Directives, expressives, and motivation

Transcription

1 Theoretical Economics 12 (2017), / Directives, exressives, and motivation Toru Suzuki Economics Disciline Grou, University of Technology Sydney When an agent s motivation is sensitive to how his suervisor thinks about the agent s cometence, the suervisor has to take into account both informational and exressive contents of her message to the agent. This aer shows that the suervisor can credibly exress her trust in the agent s ability only by being unclear about what to do. Suggesting what to do, i.e., directives, could reveal the suervisor s distrust and reduce the agent s equilibrium effort level even though it rovides useful information about the decision environment. There is also an equilibrium in which directives are neutral in exressive content. However, it is shown that neologism roofness favors equilibria in which directives are doubleedged swords. Keywords. Communication games, directives, exressives, economics and language. JEL classification. D Introduction Consider a manager who tells a worker what to do for every single task. Even if the manager s message can resolve uncertainty about the decision environment, such a message could have an adverse effect on the worker s motivation when it sounds disresectful. Words could convey the seaker s attitude toward the listener, i.e., exressive content, and affect the listener s decision in ractice. In fact, the imortance of such a sychological effect of words has been known for a long time: in Aristotle s Rhetoric, suchan effect athos is treated as one of the most imortant forces of words. The urose of this aer is to analyze the trade-off between the informational and exressive contents of messages in a simle communication game. This aer focuses on a communication roblem between a suervisor (she) and an agent (he) that is alicable to manager worker, teacher student, and arent children communications. The agent has two decisions to make: choosing (i) what to do and (ii) how much effort to ut in. The agent is most roductive when what he does matches the state of nature. Both the suervisor and the agent refer higher outut while the Toru Suzuki: toru.suzuki@uts.edu.au I am grateful to two anonymous referees and the co-editor for insightful suggestions. I also thank Francoise Forges, Piero Gottardi, Susumu Imai, Claudio Mezzetti, Masahiro Okuno-Fujiwara, Jonathan Newton, Masatoshi Tsumagari, and articiants of seminars in Queen s University in Belfast, Keio University, UNSW, University of Queensland, University of Sydney, and UTS for helful discussions and comments. I owe secial thanks to Bart Liman and Joel Sobel for detailed comments. Coyright 2017 The Author. Theoretical Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at htt://econtheory.org. DOI: /TE1843

2 176 Toru Suzuki Theoretical Economics 12 (2017) agent also takes into account the disutility of effort. The suervisor who does not know whether the agent knows the state sends a costless message to the agent. Note that there is no conflict of interest in communicating about the state. Thus, with the standard assumtion that ayoffs deend only on the action and the state, the only relevant thing to communicate is the state. As a result, the suervisor communicates efficiently about the state in the most natural equilibrium of the game, that is, she suggests what to do. However, the suervisor s roblem becomes nontrivial if the exressive content of the message could affect the agent s action. Secifically, suose that making additional effort becomes more ainful for the agent if he feels the suervisor s distrust in his ability. The suervisor then has to take into account how the agent could interret the exressive content of her message in equilibrium. I analyze the communication roblem as a chea talk game in two dimensions. The agent is either cometent or incometent deending on whether he can observe the state of nature. The suervisor who does not know the agent s cometence sends a message given the state and her belief about the agent s cometence. The agent who does not know the suervisor s belief about his cometence udates his belief about the suervisor s belief given her message. If the agent is incometent, he also udates his belief about the state given her message. The agent then makes decisions based on his udated belief. To analyze the game, this aer emloys two useful concets, directives and exressives in hilosohy of language. A directive message reveals the true state and induces the right action; an exressive message exresses the suervisor s belief about the agent and could affect the agent s motivation. In this aer, whether a message is directive (exressive) is determined in equilibrium; this aroach allows a message to be both directive and exressive. The game has an equilibrium in which the suervisor always uses a directive message as in the standard setting. However, since the suervisor has an incentive to retend that she trusts the agent s ability, a question is whether she can credibly exress her belief about the agent s cometence in equilibrium. It is shown that such an exressive equilibrium always has a handicaing roerty; that is, the suervisor who trusts the agent s ability always communicates inefficiently about the state to exress her trust. In other words, whenever the suervisor credibly communicates her belief about the agent s ability, she needs to sacrifice informativeness about the state. Since the suervisor has various ways to handica herself in the two-dimensional communication, many kinds of exressive equilibria can exist. To rovide further insights, I focus on exressive equilibria with some degree of consistency in exressive content; that is, whether the exressive content of a message is more ositive than that of another message does not deend on the receiver s cometence. Intuitively, in these equilibria, when a message means trust for the cometent agent, the message does not mean distrust for the incometent agent. This consistency roerty is aealing if we consider the use of a natural language; when the suervisor uses messages so that the equilibrium meaning resects the reexisting meaning to some degree, the equilibrium ossesses the consistency roerty. It is shown that any directive message has negative exressive content in any exressive equilibrium with the consistency roerty. The

3 Theoretical Economics 12 (2017) Directives, exressives, and motivation 177 result might exlain why directives could sound offensive in ordinary communication even though more information about the decision environment should not be harmful to the decision maker. Since there is no conflict of interest in communicating about the state, the main urose of inefficient communication seems to be to exress trust. Thus, I consider exressive equilibria in which whenever the suervisor communicates about the state inefficiently, the urose is urely exressive; that is, the use of the messages does not deend on the state. To characterize such state-indeendent exressive equilibria, I introduce an intuitive communication strategy called simle exressive: the suervisor uses directives when her trust in the agent s ability is lower than a certain level while her message is uninformative about the state if the level of her trust is higher. I show that any state-indeendent exressive equilibrium is ayoff-equivalent to a simle exressive equilibrium. I also rovide a sufficient condition for the existence of a simle exressive equilibrium; a simle exressive equilibrium exists whenever the agent s choice of what to do is sufficiently imortant and difficult. Intuitively, the condition guarantees that the suervisor can effectively handica herself to make her exressive message credible. While directives have negative exressive content in natural exressive equilibria, there also exists an equilibrium that consists entirely of ure directives, which have no exressive content. However, introsection suggests that the equilibrium with ure directives might not be so stable. Suose that the suervisor says I trust you in an organization where the use of ure directives is their convention. If this message could effectively handica herself by inducing an subotimal choice, it might be able to convey her trust overriding the equilibrium. To investigate this observation formally, I emloy the concet of neologism-roof equilibrium. It is shown that whenever a simle exressive equilibrium exists, the equilibrium that consists entirely of ure directives is not neologism-roof. Moreover, simle exressive equilibria are neologism-roof whenever the agent s ayoff function is not too sensitive to the suervisor s trust in the agent s ability. Thus, neologism roofness favors simle exressive equilibria over those with ure directives. Related literature. This aer contributes to the literature on economics and language, esecially ragmatics in economic contexts. 1 Rubinstein (2000) andglazer and Rubinstein (2001, 2006) analyzed a ersuasion roblem by extending the idea of Grice (1989) to a noncooerative situation. In their models, the seaker s statement is verifiable, and the listener chooses a ersuasion rule that secifies how the listener resonds to each statement. The otimal ersuasion rule is then determined as the solution of the listener s otimization roblem. Since the otimal ersuasion rule imlicitly catures how the listener interrets each statement, their aroach essentially ins down the meaning of statements in the noncooerative context. The current aer, on the other hand, analyzes an advising roblem in which the suervisor has to manage not only the informational but also the exressive content of a message to maximize the agent s outut. Since the suervisor s trust is rivate information and unverifiable, my 1 Pragmatics is the branch of linguistics that analyzes uses of language.

4 178 Toru Suzuki Theoretical Economics 12 (2017) model belongs to a class of chea talk games introduced by Crawford and Sobel (1982), and the main interest is in the trade-off between the informational and exressive contents of a message. Thus, the relevant concet in ragmatics is different from Glazer and Rubinstein s. This aer borrows two useful concets directives and exressives from ragmatics. These concets were originally introduced by Searle (1975) to classify effects of utterances a seaker intends, i.e., illocutionary acts. 2 A directive message is an utterance that intends to suggest or order a certain action; an exressive message is an utterance that intends to exress the seaker s feeling or attitude. Imortantly, the seaker s intended meaning of a sentence does not always coincide with the literal meaning but it is determined by the use of the sentence given a context. Thus, the current aer defines directives and exressives as roerties of a communication strategy so that whether a message is directive or/and exressive is determined in equilibrium. The distinction between directives and exressives is useful since their roles are different in this model: a directive message resolves uncertainty of decision environments while an exressive message communicates about the suervisor s attitude and affects the agent s motivation. Another toic that is closely related to my aer is oliteness. One of the major aroaches to understanding oliteness in communication is based on face-saving by Brown and Levinson (1987). According to the theory, some oliteness is intended to avoid face-threatening acts. For examle, a sensible seaker knows that a directive message could threaten a listener s face and so avoids it or uses an indirect exression. If the agent s belief about how the suervisor views the agent s cometence is a roxy for face, the current aer can be interreted as a communication game with face-saving. Unlike the original aroach, which relies on introsection to identify facethreatening acts, my aroach exlains the conditions under which directives could be face-threatening in equilibrium. The idea that the sender s rivate information interferes with factual language is similar to Blume and Board (2013). In their model, the sender s set of available messages deends on his language tye. Since the sender s language tye is rivate information, and his message reflects not only the state but also his language tye, the sender cannot communicate about the state efficiently even in the common interest game. In the current aer, the sender s rivate information is her belief about the receiver s cometence. Thus, the fact that the suervisor uses a directive message could reveal her distrust in the agent s cometence. The difference between the current aer and their aer is not only in the setting but also in the nature of the result. Efficient communication about the state is recluded in their model whereas it is an equilibrium outcome that can be suscetible to an intuitive neologism in the current aer. Thus, whether factual language emerges as a reasonable rediction deends on the situation in the current aer. There are also other chea talk models in which a otentially biased sender tries to communicate about the state, e.g., Morris (2001) and Morgan and Stocken (2003). However, 2 The term illocutionary act was introduced by Austin (1962). Searle (1975) categorized illocutionary acts into five classes: directives, exressives, commissives, assertives, and declaratives.

5 Theoretical Economics 12 (2017) Directives, exressives, and motivation 179 as in Blume and Board (2013), these models reclude efficient communication about the state. The basic idea that chea talk can credibly signal one asect by sacrificing the other asect is similar to Chakraborty and Harbaugh (2010). Unlike in their model, the sender does not talk down one asect to credibly signal the other asect. Instead, the sender makes her message credible by being uninformative about the other asect in the current aer; that is, the oacity of a message exresses the suervisor s trust. In the current aer, the sender can credibly exress her trust since she does not know the receiver s tye. The idea that multile receivers make communication informative is similar to the notion of mutual disciline in Farrell and Gibbons (1989). The difference is that unlike in their model, the two ossible receivers in the current aer have the identical ayoff function but they are different only in rivate information. Finally, this aer also contributes to the literature that studies incentive roblems beyond extrinsic motivation. Bénabou and Tirole (2003) analyze how an incentive scheme should be designed when higher extrinsic incentive could damage the worker s intrinsic motivation. This aer, on the other hand, analyzes how the suervisor can manage the worker s motivation with chea talk. In my model, a directive message could be a double-edged sword: it rovides useful information about the decision environment while it could also damage the worker s motivation because of the exressive content. 2. Model 2.1 Basics There is a suervisor (she) and an agent (he). Let be a finite set of states of nature and let π(ω) be a common rior of ω. The suervisor knows ω whereas the agent may or may not know ω. Secifically, the agent is either cometent or incometent: the agent is cometent if he knows ω whereas he is incometent if he does not know ω. Theagent has two decision roblems. The first roblem is technology choice. In this roblem, he chooses technology a from A ={a ω } ω {a 0 },wherea ω is the otimal technology in state ω, anda 0 is a safe technology. The agent s second roblem is to choose his effort level e from E =[e ē] R + given his technology choice a. The agent s outut deends on (i) the match between technology a and state ω and (ii) his effort level e. If the agent chooses the right technology given ω, i.e., a = a ω at ω, then the outut given e is y(e),wherey : E R +. It is assumed that y(e) is twice differentiable, y (e) > 0 and y (e) 0 for any e E. If the agent chooses a wrong technology, i.e., a = a ω at ω ω, his roductivity is discounted by δ [0 1). That is, the outut given e is δy(e). Finally, if the agent chooses the safe technology, i.e., a = a 0,theroductivity is discounted by λ (0 1), and the outut given e is λy(e) indeendently of ω. Itis assumed that λ>δso that the roductivity is lowest under wrong choices. Let f (e a ω) be the outut function that summarizes the above secification. The suervisor does not know whether the agent is cometent. Let be the suervisor s rior belief that the agent is cometent. That is, measures the suervisor s trust in the agent s cometence. The agent does not know how the suervisor thinks

6 180 Toru Suzuki Theoretical Economics 12 (2017) about his cometence. In otherwords, is the suervisor s rivate information. Assume that follows distribution () with continuous density ψ() and su(ψ) = P =[0 1]. Furthermore, and ω are indeendent, and is common knowledge. The agent s disutility of effort deends on the suervisor s trust in the agent s cometence. Secifically, an additional effort gives him higher disutility if he feels that he is less likely to be trusted. This might be because it is less enjoyable to work under the suervisor who does not trust the agent s cometence. 3 Formally, let c(e ) be the agent s disutility function, which is twice differentiable in each argument. 4 The marginal disutility of effort is strictly ositive and increasing in e given any, i.e., c 1 (e ) > 0 and c 11 (e ) > 0 for any (e ). Moreover, so as to cature the negative effect of distrust on the agent s motivation, assume that c 12 (e ) < 0 for any (e ). Finally, assume c 1 (ē 1)>y (ē) and c 1 (e 0)<y (e) so that the otimal effort level of the cometent agent is in the interior of E given any belief about. The agent s outut is shared by both layers, and the suervisor refers a higher outut level indeendently of ( ω). Secifically, the suervisor s ayoff given (e a ω) is f (e a ω). Since the effort is costly for the agent, the agent s ayoff given (e a ω )is f (e a ω) c(e ). 5 The game roceeds as follows: the suervisor observes ( ω) and sends message m from a sufficiently rich set M. Then, given the message and his cometence, the agent chooses technology a A and his effort level e E. Remark 1. To cature the effect of a message on the agent s motivation, this aer assumes that the agent intrinsically cares about the suervisor s belief as in Bernheim (1994), Köszegi (2006), and other sychological games. 6 Note that a message does not affect the agent s disutility directly. Instead, it affects the agent s ayoff through his equilibrium belief about the suervisor s oinion. This aroach allows us to analyze how the exressive content of a message deends on the arameters that cature the hysical setting behind the communication. The belief-deendent ayoff can be also interreted as a reduced form of a dynamic interaction with the standard ayoff function. 7 However, since the exressive content of 3 In social sychology, Enzle and Anderson (1993) found that surveillance could undermine intrinsic motivation. Moreover, the effect was more ronounced when the subjects interreted the surveillance as a stronger signal of distrust. 4 Since the agent does not observe, c(e ) is the agent s disutility of e when he knows hyothetically. Thus, the agent s decision making is based on his exected disutility. 5 The outut is imlicitly treated as a nonrivalrous benefit to the suervisor and the agent. However, the results of this aer are reserved even if the outut is a rofit, which is allocated to each layer according to a sharing rule. 6 In sychological games (Geanakolos et al. (1989)), ayoffs deend on action beliefs. As in Bernheim (1994), ayoffs deend on tye beliefs in the current aer. Battigalli and Dufwenberg (2009) introduce dynamic sychological games that can incororate tye-belief-deendent ayoffs. 7 From the ersective of evolutionary sychology, Nature might give us a belief-deendent reference to deal with comlicated social interaction roblems under a cognitive constraint.

7 Theoretical Economics 12 (2017) Directives, exressives, and motivation 181 words could affect the listener s decision even without any future interaction in ractice, the sychological interretation might be more natural. 2.2 Strategy and equilibrium The suervisor sends m M given ( ω). The suervisor s communication strategy is then a maing: 8 σ : P M The agent observes rivate signal h: if he is cometent, he observes the true state, i.e., h = ω, whereash = n if he is incometent. Let H = {n}. The agent s technology choice function is a : M H A Moreover, the agent s effort choice function is e : M H E I emloy erfect Bayesian equilibrium to analyze this game. Secifically, first, let e(m h a) be the agent s otimal effort level given (m h) when his technology is a. That is, e(m h a) solves max e E { ω } f (e a ω)μ ω (ω m h) c(e )μ ( m h) d where μ ω (ω m h) and μ ( m h) are the agent s osterior beliefs about ω and, resectively. From the assumtions about y(e) and c(e ), the exected marginal outut is decreasing in e whereas the exected marginal disutility is strictly increasing in e given any μ ω and μ.thus,forany(m h), e(m h a) is unique given a. Given {e(m h a)} a A, the agent solves the roblem { } max a A ω f(e(m h a) a ω)μ ω (ω m h) c(e(m h a) )μ ( m h) d Since the cometent agent knows the true state, his otimal technology choice is always a ω at ω. On the contrary, the incometent agent could have more than one otimal technology deending on μ ω.leta (m h) be the agent s otimal technology choice function that secifies his choice given (m h). Moreover,lete (m h) = e(m h a (m h)) be the otimal effort level given (m h). 9 The suervisor with ( ω) believes that the agent is cometent with robability. Thus, her exected ayoff from m given strategy σ is U m ( ω) = f (e (m ω) a (m ω) ω) + (1 )f (e (m n) a (m n) ω) 8 Since allowing mixed strategies does not add any insight into the results of this aer, we focus on ure strategies. 9 Even when the incometent agent has more than one otimal technology, the otimal effort level is the same under any otimal technology.

8 182 Toru Suzuki Theoretical Economics 12 (2017) Then σ ( ω) is an equilibrium communication strategy if, for any ( ω) P, for all m M. U σ ( ω) ( ω) U m ( ω) Remark 2. Since the cometent agent knows ω, some off-equilibrium message for the cometent agent can be an equilibrium message for the incometent agent. To see this, suose σ( ω ) = m but σ( ω ) m for any. If the cometent agent receives m at ω, he cannot aly Bayes rule since there is no that sends m at ω.on the contrary, if the incometent agent receives m, he can still aly Bayes rule since the message could be sent by the suervisor with ( ω ). 3. Equilibrium analysis Since the suervisor s ayoff function does not directly deend on m, therealwaysexist equilibria in which the suervisor s message conveys no information about ( ω). This section analyzes equilibria that reveal some information about ( ω). 3.1 Fully directive equilibrium With ayoff functions that do not deend on the other layer s belief, there is no conflict of interest in the communication. Consequently, the only relevant information for both layers is ω, and the suervisor reveals ω in the most natural equilibrium. Since the suervisor knows that revealing ω always induces the right technology choice, this is essentially a directive message. To formalize the notion of directives, let P σ (m ω) ={ P : σ( ω)= m} Given communication strategy σ, m M is directive if there exists ω such that P σ (m ω ) and P σ (m ω) = for any ω ω. In short, a message is directive if the agent can uniquely identify the state when he receives it. The simlest communication strategy with directives is making a suggestion at every state indeendently of. Formally, a communication strategy is fully directive if, for each ω, there exists message m ω M such that P σ (m ω ω)= P and P σ (m ω ω ) = for any ω ω. Moreover, an equilibrium is a fully directive equilibrium if the suervisor uses a fully directive strategy. In this equilibrium, each message erfectly reveals state ω while it is uninformative about ; that is, the directives have no exressive content. The first result states that the most natural equilibrium under belief-indeendent ayoffs is reserved in this model. Fact 1. There exists a fully directive equilibrium. In this equilibrium, the suervisor s ayoff is always y(ẽ), where { } ẽ = arg max y(e) c(e )ψ() d e E In short, ẽ is the effort level induced by directives without exressive content.

9 Theoretical Economics 12 (2017) Directives, exressives, and motivation Exressive equilibrium The suervisor always wants the agent to believe that is high while her ayoff function does not deend on m. Thus, it is not obvious whether the suervisor can credibly exress her trust in the agent s ability. Given a communication strategy σ, m M is exressive if P σ (m ω) P and P σ (m ω) for some ω. In short, an exressive message reveals some information about. An equilibrium is an exressive equilibrium if some ( ω) sends an exressive message. Since this is a chea talk game, there can be an exressive equilibrium in which no message can influence the agent s effort level. However, such an equilibrium is not interesting as it is ayoff-equivalent to some equilibrium that is ooling in. Thus, our interest is in exressive equilibria in which the exressive content of some message influences the agent s effort. An exressive equilibrium is influential if there exist P such that e(σ( ω) ω) e(σ( ω) ω) for some ω. Proosition 1. In any influential exressive equilibrium, there exists ( ω ) P such that σ( ω ) is not directive for all ( 1]. MostroofsaregivenintheAendix. The roosition states that any influential exressive equilibrium involves some inefficient communication about the state. The idea is that since the suervisor has an incentive to exaggerate her trust, she can credibly exress her trust only by sacrificing informativeness about the state. In other words, the suervisor needs to handica herself by communicating inefficiently about the state to exress her trust. The suervisor has various ways to handica herself in this multidimensional communication. As a result, the use of exressive messages could be comlex in some exressive equilibria; for examle, whether the exressive content of a message is ositive or negative could deend on the receiver s cometence. Since such a lack of consistency in exressive content could make the equilibrium difficult to lay, one of the roerties of natural exressive equilibria seems to be some degree of consistency in exressive content. Secifically, if the exressive content of m is more ositive than that of m for the cometent agent, the incometent agent also finds m more ositive. 10 To formalize the roerty, let e c (m n) be the solution of the roblem { max y(e) e E } c(e )μ ( m n) d That is, this is the otimal effort level when the agent knows ω but his belief about conditional on m is the same as the incometent agent s. In short, e c (m n) extracts the effect of the exressive content of m on the incometent agent s effort choice. An influential exressive equilibrium is consistent if, for any equilibrium messages m and m such that e(m ω) e(m ω)for some ω, e c (m n) e c (m n). 10 This roerty is aealing when we consider the use of a natural language: when messages have reexisting meanings, the equilibrium meaning of each message would resect the reexisting meaning to some degree in the focal equilibrium.

10 184 Toru Suzuki Theoretical Economics 12 (2017) Lemma 1. In any consistent exressive equilibrium, if σ( ω) is not directive, then σ( ω) is not directive for all >. To rovide intuition, observe that, in any consistent exressive equilibrium, the cometent agent s effort level is strictly higher than the incometent agent s whenever the suervisor sends an uninformative message. Then, since the suervisor with a low (high) believes that the agent is cometent with a low (high) robability, the suervisor s exected ayoff from an uninformative message is increasing in. On the contrary, note that the suervisor s ayoff from a directive message is constant in. Hence, the suervisor refers to send an uninformative message whenever a lower sends an uninformative message. From Lemma 1, if the suervisor with at ω sends a directive message in a consistent exressive equilibrium, the suervisor with < at ω also sends a directive message. Hence, if the suervisor sends a directive message and a nondirective message at ω, the exressive content of the directive message is always negative relative to that of the nondirective message in the consistent exressive equilibrium. Corollary 1. In any consistent exressive equilibrium, if σ( 1 ω) is directive while σ( 2 ω)is not directive, then e(σ( 1 ω) ω) < e(σ( 2 ω) ω). Since there is no conflict of interest in communicating about the state, the only motivation to use a nondirective message seems to be to exress her trust. In fact, in ordinary communication, an exressive sentence such as I trust you is often used as a ure exressive, that is, the message does not convey any information about the decision environment. Let σ (m) ={ω : σ( ω)= m for some } An influential exressive equilibrium is state indeendent if, for any m M such that σ (m) > 1, P σ (m ω) = P σ (m ω ) for any ω ω. That is, in a state-indeendent exressive equilibrium, any nondirective message does not reveal any information about ω. Clearly, any state-indeendent exressive equilibrium is a consistent exressive equilibrium. The next lemma rovides the roerty of state-indeendent exressive equilibrium: the suervisor never uses a message that can make the incometent agent choose a wrong technology. Lemma 2. In any state-indeendent exressive equilibrium, there is no ( ω) P such that a(σ( ω) n) = a ω a ω. From Proosition 1, the suervisor makes her exressive message credible by handicaing herself. If the suervisor induces the incometent agent to choose a ω at ω, the same message induces the right technology at ω. Then, since sending the message at ω is more costly than sending it at ω, the exressive content of the message should deend on the state. Thus, in any state-indeendent exressive equilibrium, the suervisor never uses a message that can induce a wrong choice.

11 Theoretical Economics 12 (2017) Directives, exressives, and motivation 185 Now I introduce an intuitive communication strategy in which the uninformative message is state indeendent. In this strategy, the suervisor with a low uses directives while the suervisor with a high sends a message that is uninformative about ω. Formally, a communication strategy is simle exressive if { m if ( ˆ 1] σ( ω)= m(ω) if [0 ˆ) where m(ω) is an injection from to M\{m }.Inshort,m(ω) is a directive message; that is, if the agent receives m(ω ), he knows that the state is ω. An equilibrium is a simle exressive equilibrium if the suervisor uses a simle exressive strategy. In this equilibrium, the uninformative message m exresses the suervisor s trust in the agent s cometence by being comletely uninformative about ω. In ordinary communication, it can be interreted as an exressive sentence whose literal meaning does not refer to any state, e.g., I trust your ability. On the contrary, all the directive messages have negative exress content; that is, they reveal that is lower than ˆ. Proosition 2. Any state-indeendent exressive equilibrium is ayoff-equivalent to a simle exressive equilibrium. The logic behind this result is as follows. From Lemma 2, ifthesuervisorusesa message at more than one state, the message has to make the incometent agent choose the safe technology. Moreover, from the state-indeendence roerty, the exressive message has to be used for all states. Then, from Lemma 1, thereexists ˆ such that the suervisor with < ˆ sends some directive message at each state while the suervisor with > ˆ sends some exressive message that makes the incometent agent choose the safe technology. When the suervisor uses a communication strategy that is convex in, this is a simle exressive equilibrium with cutoff ˆ. It can be shown that even if the suervisor uses a strategy that is nonconvex in, it is ayoff-equivalent to the simle exressive equilibrium with ˆ. The existence of a simle exressive equilibrium is not guaranteed. To rovide a sufficient condition for the existence of a simle exressive equilibrium, let e 0 = arg max{y(e) c(e 0)} e E and { } e λ = arg max λy(e) c(e )ψ() d e E That is, e 0 is the otimal effort level under the right technology when the agent is comletely distrusted. Moreover, e λ is the otimal effort level under the safe technology when the agent has no information about how the suervisor thinks about him. Let ω max arg max ω π(ω). Proosition 3. There exists a simle exressive equilibrium if (1 δ)π(ω max ) + δ λ< y(e 0 )/y(e λ ). Furthermore, if (1 δ)π(ω max ) + δ>λ, then there is no simle exressive equilibrium.

12 186 Toru Suzuki Theoretical Economics 12 (2017) Intuitively, Proosition 3 states that a simle exressive equilibrium exists if the technology choice roblem is sufficiently difficult and imortant. First, if (1 δ)π(ω max ) + δ λ, m induces the incometent agent to choose the safe technology. This condition is satisfied when the technology choice roblem is sufficiently difficult given δ and λ. To see this, suose =5. If π(ω max ) = 0 2, the technology choice roblem is difficult for the incometent agent since all states are equally likely to haen. On the contrary, when π(ω max ) = 0 9, the choice roblem is easier since the true state is very likely to be ω max. Second, if λ<y(e 0 )/y(e λ ), managing the incometent agent s technology choice is more imortant than managing the exressive content of directives. To see this, recall that y(e 0 ) is the outut level when the suervisor uses a directive message and the agent thinks that he is comletely distrusted. Moreover, λy(e λ ) is the outut level when the incometent agent chooses the safe technology without any information. Thus, if λ<y(e 0 )/y(e λ ), the loss from inducing the safe technology instead of the right technology is larger than the loss from being erceived as the worst tye, i.e., = 0. The equilibrium cutoff tye is constructed so that the cutoff tye s ayoffs from m ω and m are indifferent. Since the cutoff tye s ayoffs from m ω and m are increasing in the cutoff, the uniqueness of the equilibrium cutoff is not guaranteed. However, it can be shown that if the disutility function is not so sensitive to, the equilibrium cutoff is unique and λ<y(e 0 )/y(e λ ) becomes a necessary condition for the existence of a simle exressive equilibrium. The equilibrium effort levels in simle exressive equilibrium deend on the roductivity of the safe technology, i.e., λ. To rovide the comarative statics, suose that there exists a unique simle exressive equilibrium outcome. Let e(m h; λ) be the effort level in the simle exressive equilibrium conditional on (m h) under λ. Fact 2. Suose λ >λ and that there exists a unique equilibrium cutoff under λ and λ.thene(m ω; λ )>e(m ω; λ ) and e(m ω h; λ )>e(m ω h; λ ) for h = n ω. Fact 2 imlies that the exressive content of m and m ω is more ositive under λ. To see this, note that when the safe technology becomes less roductive, the suervisor s exected ayoff from m gets lower whereas it does not affect her exected ayoff from m ω. Thus, the lower λ encourages more to send m ω instead of m. As a result, the lower λ increases the equilibrium cutoff tye. Then, since m reveals > ˆ and m ω reveals < ˆ, the lower λ makes the exressive content of the messages more ositive. A natural question is whether a simle exressive equilibrium is more desirable than the equilibrium without exressive content, i.e., fully directive equilibrium. This is a nontrivial question because of the trade-off between informational and exressive contents; fully directive equilibria are more efficient in terms of communication about ω whereas the exressive message can boost the agent s effort level in simle exressive equilibria. The following result rovides useful insight into the trade-off. Fact 3. Suose there exists a simle exressive equilibrium with cutoff ˆ. There exists ( ˆ 1) such that the suervisor with > refers the simle exressive equilibrium

13 Theoretical Economics 12 (2017) Directives, exressives, and motivation 187 to a fully directive equilibrium, whereas the suervisor with < refers the fully directive equilibrium to the simle exressive equilibrium. To see the idea of Fact 3, note that the suervisor s ayoff from a directive message in a simle exressive equilibrium is strictly lower than that in a fully directive equilibrium because of the negative exressive content. Thus, when the suervisor sends the directive message, she refers the fully directive equilibrium to the simle exressive equilibrium. On the other hand, the cometent agent s outut level conditional on m is higher than the outut level in the fully directive equilibrium because of the ositive exressive content of m. When the agent is incometent, the outut level is lower than that in the fully directive equilibrium since it induces the safe technology. Thus, if is sufficiently high, the suervisor refers the simle exressive equilibrium whereas the suervisor with a low refers the fully directive equilibrium. Fact 3 suggests that the suervisor refers a fully directive equilibrium in the ex ante stage if the robability of > is sufficiently small. However, since deends on the distribution of, it is not clear whether the suervisor refers one equilibrium to another in general. The following examle shows that the suervisor s favorite equilibrium can be sensitive to the arameter values of the model. Examle 1. Suose y(e) = 2e, c(e ) = e 2 /2 e and () is uniform on P. Moreover, δ = 0, =10, andπ(ω) is uniform on. If λ = 0 2, the suervisor refers a simle exressive equilibrium to a fully directive equilibrium in the ex ante stage. Conversely, she refers the fully directive equilibrium in the ex ante stage if λ = 0 6. As I exlained in Fact 2, the exressive content of m is more ositive when λ is lower. On the other hand, since a lower λ increases the equilibrium cutoff tye, it reduces the ex ante robability of sending m. The net effect is not clear in general but, in thecaseofexamle 1, the first effect dominates the second effect. The agent s favorite equilibrium is also sensitive to the secification and the arameter values of the model. For examle, in the setting of Examle 1, the cometent agent always refers a simle exressive equilibrium to a fully directive equilibrium whereas the incometent agent refers the oosite. Thus, whether the agent refers one equilibrium to another in the ex ante stage deends on the agent s rior belief. Finally, the suervisor s favorite equilibrium does not always coincide with the agent s favorite equilibrium; for instance, when λ is low in the setting of Examle 1, the agent refers a fully directive equilibrium whereas the suervisor refers a simle exressive equilibrium in the ex ante stage. Recall that when there is a state that is very likely to haen, i.e., (1 δ)π(ω max )+δ> λ, there is no state-indeendent exressive equilibrium. Thus, an exressive equilibrium has to be state deendent under such situations. Unlike in state-indeendent exressive equilibria, the suervisor uses a message that can induce a wrong technology to exress in state-deendent exressive equilibria. The next roosition shows that an intuitive exressive equilibrium can exist when there is no simle exressive equilibrium.

14 188 Toru Suzuki Theoretical Economics 12 (2017) Proosition 4. Suose (1 δ)π(ω max ) + δ>λ.ifδ<y(e 0 )/y(ẽ), there exists a consistent exressive equilibrium in which the suervisor uses a communication strategy of the form σ( ω max ) = m max for any { mmax if ( ˆ 1] σ( ω) = m(ω) if [0 ˆ) where m(ω) is an injection from to M\{m max }. In this equilibrium, the suervisor sends m max indeendently of in ω max while she sends m max in ω ω max only if her trust is higher than a certain level. Moreover, as in simle exressive equilibrium, the suervisor uses a directive message in each ω ω max if her trust is lower than a certain level. Since ω max is very likely to haen, the incometent agent chooses a ωmax when the received message is not so informative about ω. Asm in a simle exressive equilibrium, m max exresses her trust by communicating inefficiently about ω. In other words, only the suervisor who strongly believes the agent s cometence can send such an uninformative message at ω ω max. Unlike m in simle exressive equilibrium, the exressive content of m max deends on ω. In fact, if the cometent agent receives m max at ω max, the message has no exressive content, i.e., P σ (m max ω max ) = P while it has ositive exressive content P σ (m max ω)= ( ˆ 1] at any ω ω max. When the incometent agent receives m max, it is not comletely uninformative about ω since it is more likely to be sent at ω max. The role of the condition δ<y(e 0 )/y(ẽ) is similar to that of λ<y(e 0 )/y(e λ ) in simle exressive equilibrium; that is, when this condition is satisfied, inducing a wrong technology is more damaging than making the agent believe = 0. When this condition is violated, the conflict of interest in communication about is too strong to credibly exress the suervisor s trust. Not surrisingly, there can be a state-deendent exressive equilibrium in which the suervisor uses more than two messages in some states. For examle, suose = {1 2 3} and ω max = 1. Then consider the class of strategies σ( 1) = m 1 for any m 1 if ( ˆ 2 1] σ( 2) = m 0 if ( ˆ 1 ˆ 2 ] m 2 if [0 ˆ 1 ] m 1 if ( ˆ 2 1] σ( 3) = m 0 if ( ˆ 1 ˆ 2 ] if [0 ˆ 1 ] m 3 If a(m 1 n)= a 1 and a(m 0 n)= a 0, this can be an equilibrium strategy under some situation. Since m 1 can induce a wrong technology at state 2 and state 3, it handicas the

15 Theoretical Economics 12 (2017) Directives, exressives, and motivation 189 suervisor more than m 0 when δ is sufficiently small. Then m 1 conveys a stronger trust than m 0 in equilibrium. 4. Aretheuredirectivesstable? In fully directive equilibria, the suervisor can communicate about the state efficiently, and all the directives are neutral in exressive content, i.e., ure directives. At first glance, fully directive equilibria seem lausible since there is no conflict of interest in communicating about the state. However, fully directive equilibria might not be so stable. Suose a suervisor works at an organization where using ure directives are the convention. If she says I have no suggestion since I trust you, deviating from the convention, she might be able to convey her trust since the focal meaning of the message could induce a subotimal technology and she would handica herself. Thus, when the suervisor and the agent haen to share the focal meaning of the off-equilibrium message, it could override the fully directive equilibrium. To investigate the above observation, I emloy the concet of neologism roofness introduced by Farrell (1993). The concet is based on the assumtion of a reexisting common language that comes from outside of the game. This assumtion makes it ossible that the listener can at least understand the meaning of a neologism. A utative equilibrium is tested according to whether the suervisor can uset the equilibrium by a credible neologism. To define a credible neologism for this game, suose that when the suervisor s neologism means that her tye belongs to D := Q Z P,theagent udates his belief according to Bayes rule. That is, μ ω (ω π(ω ) D n) = ω Z π(ω) if ω Z 0 if ω / Z ψ( ) μ ( D h) = Q ψ() d if Q 0 if / Q for any h. LetU D ( ω) be tye ( ω) s ayoff in which the agent otimally resonds to D based on the above udating rule. The term D is a self-signaling set if ( ω) D, U D ( ω) > U σ ( ω) ( ω), while U D ( ω) U σ ( ω) ( ω) if ( ω) / D. In other words, D is self-signaling when it satisfies a fixed oint roerty: the set of ( ω) who are strictly better off by claiming she belongs to D exactly coincides with D given an equilibrium. An equilibrium is neologism-roof if no self-signaling set exists. The next result shows that neologism roofness favors simle exressive equilibria over fully directive equilibria. Proosition 5. Whenever a simle exressive equilibrium exists, fully directive equilibria are not neologism-roof. Whenever a simle exressive equilibrium exists, the suervisor can uset fully directive equilibria by a neologism with D = ( 1], where is higher than the cutoff

16 190 Toru Suzuki Theoretical Economics 12 (2017) tye of the simle exressive equilibrium. Intuitively, the neologism exresses her trust while making it credible by being uninformative about ω. Even if no simle exressive equilibrium exists, fully directive equilibria can fail to be neologism-roof. In fact, fully directive equilibria are not neologism-roof whenever the safe choice is the incometent agent s otimal choice under π(ω). Note that this condition is always satisfied when a simle exressive equilibrium exists. To rank simle exressive and fully directive equilibria based on neologism roofness, I need to show that simle exressive equilibria can be neologism-roof under a reasonable condition. It can be shown that if a simle exressive equilibrium is not neologism-roof, any self-signaling set needs to take the form D =[0 q) {ω}, where q> ˆ. 11 To obtain further results, I secify the disutility function as c(e ; β) = C(e) βb(e ), whereβ (0 1]. One interretation of this secification is that the agent s net disutility of effort c(e ) is determined by his ain, i.e., C(e),andhis enjoyment that can be enhanced by the suervisor s trust, i.e., βb(e ). Then assume that C : E R + is twice differentiable and strictly convex whereas b : E P R + is twice differentiable in each argument and b 1 (e ) > 0. Moreover, assume b 11 (e ) 0, b 12 (e ) > 0,andC (e) b 1 (e ) > 0 for any (e ) so that the assumtions about c(e ) in Section 2 are guaranteed under any β (0 1]. The following result states that a simle exressive equilibrium is neologism-roof if the marginal disutility of effort is sufficiently insensitive to. Proosition 6. If β is sufficiently small, any simle exressive equilibrium is neologismroof. The idea of Proosition 6 is as follows. As I mentioned earlier, whenever a credible neologism exists, the self-signaling set takes the form of [0 q) {ω},whereq> ˆ. Note that the suervisor s ayoff from D =[0 q) {ω} with q = ˆ is the same as the cutoff tye s ayoff in the simle exressive equilibrium with ˆ.Whenβ is lower, the agent s effort becomes less sensitive to q. Then, since the suervisor s ayoff from m is strictly increasing in, the neologism with any q> ˆ cannot be more attractive than sending m in the simle exressive equilibrium. How small is sufficiently small when a simle exressive equilibrium is neologism-roof? The next result shows that the condition in Proosition 6 can be quite mild. Fact 4. Suose () is a uniform distribution on [0 1], y(e) = αe,andc(e ) = e 2 /2 βe,whereα>β>0. There exists a simle exressive equilibrium if (1 δ)π(ω max ) + δ λ and β<2(1 λ 2 )α/λ. Moreover, whenever a simle exressive equilibrium exists, it is neologism-roof. Fact 4 says that whenever β suorts a simle exressive equilibrium in the linear-quadratic-uniform setting, it is always small enough to make the equilibrium neologism-roof. 11 For more details, see the roof of Proosition 5 in the Aendix.

17 Theoretical Economics 12 (2017) Directives, exressives, and motivation 191 When there are multile simle exressive equilibrium outcomes, only those with the highest cutoff can be neologism-roof. 12 The idea of this result is as follows. If there is another simle exressive equilibrium with a higher cutoff tye, the suervisor s exected ayoff from neologism [0 q) {ω} is sensitive to q. As a result, a higher q makes the ayoff from the neologism higher than tye q s equilibrium ayoff. Since tye q s equilibrium ayoff becomes higher than that from the neologism as q goes to 1, there exists q that makes [0 q) {ω} a self-signaling set. Note that this result does not contradict Proosition 6; there exists a unique simle exressive equilibrium outcome if β is sufficiently low. Remark 3. While neologism roofness turned out to be effective to rank the two imortant equilibria, the fixed oint roerty of self-signaling sets makes obtaining more general results difficult. To see this, consider any consistent exressive equilibrium. While neologism [0 q) {ω} is a tyical candidate for a credible neologism, whether it can be a self-signaling set or not deends on the sensitivity of the disutility function to ; that is, it is a quantitative question that deends on the secification and the arameter values of the model. A disutility function that is less sensitive to can make such a neologism noncredible. However, the class of exressive equilibria could fail to exist when a disutility function becomes less sensitive to. 5. Discussion 5.1 Two-way communication This aer focuses on one-way communication. The setting catures a situation where there is a formal or informal rule that does not allow the agent to seak first. Another interretation is that the agent has no incentive to reveal his tye because of some factor that is not in the model. For examle, suose the suervisor could relace the agent before the task if she knows the agent is incometent. Then, since the incometent agent has no incentive to reveal his tye, we can analyze the communication roblem as one-way communication without loss of generality. However, one might ask how the nature of communication could be changed if the agent can seak first, i.e., two-way communication. To investigate the question, the assumtion of belief-deendent disutility needs to be clarified for two-way communication since the suervisor could udate her belief based on the agent s message. Recall that the assumtion is based on the idea that distrust reduces motivation. However, when the agent voluntarily reveals his cometence in equilibrium, there is no room for distrust. Thus, if the agent claims that he is incometent, it should not be offensive that the suervisor erceives the agent as incometent. Hence, it seems natural to stay with the assumtion that the agent s disutility deends on the suervisor s rior, i.e., her ersonal view about the agent s cometence, rather than her osterior belief that is based on the agent s message. First, consider a communication rotocol that forces the agent to seak first. Secifically, in the first stage, the agent sends a costless message about his cometence to the 12 See the Aendix for the roof of this result.