if e is small enough, there exists an equilibrium partition close to the one found in Lemma 1.
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See their Theorems 4 and 5. 4
] ]
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Lemma 1. There exists s , . . . , s
s
1 pd
with s]#s1 # ? ? ? #sp #s such that the partition of [s,s]]]
defined by the intervals I (with Ii i5[si21,s ] if 2i ,i#p, I15[s,s ] and I] 1 p115[s ,s]) characterizes anp
equilibrium of the communication game in the case where S’s and R’s preferences are identical(i.e., in the case where e50).
Proof of Lemma 1. We fix some large enough integer p. Let us define the following notations:
]
Given an interval I included in [s,s], we also define the length of I by]
l(I )5Max(I )2Min(I )
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We also define, for every s in [s,s] and every x in T, x(s) and s(x) given, respectively, by]
x(s)5Arg max U(.,s)
H
x5Arg max U [.,s(x)]Conditions (1) and (2) imply that each of these equations has a unique solution, and that the derivatives x9(s) and s9(x) are strictly positive, and bounded away from zero.
For any s,ss 9d[Q and (x,x2 9)[Z we define h(s,s2 9) and k(x,x9) by
Also, Conditions (1) and (2) imply that H and K are continuous. Therefore KOH is a continuous function mapping the convex, compact set Q into itself. Brouwer’s theorem implies the existence ofp
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a fixed point s , . . . , s
s
1 pd
of KOH in Q .p]
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the message m after observing s in I , then R’s bayesian updating leads him to pick h(si i i21,s ).i
Proof of Lemma 2. The identity≠U /≠x[x(s9),s9]50, and a second-order development of the function
U(.,s9) between x(s9) and x(s) on the one hand, between x(s9) and X(s,s9) on the other hand, imply that
This implies that as s9 converges to s,
X(s,s9)2x(s9)
]]]]]
U
x(s9)2x(s)U
converges toward 1, which together with the inequality x(s)#x(s9)#X(s,s9) implies that
X(s,s9)2x(s)
This equality and the identity s9[x(s)]x9(s)51 implies
2
which establishes the first identity. Similarly, a second-order development of U(.,s9) between x(s9) and
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Proof of Lemma 3. Lemma 2 and Taylor’s formulas imply that the existence of A.0 such that for any x, x9, s, and s9:
leading, after a few manipulations (using the fact that each li is smaller than one) to
2
which is strictly greater than 1 if p is large enough, since the last expression converges toward 2 as p
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tends to infinity. This leads to 1.1, which is impossible. Therefore the inequality Max(l ,..,l )1 p .p
exp[2B(B11)], assumed above, cannot hold if p is large enough. h
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We write hereafter A( p) for 2p exp[2B(B11)].
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a linear application from a p-dimensional vector space into itself, it has generically full rank (i.e., a
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rank equal to p) . By the implicit function theorem, this implies that for any neighborhood V of
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inequality Max(l ,..,l )1 p #A( p) / 2 from Lemma 4 implies that Max(l , . . . , l )e1 ep #A( p). This
com-pletes the proof of the proposition. h
References
