Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium Member Libraries http://www.archive.org/details/credibleneologisOOfarr HB31 M415 • Dewey 7tfi»3^ working paper department dec f of economics Credible Neologisms in Games of Communication Joseph Farrell No. 386 June 1985 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 \m Credible Neologisms in Games of Communication Joseph Farrell No. 386 June 1985 Credible Neologisms in Games of Communication by Joseph Farrell GTE Laboratories 40 Sylvan Road Waltham, MA 02254 claA December 1983, M.\.T. revised June 1985 Thanks are due to many people for the much-delayed revision of this paper. Vincent Crawford, Jerry Green, Eric Maskin, Andrew McLennan, and especially Roger Myerson and Joel Sobel have contributed ideas and encouragement. A special debt is due to Suzanne Scotchmer and Joel Sobel, both of whom kept encouraging me. I also wish to thank Kathleen Stevens for programming assistance Introduction 1. Little attention has been paid to communication in games: the most part, for game theorists think as if no player would ever trust another (as is the case with two-person zero-sum games). But an important role of language is coordination of actions, and game theory is the natural discipline in which to study language from that point of view, In [3], showed that as Lewis [9] showed. we defined communication equilibrium in games, it is a strict and useful extension of Bayesian like that used by Crawford equilibrium. However, and Sobel and Green and Stokey [4], [2] that notion, the way language is interpreted, In and requires only that given there is no incentive to lie . the present paper we extend the analysis by examining incentives to introduce credible neologisms. Our principal interpretation This can be viewed in two ways. is that it is an equilibrium condition that nobody should have an incentive to introduce has, credible neologism: for, if someone then the purported equilibrium does not describe how the game will be played. a a We elaborate on this below, neologism-proof equilibrium. when we define Another related approach, which we only hint at here in Section 5, is to ask when an evolving language will develop neologisms that will not immediately be discredited by lies: truthfully, at a self-signaling neologism will be used least at first. More technically, the paper can be viewed as examining the implications of in a certain restriction on disequilibrium beliefs a The restriction is that, sequential equilibrium. set X to what if there is of types that strictly prefer being treated as a group they get in equilibrium, to treated as X and if no other types would wish then it must be possible for the message-sender X, to induce the belief X. This restriction is similar to that independently proposed by Grossman and Perry [5]. However, they, like the literature on stability of signaling equilibrium (Banks and Sobel [1], Kohlberg and Mertens payoffs. Kreps [6], [7]) consider signals that directly affect Their criteria, being based on dominance arguments, have no force if, as here, signals do not directly affect payoffs. The paper is organized as follows. simple communication game, (an X with the properties there is such is a Section 2, we define a and a sequential equilibrium in such a We introduce the notion of game. In a self-signaling set of types just described), and argue that if set then the equilibrium is unpersuas ive vulnerable to the introduction of a , as it credible neologism. An equilibrium not subject to such an objection we call neologismproof . In Section 3, concept, we give four examples to illustrate our and relate it to Myerson's notion mechanism. [10] of core We show by example that neologism-proof equilibrium need not exist In Section 4, proof equilibrium. we prove some existence results for neologismIn Section 5, we consider what happens if agents repeatedly play (adjusting towards best responses) a that has no neologism-proof equilibrium. the our example, In game average result depends on relative speeds of adjustment, and we suggest that the usual game- theoret ic paradigm (in which all those speeds are assumed infinite) may be misleading. Section 6 concludes 2 Assumptions and Definitions . 2.1 Simple Communication Games First, we define our object of analysis: communication game G. and the Receiver (R). finite, (S) has payoff-relevant private information which we sometimes call his type t€T, imple the Sender players: There are two S a s and that R's prior -n on T, . We assume that which is T is common knowledge, assigns strictly positive probability to each type t. The only choice that enters directly into payoffs is R's choice of an action a on t. u e A, where is a payoffs also depend fixed finite set; Players maximize their expected payoffs (a, t) S, possible messages; as u (a,t) and . The Sender, of M A chooses R a message m from a class M then learns m before choosing infinite but discrete - for instance, (arbitrarily long) utterances in English. a. of We think the set of all 2.2 Sequential Equilibrium sequential equilibrium A G = * R S (T,A,M ,u (•,«),u («,•), function A (i) (Kreps and Wilson, T s: -» A(M ti ) ) A function p : which assigns to each teT probability distribution p(m) on (iii) A function r : probability distribution r(m) on The functions s and to each meM a to each meM a T. A. are S's and R's Bayesian r strategiesrespectively. a . A(A) which assigns - M M A(T) which assigns - M of the game consists of the following: probability distribution s(t) on (ii) [8]) The distribution p(m) R's posterior beliefs about t€T, on T represents after hearing the message m. These functions must satisfy three conditions: First, for each teT and each meM n(t)s(t)(m) = p(m)(t) t |: T , -n(t)s(t) (m) (2.1) This condition says that the probability distributions on T*M described by the same. -n ( • ) and s(»), In other words, and by p(») and s(»), must be given s(»), R's posteriors must be consistent with his priors on We can re-write (2.1) t. more in familiar Bayes' rule form by dividing by the prior probability of m, q(m) = . £_"n ( t ) s ( t ) (m) if q(m) , 0, > but so as to allow for zero-probability messages we use the form (2.1), which can be rewritten as : T,(t)s(t)(m) = q(m)p(m)(t) (2.2) The second condition is that R's choice of action be optimal Write given his beliefs. v R (m) sup = [ , E.p m) ( ( t ) optimized expected payoff when he hears message u R m. (a,t)] for R's Then we require that: r(m)(a) > t Z p(m) T R ( t ) ( a, t Our third requirement is that, r(#), ) = v R given R's response function S's choices of messages should be optimal. syn^t Z.r(m)(a)u (a,t)] he observes t. s(t)(m) > Having defined a a S r(m)(a)u A Define v g (t) = to be S's optimized expected payoff when Then we require - (2.3) (m) 2 S (a,t) = v S (t) (2.4) sequential equilibrium, we prove sequential equilibrium always exists: that a Propos i t i on 1 Every simple communication game has : a sequential equil ibr ium. Proof f(») : Let a A be € optimal for R given his prior beliefs be any probability distribution on M s(t) = f(») for all t€T p(m) = n for all meM ( • ) "1 (m)(a) = (J 10 if L ?. a a = Define: . In this equilibrium, communicated: we call of it the we define a notion of equivalence of equilibria. We t Let 3 otherwise no information about . for all meM . This is a sequential equilibrium. course, * -n is uncommunicative equilibrium. 2.3 Outcomes and Equivalence of Equilibria Next, are not really concerned with the choice of messages; care about how communication affects the outcome of G. the outcome w of an equilibrium (s,p,r) from T to A( A ) given by: of G to be the rather, we We define function vv w(t)(a) probability that = Z meM t is a chosen if t is the state; s(t) (m)r(m) (a) (2.5) Two equilibria are equivalent if they have the same outcome. While there are always infinitely many sequential equilibria there may be many fewer equivalence classes, language slightly by saying (e.g.) that (s,p,r) unique, when is (s,p,r). Further Notation 2.4 Let X be a non-empty subset of T. probability distribution on T "(t |X) f = t X We write n ( • | X for the ) given by (t) if teX Tt(t') otherwise We write a (X) ( • , and we often abuse we mean that every equilibrium is equivalent to n 4 | X Since ) , T for R's optimal action if his beliefs are and assume that a*(X) and A are finite, is unique for each this is generically true. holds if R's preferences are strictly concave in Crawford and Sobel s a, also It as in [2]. We write v(X,t) v(X,t) X. for t's payoff if R has beliefs u S (a*(X),t). tt ( • | X ) : Given a response rule (r,t) = sup^ meM and recall that, v(r,t) is a £A if achieved. on R's part, r r(m)(a)u (a,t) part of is r we define t's payoff: a sequential equilibrium, then We define the set P(X,r) of types who strictly prefer to be thought of as unknown members of be treated according to P(X,r) = {teT | than to r: v(X,t) v(r,t)} > (2.6) When there is no danger of confusion, we write P(X) 2.5 X for P(X,r) Self-Signaling Subsets and Neologism-Proof Equilibrium We say that the subset P(X) = is X self-signaling if (2.7) X Now suppose that in a sequential equilibrium (s,p,r), confidently believes that and suppose that, S expects to be treated according to instead of sending probability in equilibrium, S "t to expect that R should believe that in X." If a r; message sent with positive sends an unexpected message or neologism is R (2.7) holds, t then it seems reasonable is in X. For if teX, then (by (2.7)) S would not try to persuade every type t with teX would try to persuade submit, a self-signaling message Moreover, that t€X. R R that teX. a we highly compelling. is Based on this, we give the following definition: equilibrium (s,p,r) of Thus, A sequential simple communication game is neo logism- proof if there is no non-empty subset X of T that is self- signaling relative to the payoffs v(r,t). sequential equilibrium is A in equilibrium, lying. language (set of messages used a and their interpretations p) that will not induce Our concept of neologism-proof equilibrium not only requires that (we start with a sequential equilibrium), but also requires that no credible neologisms (new messages, literally "new words") will be used. There is always more language available than is used in non-equilibrium messages exist given game: 5 Moreover, . non-equilibrium messages have focal meanings: meanings. Thus it is unreasonable to assume, sequential equilibrium does, there is that t is in a these their literal as the concept of that R will not understand neologisms. We will assume that, T, a for every non-empty subset X of neologism n(X) that (it is common knowledge) claims X. We will focus on whether n(X) While we certainly should not assume that is R credib le is . credulous enough to believe arbitrary statements, we also should not assume that he will not believe a convincingly credible neologism. Because a self-signaling neologism is knowledge focal meaning is such that a S message whose commonwould like R to believe it 10 if and only if it is we assert that only true, Receiver would refuse to believe Accordingly, we assume that, is available, that t is in then X. S S paranoid self-signaling neologism. a if a self-signaling neologism n(X) can (and, if t is in X, will) make R believe since all and only types Moreover, t would X € R's beliefs ought to be simply use the neologism n(X), Therefore, a ti | knows that by using n(X) he can induce beliefs and action a*(X). So it is a Myerson [10] has discussed "core mechanism") the present paper. | X a in which fewer X. related equilibrium concept (the neologisms are credible than in He requires that a neologism (in his terms, proposed alternative mechanism) be preferred by some set and be incentive-compatible (have R making to his beliefs) -n compelling restriction on sequential equilibrium that there be no self-signaling subset types, X whatever R's beliefs "between" tt a and X a of best response n | X . This stronger requirement on neologisms makes fewer neologisms credible, proof. and thus makes more sequential equilibria neologism- Myerson obtains an existence theorem (his Theorem 4), while we show below in a very simple example that neologism-proof equilibrium need not exist. 11 3 . Some Simple Examples we analyse the sequential and the neologism- In this section, proof equilibria of some simple communication games. We show how the requirement of neologism-proof ness eliminates many "unsatisfactory" sequential equilibria. even in non-pathological games, fail to exist Example We will also see that, neologism-proof equilibrium may . Eliminating an Uncommunicative Equilibrium 1: there As we saw above, is always a sequential equilibrium in which different types are treated alike, and any attempt to communicate 1 ignored, is even if it is entirely credible. Example shows how our condition of neologism-proof ness may rule out this equilibrium. We define a simple communication game as follows: T (t = TT(t 1 A = ) { 1 ,t = a, 2} T,(t , 2 a„ a„ , 1/2 ) } Payoffs can be concisely described as follows 12 S R ' ' s s payoff 1 2 1 -2 -2 -1 payoff : 2 3 2 R's best responses in terms of his posterior probability p that is t , are S 13 " a*(p) = if p < 2 a, 1/3 if p > 2/3 a if 1/3 a 3 p < 2/3 < the uncommunicative one, There are two sequential equilibria: and one in which equilibrium is S reveals his type to better (ex-ante) for S. R. The uncommunicative However, even if S can propose behavior ex-ante, he cannot get that allocation, because if he turns out to be of type 1 neologism. That is, is payoff S X = {t,} he will use a self-signaling self-signaling relative to the gets in the uncommunicative equilibrium. equilibrium is not neologism-proof. Therefore that The revealing equilibrium is neologism-proof. Example 2: Eliminating an Informative Equilibrium This game also has an informative and an un inf ormat i ve sequential equilibrium. However, in this case it is the informative equilibrium that fails to be neologism-proof. The game is the same as in Example is changed from 1 to -1. 1, except that u (a.,t.) Now S's payoffs are: 14 '2 a -1 -2 -2 1 l a a 2 ° 3 If R responds others with a9 However, is it , you the state; to some equilibrium messages with a, and to all then truth-telling is a sequential equilibrium. vulnerable to the neologism n(T): since this better is (0 won't tell and for -1) for t That is, X > you shouldn't infer anything about t." "I -, = T therefore the revealing equilibrium Example In 3: is 9 , self- is signaling relative to payoffs resulting from revelation, t and not neologism-proof. No neologism-proof equilibrium this example, we produce a game that has no neologism- proof equilibrium. As A = in examples {a, ,a2,a„}. and {t,,t ? }, 1/2 = -n(t 9 ), R's payoffs are also as in examples 1 and 1 S's payoffs are given by: 2, T = n ( t , ) = 2. and 15 a a 1 2 2 -1 -1 -2 2 ° 3 There is now just one sequential equilibrium outcome: a„ is chosen with probability one in both states. self-signaling relative to equilibrium payoffs. is no But action {t,} Therefore, is there neologism-proof equilibrium. Example 4: The C rawf ord-Sobe 1 Game We now show that the example discussed by Crawford and Sobel ([2], Section 4) often fails to have equilibrium, a neologism-proof and never has a neologism-proof equilibrium with communication. The game has infinite type and action spaces, but concavity ensures that it is well-behaved. prior ti ( • ) Payoffs are is uniform. T = [0,1], The action space A is also and the [0,1]. 16 u u R S 2 (a,t) - ~(a-t) (a,t) = -(a-(t+b)) 2 (b Crawford and Sobel show that, N satisfying 2N(N-l)b < 1, 0) > if N(b) integer the largest is then there is one sequential equilibrium for each integer N £ ) u...u [X i The N-equi 1 ibr ium is N(b). 1 described by the partition [0,1] [X = Q ,X ,X [X u ) 1 1 2 N_ ,X 1 N where X. i/N = 2bi(i-N) + If te[X. ,,X.), l-l i (0 < i IN). then R chooses the action a = (X. ,+X.)/2 in l-l l equi librium. Since always wants a to be higher than R wants S might expect that, if signaling neologism For such an t = is very close to 1, S S y, is X = (y,l], where y to be self-signaling, X > a self- X.,,. it must be the case that and using his equilibrium message (thus a = [l+y]/2) inducing a = [1+X„ ,]/2). This implies that y+b , just half way between those two possible actions: - one 0), indifferent between using the neologism (thus inducing (l+y)/2 > will try to reveal We therefore investigate whether there is the fact. when t (b [y+b] - y+b - (l+X _ )/2 N 1 S's ideal, is 17 This implies that + y - 4y - 4b 1 3y + = 2 - 4b + X = 2 - 4b + [N-l]/N = 3- 1/N - 2b(N+l) X N-1 1 ' N-l N_ 1 To check whether this ( + 1 - 2b(N-l) describes value of a y the range in we write ') ' y = [l/N+2b(N+l) ]/3 - 1 1 < and 3(y-X _ N 1 Thus, if N > and only if 3/N = ) 6b(N-l) + = 2/N + 4bN = 2/N + 4b(N-2) y € (X 2, b < 1/2. - - 1/N - 2b(N+l) 8b ,,1). Hence, then y If N = 1, if b proof equilibrium in this example; < 1/2, if b > € there is 1/2, (X N _ 1> n_o 1) neologism- the uncommunicative equilibrium is the unique neologism-proof equilibrium. (It is if also the unique sequential equilibrium.) 18 Discussion of the Examples Example 1 : For the uncommunicative outcome to be equilibrium requires beliefs p(m) in M it (i) is implausible that all the messages will be used for meaningless babble; i.e. S will be something like "Really, since such and (iii) message has a would want believed by that R will believe it. R a focal we assume that (ii) some message not used in equilibrium will have meaning, , We find such beliefs whether or not used in equilibrium. implausible because sequential for all meM [1/3,2/3] € a t, t, as focal has occurred"; (literal) meaning that if and only if it is true, we assume Thus the possibility of "unauthorized" communication rules out the outcome in which R always takes action a„ : it is a sequential equilibrium but is not neologism- proof. Example 2 : The notion of neologism-proof equilibrium does not simply mean selecting a most-communicative sequential equilibrium, as one might suspect. 19 Example 3 : We do not propose as a solution concept that type-l's can induce action and type-2's can induce action a„ a, is The fact that the sequential plainly not an equilibrium. equilibrium (in which all get a„) type-l's to get This . is blocked by the ability of does not imply that there is an equilibrium in a, which they do so. This language suggests Type t, can get payoff of a core-like approach to the problem. a 2 can separate himself. if he The usual core concept would automatically allow him to do so, that the types will be treated separately. context, our in and being so treated is if pretending to be t, attractive to t~, However, so relative to his alternative, then he cannot be "kept out" by t,. In Myerson's in [10] the uncommunicative equilibrium framework, this game is a core mechanism. According to the concept of core mechanism, type-1 cannot say "I am of type you take action a, if I ; notice that were in fact of type 2": credibility of a I 1; I propose that would not wish to propose this Myerson's requirement for neologism insists that R's proposed action be a best response to his beliefs even if his beliefs do not shift as a result of the neologism. Another way of seeing the contrast is as follows: Myerson's framework, In the neologism consists of proposing allocation that is an equilibrium if R makes inferences appropriate direction) from the proposal, and is also an a (in new the 20 our framework, equilibrium if he does not. In not worry about what happens if R fails the proposer need to make the appropriate inference, because we assume that the message is so credible that will make the inference. R What will happen in this game, given that there is no Clearly, we cannot give convincing equilibrium? equilibrium argument for any outcome. 4 . it is interesting happens if agents adjust at finite speed towards best to see what See Section responses. However, convincing a below. 5 Existence Results Examples and 3 above show that we cannot generally expect 4 neologism-proof equilibrium to exist. a this section we provide In some sufficient conditions for existence. Our first result concerns games in which S's preferences over R's beliefs are unrelated to S's true type. for instance, case, in This is often the labor-market models in simple which private information concerns "ability." say that S's preferences are uniform if, all a,,a„ u S (a for all t, , t~ e T A, e ,t 1 Formally, we ) 1 > u S (a 2 ,t ) 1 - u S (a ,t : 2 ) > u S (a 2 ,t 2 ) (4.1) and 21 Since we assume that is a * " R's best response to beliefs (X), unique for each non-empty subset X £ T, (4.1) -n implies that | X T itself is the only possible self-signaling subset (since in equilibrium each type must be indifferent among all equilibrium messages). It Propos it ion 2 follows that: If S's : preferences are uniform, uncommunicative equilibrium then the neologism-proof. is Our second result concerns games of pure conflict: that is, the case where S's and R's preferences over actions are always diametrically opposed: prefers to a. b if, given t, S prefers a to b includes the zero-sum case.) (This In then R this case, there can be no self-signaling subset relative to the no- communication equilibrium. signaling subset. v(X, t) By assumption, u R > To see this, Then we know that v(T,t) this (a*(X),t) if teX implies: < u R (a*(T),t) and hence R's expected payoff, action a (X) than from action definition of a (X). Hence we get: let X be a self- if teX; given teX, a (T). is strictly lower from But this contradicts the 22 Propos i t i on 3 equilibrium is (In [3], In : a game of pure conflict, the uncommunicative neologism-proof. we also showed that every sequential equilibrium in a two-person zero-sum game has the same payoffs as the uncommunicative equilibrium.) These two results concern cases in which there can be no important communication (although it is possible to have equilibria with "inessential" communication ). We now turn to the harder task of providing partial existence results when communication matters. in First, we note that, in the case of a game which S's and R's interests coincide completely, revealing equilibrium (which always exists) neologism-proof. is the fully- necessarily (We would like to be able to state that this the unique neologism-proof equilibrium in such a game. is T true when T has only one, has four elements: two or three elements, see the Appendix.) it is While this fails when 23 We now give some less intuitive conditions for existence. In the process, we see why equilibrium fails to exist Example 3: this, in and similar insight, our motivation for is inquiring into some intrinsically unappealing conditions. We say that S's preferences are quas i-linear if, and all non-empty A,B min[v(A, v(B,b) t) ] c^ T, < v(AuB,t) Notice that Example t„, A = {t,}, B = 3 {t„}). < : If S's [ v ( A , t ) , v ( B , t ) (4.2) ] violates this condition (look at t = The condition is a combination of quas i-concavi ty and quas i-convexi ty Propos it ion 4 max for all teT , hence the name. preferences are quas i- 1 inear , then either the uncommunicative sequential equilibrium is neologism-proof, there is Proof is a : a In or communicative sequential equilibrium. uncommunicative equilibrium, self-signaling neologism. t gets v(T,t). Then by definition Suppose X 24 v(X,t) v(X,t) When t e v(T\X, and when v(X' , But > v(T, t) i v(T,t) X, (4.3) t) < if t€X v(T, t) € X' t) > v(T,t) equilibrium in v(X, t) < 1 (4.4) they imply that T\X, t (4.4) (4.3) imply that and (4.2) = "I otherwise.] v(X, t) and (4.5) (4.5) imply that there is a sequential which some equilibrium messages mean "t all others mean "t € X'." neologism-proof, X" and This proves Proposition 4. We would like to be able to extend Proposition conditions under which, 6 4 by finding if a sequential equilibrium is not then there exists a more-communicative sequential equilibrium. That would imply the existence of a neologism-proof equilibrium, given that T is finite. (Evidently, the conditions would have to be violated in Example 2.) However, I have been unable to find natural sufficient conditions for that, extension. One result along these lines, but with very stronTT assumptions, is: 25 Propos it ion 5 Consider : sequential equilibrium in pure a strategies which partitions T into u...u T, T, . Let signaling subset wholly contained within one of the Suppose that X' = T,\X is also self-signaling. T. Proof (a) : u X u ' Tn 2 u. . . u a say i=l. : T into T. k We need to show three things: then v(X,t) If teX, v(X,t) 1 v(X',t) and 1 v(T.,t), where j>l. > v(T,,t) since teX and We have v(X,t) vCT^t) > v(X' v(T,,t) 2 v(T.,t) since teT,, t) since t^X' self- Then there is sequential equilibrium in pure strategies partitioning X be X X is self-signaling; and X' is self-signaling; and we began at a sequential equilibrium. a 26 (b) Similarly, we prove that if tcX', and v(X' (c) , v(T 1 t) ., v(X,t) and v(T ,t) > > v(X',t) follow J J J v(X,t) 1 t) then v(T.,t) If teT., v(X',t) immediately from the fact that This proves Proposition X and X' are self-signaling. 5 We can also prove another partial result to the effect that a self-signaling a contained in X partition set can be made part of a finer sequential equilibrium. we assume quasi-linear For this, and an extra condition that ensures that, preferences, if a vague then so is a purportedly more precise lie. lie is unprofitable, We defer the precise condition until the reader has seen what is needed. Propos it ion 6 : Consider strategies, so that each i, all types T in a partitioned into is T. l sequential equilibrium in pure T, induce action a (T.)- u...u T, , If there is X which lies wholly within (say) S's preferences are quasi-linear, X u (T, \X) u 1 u. . . u T, , then the following partition defines another sequential equilibrium in pure strategies: = a l sel'f-sjgna 1 ing neologism T for and, T, . and 27 provided that an extra condition Proof We need to check that none of three classes of types will : want to lie: some T satisfied (see below). is . j , those in those in X, but not in T X, and those in *1 J (a) If teX, then v(X,t) v(T,,t) > 1 will not claim falsely to be in T., are quasi-linear and v(X,t) < v(X,t), (b) £ so that If teX', vCTj.t), it t v(X',t) be in X (c) If teT., Also, Hence t since preferences v(T,,t), we have v(X',t) £ v(T,,t) then since preferences are quasi-linear and v(X,t) follows that v(X',t) for j*l. Also, j*l. for j*l. will not falsely claim to be in X'. original partition was v(T.,t) > v(T.,t) 1 1 vCT-.t). But since the sequential equilibrium, v(T,,t) a > Hence t will not falsely claim to be in T.. v(T,,t) > v(X,t); then we know that v(T ,t) > v(T lf t) and that v(X,t) We need to show that < v(T t) so t will not falsely claim to 28 v(X' t) i v(T,t) Unfortunately, our assumptions so far do not imply this: is possible that teT. would like to lie and claim to be it X'. in J although This would be a somewhat "surprising" case: to tell the truth than to tell the lie T,, If proof of Proposition complete. 6 is A prefers he strictly prefers a this can be ruled out, more precise lie X*£ T,. T then the sufficient assumption is the following: If teT. v(T ,t) and A £ or v(B,t) Next, > B, and v(T , t) B T. n = #, then either v(/>,,t) < . we prove a result that guarantees, assumption than quas i- 1 inear i ty , under a weaker the full-revelation that sequential equilibrium (if it exists) will be neologism-proof. Propos i t ion all teT, u S 7 If S's preferences are quasi-convex, : if for A.BcT, (a*(A u B), t) S < max[u (a*(A) then no self-signaling set In particular, is i.e. if full X t), u S (a*(B),t)] can be a union of partition sets T.. revelation also neologism-proof. , is a sequential equilibrium, it 29 Proof Let : = X u...u T. T, £ T = u...u Tj T . k Then by quasi- convexity applied (j-1) times u so that S X (a*(X),t) partitions Note ) . . . ,u S (a*(T ),t)] j cannot be self-signaling. But now if sets S max[u (a*(T 1 ,t), i T sequential equilibrium in pure strategies a then there can be no self-signaling into singletons, X : Observe how quas i-convexi ty is violated in Example 2 above A further existence result has been obtained by Joel Sobel (personal communication). flO] concept of a This is phrased in terms of Myerson's neutral optimum for class of games which a includes our simple communication games. optimum is a Recall that a neutral sequential equilibrium (Myerson uses more general terminology) which maximizes a weighted sum X\(t)v sequential equilibria, where the weights x(t) 2 among (t) must satisfy additional conditions which generically determine them. Sometimes, in a neutral optimum some of the weights x(t) will be zero. Sobel defines optimum that has x(t) existence of shows that, a strict neutral optimum to be a for all > teT. neutral optimum, but if there is a neologism-proof. However, it is neutral Myerson proves the may not be strict. strict neutral optimum, there a Sobel then it is no reason to believe that a 30 strict neutral optimum will "usually" exist. These partial existence results do not imply that neologism- proof equilibrium will "usually" exist. What is one to make of that fact? There is equilibrium, a core-like quality to the notion of neologism-proof as Myerson "core mechanisms." recognised Types, in naming his related concept or groups of types, can block an allocation if they can do better by distinguishing themselves, and if their distinguishing claim is credible. While coalitions in the usual core concept can keep out undesirable members, is not possible here said), so that this (the undesirable members can mimic anything the set of neologism-proof equilibria is not just the core of some related game. However, problems have the same flavor, the non-existence and in the same way they are not to be dismissed as exceptional. What can we predict for games without equilibrium? possibility is to look at how behaves over time. a In Section 5, One reasonable dynamic process we explore that approach. 31 5 . Evolution of Language and Lying In this section we re-examine Example 3 above under the assumption that agents do not instantly optimize against their strategies, but rather move in that direction. opponents' assume a finite speed of introduction of neologisms, and We a finite rate of imitation by type-2's when enough type-l's are using a neologism that using the old strategy means being recognised as type-2 (which, gives the lowest payoff). recall, consists of the positive integers We suppose that M Initially, {1,2,3,...}. time, a types of all S send message and R always responds with action a„. 1 all the This is the (unique) sequential equilibrium. Now, few (f << every k the next neologism is discovered by a periods, of the type l's, 1) who then use it effectively to distinguish themselves from the other S's. simplicity that R's learning to how fast S can change). is We assume for instantaneous (very fast compared Thus, type-l's using discovered neologism induce action a,. a newly Those using an older message induce R's best response to the mix of types using that message In each period, both type-l's and type-2's change their message-sending behavior Thus, if old messages in response to differences in payoffs. are inducing action a„, then type-2's continue to send old messages, but type-l's move (at rate) into sending a new message, if one is a finite available. After 32 enough type-l's have Moved away from the old messages, those messages begin to induce action a, (since most of the S's sending so now it pays them are now type-2's), to for the type-2's Thus the old messages get abandoned, imitate the type-l's. and the new ones become "discredited" by type-2's l's (in preference to being perhaps, while, there is to begin imitating type- identified as type-2's). For some no effective communication. next neologism is discovered, Then the communication by type-l's and begins again. He wrote a simple computer program (available on request) to simulate this process. The proportions of type-l's and of type- 2's that get treated with each of R's three actions on -Figure 1, for different values of k are plotted (the number of periods change to (the rate at which type-l's between neologisms), r better moves), (the corresponding rate for type-2's). Table 1, and In summary statistics are given for various parameter In Table 2, itaiues. s we show the results of regressing the average proportions of the two types treated with the three actions on S*e parameter values. As can be seen, increasing the ...... and s r speed" parameters does net lead to a predominance of action a„ " j . f, 1/k, The fuliy- optimizing model of game theory corresponds to ail parameters being very large: compare, but without more knowledge of how they we cannot properl}' say that the uncommunicative equilibrium correctly describes limiting behavior as the parameters grow large. r 33 6 . Conclus ion In a sequential introduced equilibrium, no agent can profitably lie. We plausible condition expressing the further a equilibrium requirement that no agent can profitably deviate by coining a credible neologism. We showed that implausible sequential equilibria; out all sequential equilibria. behavior in a this rules out some and that sometimes it rules We then simulated adaptive game with no neologism-proof equilibrium, that the predictions of sequential equilibrium were not fulfilled. and found Appendix: The Case full revelation of In this case, u R S = u is a neologism-proof equilibrium, t Pareto dominates every other sequential equilibrium (generically is the |T| = or 1 but if we will show, For = |T| easy to see is |T| =4 suppose 3, the full-revelation equilibrium that If |T| = 3, tins still true, is |T| equally S u (a*(T), then it need not be true. equilibrium was sequential other some neologism- types (otherwise some two types would find themselves in the =2). By scaling payoffs, likely and = t) = u that R (a*(T), can suppose we equilibrium the is that all three types uncommunicative. S Normalize t) for all > t: otherwise, some never appear in P(X) for any X, and we would be reduced to the case S Normalize so that u (a*({t}), t) = Choose t E T. Write a = u S 1 (a*(t is not self-signaling, we know max (a, definition of a*(T), we know c = u d = u e = u (a*({t S S (a*({t (a*({t , t r t r t 2 2 }), t }), t >), t ) 2 3 ) ). t |T| would = 2. for all t. 1 Now write are for all t. t) We can assume that u (a*({t}), "By as Each message used in equilibrium must be used with positive probabil- ity by all three case it 2, only neologism-proof equilibrium. proof. up to equiv- But it need not be the only neologism-proof equilibrium. alence). If , also 1 S ), t ), b = u (a*(t b) > 0: say a + a + b < 0. > 0. ), t ). Since {t } Now since {t d) < 0. t , } is not self-signaling, If min (c, definition of a*({t {t so t }, , than a*({t that a*({t , t }) proves the result for then < d) t }), t }). c+d<l<l+a; since If says it mm =3. d) (c, does better than |T| we know that either that a*({t and > but e > given T: a*(T) or min (c, this contradicts the e }) does 0, then a > given better, + c d + e contradiction. > 0, This Now we show by example that the uncommunica- tive equilibrium can be neologism-proof when |T| =4. (This also shows, inci- dentally, that neologism-proof equilibrium need not be unique, since full-rev- elation is also neologism-proof.) There are four equally likely types, labelled 1, 3, 2, 4. There are fifteen actions, each of which is the optimal action a*(X) for just one non-empty subset X of T. Payoffs can be described as follows: Type Action: a = a*({ }) = 1 1 .1 10 2 1 .1 -10 -10 3 -10 -10 •10 1 .1 4 .1 •10 10 1 12 23 34 41 13 24 .5 .6 .1 -10 -10 .6 .6 .1 .1 10 6 .6 .6 .1 10 .6 .6 10 .6 .1 10 .6 - 1 . 123. - .1 .9 .9 -10 234 -10 .9 .9 341 412 .9 .1 10 .1 .9 .9 .9 10 -.1 1234 easy to It is For example, proof. P({t 2 - (t 3 . check = )) t 4 (One {t t , }. that the uncommunicative equilibrium is neologism- the neologism {t_} fails because The neologism {t 2< t , 3 t 4 ) fails because P({t , t^, t^}) 2 >. could (even though not including not imagine in fact, a neologism {t } would be believed self-signaling) because R v:ould think of S's alternative as only the This line of argument low in qeneral. that, equilibrium but also some other possible neologisms. seems persuasive in this case, but is difficult to fol- Figure 1: Two graphs of the proportion of type-l's inducing action a . In The relatively small disconthe first, we see discontinuities of two types. tinuity at T = 250 is due to the introduction of a neologism (on the part of The other discontinuities, including the large one, re8% of the type-l's). The sult from a critical change in the mixture of types using a message. downward discontinuity results from a neologism becoming discredited by the influx of type-2's. In the second graph, we applied a smoothing algorithm and used value of k. We see the oscillatory behavior clearly. a much smaller TABLE .15 (35.9) XM (-4.7) .44 (50.1) YL (8.7) -.05 (-2.6) -.13 (-6.1) (5.3) YM .48 YH (5.1) .18 .54 .29 -.28 (-13.6) .50 .70 (11.3) -.36 (-20.1) -.25 (-13.2) .38 .70 (20.0) .05 .04 (22.9) -.02 (-5.2) .05 (11.0) (10.9) .51 Explanation The variable named e.g., XH denotes the fraction of type-l's (X's) treated as being of high (H) probability of being type 1 (and thus getwho induce action a ting action a ). Thus YM is the fraction of type 2's : (corresponding to middling probability of type 1). Parameters result from regression using 256 data points corresponding to 10, 20, 40, 80; f = .01, Numbers under .04, .08. .02, .04, 08, r = .01, .02, parameters are t-statistics 2 (14.6) .20 .10 (59.5) R -.01 (-1.1) .11 -.14 (-8.1) .48 (63.1) Log(s/.02) (-5.8) .17 .41 (47.6) XH Log(r/.02) -.06 Log(K/40) -.05 Intercept Variable XL 1 .04, .08; s = .01, k = .02, FXL rxM FXH FYL FYM TYH 15 .25 .60 .60 .25 .15 .30 .20 .61 . 19 .41 .55 .04 .30 .60 '.09 .40 .51 .57 .43 .00 .10 .30 . 15 . 13 .24 .63 .60 .25 .15 7 .10 .30 .30 . 18 .60 .22 .41 .55 .04 7 . 10 .30 .60 .08 .42 .50 .56 .44 .00 10 .08 .01 .01 .20 .38 .42 .50 .50 .00 10 .08 .01 .02 .22 .40 .39 .46 .54 .00 10 .08 .01 .04 12 .50 .38 .27 .73 .00 10 .08 .02 .01 .21 .28 .51 .64 .35 .00 10 .08 .02 .02 .25 .31 .44 .60 .38 .02 10 .08 .02 .04 .17 .43 .40 .39 .60 .01 10 .08 .04 .01 .16 .22 .63 .74 .26 .00 10 .08 .04 .02 . 19 .25 .56 .69 .30 .01 10 .08 .04 .04 .25 .32 .44 .61 .36 .03 15 .01 .30 .15 .07 .70 .23 .37 .58 .05 15 .01 .30 .30 .10 .81 .09 .20 .78 .02 15 .01 .30 .60 .08 .59 .33 .39 .59 .02 15 . 10 .30 .15 .09 .71 .20 .37 .58 .05 15 . 10 .30 .30 .09 .81 . 10 .20 .78 .02 15 .10 .30 .60 .08 .58 .34 .39 .58 .02 no .08 .01 .01 .26 .31 .43 .60 .37 .03 40 .08 .01 .02 .15 .48 .37 .33 .65 .02 40 .08 .01 .04 .15 .45 .39 .35 .63 .03 40 .08 .02 .01 .17 .28 .55 .64 .31 .05 40 .08 .02 .02 .26 .34 .41 .58 .38 .03 40 .08 .02 .04 .16 .46 .38 .34 .63 .03 40 .08 .04 .01 .10 .23 .67 .71 .24 .05 40 .08 .04 .02 .15 .30 .54 .61 .32 .08 40 .08 .04 .04 .21 .53 .26 .45 .51 .04 50 .08 .01 .01 .25 .36 .40 .57 .39 .04 80 .08 .01 .01 .25 .34 .40 .58 .39 .03 80 708 .01 .02 .15 .48 .37 .32 .65 .03 80 .08 .01 .04 .15 .48 .37 .32 .66 .02 80 .08 .02 .01 .15 .31 .54 .60 .32 .08 80 .08 .02 .02 .21 .53 .26 .44 .52 .04 80 .08 .02 .04 . 10 .62 .28 .22 .73 .05 80 .08 .04 .01 .08 .24 .67 .61 .25 . 80 .08 .04 .02 . 1 1 .57 .32 .45 .48 .08 80 .08 .04 .04 . 13 .74 .13 .27 .71 .02 K F R S 7 .01 .30 .15 . 7 .01 .30 7 .01 7 I . TAR I F ? 14 FOOTNOTES 1. With more than two players, communication becomes complicated affair. to each other, For example, a much more two players could whisper keeping information secret from a third. Or they could merely pretend to whisper to each other, without actually passing information. We assume that there are two players in order to avoid these problems: thus, whatever is announced is publicly heard. 2. (2.4) implies that the supremum in the definition of attained, which is not otherwise guaranteed since M infinite. attained, If r(») then (t) is is were such that the supremum could not be would have no best response, S v and r(») could not be part of a sequential equilibrium. 3. Since the strategy spaces are infinite, and Wilson 4. Let p [8] permute M do not apply directly. . equilibrium, so is 5. the theorems of Kreps Then if (s,p,r) (p o s,p o o ,r o p sequential is a ), in obvious notation This is why we take M* to be infinite. With A and T finite, it is possible to show that every sequential equilibrium is equivalent to one in which only finitely many messages are used. For example, let u R (a,t) = = u S Then any pattern of revelation is equi 1 ib r ium (a,t) a for all (a, t)eAxT. neologism-proof References Banks, J., and J. Sobel, "Equilibrium Selection in Signaling Games," mimeo, San Diego, 1985. [2] Crawford, V., and J. Sobel, "Strategic Information Transmission," Economet r ica 50 (1982), pp. 1431-1451. [1] [3] Farrell, J., "Communication Equilibrium in Games," mimeo, GTE Labs, 1985. [4] Green, [5] Grossman, S., and M. Perry, mimeo, Chicago, 1985. [6] Kohlberg, E., and J.-F. Mertens, "On the Strategic Stability of Equilibria," mimeo, Harvard, 1984. [7] J., and N. Stokey, "A Two-Person Game of Information Transmission," mimeo, Harvard University, 1980. "Perfect Sequential Equilibrium," Kreps, D., "Signaling Games and Stable Equilibria," mimeo, Stanford, 1984. [8] Kreps, [9] Lewis, [10] D., and R. Wilson, "Sequential Equilibria," Econometrica 50 (1982), pp. 863-894. D., Convention 1969. , Cambridge: Harvard University Press, Myerson, R., "Mechanism Design by an Informed Principal," Econometrica 51 (1983), pp. 1767-1797. ^353" 002 MIT LIBRARIES 3 %Q&D ODM SD^J ObO • v -afissSHifi »!sv S£3Wlra5»l2Ka v.. Date Due g^t'81 %n n'@$ NOV tHJL 2191 19 9 199 I Lib-26-67 7)0K Ca& 0^-