Subjective Evaluation in the Agency Contract and the Reputation of the Principal

Transcription

1 Subjective Evaluation in the Agency Contract and the Reputation of the Principal Shinsuke Kambe*** Faculty of Economics Gakushuin University Mejiro Toshima-ku, Tokyo Japan October 2005 * Correspondence: Faculty of Economics, Gakushuin University, Mejiro, Toshimaku, Tokyo , Japan. shinsuke.kambe@gakushuin.ac.jp ** This paper is previously titled as Trust in Leadership and the Subjective Evaluation of a Subordinate. I am grateful for the comments from Munetomo Ando, Masaki Aoyagi, Kyota Eguchi, Hideshi Ito and Tadashi Sekiguchi. I also thank the seminar participants at the Autumn-2003 meeting of the Japanese Economic Association, the CTW held in the KIER, the ISER of Osaka University, and Tsukuba University for their helpful discussions. All remaining errors are mine.

2 Abstract We study whether and how a principal uses her subjective evaluation of the agent s performance in a credible way. There are two types of the principal: trustworthy and untrustworthy. When a principal is believed to be a trustworthy type by the agent, she will obtain a higher payoff in the future transaction with him. It creates the reputational effect. In equilibrium, the trustworthy type of the principal gives high ratings to the agent more frequently than the untrustworthy one. In order to make the agent believe that she is the trustworthy type, the principal gives an honest rating at the evaluation, even when it leads to a higher wage. It is shown that the reputational effect functions in a robust way only when the accuracy of the observation about the agent s performance is different between the two types. JEL classification : D23, D82 Keywords : subjective evaluation, incentive contract, reputation 2

3 1. Introduction In the workplace, the employer (the principal) often evaluates the worker (the agent) by using her subjective observation. Since it can be easily falsified, the question arises as to how it is made credible. We consider the reputation of the principal as the possible explanation. The agent often trusts the principal who evaluates him accurately as the leader. When the principal values the agent s trust in leadership, she wants to maintain the reputation to be the trustworthy one. It may be utilized to induce the principal to tell the truth in her evaluation. In order to formulate this idea, we introduce two types of the principal: the trustworthy one (A-type) and the untrustworthy one (B-type). The type is known to the principal herself and is not known to the agent at any time. In the beginning, the principal and the agent agree a contract, which specifies the wages as a function of the principal s ratings at the evaluation. Then, the agent privately makes some effort. His effort stochastically influences the principal s observation of his performance. The principal s observation is assumed to be her private information. It is also assumed that the principal s observation is the sole measure of the agent s performance. In this paper, the evaluation based on the principal s private information of the agent s performance is called thesubjectiveevaluation. Attheevaluation,thedifferent types of the principal may give different ratings and thus the agent updates his belief based on the received rating. After the rating is given, there is another transaction between them. When the agent s updated belief that the principal is trustworthy is higher, the agent exerts more efforts in the second transaction and thus the principal s benefit is higher. Hence, both types want the agent to believe them to be A-type, which is the source of the principal s incentive to tell the truth. When the principal gives the rating, she is not influenced by its effect on the agent s incentive in the first period since the agent s effort is already taken. (How the principal evaluates the agent affects the agent s ex ante incentive.) Instead, her decision is influenced by the following two factors. One factor is the wage that is linked to the rating. In an incentive contract, a higher rating is generally linked to a higher wage. The other factor is the effect on the agent s belief. If the untrustworthy type tends to give out lower ratings more frequently, higher ratings make the agent update upward his belief that the principal is trustworthy, which benefits the principal in the future. Because of the latter effect, the principal may want to give an honest rating even when doing so is costly. 3

4 This paper, using a simple example, first demonstrates that the reputation functions as the device to induce the principal s honest evaluation. Then, we discuss the property and the robustness of the equilibria with reputation. In particular, we show that the reputational effect functions in a robust way only when the accuracy of the observation about the agent s performance is different between the two types. As this paper studies how the reputation enables the principal to use the subjective evaluation, it is related to the two lines of literature: the theory of incentive contracts and the reputation in the repeated games. The issue of subjective evaluation has been studied in the theory of incentive contracts. One typical situation where the subjective evaluation can be honestly utilized is the tournament 1 where multiple agents compete with each other. With a fixed sum of prizes, the principal has no incentive to falsify her evaluation. Our model differs from that of the tournament in that ours has only one agent. A more closely related research can be found in the theory of promotion. 2 In the model of promotion, the principal s private observation of the agent s performance in the first period indicates the agent s productivity in the second period. If the agent is more productive, it is more efficient to place him at a higher position. Then, by tying the pay raise to promotion, the principal is able to commit herself not to abuse the subjective evaluation. Our model has the opposite informational asymmetry. It is not the agent s type but the principal s type that is uncertain. In another model that is closely related to ours, Kambe (2002) and MacLeod (2003) have studied the effect of the additional information on the utilization of the subjective evaluation. Kambe (2002) has shown that the option to use the verifiable information after low ratings is the effective means to utilize the subjective evaluation of the principal while Macleod (2003) has shown how the agent s information can be used to induce truth-telling by the principal more efficiently. Our model complements these studies and shows that the reputation to be trustworthy functions as another motivation for the honest evaluation. The other literature that has a close connection to this paper is that of reputation. When the agent also observes what the principal observes but the outsiders do not, the enforcement of the (implicit) contract becomes a critical issue. In the dynamic setting 1 See Green and Stokey (1983), Lazear and Rosen (1981) and Nalebuff and Stiglitz (1983) for the early studies. 2 See Kahn and Huberman (1988), Prendergast (1993) and Waldman (1984). 4

5 with low discount rates, the reputation typically works as an enforcement mechanism. 3 When the information about the agent s performance is shared, it becomes mutually known whether the principal does not live up to her good reputation. In contrast, the reputation in our model needs to be maintained without shared information. 4 When the principal gives a low rating at the evaluation and pays a low wage, it will never be known whether the principal has willfully deceived the agent or she has merely behaved honestly. In the theory of repeated games, the private information of the players is known to undermine the power of reputation. (It typically occurs when there is no publicly observable signals about the past behaviors and the players make imperfect, private monitoring of the other players past actions. See Kandori (2002) for a recent survey on this subject.) Our model is similar to this line of literature in that both try to find out whether and how the private information can be utilized to enhance efficiency. There are two kinds of differences. First, we use the reduced form when we formulate the future transactions. Namely, our model supposes that any reputational benefit is represented by a value function that depends on the agent s belief only. Second and more importantly, we suppose that there are two types of the player for the one who makes the private observation. In our model, the reputation is caused not by what the player has done (and is expected to do in the future) but by what her type is. In the next section, we describe the basic model. Section 3, using a simple example, demonstrates how the reputational effect enables the principal to use her subjective evaluation in a credible way. Section 4 then characterizes the equilibria with reputation where the strategies of the principal are (almost surely) strict best responses and monotone in observation. In Section 5, we introduce the modified preference of the principal and show that the reputational systems that are not affected by the modification have to satisfy the property described in Section 4. It is also argued that the difference of the observation 3 For the role of implicit contracts in the principal-agent theory, see Baker, Gibbons, and Murphy (1994), Levin (2003), MacLeod and Malcomson (1989) and Pearce and Stacchetti (1998). 4 A similar situation is studied in Ely and Välimäki (2003). They study why it is difficult to establish a good reputation without shared information. The difference is that the payoffs can be manipulated through contracting in our model while they are given exogenously in their model. Due to this difference, we show that, with an appropriate contract, a good reputation can be established to increase the efficiency, which provides an interesting contrast to their impossibility result. 5

6 technologies between the two types is essential for the reputation to function in a robust way. Section 6 makes concluding remarks. We relegate all the proofs to the appendix. 2. Basic Model A principal hires an agent for two periods. For exposition, we suppose that the principal is female and the agent is male. There are two types of the principal. One type, whom we call A-type, is the trustworthy (able) one. The other type, whom we call B-type, is the untrustworthy (bad) one. We denote the set of the types by P: P = {A, B}. When the agent believes the principal to be trustworthy after the first transaction, the principal obtains the higher payoff in the second transaction. This is the source of the reputational effect. The type is known to the principal herself from the beginning but is never known to anyone. The ex ante probability that the principal is A-type is commonly known and is given by θ 0. We assume that 0 < θ 0 < 1. The timing is as follows. In the beginning, the principal proposes a contract, which specifies the set of ratings and the associated wages. If the agent refuses it, the game ends and he receives his reservation utility U. When the agent accepts the contract, the game continues and the agent privately chooses his effort e E = {e 1,e 2 }. We assume that e 1 <e 2. (The assumption that there are only two levels of efforts is made for the ease of exposition. In the concluding remarks, we mention how we can extend the model to the case where there are more than two levels of efforts.) When the agent is induced to take the effort e 2, there is an exogenous chance of shirking. Specifically, with probability µ, hetakes the lower effort e 1 by some exogenous reason 5 such as sickness. Nobody knows whether the agent will do so at the time of contracting, and, at the evaluation, the principal does not know whether he has done so. We assume that 0 µ<1. This assumption reflects the reality that, in most of workplaces, there is a distribution of efforts. His effort is not observable to anyone. The principal privately observes his performance imperfectly. The principal s private observation of the agent s performance is denoted by x X [0, 1]. Neither the agent nor the public knows what the principal observes at any time. We assume that there is no other information available about the agent s performance. Given the observation, the principal publicly gives a rating y i ( Y {y 1,...,y I }). We assume 5 We make the agent s randomization of efforts endogenous in the appendix. 6

7 that the set of the ratings, Y,isfinite and that Y < I. The subjective evaluation is the combination of the principal s private observation of the agent s performance and her public announcement of the rating to the agent s performance. Given the rating y i, the principal pays the wage w(y i ) according to the agreed-upon contract. The (behavior) strategy of A- type at the evaluation is described by the probability of the rating y i given the observation x, s A (y i x), and that of B-type is described by s B (y i x). Facedwiththeratingy i,the agent who has taken the effort e updates his belief that the principal is A-type to θ(y i e). The players engage in another transaction in the next period, where the benefit to the principal depends on the agent s belief θ at the beginning of the second period. We suppose that, in the second period, the agent cooperates more with the principal when he believes that the principal is more likely to be the trustworthy type. Formally, we assume that the principal s benefit from the second transaction is given by a continuous, strictly increasing function of the agent s belief, π(θ): π(θ 0 ) < π(θ) for any θ 0 and θ such that θ 0 < θ. Since this benefit is solely dependent on the agent s belief, we call it reputational benefit. This is the main difference 6 between the two types of the principal. The principal is better off when the agent believes her to be trustworthy with a higher probability. On the other hand, we assume that the agent s utility in the second transaction is independent of what happens in the first transaction. Namely, it is independent both of his action in the first period and of his belief 7 at the beginning of the second period. For the first period, the expected utility function of the agent is given by u(w) e, where w is his wage and e is his effort. (Since either the agent s action in the first period or his belief at the beginning of the second period does not affect his utility in the second period, the agent maximizes his overall utility by maximizing his utilities period by period. Since our interest is what contract is offered in the first period, we abstract from the agent s behavior in the second transaction.) We assume that u(w) is twice continuously differentiable, u 0 (w) > 0andu 00 (w) 0 for any w (0, ). (We do not exclude the agent s risk neutrality.) Moreover, we assume that lim w 0 u(w) = and lim w u(w) =. 6 Unlike some other models on reputation (e.g., Kreps et al.,1985), the sets of behavior strategies are common between the two types of the principal. On the other hand, their observation technologies are different in our model, which turns out to be essential for the reputation to function in a robust way. 7 If the agent s belief affects his own utility in the second transaction, then some player may want the initial contract to cause more (or less) accurate revelation about the principal s type at the end of the first period. We do not analyze this effect in the current model. 7

8 The principal s payoff for the two periods is given by R(e) w + π(θ), where e is the agent s effort in the first period, w is the wage in the first period, and θ is the agent s belief that the principal is A-type at the beginning of the second period. The payoff function of the principal is independent of the type. When the principal is deciding which rating to give, the relevant payoff at the continuation game is w + π(θ) since the effort is already chosen. We call it the interim payoff. The effort stochastically influences the principal s private observation. The set of the principal s observations is given by the interval X [0, 1]. When the effort is e, the density function of observing x( X) for the trustworthy principal is given by f A (x e) andthat for the untrustworthy one is given by f B (x e). These functions define the observation technologies of the principal. The two types of the principal may have different observation technologies. The cumulative distribution functions are denoted by F A (x e) andf B (x e) respectively. We assume that both f A (x e) andf B (x e) arepositiveforanyx X and any e E. Due to this assumption, the principal cannot tell exactly what effort the agent has chosen (or has not chosen) after any observation. Moreover, we assume that the density functions satisfy the monotone likelihood ratio property (the MLRP hereafter): f p (x e 2 )/f p (x e 1 ) is strictly increasing in x X for p P. For p P, we use h p (e x) to denote p-type s belief that the agent has taken the ³ effort e when she observes x. It holds that h p (e 2 x) = 1+ ν f p (x e 1 ) 1 ν f p (x e 2 ) 1, where ν is the probability of the lower effort. By the MLRP, when the agent takes both efforts with positive probabilities, for p P, h p (e 2 x) is strictly increasing in x( X). agent generally has the different beliefs when he chooses different efforts. Define π(y i ) π(θ(y i e 2 )) π(θ(y i e 1 )). This is the difference of the reputational benefits from a given rating between the two efforts. It affects the interim payoffs of the two types of the principal at the evaluation. For the p-type (p P) with the observation x, theinterimpayoff is given by w(y i )+ P e E h p(e x)π(θ(y i e)) = w(y i )+π(θ(y i e 1 )) + h p (e 2 x) π(y i ). Hence, when π(y i ) is not zero, the principal s belief about the agent s effort affects the interim payoff. In the following analysis, we investigate whether both the higher effort of the agent and the truth-telling of the principal can be induced by studying the continuation equilibria 8 The

9 given a contract. 8 Whenever we say an equilibrium, it implicitly refers to a continuation equilibrium for a given contract. We use the sequential equilibrium as our equilibrium concept. Namely, both the agent and the principal take the best responses at their moves (sequential rationality), and the principal uses the Bayes rule to compute her belief about the agent s effort given her private observation and the agent does the same to update his belief about the principal s type given the received rating (consistent belief). Throughout the analysis, we assume that the agent does not update his belief after any contract. Since B-type never wants to reveal her type, given any contract, we deem that the natural belief of the agent about the principal s type should remain unchanged from the prior and make the following assumption. Assumption Given any contract that the principal offers in the beginning, the agent s belief is unchanged from θ 0. Formally, let c =(Y,{w(y i ):y i Y }) be a contract and C be its set. Given a contract c, the continuation game can be described by the agent s decision about the acceptance of the contract, the probability of the higher effort 1 ν, the principal s strategies at the evaluation {s A (y i x),s B (y i x)} yi Y,x X, and the agent s belief {θ(y i e)} yi Y,e E. Note that the agent s belief is not a function of the contract because of the assumption made above. When there exists a sequential equilibrium in the continuation game given a contract c and when the agent accepts the contract and the agent takes the effort e 2 with the probability 1 ν (ν < 1) in the equilibrium, we denote the representative one by E(c). Let C (ν)( C) be the set of the contracts given which there exists this kind of a continuation equilibrium. For any c C (ν), the credible use of the subjective evaluation cannot be possible without the reputation. In this sense, we call the combination of the contract c and its continuation equilibrium E(c) areputational system. We study whether C (ν) exists and, if it exists, what forms the contracts in C (ν) and their corresponding continuation 8 Once the continuation equilibria given contracts are characterized, we can proceed to the analysis of the equilibria for the entire game. Then, we can ask which contract is optimal and also can study which effort (or the mixture of the efforts) should be induced. Since its analysis will not add much new insight to the current analysis, it is not pursued in this paper. 9

10 equilibria take. Except in the appendix II, we focus on the case that the agent takes the higher effort with the probability 1 µ: ν = µ. 3. An Example This section, by using a simple example, demonstrates how some contract is able to induce the principal to evaluate the agent s performance honestly and also to induce the agent to take the higher effort with a positive probability. For the example in this section, we assume that the agent exogenously chooses the lower effort with a positive probability (µ >0) and that the set of ratings is given by Y = {y 1,y 2 }. We suppose that the likelihood ratio evaluated at the highest observation is higher for A-type than for B-type: f A (1 e 2 )/f A (1 e 1 ) >f B (1 e 2 )/f B (1 e 1 ). It implies that, for the higher observations, the observation of A-type is more informative than that of B-type in terms of predicting the agent s effort. Let ˆx be an observation where ˆx <1andf A (ˆx e 2 )/f A (ˆx e 1 ) >f B (1 e 2 )/f B (1 e 1 ). (By the above supposition, we can always find such an observation.) We construct a two-tier contract with ˆx as the cutoff point, which is the lowest observation for the higher rating. Consider the following strategies at the evaluation. A-typegivestheratingy 2 (and pays the higher wage w(y 2 )) when her observation x is no smaller than ˆx (x ˆx), and gives the rating y 1 (and pays the lower wage w(y 1 )) when her observation x is smaller than ˆx (x <ˆx). B-type gives the rating y 1 (and pays the lower wage w(y 1 )) with the probability one. Namely, s A (y 2 x) =1ifandonlyifx [ˆx, 1], and s B (y 1 x) = 1 for any x X. For θ e E, wedefine θ(y 1 e) = 0 F A (ˆx e) θ 0 F A (ˆx e)+(1 θ 0 ). This gives the agent s belief that the principal is A-type when the agent has taken the effort e and is given the rating y 1 under the above strategies. Note that, under the specified strategies, the agent s belief is one when she is given the rating y 2 : θ(y 2 e) = 1 for any e E. Suppose that the contract specifies the wages so that they satisfy the following system of equations. w(y 2 ) w(y 1 )=π(1) X e E h A (e ˆx)π(θ(y 1 e)), and (1) θ 0 (1 F A (ˆx e 2 ))u(w(y 2 )) + 1 θ 0 (1 F A (ˆx e 2 )) u(w(y 1 )) e 2 = U. (2) The first equation is concerned with the difference of the wages while the second equation is concerned with the general levels of the wages. Since the agent s utility is assumed to 10

11 take the value from the minus infinity to the infinity, we can find the pair of wages that satisfies the above equations. We would like to claim that these strategies, together with the agent s acceptance of the contract and the agent s choice of the higher effort (with the probability 1 µ), form a continuation equilibrium given the contract when the difference in cost between the two efforts is sufficiently small. First, we examine the agent s behavior. Note that, under the described strategies, the agent receives the higher rating y 2 if and only if the principal is A-type and her observation is no smaller than ˆx. Thus, the equation (2) states that the agent s expected utility from the supposed equilibrium is equal to U. Namely, the agent s individual rationality constraint is satisfied and the agent s acceptance is his best response. Since π(1) > π(θ(y 1 e)) for any e E, the equation (1) implies that w(y 2 ) >w(y 1 ). Since the higher wage is paid only for the higher observations, the MLRP implies that the agent has the incentive to choose the higher effort when the difference in cost between the two efforts is sufficiently small. Next, we examine the principal s behavior. When the principal gives the rating y 2,given the described strategies, the agent believes her to be A-type. Thus, her interim payoff from the rating y 2 is given by w(y 2 )+π(1). On the other hand, when she gives the rating y 1,the agent s belief is θ(y 1 e) whenhiseffort is e. Thus, when the observation is x, theinterim payoff of p-type (p P) from the rating y 1 is given by P e E h p(e x)π(θ(y 1 e)) w(y 1 ). Hence, the equation (1) states that, given the above contract and the agent s belief, the interim payoffsofa-type for both ratings are identical when her observation is ˆx. Due to the monotone likelihood ratio property, it holds that F A (x e 2 ) <F A (x e 1 ) for any x (0, 1). It implies that the agent s belief given the rating y 1 is lower when he has taken the effort e 2 than when he has taken the effort e 1.Orwehaveθ(y 1 e 2 )) < θ(y 1 e 1 ). Hence, it holds that π(y 1 )=π(θ(y 1 e 2 )) π(θ(y 1 e 1 )) < 0. Observe that the interim payoff of p-type from the rating y 1 can be rewritten as w(y 1 )+π(θ(y 1 e 1 ))+h p (e 2 x) π(y 1 ). Since h A (e 2 x) is strictly increasing because of the MLRP, it is strictly decreasing in x. ThefactthatA-type with the observation ˆx is indifferent between the two ratings implies that A-type with the observation higher (lower) than ˆx strictly prefers giving the rating y 2 (y 1 respectively). For B-type, by assumption, it holds that f B (1 e 2 )/f B (1 e 1 ) <f A (ˆx e 2 )/f A (ˆx e 1 ). It implies that h B (e 2 x) <h A (e 2 ˆx) for any x X. The same logic as above then implies that B-type strictly prefers giving the rating y 1 no matter what observation she obtains. 11

12 Therefore, when the difference in cost between the two efforts is sufficiently small, the stated strategies form a continuation equilibrium given the above contract. Namely, the above contract is in C (µ). This example illustrates three points in how the reputational effect enables the principal to utilize the subjective evaluation in a credible way. First, for the reputation to function, the two types of the principal need to have different observation technologies. It causes the two types of the principal to behave differently,whichinturncausestheagenttoupdate his belief after the ratings are given. Faced with the higher rating, the agent believes that it is given by A-type. On the other hand, he believes that it is given by B-type with a positive probability when he is given a lower rating. (In more general instances studied later, when the reputation is successfully utilized, the rating that is associated with a higher (lower) wage is given by the trustworthy type with a relatively higher (lower respectively) probability than by the untrustworthy one.) This difference in the agent s belief is utilized to induce the principal to evaluate honestly. This is the second point. Namely, the rating that is linked to the higher wage leads to the higher belief by the agent. Third, as the principal gives the higher rating after the higher observations, this wage difference creates the agent s incentive to take the higher effort. For the higher wage to be associated with the higher observations, it is necessary that A-type is more informed of the agent s performance given the higher observations, which we have assumed for this example. 4. Strict Partition Property The interesting aspect of the equilibrium described in the above example is that both types of the principal (almost surely) have the strict best responses for the ratings at the evaluation and also that their responses are (weakly) monotonic in the observations. Let us generally define the property that we have found in the example. Strict Partition Property Suppose that the set of ratings is Y = {y 1,...,y I } and consider the sequences of cutoff points {ˆx A i }I i=0 and {ˆxB i }I i=0. We say that the reputational system has the strict partition property (the SPP hereafter) when, for any p P, i) ˆx p 0 =0,ˆxp I =1,andˆxp i 1 ˆxp i for i =1,...I,and 12

13 ii) when p-type s observation x is strictly between ˆx p i 1 and ˆxp i (ˆx p i 1 <x<ˆxp i ), she strictly prefers giving the rating y i in the corresponding continuation equilibrium for i =1,...,I. Here, the strictness refers to that of the best responses. In the sense that the higher observation is unambiguously linked to the higher rating, this is a natural property that we expect in an incentive contract. Within the set C (µ), we denote the set of contracts given which the corresponding reputational system has the SPP by C (µ)( C (µ)). This section first characterizes the reputational systems with the SPP. Using the characterization, it then studies how the observation technologies have to be different between the two types of the principal in order for the reputational system with the SPP to exist Reputational System with the Strict Partition Property This subsection characterizes the reputational system with the SPP by providing the necessary condition as well as the sufficient one. Observe that π(y i ) π(θ(y i e 2 )) π(θ(y i e 1 )) and that the interim payoff of p- type (p P) isgivenby w(y i )+π(θ(y i e 1 )) + h p (e 2 x) π(y i ) when she observes x and ³ gives the rating y i.sinceh p (e 2 x) = 1+µf p (x e 1 )/ (1 µ)f p (x e 2 ) 1 by the Bayes rule, the two types have the same best responses at the evaluation when their observations correspond to the same likelihood ratio. In the reputational system with the SPP, the principal has the unique best response (almost surely). Hence, it implies that the two types whose observations correspond to the same likelihood ratio (almost surely) behave in the same way at the evaluation. If the agent takes the lower effort with the probability zero, the conditional probability h p (e 2 x) is one and the principal s interim payoff is not dependent on the observation. Then, it is not possible that the principal has the strict best response at the evaluation in any reputational system. Namely, C (0) =. For the reputational system to have the SPP, the agent needs to take the lower effort with a positive probability. In the following, we suppose that µ>0. We now consider the property of the cutoff points that are associated with the SPP. By symmetry, we first examine the best response of A-type at the evaluation. Consider the case that ˆx A i 1 < ˆxA i < ˆx A i+1. Choose x and x0 so that ˆx A i 1 <x<ˆxa i <x 0 < ˆx A i+1. By 13

14 the definition of the SPP, A-type with the observation x strictly prefers the rating y i to the rating y i+1 while A-type with the observation x 0 strictly prefers the rating y i+1 to the rating y i. Hence, the following inequalities hold. w(y i )+π(θ(y i e 1 )) + h A (e 2 x) π(y i ) > w(y i+1 )+π(θ(y i+1 e 1 )) + h A (e 2 x) π(y i+1 ), and w(y i+1 )+π(θ(y i+1 e 1 )) + h A (e 2 x 0 ) π(y i+1 ) > w(y i )+π(θ(y i e 1 )) + h A (e 2 x 0 ) π(y i ). Combining these, we obtain ha (e 2 x 0 ) h A (e 2 x) π(y i+1 ) π(y i ) > 0. Since h A (e 2 x 0 ) h A (e 2 x) > 0 by the MLRP, it has to hold that π(y i+1 ) > π(y i ). Namely, π(y i ) is increasing in i. As the rating becomes higher, the difference between the reputational benefit givene 2 and the one given e 1 becomes larger. (This is often called the single crossing property because the interim payoffs for different ratings cross each other only once when they are regarded as the functions of the observation x.) Moreover, when we take both x and x 0 closer to ˆx A i, the continuity of h A(e 2 x) withrespecttox implies that w(y i )+π(θ(y i e 1 )) + h A (e 2 ˆx A i ) π(y i ) = w(y i+1 )+π(θ(y i+1 e 1 )) + h A (e 2 ˆx A i ) π(y i+1 ). Hence, A-type with the observation ˆx A i has to be indifferent between the rating y i and the rating y i+1. (Given the single crossing property, this condition constitutes the truth-telling condition for the principal.) Now, we also suppose that ˆx B i 1 < ˆxB i < ˆx B i+1. By applying the same logic to B-type with the observation ˆx B i, we can conclude that she has to be indifferent between the rating y i and the rating y i+1 : w(y i )+π(θ(y i e 1 )) + h B (e 2 ˆx B i ) π(y i ) = w(y i+1 )+π(θ(y i+1 e 1 )) + h B (e 2 ˆx B i ) π(y i+1 ). Since π(y i+1 ) > π(y i ), it implies that h A (e 2 ˆx A i )=h B(e 2 ˆx B i ). It means that, on the corresponding cutoff points, the likelihood ratios of the two types have to be identical: f A (ˆx A i e 2 )/f A (ˆx A i e 1 )=f B (ˆx B i e 2 )/f B (ˆx B i e 1 ). 14

15 There may be a rating that is given only by one type. (Without loss of generality, we ignore the rating that is never given by either type.) From the above, we know that both types behave in the same way when the likelihood ratios given their observations are equal. Hence, if the two players behave differently, it occurs either for the highest rating or for the lowest rating. When the rating y i is given only by one type, it holds that π(y i )=0. Hence, by the single crossing property, there exists at most one rating that is given only by one type. Consider the case that the lowest rating is given only by A-type: ˆx A 0 < ˆx A 1 and ˆx B 0 =ˆx B 1. Then, the fact that A-type with the observation ˆx A 1 is indifferent between the rating y 1 and the rating y 2 and B-type never prefers giving the rating y 1 implies that f A (ˆx A 1 e 2 )/f A (ˆx A 1 e 1 ) f B (0 e 2 )/f B (0 e 1 ). Similarly, consider the other case that the highest rating is given only by A-type: ˆx A I 1 < ˆxA I and ˆx B I 1 =ˆxB I. Then, it has to hold that f A (ˆx A I 1 e 2 )/f A (ˆx A I 1 e 1 ) f B (1 e 2 )/f B (1 e 1 ). We can obtain the conditions in a similar way when there exists one rating that is given only by B-type. Let us summarize the above arguments in two conditions. Condition CP Take the cutoff points ({ˆx A i }I i=0, {ˆxB i }I i=0 ) such that ˆxp i 1 ˆxp i any i {1,...,I}. for any p P and for i) At most one of the four equality holds among ˆx A 0 =ˆxA 1,ˆxB 0 =ˆxB 1,ˆxA I 1 =ˆxA I,and ˆx B I 1 =ˆxB I. The rest of the cutoff points are distinct. ii) For i {1,...,I 1}, whenˆx A i 1 < ˆxA i < ˆx A i+1 and ˆxB i 1 < ˆxB i < ˆx B i+1,itholds iii) that f A (ˆx A i e 2)/f A (ˆx A i e 1)=f B (ˆx B i e 2 )/f B (ˆx B i e 1 ). Take p and p 0 from P such that p 6= p 0. When ˆx p 0 =ˆxp 1, it holds that f p 0(ˆxp0 1 e 2)/f p 0(ˆx p0 1 e 1) f p (0 e 2 )/f p (0 e 1 ). When ˆx p I 1 =ˆxp I,itholdsthatf p 0(ˆxp0 I 1 e 2)/f p 0(ˆx p0 I 1 e 1) f p (1 e 2 )/f p (1 e 1 ). When it is believed by the agent that the principal s strategies are characterized by the cutoff points, it holds that π(y i ) < π(y i+1 ) for any i {1,...,I 1}. Truth-telling Condition 15

16 Take the cutoff points ({ˆx A i }I i=0, {ˆxB i }I i=0 ) and the wages {w(y i)} I i=1. Suppose that the agent believes the principal s strategies to be characterized by the cutoff points. ˆx p i 1 < ˆxp i < ˆx p i+1 for p P, it holds that w(y i)+π(θ(y i e 1 )) + h p (e 2 ˆx p i ) π(y i)= w(y i+1 )+π(θ(y i+1 e 1 )) + h p (e 2 ˆx p i ) π(y i+1). The above analysis has shown that the reputational system with the SPP has to satisfy these conditions. Condition CP is solely concerned with the cutoff points. In particular, it requires that i) there is at most one rating that is given only by one type, ii) the likelihood ratios on the corresponding cutoff points are same between the two types except possibly at the end where only one type may give the rating, and iii) the single crossing property holds. On the other hand, the truth-telling condition is concerned with the wages as well as the cutoff points. It requires that the principal whose observation corresponds to a cutoff point is indifferent between the two adjacent ratings. In addition to these conditions, the reputational system with the SPP has to satisfy the agent s incentive compatibility constraint (IC) as well as his individual rationality constraint (IR): If IX IX u(w(y i ))Prob(y i e 2 ) e 2 u(w(y i ))Prob(y i e 1 ) e 1, and i=1 i=1 IX u(w(y i ))Prob(y i e 2 ) e 2 U, i=1 (IC) (IR) where Prob(y i e) =θ 0 FA (ˆx A i e) F A(ˆx A i 1 e) +(1 θ 0 ) F B (ˆx B i e) F B (ˆx B i 1 e). The next proposition provides the necessary condition for the reputational system with the SPP. Proposition 1 We focus on the reputational system where all the ratings are used with positive probabilities: P R p sp (y i x) µdf p (x e 1 )+(1 µ)df p (x e 2 ) > 0 for any y i Y. In any reputational system with the SPP, the agent takes both efforts with positive probabilities. Moreover, any reputational system with the SPP has to satisfy both Condition CP and the truth-telling condition in addition to the agent s IC and IR. On the other hand, the above argument also implies that, when both Condition CP and the truth-telling condition are satisfied, the best response of p-type at the evaluation is to 16

17 report y i when her observation is between ˆx p i 1 and ˆxp i. Namely, the principal is induced to tell the truth. By increasing the level of wages, the agent s individual rationality constraint can be satisfied without affecting these conditions. Under the MLRP, if the wages are increasing in the ratings, there is some incentive for the agent to take the higher effort. Hence, given the cutoff points and the wages, if both Condition CP and the truth-telling condition are satisfied, if the wages are increasing in the ratings, and if the difference in cost between the two efforts is sufficiently small, we can construct a reputational system with the SPP. Proposition 2 Suppose that the contract (wages and ratings) together with the principal s strategies and the corresponding agent s beliefs satisfy Condition CP, the truth-telling condition, and the increasing wages (w(y i ) w(y i+1 ) for any i {1,...,I 1} and w(y i ) <w(y i+1 ) for some i {1,...,I 1}). Then, when the difference in cost between the two efforts is sufficiently small, we can construct a reputational system with the SPP Relative Accuracy of Observation Technologies between the Two Types Using the characterization derived above, this subsection investigates how the observation technologies have to be different between the two types of the principal in order for the reputational system with the SPP to exist. When the observation technologies of the two types are same, the third part of Condition CP is never satisfied. For the reputational system with the SPP to exist, it is necessary that the observation technologies are different between the two types. In this subsection, using the binary evaluation (the evaluation where there are only two ratings), we illustrate how the accuracy of observations has to be different between the two types, and, in particular, how A-type has to have more accurate observation of the agent s performance in addition to being trustworthy. In this subsection, we suppose that Y = {y 1,y 2 }. 1) A Necessary Condition for the Binary Evaluation First, we show that, given the higher effort, A-type is more likely to obtain the higher observations and to give the higher rating in any reputational system with the SPP than B-type is: F A (ˆx A 1 e 2 ) <F B (ˆx B 1 e 2 ). Namely, when the agent takes the higher effort, the higher observations are more likely for A-type than for B-type. 17

18 As shown in the third part of Condition CP, it has to hold that π(y 2 ) > π(y 1 ). Moreover, the incentive compatibility constraint of the agent implies that w(y 2 ) >w(y 1 ). The former requires either that A-type gives the higher rating more frequently given the higher effort or that A-type gives the lower rating more frequently given the lower effort. On the other hand, the latter requires, roughly speaking, that the higher wage given the higher rating has to be compensated by the higher belief by the agent. It implies that A-type is more likely to give the higher rating. Combining these two factors, we can understand, in the reputational system with the SPP, how A-type needs to obtain the higher observations with the higher probability when the agent takes the higher effort. Result 1 Suppose that Y = {y 1,y 2 }. Let {ˆx A i }2 i=0, {ˆxB i i=0 }2 be the cutoff points of a reputational system with the SPP. Then, it has to hold that F A (ˆx A 1 e 2) <F B (ˆx B 1 e 2). 2) Marginally Informed A-type and the Impossibility of the Reputational System Next, consider the case that B-type is more informed of the agent s performance than A-type is. In particular, we suppose that there exists δ(> 0) such that 1 δ < f A (x e 2 )/f A (x e 1 ) < 1+δ for any x [0, 1]. Given any observation, the likelihood ratio for A-type is close to one, which indicates that A-type updates her belief about the agent s effort only by a small degree. In case of the binary evaluation, when δ is sufficiently small, there is no way that the reputational system with the SPP is feasible. To understand the reason, let us suppose the contrary. When δ is small, the likelihood ratio f A (ˆx A 1 e 2 )/f A (ˆx A 1 e 1 ) is close to one. Then, by the second part of Condition CP, the likelihood ratio of B-type f B (ˆx B 1 e 2 )/f B (ˆx B 1 e 1 ) is also close to one. Then, for sufficiently small δ, the MLRP implies that F B (ˆx B 1 e 2 ) < F B (ˆx B 1 e 1 ). Since F A (ˆx A 1 e 2 ) F A (ˆx A 1 e 1 ) for small δ s, it holds that θ(y 2 e 2 ) < θ(y 2 e 1 ) and also that θ(y 1 e 2 ) > θ(y 1 e 1 ). This implies that π(y 2 ) < 0 < π(y 1 ), which violates the third part of Condition CP. When B-type is more informed, the one who is able to reveal one s type is B-type. However, B-type does not want to reveal her type. Thus, the reputation cannot be used to induce the truth-telling from the principal. In the example at Section 3, A-type, who is 18

19 willing to reveal her own type, is more informed. For the reputation to work successfully, the trustworthy type has to be more informed and is able to reveal one s type (at least more frequently than B-type). Result 2 Suppose that Y = {y 1,y 2 }.FixB-type s observation technology f B (x e). Then, there exists δ such that, when 1 δ <f A (x e 2 )/f A (x e 1 ) < 1+δ for any x [0, 1], it holds that C (µ) = for any µ. 5. The Robustness of the Reputational System In this section, we consider the modification to the basic model by assuming that, when the interim payoffs are same, the principal prefers the rating associated with the highest wage among those that bring out the same interim payoff. The reputational system that remains so after the modification is called robust. Game with Lexicographic Ordering Suppose that w(y i )+ P e E h p(e x)π(θ(y i e)) = w(y j )+ P e E h p(e x)π(θ(y j e)) and that w(y i ) <w(y j ). Then, p-type (p P) never gives the rating y i when her observation is x. (The principal still prefers the rating that brings the higher interim payoff when the expected interim payoffs aredifferent.) The rest is identical to the basic model described in Section 2. The modified game described above is called the game with lexicographic ordering while the game described in Section 2 is called the original game. The supposition behind this modification is that there might be a small psychological cost associated with giving the lower wages. (Since the higher wage increases the utility of the worker, the assumption states that the principal prefers the weakly Pareto improving change.) We say that the reputational system is robust when the introduction of the lexicographic ordering does not undermine it. Robust Reputational System Take a reputational system for the original game: c C (µ) ande(c ). If the strategies and the beliefs in E(c ) constitute a continuation equilibrium given c in the game with lexicographic ordering, we say that the reputational system is robust. 19

20 As we will explain in the second part of this section, some reputational system in C (µ) may be critically dependent on the indifference of the principal among ratings. We deem them to be artificial and fragile. They cannot be maintained once the lexicographic ordering is introduced. The first subsection shows that the robustness of reputational systems is equivalent to the strict partition property. The second subsection then examines when the robustness holds by looking at the reputational systems that are not robust (and do not have the SPP) The Robustness and the Strict Partition Property When the reputational system has the SPP, the two types of the principal have the unique optimal choices of ratings except when their observations are exactly at the cutoff points. Since the principal whose observation is at the cutoff point is indifferent between the adjacent ratings and also since the probability of such instances is zero, the (almost same) outcome of a given reputational system with the SPP can be supported 9 in a robust reputational system. What we want to show in the remaining of this subsection is the reverse. Namely, we show that the robust reputational system has to have the SPP. Without loss of generality, in the following analyses, let us exclude the ratings that are never used with a positive probability by either type from the set Y. We say that the ratings y i and y j are equivalent when it holds that w(y i )=w(y j )and θ(y i e) =θ(y j e) for any e E. Lety k be a new rating, which is the combination of these ratings (i.e., the rating y k is reported when either y i or y j is supposed to be reported). We choose its wage so that w(y k ) w(y i ). As a standard property of the conditional probability, it holds that θ(y k e) =θ(y i e) =θ(y j e) for any e E. Hence, the replacement of the two ratings with the newly created rating can be supported in the continuation equilibrium. In this sense, we regard the combination of the equivalent ratings as one rating. 9 In order to transform the reputational system with the SPP to the robust one, we need to modify only the strategy of the principal whose observation is at a cutoff pointsothatshe chooses the rating associated with the higher wage between the adjacent ratings. Since she is indifferent between the adjacent ratings, it is also an optimal strategy for her. Since the probability that the principal s observations lie exactly at the cutoff points is zero, changing her behavior strategies in this way does not change the belief of the agent. Thus, when we make the change described above, the players strategies form a continuation equilibrium both in the original game and in the game with lexicographic ordering. Namely, whenever there is a reputational system with the SPP, the robust reputational system with the almost same outcome can be created by the same contract. 20

21 We claim that, whenever π(y i )= π(y j ), the two ratings are equivalent in the robust reputational system. Suppose that p-type (p P) givestheratingy i after the observation x and and gives the rating y j after the observation x 0.(WhenonlyA-typegivestheratingy i with a positive probability and only B-typegivestheratingy j with a positive probability, we can easily obtain the same conclusion.) Then, the principal s truth-telling requires that the following inequalities hold. w(y i )+π(θ(y i e 1 )) + h p (e 2 x) π(y i ) w(y j )+π(θ(y j e 1 )) + h p (e 2 x) π(y j ), w(y j )+π(θ(y j e 1 )) + h p (e 2 x 0 ) π(y j ) w(y i )+π(θ(y i e 1 )) + h p (e 2 x 0 ) π(y i ). Since π(y i )= π(y j ), it implies that w(y i )+π(θ(y i e 1 )) = w(y j )+π(θ(y j e 1 )) and that the two types of the principal are indifferent between the rating y i and the rating y j no matter what observation they obtain. In the game with lexicographic ordering, it has to hold that w(y i ) = w(y j ), which implies that π(θ(y i e 1 )) = π(θ(y j e 1 )). Since π(y i )= π(y j ), we have π(θ(y i e 2 )) = π(θ(y j e 2 )) as well. Hence, these ratings are equivalent. It implies that different ratings have to have different π(y i ) s in the robust reputational system unless they are equivalent. Since there has to be a variation in wages for the agent s incentive compatibility constraint to be satisfied, there need to be at least two π(y i ) s in equilibrium. Moreover, since the interim payoffs from these ratings have to be different, it has to hold that µ>0. Let Y = {y 1,y 2,...,y I }. We put the indices to y i so that π(y i ) < π(y i+1 ) for any i {1,...,I 1}. In any robust reputational system, we claim that the higher rating needs to correspond to the higher likelihood ratio. To understand its logic, let us take the ratings y i and y j such that π(y i ) > π(y j ). Bysymmetry,westudythebehaviorofA-type. First, suppose that A-type with the observation x gives the rating y i in equilibrium. Then, it needs to hold that π(θ(y i e 1 )) + h A (e 2 x) π(y i ) w(y i ) π(θ(y j e 1 )) + h A (e 2 x) π(y j ) w(y j ). Now consider the interim payoff of A-type when she observes x 0 (>x). Since h A (e 2 x) is strictly increasing in x and π(y i ) > π(y j ), the above inequality will hold strictly when we replace x with x 0. It implies that A-type with the observation x 0 strictly prefers giving the rating y i to giving the rating y j. (Observe that the above is caused by the single crossing property.) Next, suppose that A-type with the observation x gives the rating y j in equilibrium. Then, by the same logic, A-type strictly prefers giving y j when her observation 21

22 is lower than x. The next lemma proves this monotonicity property about p-type s strategy at the evaluation. Lemma 1 Suppose that µ>0. Take y i and y j from Y such that π(y i ) > π(y j ). 1) If p-type (p P) with the observation x gives the rating y i in equilibrium, p-type with the observation higher than x strictly prefers giving the rating y i to giving the rating y j.on the other hand, if p-type with the observation x gives the rating y j in equilibrium, p-type with the observation lower than x strictly prefers giving the rating y j to giving the rating y i. 2) Between y i and y j,ifp-type (p P) with the observation x 0 prefers the rating y i and the same type with the observation x 00 prefers the rating y j,thereexistsxsuch that the same type with this observation is indifferent between the two ratings. Moreover, the other type with the same likelihood is also indifferent. This lemma implies that the equilibrium has the partition property and that the likelihood ratios at the cutoff points are same between the two types. As the direct consequence of the lemma, we have the following proposition. Proposition 3 When we regard the combination of equivalent ratings as one rating, any robust reputational system has the strict partition property. In this sense, our focus on the reputational system with the SPP in the previous section is justified Conditions for the Robust Reputational System By studying when there are no reputational systems with the SPP, this subsection illustrates what is essential in having the robust reputational system. First, consider the case that the agent takes the higher effort with the probability one (ν = µ = 0). As we have shown in Section 4, there is no reputational system with the SPP in such a case and thus the above result implies that there is no robust reputational system. Yet, in this case, some reputational system without the SPP is always feasible as long as the difference in cost between the two efforts is sufficiently small. Let Y = {y 1,y 2 } and suppose 22