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Pure communication between agents with close preferences
Economics Letters 66 (2000) 171–178 www.elsevier.com / locate / econbase
Pure communication between agents with close preferences 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 increases when the distance between S’s and R’s preferences decreases 1 . 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: [email protected] (D. Spector) 1 This statement results from their Lemma 6 and Theorems 4 and 5. 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 99 )00218-9
2000 Elsevier Science S.A. All rights reserved.
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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 ] [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 ] 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 ≠ 2U ]] ,0 ≠x 2
(1)
≠ 2U ]] . 0 ≠x ≠s
(2)
and
Similarly, R’s preferences are characterized by a utility function (U 1 eV )(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]x,s]d , 0 , ]sx,sd ≠x ≠x ] ]
(3)
2.2. Messages and equilibria For each p we consider the game where the message is constrained to belong to some p-element set
hm 1 , . . . ,mpj. The equilibrium concept is the Nash Bayesian equilibrium: given the state of the world 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 ] characterized by a partition of [s,s] ] into q intervals I1 ,.., Iq , such that the message m i is sent with probability one if s belongs to the interior of Ii . Therefore an equilibrium can be summarized by such ] 2 a partition of [s,s] ] . 2
This ignores ‘‘duplicate’’ messages, conveying the same information: without loss of generality, it is possible to restrict the analysis to equilibria where two different messages have different meanings, i.e., convey different information about the state of the world. See C–S for more details on this.
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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 1 , . . . , m p , and S sends the message m h after observing a signal in fs h 21 ( p,e ),s h ( p,e )d, with s] 5 s 0 ( p,e ) # s 1 ( p,e ) # ? ? ? # s p 21 ( p,e ) # s p ( p,e ) 5s] In addition, there exists a sequence A( p) with Lim p ` A( p) 5 0 such that for any such equilibrium partition, Maxfs h ( p,e ) 2 s h21 ( p,e )g , A( p) if e is close enough. Proof of the proposition. See Appendix A. Corollary. For every h.0 there exists e0 .0 such that if e , 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 length l, then the variance of his belief is smaller than l 2 . Therefore, applying the proposition above to an integer p such that A( p) #œ] h proves the result. h
4. Conclusion In their discussion of the equilibrium selection problem, C–S argue 3 that one should select the equilibrium with the largest number of messages, because it maximizes R’s and S’s utility ex ante 4 . 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 (e 5 0). 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. 3
See their Theorems 4 and 5. Farrell (1993) challenges the robustness of this argument. He defines a criterion called ‘‘neologism-proofness’’, and shows that the equilibria of C–S do not satisfy this criterion. 4
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] Lemma 1. There exists ss *1 , . . . , s p*d with s] # s *1 # ? ? ? # s *p #s] such that the partition of [s,s] ] ] defined by the intervals Ii (with Ii 5[s i21 ,s i ] if 2 , i # p, I1 5 [s,s ] 1 ] and Ip 11 5 [s p ,s]) characterizes an equilibrium of the communication game in the case where S’ s and R’ s preferences are identical ( i.e., in the case where e 5 0). Proof of Lemma 1. We fix some large enough integer p. Let us define the following notations: Q p 5hss 1 , . . . , s pd such that s] # s 1 # ? ? ? # s p #s]j ] ] T 5hx [ [x,x] ] such that x 5 Arg max U(x,s) for some sj 5 [Arg maxU(.,s), ] Arg max U(.,s)] Zp 5hsx 1 , . . . , x pd [ T p such that x 1 # ? ? ? # x pj Given ss 1 , . . . , s pd [ Q p , we define ] Ii 5 [s i21 ,s i ] if 2 , i # p, I1 5 [s,s ] 1 ] and Ip 11 5 [s p ,s] ] we also define the length of I by Given an interval I included in [s,s], ] l(I) 5 Max(I) 2 Min(I) ] We also define, for every s in [s,s] ] and every x in T, x(s) and s(x) given, respectively, by x(s) 5 Arg max U(.,s) x 5 Arg max U [.,s(x)]
H
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 ss,s9d [ Q 2 and (x,x9) [ Z2 we define h(s,s9) and k(x,x9) by s9
5
E
h(s,s9) 5 Arg max U(.,s )m (s ) ds s
U [x,k(x,x9)] 5 U [x9,k(x,x9)]
We define then, for (s 1 , . . . , s p ) [ Q p and sx 1 , . . . , x p11d [ Zp11 : ] H(s 1 , . . . , s p ) 5fh(s,s ] 1 ), h(s 1 ,s 2 ), . . . , h(s p 21 ,s p ), h(s p ,s)g K(x 1 , . . . , x p 11 ) 5fk(x 1 ,x 2 ), . . . , k(x p ,x p 11 )g
H
Clearly H maps Q p into Zp 11 . Similarly, if sx 1 , . . . , x p11d [ Zp11 then Conditions (1) and (2) imply that s(x i21 ) # k(x i21 ,x i ) # s(x i ), implying k(x i21 ,x i ) # k(x i ,x i 11 ). Therefore K maps Zp 11 into Q p . Also, Conditions (1) and (2) imply that H and K are continuous. Therefore KOH is a continuous function mapping the convex, compact set Q p into itself. Brouwer’s theorem implies the existence of a fixed point ss *1 , . . . , s *p d of KOH in Q p . ] * * Finally, the partition of [s,s] ] induced by ss 1 , . . . , s p d is an equilibrium partition if e 5 0: if S sends
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the message m i after observing s in Ii , then R’s bayesian updating leads him to pick h(s i21 ,s i ). Conversely, if for each i the message m i causes R to choose x i , then it is optimal for S to send m i only when observing s in fk(x i21 ,x i ),k(x i ,x i 11 )g. Therefore ss 1 , . . . , s pd [ Q p defines an equilibrium partition if and only if it is a fixed point of KOH. h Lemma 2. If for any ss,s9d [ Q 2 , we define the functions X(s,s9) and S(s,s9) implicitly, by the equalities ≠ ≠ s9 5 k[x(s),X(s,s9)] and x(s9) 5 hhs,s[S(s,s9)]j, then ]fS(s,s9)g 5 ]hs[X(s,s9)]j 5 2. ≠s9 s95s ≠s9 s95s
U
U
Proof of Lemma 2. The identity ≠U / ≠x[x(s9),s9] 5 0, 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
U
U UD
≠ 2U ]] U [x(s),s9] 2 U [X(s,s9),s9] 2hfx(s9) 2 x(s)g 2fx(s9) 2 X(s,s9)g j 2 [x(s9),s9] ≠x 3 3 3 3 (s9 2 s) ≠U ≠U ≠X fX(s,s9) 2 x(s)g # ]]]]]Max ]] # ]]]Max ]] Max ]ss 1 ,s 2d 3 3 3 3 ≠s9 (s ,s )[Q ≠x ≠x 1 2 2 2
U U
2
U US
U
3
or
U S
U
3
U U
(s9 2 s) ≠ 2U ≠ 3U ]] ]]] ]] Max hfx(s9) 2 x(s)g 2fx(s9) 2 X(s,s9)g j 2 [x(s9),s9] # 3 3 ≠x ≠x 3 ≠X Max ]ss 1 ,s 2d (s 1 ,s 2 )[Q 2 ≠s9 2
U
2
UD
This implies that as s9 converges to s, X(s,s9) 2 x(s9) U]]]]] x(s9) 2 x(s) U converges toward 1, which together with the inequality x(s) # x(s9) # X(s,s9) implies that X(s,s9) 2 x(s) ]]]] x(s9) 2 x(s) converges toward 2, or X(s,s9) 5 x(s) 1 2[x(s9) 2 x(s)] 1 Of(s9 2 s)2g This equality and the identity s9[x(s)]x9(s) 5 1 implies s[X(s,s9)] 5 shx(s) 1 2[x(s9) 2 x(s)] 1 Of(s9 2 s)2gj 5 s[x(s)] 1 2s9[x(s)][x(s9) 2 x(s)] 1 Ofss9 2 sd 2g 5 s 1 2s9[x(s)]hx9(s)(s9 2 s) 1 Ofss9 2 sd 2gj 1 Ofss9 2 sd 2g 5 s 1 2(s9 2 s) 1 Ofss9 2 sd 2g which establishes the first identity. Similarly, a second-order development of U(.,s9) between x(s9) and x(s) on the one hand, between x(s9) and x[S(s,s9)] on the other hand, implies the second identity. h
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Lemma 3. There exists B .0 such that if ss *1 , . . . , s *p d is a fixed point of KOH, then, writing l i for l(Ii ), the inequality 2
2
ul i 11 2 l iu , B ? Minsl i ,l i 11d
holds for 1 # i # p 2 1. 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:
H
usS[s,s9) 2 s9d] 2 (s9 2 s)u # Ass9 2 sd 2
(A.1)
uhsfX(s,s9)g 2 s9j 2 (s9 2 s)u # Ass9 2 sd 2
Assume that ss *1 , . . . , s p*d is a fixed point of KOH. Then, the definitions of h and k imply that for ] * * * * * * * * * 1 # i # p [writing s *0 for s, ] s p 11 for s, and x i for hss i21 ,s i d], s i21 # s(x i ) # s i # s(x i 11 ) # s i 11 , and * * * * * * * * * s i 5 Sfs i21 ,ssx i dg, x i 11 5 Xfssx i d,s i g, and s i 11 5 Sfs i ,ssx i 11dg. These identities and Condition (A.1) imply that, with the notations l1 5 s(x i* ) 2 s *i21 , l2 5 s *i 2 s(x *i ), l3 5 s(x i*11 ) 2 s *i , and l4 5 s *i 11 2 s(x *i 11 ): 2
2
2
u l2 2 l1u # Al 1 ; u l3 2 l2u # Al 2 ; u l4 2 l3u # Al 3
leading, after a few manipulations (using the fact that each li is smaller than one) to Max u li 2 l1u # Kl 12 with K 5 Ah1 1 (1 1 A)2g 1 A2f1 1 A[1 1 (1 1 A)2 ]j 2
2#i #4
This and the identities l i 21 5 l1 1 l2 and l i 5 l3 1 l4 imply therefore 2
2
ul i 11 2 l iu # 3Kl 1 , 3Kl i 2
Similarly, ul i 11 2 l iu , 3Kl i 11 , yielding the result with B53K. h Lemma 4. If p is large enough, then Max(l 1 ,..,l p ) # p 21 exp[2B(B 1 1)]. Proof of Lemma 4. We consider some p such that p . (B 1 1) exp[2B(B 1 1)]. The identity l 1 1 ? ? ? 1 l p 5 1 implies the existence of i 1 such that l i 1 # 1 /p. Let us assume that Max(l 1 , . . . ,l p ) . p 21 exp[2B(B 1 1)] and (for example) that for some i* , i 1 , l i * . p 21 exp[2B(B 1 1)]. We define i 0 5 Maxhiui , i 1 and l i . p 21 exp[2B(B 1 1)]j. For every i [si 0 ,i 1g, one can write l i 5 n i21 with n i $ p exp[22B(B 1 1)] . B 1 1. Lemma 3 implies that for i [si 0 ,i 1g, 2
l i 11 $ l i 2 Bl i or
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n 2i Bn i n i 11 # ]] 5 n i 1 ]] # n i 1 B(B 1 1) ni 2 B ni 2 B since the function (xB) /(x 2 B) is a decreasing function of x and n i . B 1 1. Therefore, q5Int
i1
1$
S
n i 2n i 1 0 ]] B(B 11 )
Ol $ O i
i 5i 0
q 51
D
O
1 1 ]]]]] 5 ]]] n i 0 1 qB(B 1 1) B(B 1 1) n integer
n i ,nB(B 11 ),n i
F S
$fB(B 1 1)g 21
ni 1 log ]]]]] n i 0 1 B(B 1 1)
DG
0
S
1 ] n 1
p $fB(B 1 1)g 21 log ]]]]]]]]]] p exp[22B(B 1 1)] 1 B(B 1 1)
D
which is strictly greater than 1 if p is large enough, since the last expression converges toward 2 as p tends to infinity. This leads to 1.1, which is impossible. Therefore the inequality Max(l 1 ,..,l p ) . p 21 exp[2B(B 1 1)], assumed above, cannot hold if p is large enough. h We write hereafter A( p) for 2p 21 exp[2B(B 1 1)]. Lemma 5. If e .0 is small enough, then there exists ss e*1 , . . . , s e*pd [ Q p such that ( i) the partition * * * * ] fs,s ] e 1d, fs e 1 ,s e 2d, . . . , fs e p ,sg is an equilibrium partition of the game where] R’ s utility function is (U 1 eV ). ( ii) If we define le i 5 s *e i 2 s e*i 21 (writing s e*0 for ]s and s e*p 11 for s), then Max(le 1 ,..,le p ) # A( p) Proof of Lemma 5. We define, for ss,s9d [ Q 2 and (s 1 , . . . , s p ) [ Q p s9
E
he (s,s9) 5 Arg max sU 1 eVd(.,s )m (s ) ds s
and ] He (s 1 , . . . , s p ) 5fh e (s,s ] 1 ),he (s 1 ,s 2 ), . . . , he (s p ,s)g Consider the function Le 5 Id 2 KOHe , from Q p into R p . Since the differential of L0 at ss *1 , . . . , s *p d is a linear application from a p-dimensional vector space into itself, it has generically full rank (i.e., a rank equal to p)5 . By the implicit function theorem, this implies that for any neighborhood V of ss *1 , . . . , s *p d, there exists e0 . 0 such that if e , e0 , the equation Le (s 1 , . . . , sp ) 5 0 has a solution ss *e 1 , . . . , s *e pd in V. But the equality Lef(s *e 1 , . . . , s *e p )g 5 0, equivalent to (KOHe )f(s e*1 , . . . , s *e p )g 5 (s *e 1 , . . . , s *e p ), means that (s *e 1 , . . . , s *e p ) defines an equilibrium partition in the case where R’s utility function is (U 1 eV ). If V is small enough, then each l i 2 le i is close enough to zero, so that the
5
One can in fact prove that this differential is of rank p for sure, if p is large enough (proof available from the author).
178
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inequality Max(l 1 ,..,l p ) # A( p) / 2 from Lemma 4 implies that Max(le 1 , . . . , le p ) # A( p). This completes the proof of the proposition. h
References Crawford, V., Sobel, J., 1982. Strategic information transmission. Econometrica 50, 1431–1451. Farrell, J., 1993. Meaning and credibility in cheap-talk games. Games Economic Behav. 5, 514–531.
Pure communication between agents with close preferences 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 increases when the distance between S’s and R’s preferences decreases 1 . 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: [email protected] (D. Spector) 1 This statement results from their Lemma 6 and Theorems 4 and 5. 0165-1765 / 00 / $ – see front matter PII: S0165-1765( 99 )00218-9
2000 Elsevier Science S.A. All rights reserved.
D. Spector / Economics Letters 66 (2000) 171 – 178
172
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 ] [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 ] 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 ≠ 2U ]] ,0 ≠x 2
(1)
≠ 2U ]] . 0 ≠x ≠s
(2)
and
Similarly, R’s preferences are characterized by a utility function (U 1 eV )(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]x,s]d , 0 , ]sx,sd ≠x ≠x ] ]
(3)
2.2. Messages and equilibria For each p we consider the game where the message is constrained to belong to some p-element set
hm 1 , . . . ,mpj. The equilibrium concept is the Nash Bayesian equilibrium: given the state of the world 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 ] characterized by a partition of [s,s] ] into q intervals I1 ,.., Iq , such that the message m i is sent with probability one if s belongs to the interior of Ii . Therefore an equilibrium can be summarized by such ] 2 a partition of [s,s] ] . 2
This ignores ‘‘duplicate’’ messages, conveying the same information: without loss of generality, it is possible to restrict the analysis to equilibria where two different messages have different meanings, i.e., convey different information about the state of the world. See C–S for more details on this.
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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 1 , . . . , m p , and S sends the message m h after observing a signal in fs h 21 ( p,e ),s h ( p,e )d, with s] 5 s 0 ( p,e ) # s 1 ( p,e ) # ? ? ? # s p 21 ( p,e ) # s p ( p,e ) 5s] In addition, there exists a sequence A( p) with Lim p ` A( p) 5 0 such that for any such equilibrium partition, Maxfs h ( p,e ) 2 s h21 ( p,e )g , A( p) if e is close enough. Proof of the proposition. See Appendix A. Corollary. For every h.0 there exists e0 .0 such that if e , 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 length l, then the variance of his belief is smaller than l 2 . Therefore, applying the proposition above to an integer p such that A( p) #œ] h proves the result. h
4. Conclusion In their discussion of the equilibrium selection problem, C–S argue 3 that one should select the equilibrium with the largest number of messages, because it maximizes R’s and S’s utility ex ante 4 . 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 (e 5 0). 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. 3
See their Theorems 4 and 5. Farrell (1993) challenges the robustness of this argument. He defines a criterion called ‘‘neologism-proofness’’, and shows that the equilibria of C–S do not satisfy this criterion. 4
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] Lemma 1. There exists ss *1 , . . . , s p*d with s] # s *1 # ? ? ? # s *p #s] such that the partition of [s,s] ] ] defined by the intervals Ii (with Ii 5[s i21 ,s i ] if 2 , i # p, I1 5 [s,s ] 1 ] and Ip 11 5 [s p ,s]) characterizes an equilibrium of the communication game in the case where S’ s and R’ s preferences are identical ( i.e., in the case where e 5 0). Proof of Lemma 1. We fix some large enough integer p. Let us define the following notations: Q p 5hss 1 , . . . , s pd such that s] # s 1 # ? ? ? # s p #s]j ] ] T 5hx [ [x,x] ] such that x 5 Arg max U(x,s) for some sj 5 [Arg maxU(.,s), ] Arg max U(.,s)] Zp 5hsx 1 , . . . , x pd [ T p such that x 1 # ? ? ? # x pj Given ss 1 , . . . , s pd [ Q p , we define ] Ii 5 [s i21 ,s i ] if 2 , i # p, I1 5 [s,s ] 1 ] and Ip 11 5 [s p ,s] ] we also define the length of I by Given an interval I included in [s,s], ] l(I) 5 Max(I) 2 Min(I) ] We also define, for every s in [s,s] ] and every x in T, x(s) and s(x) given, respectively, by x(s) 5 Arg max U(.,s) x 5 Arg max U [.,s(x)]
H
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 ss,s9d [ Q 2 and (x,x9) [ Z2 we define h(s,s9) and k(x,x9) by s9
5
E
h(s,s9) 5 Arg max U(.,s )m (s ) ds s
U [x,k(x,x9)] 5 U [x9,k(x,x9)]
We define then, for (s 1 , . . . , s p ) [ Q p and sx 1 , . . . , x p11d [ Zp11 : ] H(s 1 , . . . , s p ) 5fh(s,s ] 1 ), h(s 1 ,s 2 ), . . . , h(s p 21 ,s p ), h(s p ,s)g K(x 1 , . . . , x p 11 ) 5fk(x 1 ,x 2 ), . . . , k(x p ,x p 11 )g
H
Clearly H maps Q p into Zp 11 . Similarly, if sx 1 , . . . , x p11d [ Zp11 then Conditions (1) and (2) imply that s(x i21 ) # k(x i21 ,x i ) # s(x i ), implying k(x i21 ,x i ) # k(x i ,x i 11 ). Therefore K maps Zp 11 into Q p . Also, Conditions (1) and (2) imply that H and K are continuous. Therefore KOH is a continuous function mapping the convex, compact set Q p into itself. Brouwer’s theorem implies the existence of a fixed point ss *1 , . . . , s *p d of KOH in Q p . ] * * Finally, the partition of [s,s] ] induced by ss 1 , . . . , s p d is an equilibrium partition if e 5 0: if S sends
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the message m i after observing s in Ii , then R’s bayesian updating leads him to pick h(s i21 ,s i ). Conversely, if for each i the message m i causes R to choose x i , then it is optimal for S to send m i only when observing s in fk(x i21 ,x i ),k(x i ,x i 11 )g. Therefore ss 1 , . . . , s pd [ Q p defines an equilibrium partition if and only if it is a fixed point of KOH. h Lemma 2. If for any ss,s9d [ Q 2 , we define the functions X(s,s9) and S(s,s9) implicitly, by the equalities ≠ ≠ s9 5 k[x(s),X(s,s9)] and x(s9) 5 hhs,s[S(s,s9)]j, then ]fS(s,s9)g 5 ]hs[X(s,s9)]j 5 2. ≠s9 s95s ≠s9 s95s
U
U
Proof of Lemma 2. The identity ≠U / ≠x[x(s9),s9] 5 0, 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
U
U UD
≠ 2U ]] U [x(s),s9] 2 U [X(s,s9),s9] 2hfx(s9) 2 x(s)g 2fx(s9) 2 X(s,s9)g j 2 [x(s9),s9] ≠x 3 3 3 3 (s9 2 s) ≠U ≠U ≠X fX(s,s9) 2 x(s)g # ]]]]]Max ]] # ]]]Max ]] Max ]ss 1 ,s 2d 3 3 3 3 ≠s9 (s ,s )[Q ≠x ≠x 1 2 2 2
U U
2
U US
U
3
or
U S
U
3
U U
(s9 2 s) ≠ 2U ≠ 3U ]] ]]] ]] Max hfx(s9) 2 x(s)g 2fx(s9) 2 X(s,s9)g j 2 [x(s9),s9] # 3 3 ≠x ≠x 3 ≠X Max ]ss 1 ,s 2d (s 1 ,s 2 )[Q 2 ≠s9 2
U
2
UD
This implies that as s9 converges to s, X(s,s9) 2 x(s9) U]]]]] x(s9) 2 x(s) U converges toward 1, which together with the inequality x(s) # x(s9) # X(s,s9) implies that X(s,s9) 2 x(s) ]]]] x(s9) 2 x(s) converges toward 2, or X(s,s9) 5 x(s) 1 2[x(s9) 2 x(s)] 1 Of(s9 2 s)2g This equality and the identity s9[x(s)]x9(s) 5 1 implies s[X(s,s9)] 5 shx(s) 1 2[x(s9) 2 x(s)] 1 Of(s9 2 s)2gj 5 s[x(s)] 1 2s9[x(s)][x(s9) 2 x(s)] 1 Ofss9 2 sd 2g 5 s 1 2s9[x(s)]hx9(s)(s9 2 s) 1 Ofss9 2 sd 2gj 1 Ofss9 2 sd 2g 5 s 1 2(s9 2 s) 1 Ofss9 2 sd 2g which establishes the first identity. Similarly, a second-order development of U(.,s9) between x(s9) and x(s) on the one hand, between x(s9) and x[S(s,s9)] on the other hand, implies the second identity. h
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Lemma 3. There exists B .0 such that if ss *1 , . . . , s *p d is a fixed point of KOH, then, writing l i for l(Ii ), the inequality 2
2
ul i 11 2 l iu , B ? Minsl i ,l i 11d
holds for 1 # i # p 2 1. 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:
H
usS[s,s9) 2 s9d] 2 (s9 2 s)u # Ass9 2 sd 2
(A.1)
uhsfX(s,s9)g 2 s9j 2 (s9 2 s)u # Ass9 2 sd 2
Assume that ss *1 , . . . , s p*d is a fixed point of KOH. Then, the definitions of h and k imply that for ] * * * * * * * * * 1 # i # p [writing s *0 for s, ] s p 11 for s, and x i for hss i21 ,s i d], s i21 # s(x i ) # s i # s(x i 11 ) # s i 11 , and * * * * * * * * * s i 5 Sfs i21 ,ssx i dg, x i 11 5 Xfssx i d,s i g, and s i 11 5 Sfs i ,ssx i 11dg. These identities and Condition (A.1) imply that, with the notations l1 5 s(x i* ) 2 s *i21 , l2 5 s *i 2 s(x *i ), l3 5 s(x i*11 ) 2 s *i , and l4 5 s *i 11 2 s(x *i 11 ): 2
2
2
u l2 2 l1u # Al 1 ; u l3 2 l2u # Al 2 ; u l4 2 l3u # Al 3
leading, after a few manipulations (using the fact that each li is smaller than one) to Max u li 2 l1u # Kl 12 with K 5 Ah1 1 (1 1 A)2g 1 A2f1 1 A[1 1 (1 1 A)2 ]j 2
2#i #4
This and the identities l i 21 5 l1 1 l2 and l i 5 l3 1 l4 imply therefore 2
2
ul i 11 2 l iu # 3Kl 1 , 3Kl i 2
Similarly, ul i 11 2 l iu , 3Kl i 11 , yielding the result with B53K. h Lemma 4. If p is large enough, then Max(l 1 ,..,l p ) # p 21 exp[2B(B 1 1)]. Proof of Lemma 4. We consider some p such that p . (B 1 1) exp[2B(B 1 1)]. The identity l 1 1 ? ? ? 1 l p 5 1 implies the existence of i 1 such that l i 1 # 1 /p. Let us assume that Max(l 1 , . . . ,l p ) . p 21 exp[2B(B 1 1)] and (for example) that for some i* , i 1 , l i * . p 21 exp[2B(B 1 1)]. We define i 0 5 Maxhiui , i 1 and l i . p 21 exp[2B(B 1 1)]j. For every i [si 0 ,i 1g, one can write l i 5 n i21 with n i $ p exp[22B(B 1 1)] . B 1 1. Lemma 3 implies that for i [si 0 ,i 1g, 2
l i 11 $ l i 2 Bl i or
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n 2i Bn i n i 11 # ]] 5 n i 1 ]] # n i 1 B(B 1 1) ni 2 B ni 2 B since the function (xB) /(x 2 B) is a decreasing function of x and n i . B 1 1. Therefore, q5Int
i1
1$
S
n i 2n i 1 0 ]] B(B 11 )
Ol $ O i
i 5i 0
q 51
D
O
1 1 ]]]]] 5 ]]] n i 0 1 qB(B 1 1) B(B 1 1) n integer
n i ,nB(B 11 ),n i
F S
$fB(B 1 1)g 21
ni 1 log ]]]]] n i 0 1 B(B 1 1)
DG
0
S
1 ] n 1
p $fB(B 1 1)g 21 log ]]]]]]]]]] p exp[22B(B 1 1)] 1 B(B 1 1)
D
which is strictly greater than 1 if p is large enough, since the last expression converges toward 2 as p tends to infinity. This leads to 1.1, which is impossible. Therefore the inequality Max(l 1 ,..,l p ) . p 21 exp[2B(B 1 1)], assumed above, cannot hold if p is large enough. h We write hereafter A( p) for 2p 21 exp[2B(B 1 1)]. Lemma 5. If e .0 is small enough, then there exists ss e*1 , . . . , s e*pd [ Q p such that ( i) the partition * * * * ] fs,s ] e 1d, fs e 1 ,s e 2d, . . . , fs e p ,sg is an equilibrium partition of the game where] R’ s utility function is (U 1 eV ). ( ii) If we define le i 5 s *e i 2 s e*i 21 (writing s e*0 for ]s and s e*p 11 for s), then Max(le 1 ,..,le p ) # A( p) Proof of Lemma 5. We define, for ss,s9d [ Q 2 and (s 1 , . . . , s p ) [ Q p s9
E
he (s,s9) 5 Arg max sU 1 eVd(.,s )m (s ) ds s
and ] He (s 1 , . . . , s p ) 5fh e (s,s ] 1 ),he (s 1 ,s 2 ), . . . , he (s p ,s)g Consider the function Le 5 Id 2 KOHe , from Q p into R p . Since the differential of L0 at ss *1 , . . . , s *p d is a linear application from a p-dimensional vector space into itself, it has generically full rank (i.e., a rank equal to p)5 . By the implicit function theorem, this implies that for any neighborhood V of ss *1 , . . . , s *p d, there exists e0 . 0 such that if e , e0 , the equation Le (s 1 , . . . , sp ) 5 0 has a solution ss *e 1 , . . . , s *e pd in V. But the equality Lef(s *e 1 , . . . , s *e p )g 5 0, equivalent to (KOHe )f(s e*1 , . . . , s *e p )g 5 (s *e 1 , . . . , s *e p ), means that (s *e 1 , . . . , s *e p ) defines an equilibrium partition in the case where R’s utility function is (U 1 eV ). If V is small enough, then each l i 2 le i is close enough to zero, so that the
5
One can in fact prove that this differential is of rank p for sure, if p is large enough (proof available from the author).
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inequality Max(l 1 ,..,l p ) # A( p) / 2 from Lemma 4 implies that Max(le 1 , . . . , le p ) # A( p). This completes the proof of the proposition. h
References Crawford, V., Sobel, J., 1982. Strategic information transmission. Econometrica 50, 1431–1451. Farrell, J., 1993. Meaning and credibility in cheap-talk games. Games Economic Behav. 5, 514–531.