A dynamic model of authority in organizations

Li, Bingbing; Förster, Manuel Working Paper A dynamic model of authority in organizations Center for Mathematical Economics Working Papers, No. 753 Provided in Cooperation with: Center for Mathematical Economics (IMW), Bielefeld University Suggested Citation: Li, Bingbing; Förster, Manuel (2025) : A dynamic model of authority in organizations, Center for Mathematical Economics Working Papers, No. 753, Bielefeld University, Center for Mathematical Economics (IMW), Bielefeld, https://nbn-resolving.de/urn:nbn:de:0070-pub-30071276 This Version is available at: https://hdl.handle.net/10419/333506 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ 753 September 2025 A dynamic model of authority in organizations Bingbing Li and Manuel Foerster Center for Mathematical Economics (IMW) Bielefeld University Universit¨atsstraße 25 D-33615 Bielefeld ·Germany e-mail: [email protected] uni-bielefeld.de/zwe/imw/research/working-papers ISSN: 0931-6558 Unless otherwise noted, this work is licensed under a Creative Commons Attribution 4.0 International (CC BY) license. Further information: https://creativecommons.org/licenses/by/4.0/deed.en https://creativecommons.org/licenses/by/4.0/legalcode.en A dynamic model of authority in organizations Bingbing Li * and Manuel Foerster  September 29, 2025 Abstract In our principal-agent model, the principal can repeatedly delegate authority to an agent with uncertain preferences or take the decisions himself. The principal learns the state at the end of each period and then updates his belief about the agent’s bias based on the decision implemented if he delegated authority. We demonstrate that equilibria are characterized by an “imitation” interval of agent types (biases) who mimic less biased types in order to be retained. Interestingly, the principal generally benefits from the agent’s imitation compared to a benchmark. Furthermore, comparative statics reveal that, surprisingly, the principal may be worse off with better information. Finally, an extension to finitely many periods shows that the imitation interval gradually shifts, such that agent types within the interval imitate less biased types. JEL classification: D23; D82; D83; D73; C72. Keywords: Delegation, preference uncertainty, private information, dynamic game, organizational design. 1 Introduction In various organizational contexts, decision-makers engage better-informed experts lower in the hierarchy for assistance or advice. For instance, policy-makers seek informational support from subordinate bureaucrats when formulating draft policies. However, bureaucrats often have a partisan orientation, which may not * Center for Mathematical Economics, Bielefeld University, PO Box 10 01 31, 33501 Bielefeld, Germany. Email: [email protected].  Center for Mathematical Economics, Bielefeld University, PO Box 10 01 31, 33501 Bielefeld, Germany. Email: [email protected]. 1 align with that of the current policy-maker and affect their behavior.1Crucially, the policy-maker—at least initially—faces uncertainty regarding these partisan orientations. When delegating decision-making authority to subordinates, the policy-maker therefore has to take into account both their expertise and potential conflicts of interest. Over time, the chosen subordinate may then reveal her partisan preferences through her decisions. Another important aspect therefore is the dynamic nature of the delegation process (under preference uncertainty): The policy-maker will not only get to know the environment in her policy domain but also learn the preferences of subordinates, allowing her to reassess her choices. The trade-off between keeping authority and delegating it to a specialist with different objectives has been extensively discussed in the organizational economics literature. Yet, the preference conflict between principal and agent has been considered common knowledge (e.g., Aghion and Tirole,1997;Dessein,2002;Alonso and Matouschek,2008). In this paper, we build a theoretical model in which a principal repeatedly decides between keeping authority and granting decision rights to an agent with uncertain preferences. We demonstrate how the prospect of repeated influence on the decision-making process can discipline experts with partisan interests to behave like “good” ones whose objectives are roughly in line with the principal’s. In our model, a principal (he) has to take a decision whose payoff depends on the evolving state of the world in each of two periods. He can either keep authority and decide himself or delegate authority to a better-informed agent (she) with uncertain preferences. At the end of the first period, the principal observes the true state of the world, thereby acquiring partial insights into the state of the next period (i.e., the state is correlated across periods). If the principal has delegated authority in the first period, he forms expectations regarding the agent’s type based on this information and the decision implemented. At the beginning of the second period, the principal then decides again whether to keep authority or delegate it to a better-informed agent—if he delegated authority in the first period, the principal may either retain or replace the current agent. Observe that a principal who prefers to keep decision-making authority in the 1In the U.K., bureaucrats have tried to “frustrate this process of Brexit because it goes against the grain so fundamentally”, see https://www.bbc.com/news/uk-42782637?utm_source, accessed August 8, 2025. For similar cases in the U.S., see https://www.washingtonpost.com/ politics/2025/02/07/trump-resistance-federal-workers/?utm_source, accessed August 8, 2025. See also Brehm and Gates (1999), who conclude that “the overwhelming evidence [. . . ] indicates that the bureaucrat’s own preferences have the greatest effect on behavior.” 2 first period will do so also in the second period, as he then is better informed. We thus focus on the case when the principal delegates decision-making authority in the first period. We first establish that, when the discount rate is low, the unique equilibrium is characterized by an imitation interval. While agents who have a low preference conflict with the principal choose their bliss point and are subsequently retained, agents within this interval imitate an agent of the former group in order to be retained as well. Having an intermediate conflict of interest with the principal, they trade off an immediate loss due to a deviation from their bliss point for a future gain due to being retained and thus able to choose their bliss point in the next period. In turn, the principal retains the agent despite the imitation because the expected preference conflict is small enough. Second, when the discount rate is high, the value of retention for the agents is high, as it allows them to implement their bliss point in the second period. In equilibrium, the principal will then retain the agent upon observing potential mimicking only with a certain probability, and in turn, fewer agent types mimic. We then show that the principal generally benefits from the agent’s imitation. Disciplining the agent, it yields a gain from a better decision in the first period compared with a benchmark in which all agents choose their bliss points. In the second period, it then yields a potential loss from retaining rather biased agents that the principal would not retain if informed about their type. Interestingly, we establish that the loss is, generally, second order. Next, we examine whether the principal benefits from a more slowly changing environment, such that he has better information about the state in the second period. Our findings reveal that, surprisingly, the principal may be worse off with better information. Some relatively biased agents then cease mimicking a less biased one and instead implement their bliss point in the first period, which reduces the principal’s instantaneous, and potentially his total, utility. The reason is that imitation becomes less attractive compared to the alternative of the principal taking the decision himself, as the latter becomes better informed. Finally, we extend the model to finitely many periods. We establish that the principal is less and less willing to retain the agent upon observing potential mimicking as opportunities to mimic in the future vanish. In equilibrium, the imitation interval thus gradually shifts, such that the agent types within the interval imitate less biased types. Related literature. This paper is situated within the literature on the delegation of formal authority going back to the seminal work by Aghion and Tirole 3 (1997). They show that the principal may delegate authority in order to give the agent better incentives to acquire information. Dessein (2002) takes the information structure as given and investigates the principal’s trade-off between keeping authority and delegating it to the agent depending on their preference conflict. While Dessein shows that the principal prefers delegation over communication if objectives are sufficiently aligned, Deimen and Szalay (2019) show the reverse may hold if the agent has to decide on the amount of information she observes about each of two states. Other contributions have extended the framework to optimal delegation mechanisms (Alonso and Matouschek,2008), transfers (Kr¨ahmer,2006; Lim,2012), or both (Foerster and Habermacher,2025a).2In contrast to these papers, we consider uncertainty regarding the preference conflict and show how the prospect of repeated influence disciplines experts with partisan interests. The disciplining effect of repeated interaction relates our paper to the political agency literature, which has shown that elections can serve the purpose of disciplining a “bad” incumbent politician (agent) to behave like a “good” one to get re-elected by voters (the principal) (Berganza,2000;Besley and Smart,2007; Foerster and Voss,2022). We show that the same mechanism applies to organizations when a principal faces uncertainty regarding the agent’s preferences and can re-allocate authority depending on her decisions. From a modeling point of view, our model differs from these papers in various aspects. First and foremost, we consider a continuum of types but abstract from differences in ability as in Foerster and Voss (2022). This has an interesting consequence: Since biased agents in Foerster and Voss (2022) imitate less able ones, the disciplining effect may, unlike in our model, be negative from the principal’s point of view. Second, we allow for a correlation of the state across periods, such that the principal may have better information in the second period. It turns out that, surprisingly, more information can sometimes harm the principal. Third, we extend the model to finitely many periods and investigate correlations of the state across periods. Moreover, the extension of our model to finitely many periods shares some features with Banks and Duggan (2008). Instead of a principal-agent model, they consider a general model of repeated elections in which the challenger in each period is randomly chosen from the set of voters, who are assumed to have private policy preferences. Banks and Duggan show that incumbents choose policy compromises in order to get re-elected—reminiscent of experts imitating less biased ones in order to be retained in our model. 2Foerster and Habermacher (2025b) extend the framework to Bertrand competition between (policy) experts. 4 In another related paper, Prendergast (2007) studies a principal who may hire a bureaucrat motivated by own objectives to exert effort. In an extension to preference uncertainty, he shows that the wage offered by the principal may serve as a selection device. In contrast, we focus on the allocation of decision rights within an organization and consider selection through repeated interaction. To the best of our knowledge, we are the first to study the repeated allocation of authority to an agent with uncertain preferences in an organizational context. Our results show how the prospect of repeated influence on the decision-making process can discipline agents with partisan interests to behave like—and thereby imitate—agents whose objectives are roughly in line with the principal’s. The rest of the paper proceeds as follows. In Section 2, we set up the model. Section 3derives the pure and mixed equilibria and analyzes the benefits of imitation. Section 4derives comparative statics on the correlation of the state. Section 5extends the model to finitely many periods. Section 6concludes. 2 Model and notation We consider an economy populated by many ex ante identical agents and a principal. In period t= 1 the unknown state (of the world) θ1∈Θ = Ris distributed according to a commonly known distribution Fon Θ with expected value µand variance σ2. In t= 2, the state θ2∈Θ is partially correlated with θ1at a rate λ∈[0,1), i.e., θ2=λθ1+ (1 −λ)˜ θ2with ˜ θ2and θ1i.i.d. In each period t= 1,2, the principal P(he) has to take a decision yt∈R. In the first period, P(who does not observe the state) decides whether to delegate authority over the decision y1to an agent A1(she), whose type b1∈B=Ris her private information. We assume that b1is randomly drawn from a commonly known distribution Gwith continuous, strictly positive, and symmetric density g on Rwith expected value µG= 0 and variance σ2 G.3If A1is selected, she observes the true state θ1and then takes the decision y1. If Pretains the authority, he takes the decision himself. At the end of the first period, Pobserves the state θ1 and the outcome y1. At the beginning of the second period, Pdecides whether to retain A1(if he has delegated the decision to A1in the first period), to choose another agent, A2, from the pool, or to take the decision himself. A2’s type b2∈Bis her private information and randomly drawn from G(independently of b1). 3We assume symmetry to simplify the exposition. Our results are qualitatively robust to non-symmetric distributions. 5 The utility of Pis given by v(y1, y2|θ1, θ2) = − 2 X t=1 δt−1(θt−yt)2, where 0 < δ < 1 denotes the discount factor. That is, Phas bliss point θtin period t= 1,2. Similarly, the utility of an agent of type b, who may or may not have authority over the decision in one or both periods, is u(y1, y2|b, θ1, θ2) = − 2 X t=1 δt−1(θt+b−yt)2, i.e., the agent of type bhas bliss point θt+bin period t= 1,2. To summarize, the timing of events is as follows: 1. Nature draws the state θ1and the type b1of A1. 2. Pdecides whether to delegate authority to A1. 3a. A1takes the decision y1if Phas delegated authority to her. 3b. Ptakes the decision y1himself if he keeps authority. 4. Plearns θ1and y1at the end of period 1. 5. Pdecides whether to retain A1, to choose another agent A2, or to take the decision himself at the beginning of period 2. 6. Nature draws the state θ2. If Phas delegated authority to another agent A2, nature draws her type b2. 7. P(A1or A2) takes the decision y2. 8. Payoffs realize. The solution concept we employ is perfect Bayesian equilibrium. 3 Equilibrium analysis We first determine the second-period decision depending on authority and then the allocation of authority. Second, we determine A1’s decision in the first period. Note that we will restrict attention to the interesting case where Pdelegates authority to A1at the beginning of the first period. To conserve on notation, we 6 do not explicitly define the strategy of P, who decides whether to retain A1based on the observed decision ˆy1and the state θ1at the end of period 1. 3.1 Second-period decision and allocation of authority If Pkeeps the authority in the second period, he maximizes his one-period expected utility given the first-period state θ1, and thus chooses y∗ 2,P = arg max y2 E[−(θ2−y2)2|θ1] = E[θ2|θ1] = λθ1+ (1 −λ)µ. (1) The residual variance is (1 −λ)2σ2. If Phas delegated authority to agent A2of type bin the second period, the latter maximizes her one-period expected utility, as revealing bias then is irrelevant, and thus chooses her bliss-point decision y∗ 2,A(b, θ2) = arg max y2 E[−(θ2+b−y2)2|θ2] = θ2+b. (2) The residual variance is zero. We now turn to the decision of Pwhether to delegate authority at the beginning of period 2. Suppose for the moment that Phas kept authority and thus taken the decision himself in the first period. He will then continue to do so in the second period because the state is correlated, such that she has better information than in the first period. Therefore, we henceforth assume that the prior variance is large enough such that Pdelegates authority to A1in the first period: Assumption 1. σ2≥σ2 G. If Phas delegated authority to A1in the first period and observed the decision ˆy1and the state θ1, he updates his prior belief about A1’s type to the posterior ˜ G=˜ G(·; ˆy1, θ1). Proposition 1. Suppose that Phas delegated authority to A1in the first period. It is (weakly) optimal for Pto retain A1at the beginning of period 2 if his posterior belief ˜ Gis such that E˜ G[b2]≤min{(1 −λ)2σ2, σ2 G}.(3) Otherwise, it is weakly optimal to choose another agent from the agent pool if σ2 G≤(1 −λ)2σ2.(4) Otherwise, it is optimal for Pto take the decision himself. 7 retained. The following example illustrates the change in the imitation interval as λincreases. Figure 2: Imitation interval (¯ b(λ),¯ β(λ)] depending on λin Example 1. Example 1. Suppose that b∽N(0,1), i.e., σ2 G= 1,σ2= 4, and δ= 0.6. First, notice that Assumption 1holds since σ2 G= 1 <4 = σ2, such that Palways chooses one agent in the first period. Second, we have δ≤δ∗for all λ∈[0,1], such that a pure LCSE always exists. When λ≤0.5, the condition σ2 G≤(1−λ)2σ2is satisfied, such that ¯ b(λ)≈0.3and ¯ β(λ)≈2are constant in λ. However when λ > 0.5, the condition σ2 G>(1 −λ)2σ2holds. In this range, both ¯ b(λ)and ¯ β(λ)decrease as λ increases, ultimately reaching 0when λ= 1. In particular, the imitation interval vanishes as λ→1, see Figure 2for an illustration. Note that although ¯ b(λ)may in principle increase in λ, this requires particular prior distributions G. After having determined the effect of changes in the correlation of the state across periods on the imitation interval, we next analyze the effect on P’s equilibrium payoff. Observe that by Proposition 2, the unique pure LCSE is characterized by the imitation interval. It thus follows from Lemma 3that P’s expected utility remains constant as λincreases when the correlation of the state across periods is low. When it is high, however, his ex ante expected utility in equilibrium can be calculated as −2 Z¯ b 0 (b2+δb2)dG +Z¯ β ¯ b (¯ b2+δb2)dG +Z∞ ¯ β (b2+δ(1 −λ)2σ2)dG!. 14 First, the agents b1∈[0,¯ b] choose their bliss points and are retained by P. Second, the agents b1∈[¯ b, ¯ β] imitate agent ¯ bin the first period and choose their bliss points in the second period. All other agents b1>¯ βchoose their bliss points in the first period, upon which Pdoes not retain them and instead takes the decision himself in the second period. On the one hand, P’s expected payoff may decrease as λ increases because some relatively biased agents who did imitate b′ 1=¯ bbefore will not do so anymore and instead choose their bliss points b1>¯ b(¯ β(λ) decreases by Lemma 3). On the other hand, P’s expected payoff may increase when the gain from a better decision in the second period, in case he decides by himself, dominates the former loss. Proposition 5. (i) When λ≤1−qσ2 G σ2,P’s ex-ante expected utility is constant in λ. (ii) When λ > 1−qσ2 G σ2,P’s ex-ante expected utility may increase or decrease in λ. The following example illustrates the change in P’s payoff due to changes in the correlation of the state. Example 2. Suppose that b∽N(0,1), i.e., σ2 G= 1,σ2= 4, and δ= 0.6. Figure 3illustrates P’s payoff as a function of λ. The payoff remains constant when λ≤0.5. For λ > 0.5, we have that ¯ b(λ)and ¯ β(λ)are decreasing in λ(cf. Example 1). P’s expected payoff is decreasing for λ∈(0.5,0.83), and increasing for λ∈[0.83,1]. Example 2reveals a possible counterintuitive outcome: Pmay be worse off in equilibrium when the environment changes less across periods, although he then is better informed. Note that we can also construct such examples in case δ > δ∗ for the mixed equilibrium considered in Proposition 3. 5 Extension to finite horizon So far, we have analyzed how the prospect of influence in the second period affects A1’s behavior in the first period. We now extend the model to finitely many periods T≥3. In period t= 1 the unknown state θ1∈Θ = Ris distributed according to a commonly known distribution Fon Θ with expected value µand variance σ2. In t= 2,3, . . . , T, the state θt∈Θ is partially correlated with the previous period at a rate λ∈[0,1), i.e., θt=λθt−1+ (1 −λ)˜ θtwith θ1and (˜ θt)T t=2 i.i.d. 15 Figure 3: Change of P’s expected payoff in Example 2. The game otherwise proceeds as described in Section 2; in particular, Pobserves the state of the current period and the decision implemented at the end of each period. At the beginning of the next period, Pthen decides whether to retain the agent, choose a new agent from the pool, or take the decision himself. We restrict attention to Markov perfect Bayesian equilibrium, where at the beginning of each period t,Pmakes the authority delegation decision based solely on the state and the decision implemented in the last period t−1 (ignoring previous periods t−2, t −3, . . .). Let ut(ˆyt, yt+1, . . . , yT|bt, θt, θt+1, . . . , θT) = − T X t′=t δt′−1(θt′+bt−yt′)2 denote the restriction of agent bt’s utility function to periods t′≥t. Definition 3 (Markov perfect Bayesian equilibrium).A strategy profile y∗ tis a Markov (perfect Bayesian) equilibrium of the signaling game in period tif, for some posterior belief ˜ G=G(·|θt,ˆyt)in period t+ 1 that is consistent with y∗ tand for each agent type bt∈Band each state θt∈Θ, y∗ t(bt, θt)∈∆argmax ˆyt E˜ G[ut(ˆyt, y∗ t+1, . . . , y∗ T|bt, θt, θt+1, . . . , θT)|θt],∀t≤T−1. It is straightforward to extend the definition of LCSE (Definition 2) to Markov perfect Bayesian equilibria. Similarly to the baseline model, we proceed by backward induction. We restrict attention to small enough discount factors δsuch that 16 we can apply the results from Proposition 2and let [¯ b, ¯ β] denote the imitation interval in the pure LCSE of the two-period game. We establish that the boundaries of the imitation interval are decreasing over time and eventually coincide with the imitation interval of the two-period game. Proposition 6. (Finite-horizon pure Markov LCSE) If δ≤δ∗, then there exists a unique pure Markov LCSE such that in each period t= 1,2,...T −1, (i) agent btwho got the decision delegated imitates ˜ btif bt∈(˜ bt,˜ βt]and chooses her bliss point θt+btelse; (ii) Pretains the agent if and only if yt∈[θt, θt+˜ bt]; with [˜ bT−1,˜ βT−1] = [¯ b, ¯ β]and ˜ bt<˜ bt−1<˜ βt<˜ βt−1for 2≤t≤T−1. In the second but last period, strategic incentives are identical to the twoperiod game under Markov strategies, so that we obtain the imitation interval [¯ b, ¯ β] as in the baseline model. In earlier periods, Pbenefits more from retaining an agent because the latter may continue to imitate in the future. This shifts the lower bound and as a result also the upper bound of the imitation interval to the right as we move back to the beginning of the game. Hence, Ptightens the requirements for retaining the current agent over time by requiring decisions that are less biased. Example 3. Suppose that b∽N(0,1), i.e., σ2 G= 1,σ2= 4,δ= 0.6,λ= 0.8, and T= 15. Figure 4illustrates the specific changes of the imitation interval’s boundaries over time. Note that λis large enough that Ptakes the decision forever if he doesn’t retain the selected agent. The boundaries ˜ btand ˜ βtare decreasing over time until being equal to ¯ band ¯ β(cf. Example 1) at period 14 (the last period where imitation takes place). Observe also that the changes are becoming increasingly rapid over time. 6 Conclusion In our model, the principal faces a trade-off between keeping authority and delegating it to an informed agent with uncertain preferences, while the selected agent compares the future benefits of repeated influence on the decision-making process with the instantaneous loss of deviating from her bliss point. We focus on the case where the principal is willing to delegate authority in the first period. When the 17 Figure 4: Imitation interval (˜ bt,˜ βt] over time in Example 3. discount rate is low, the unique LCSE is characterized by an imitation interval such that the agents within this interval imitate the “good” ones whose preferences are roughly aligned with the principal. When the discount rate is high, there is a mixed equilibrium in which the principal retains the agent upon observing potential mimicking only with a certain probability. Compared with the benchmark in which all agents choose their bliss points and reveal their bias in the first period, imitation allows the principal to obtain higher ex ante expected utility. Furthermore, we demonstrate that the principal may be worse off with better information, as he incurs a loss from some relatively biased agents ceasing to mimic a less biased one in the first period. Interestingly, a principal may thus have an incentive to strategically limit their access to information, thereby avoiding the adverse incentive effects that full observability can generate. In other words, choosing to remain partially uninformed can serve as a commitment device that induces more favorable behavior from the agent. Finally, we extend the two-period model to finitely many periods and show that the imitation interval gradually shifts over time. In the future, this paper can be extended in the following ways. First, we can examine how the equilibrium changes when we incorporate cheap talk and monetary transfers, which, of course, complicates the analysis. Second, we can also design a mechanism that allows the principal to improve his payoff as he knows more, such as committing to the behavior of agents during the authorization decision stage in the decision-making process. 18 Acknowledgements We would like to thank Ole Jann, Andreas Kleiner, Sebastian Ertner, and seminar participants at Bielefeld University, the QED Jamboree 2025, the Lisbon Meetings in Game Theory and Applications 2025, and the 14th Conference on Economic Design 2025 for helpful comments and discussions. We gratefully acknowledge research funding by the Deutsche Forschungsgemeinschaft (DFG) through the Research Project Training Group (RTG 2865)—Coping with Uncertainty in Dynamic Economics (CUDE). References Aghion, P. and J. Tirole (1997). Formal and real authority in organizations. 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Authority and communication in organizations. The Review of Economic Studies 69(4), 811–838. Foerster, M. and D. Habermacher (2025a). Authority, communication, and internal markets. SSRN Working Paper. 19 Foerster, M. and D. Habermacher (2025b). Policy-advising competition and endogenous lobbies. Journal of Public Economics 245. Foerster, M. and A. Voss (2022). Believe me, I am ignorant, but not biased. European Economic Review 149. Kr¨ahmer, D. (2006). Message-contingent delegation. Journal of Economic Behavior & Organization 60(4), 490–506. Lim, W. (2012). Selling authority. Journal of Economic Behavior & Organization 84(1), 393–415. Prendergast, C. (2007). The motivation and bias of bureaucrats. American Economic Review 97(1), 180–196. Tirole, J. (1990). In honor of david kreps, winner of the john bates clark medal. Journal of Economic Perspectives 4(3), 149–170. A Appendix: Proofs Proof of Proposition 1.Suppose Phas selected an agent in the first period. Given posterior ˜ G, his expected utility from retaining the selected agent is E˜ G[−(θ2−y∗ 2,A(b, θ2))2] = −E˜ G[b2]. The expected utility of Pfrom choose a new agent from the agent pool is EG[−(θ2−y∗ 2,A(b, θ2))2] = −EG[b2] = −σ2 G. Finally, the expected utility of Pfrom taking the decision himself is −(1 −λ)2σ2, which proves the claim. Proof of Lemma 1.The selected agent b2would choose her bliss point y∗ 2,A(b2, θ2) = θ2+b2in the second period, and is thus ceteris paribus weakly better off when she is being retained than when she is not being retained. Since Pretains the agent if and only if y1∈[θ1+b, θ1+¯ b], choosing her bliss point y∗ 1,A(b1, θ1) = θ1+b1in the first period is optimal if b1∈[b,¯ b]. 20 If b1>¯ b(b1< b is analogous), then the agent has to take a decision y1∈ [θ1+b, θ1+¯ b] in order to be retained. To establish the claim, observe that arg min y1∈[θ1+b,θ1+¯ b](θ1+b1−y1)2=θ1+¯ b. Proof of Lemma 2.Fix any decision y1of the selected agent in the first period. Suppose first that (1 −λ)2σ2≥σ2 G, such that by Proposition 1Pchooses another agent from the agent pool if he does not retain the selected agent. Using (2), the value of retention for agent A1then is: R(b1) =E[u(y1, y∗ 2,A(b1, θ2)|b1, θ1, θ2)|b1,retained] −E[u(y1, y∗ 2,A(b2, θ2)|b1, θ1, θ2)|b1,not retained] =0 −δEG[−(θ2+b1−y∗ 2,A(b2, θ2))2|b1] =δ(b2 1+σ2 G). Second, suppose that (1 −λ)2σ2< σ2 G, such that by Proposition 1Ptakes the decision himself if he does not retain the selected agent. Using (1), the value of retention for agent A1then is: R(b1) =E[u(y1, y∗ 2,A(b1, θ2)|b1, θ1, θ2)|b1,retained] −E[u(y1, y∗ 2,P (λ, θ1)|b1, θ1, θ2)|b1,not retained] =0 −δEF[−(θ2+b1−y∗ 2,P (λ, θ1))2|b1] =δ(b2 1+ (1 −λ)2σ2). Proof of Proposition 2.Let (1 −λ)2σ2< σ2 G, so that Pprefers to take the decision himself in the second period instead of choosing another agent from the pool ((1 −λ)2σ2≥σ2 Gis analogous). Suppose that Pretains the agent if and only if y1∈[θ1, θ1+¯ b], where ¯ b > 0. By Lemma 1, agents b1∈[0,¯ b] then choose their bliss point, while agents b1>¯ btake the decision y1=θ1+¯ bif they want to be retained. We first prove the following lemma. Lemma 4. If δ≤δ∗, then there exist unique ¯ b=¯ b(δ)>0and ¯ β=¯ β(δ,¯ b)>¯ b such that EG[b2|b∈[¯ b, ¯ β]] = min{σ2 G,(1 −λ)2σ2}. 21 Proof. Fix any ¯ b > 0. We first prove that there exists a unique ¯ β > ¯ bsuch that NR(¯ β,¯ b) = 0, i.e., agents b1∈(¯ b, ¯ β] prefer to choose y1=θ1+¯ band are retained. Let b1>¯ b, then Lemma 1and Lemma 2yield NR(b1,¯ b) = R(b1)−c(θ1+¯ b|b1, θ1)=(δ−1)b2 1+ 2¯ bb1+δ(1 −λ)2σ2−¯ b2. which is a concave quadratic function. And because NR(b1,¯ b) = R(b1) = δ((1 −λ)2σ2+b2 1)>0 for b1∈[0,¯ b], there exists a unique ¯ β=¯ β(δ,¯ b)>¯ bsuch that NR(¯ β,¯ b) = 0 by continuity. So the agents b1≤¯ βprefer to imitate b′ 1=¯ bin order to be retained in the second period, while the agents b1>¯ βinstead prefer to choose their bliss point (according to Lemma 1). Second, we prove that there exists ¯ b > 0 such that EG[b2|b∈[¯ b, ¯ β]] = (1 −λ)2σ2if δ≤δ∗. By definition of ¯ βand given δ=δ∗, we have NR(b1,0) = (δ∗−1)( ¯ β(δ∗,0))2+δ∗(1 −λ)2σ2= 0 for b1=¯ β(δ∗,0). Then we get ¯ β(δ∗,0) = qδ∗ 1−δ∗(1 −λ)σif δ∗<1 and ¯ β(1,0) = +∞, where ∂¯ β(δ∗,0)/∂δ > 0. By definition of δ∗and the latter, EG[b2|b∈[0,¯ β(δ, 0)]] ≤(1 −λ)2σ2if δ≤δ∗. Next, note that, given any ¯ b,¯ β=¯ β(δ,¯ b) satisfies (δ−1)¯ β2+ 2¯ b¯ β+δ(1 −λ)2σ2−¯ b2= 0, which yields ¯ β(δ,¯ b) = ¯ b+pδ¯ b2+δ(1 −δ)(1 −λ)2σ2 1−δ and thus ∂¯ β(δ,¯ b)/∂¯ b > 0. Therefore, EG[b2|b∈[¯ b, ¯ β(δ,¯ b)]] is increasing in ¯ b, which establishes the claim. Note that there cannot be two two solutions ¯ b < ¯ b′, as then ¯ β(δ,¯ b)<¯ β(δ,¯ b′) and thus (1 −λ)2σ2=EG[b2|b∈[¯ b, ¯ β(δ∗,¯ b)]] < EG[b2|b∈[¯ b′,¯ β(δ∗,¯ b′)]] = (1 −λ)2σ2, which is a contradiction. 22 We next establish the equilibrium. If δ≤δ∗, then the considered strategies constitute an equilibrium by Lemma 1and definition of ¯ b > 0 and ¯ β(δ,¯ b)>¯ b. In particular, all agents b1<¯ bseparate at least cost and there cannot be another equilibrium in which a larger mass of agents separates by definition of ¯ b, which implies that it is the unique LCSE. Next, consider δ > δ∗and suppose that there is a closed interval [b′,¯ b′], b′≤¯ b′, such that Pretains the agent upon ˆy1∈[θ1+b′, θ1+¯ b′] and that there is no decision ˆy1> θ1+¯ b′for which this is the case. Then each agent b1∈[b′,¯ b′] chooses her bliss point and is retained by Lemma 1. Recall that by definition of δ∗,EG[b2|b∈[0,¯ β(δ, 0)]] >(1 −λ)2σ2. Since further ∂¯ β(δ,¯ b)/∂¯ b > 0, we have EG[b2|b∈[¯ b′,¯ β(δ,¯ b′)]] >(1 −λ)2σ2for any ¯ b′>0, a contradiction. Finally, suppose that Pdoes not retain any agent. Then all agents choose their bliss points and disclose their type in the first period. But then Pretains the selected agent when observing y1∈[θ1, θ1+ (1 −λ)σ], as then b1∈[0,(1 −λ)σ] and thus b2 1≤(1 −λ)2σ2, a contradiction. This establishes that there is no pure equilibrium if δ > δ∗, which finishes the proof. Proof of Proposition 3.If δ > δ∗(in this case, σ2 G>(1−λ)2σ2, see Remark 1), we consider the mixed equilibrium that Pretain A1only when observing y1=θ1 with a positive probability PR, otherwise he take the decision himself. Similar to pure equilibrium, the net value function of agent b1to imitate the neutral one is NR(b1, δ) = −b2 1+PR·δ((1 −λ)2σ2+b2 1). Then we can find the indifferent agents ¯ β′(δ) = qPR·δ 1−PR·δ(1−λ)σ < qδ 1−δ(1−λ)σ= ¯ β(δ, 0) from NR(¯ β′(δ), δ) = 0. The agent b1∈(0,¯ β′] imitates the neutral one because NR(b1, δ)≥0, while the agent b1∈(¯ β′,∞) chooses her bliss points (the proof is analogous to Proposition 2) given PR. When determining the strategies of A1, the expected utility of Pto retain A1 equals that to take the decision himself when observing y1=θ1, which is Z¯ β′(δ) 0 −b2dG(b) + Z+∞ ¯ β′(δ) −δ(1 −λ)2σ2dG(b) = −(1 −λ)2σ2. Then we can get ¯ β′(δ) from EG[b2|b∈[0,¯ β′]] = (1−λ)2σ2and PR=¯ β′ δ(¯ β′+(1−λ)2σ2)= ¯ β′ R(¯ β′)from NR(¯ β′, δ) = 0, which finishes the proof. 23 Let vt retain(bt−1) denote the continuation payoff of Pat period t≥2 if he retains At−1conditional on her bias bt−1. Since Pin equilibrium will be indifferent between retaining A2and deciding himself in subsequent periods upon observing ˆy2=θ2+˜ b2, we obtain E[v3 retain(b2)|b2∈(˜ b2,˜ β2]] = −1−δm−1 1−δ(1 −λ)2σ2. Note that vt retain(b′ t−1)< vt retain(b′′ t−1) if b′ t−1> b′′ t−1. Let ut retained(bt−1) denote the continuation payoff of At−1with bias bt−1if she is retained at period t≥2. Next, consider the first period and suppose that the indifferent agent b1=˜ β1 is selected. Then, in equilibrium, we have δ−δm+1 1−δ((1−λ)2σ2+˜ β2 1) =          (˜ β1−˜ b1)2−δ2u3 retained(˜ β1), if ˜ β1≤˜ b2<˜ β2 (˜ β1−˜ b1)2+δ(˜ β1−˜ b2)2−δ2u3 retained(˜ β1), if ˜ b2<˜ β1≤˜ β2 (˜ β1−˜ b1)2+δ2−δm+1 1−δ((1 −λ)2σ2+˜ β2 1), if ˜ β1>˜ β2 because agent ˜ β1is indifferent between choosing her bliss point in the first period (left-hand side) and imitating the agent ˜ b1(right-hand side). If ˜ β1≤˜ b2<˜ β2, we have 1−δm 1−δ(1 −λ)2σ2 =EG[b2 1|b1∈[˜ b1,˜ β1]] −δE[v3 retain(b2)|b2∈[˜ b1,˜ β1]] <(1 −λ)2σ2−δE[v3 retain(b2)|b2∈[˜ b2,˜ β2]] =(1 −λ)2σ2+δ1−δm−1 1−δ(1 −λ)2σ2 =1−δm 1−δ(1 −λ)2σ2, (10) a contradiction. The first equality in (10) holds since Pis indifferent between making the decisions himself and retaining A1if he observes y1=θ1+˜ b1in the first period. Suppose Pretains A1in the first period. In that case, A1chooses her bliss point in the second period and is retained again in the third period because ˜ β1≤˜ b2. So the expected utility of Pto retain the agent A1at the beginning of period 2 is −EG[b2 1|b1∈[˜ b1,˜ β1]] + δE[v3 retain(b2)|b2∈[˜ b1,˜ β1]]. The second equality in (10) holds because E[v3 retain(b2)|b2∈[˜ b2,˜ β2]] = −1−δm−1 1−δ(1 −λ)2σ2. The final equality in (10) holds because of the calculation. Next, we prove the inequality in (10) holds. Obviously, E[v3 retain(b2)|b2∈[˜ b1,˜ β1]] > E[v3 retain(b2)|b2∈[˜ b2,˜ β2]], since 30 ˜ b1<˜ β1≤˜ b2<˜ β2. Suppose EG[b2 1|b1∈[˜ b1,˜ β1]] >(1 −λ)2σ2, then we have −1−δm−1 1−δ(1 −λ)2σ2> E[v3 retain(b2)|b2∈[˜ b1,˜ β1]] > E[v3 retain(b2)|b2∈[˜ b2,˜ β2]] = −1−δm−1 1−δ(1 −λ)2σ2. The first inequality holds since EG[b2 1|b1∈[˜ b1,˜ β1]] >(1 −λ)2σ2and 1−δm 1−δ(1 − λ)2σ2=EG[b2 1|b1∈[˜ b1,˜ β1]] −δE[v3 retain(b2)|b2∈[˜ b1,˜ β1]]. There is a contradiction, so EG[b2 1|b1∈[˜ b1,˜ β1]] ≤(1 −λ)2σ2, such that the inequality in (10) holds. Similarly to the three-period case, we can exclude other cases and prove that ˜ b2<˜ b1<˜ β2<˜ β1, which establishes the claim. Now, we have proven that when δis small enough, we get ˜ bt<˜ bt−1<˜ βt<˜ βt−1 (2 ≤t≤T−1) for any T≥3. Finally, note that when δ=δ∗, we have 0 = ˜ bT−1<˜ βT−1and 0 <˜ bt<˜ βtfor any 1 ≤t≤T−2. This implies that there exists a unique Markov LCSE when δ≤δ∗. Finally, we prove that if Pdoesn’t retain the agent, there exists 1 ≤t∗≤T such that he chooses another agent from the pool if t < t∗, or he takes the decision himself if t≥t∗. Let vt pool(bt−1) denote the continuation payoff of Pat period t≥2 if he doesn’t retain At−1, then the equivalent proof is that: 31