Uncertainty-driven cooperation

Cetemen, Doruk; Hwang, Ilwoo; Kaya, Ayça Article Uncertainty-driven cooperation Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Cetemen, Doruk; Hwang, Ilwoo; Kaya, Ayça (2020) : Uncertainty-driven cooperation, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 15, Iss. 3, pp. 1023-1058, https://doi.org/10.3982/TE3616 This Version is available at: https://hdl.handle.net/10419/253473 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-nc/4.0/ Theoretical Economics 15 (2020), 1023–1058 1555-7561/20201023 Uncertainty-driven cooperation Doruk Cetemen Collegio Carlo Alberto Ilwoo Hwang Department of Economics, University of Miami Ayça Kaya Department of Economics, University of Miami We consider dynamic team production in the presence of uncertainty. Team members receive interim feedback that depends on both their current effort level and the project’s uncertain prospects. In this environment, each member can encourage the others by making them more optimistic about the project’s prospects. We study the extent to which this incentive counters the usual free-riding incentive. Restricting the agents’ access to feedback can increase their equilibrium effort levels by mitigating the ratchet effect. In this case, using joint performance measures can be beneficial even when individual measures are available. Keywords. Team production, free-riding, uncertainty, learning. JEL classification. C72, C73, D23, D83. 1. Introduction Teams, as agile and adaptable units of production, are commonly utilized in the modern workplace. Businesses are increasingly moving away from traditional hierarchical structures to networks of project-based teams (Deloitte 2016). According to Lazear and Shaw (2007), the percentage of large firms utilizing self-managed teams in the United States is close to 80%. Similarly, Bandiera et al. (2013) cite evidence that 47% of British establishments have more than 90% of their workers organized in teams.1 What distinguishes a team from a mere group of workers is “shared responsibility of work outcomes” (Hackman 1987). The economics literature has long recognized that Doruk Cetemen: [email protected] Ilwoo Hwang: [email protected] Ayça Kaya: [email protected] We are grateful to the anonymous referees for their insightful suggestions. We thank Paulo Barelli, Dan Bernhardt, Simon Board, Ralph Boleslavsky, Alessandro Bonatti, Yeon-Koo Che, Gonzalo Cisternas, George Georgiadis, Ben Golub, Hari Govindan, Yingni Guo, Marina Halac, Navin Kartik, George Mailath, Dmitry Orlov, Romans Pancs, Heikki Rantakari, Vasiliki Skreta, Takuo Sugaya, Curtis Taylor, Yuichi Yamamoto, Huseyin Yildirim, and various seminars and conferences audiences for insightful comments. This paper is a revised version of Chapter 1 of Cetemen’s Ph.D. dissertation. 1The cited source is “Workplace Employment Relations Survey” (2004). See footnote 1 in Bandiera et al. (2013) for a description of the source. ©2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE3616 1024 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) such arrangements are problematic due to inherent free-riding incentives (Hölmstrom 1982). On the positive side, with shared responsibility, each member of a team benefits from the hard work of others and, therefore, would like to encourage his teammates to contribute. This feature may be an advantage of using teams and potentially a reason why businesses use teams so often. When the environment affords tools and channels by which the team members can indeed encourage each other, the cost of free-riding can be more than justified by the additional value thereby created. In this paper, we show that such a counter force exists if the team interacts over time and the environment features uncertainty.2Moreover, we show how the organization can further exploit the encouragement mechanism by designing performance measures and feedback rules. We consider a team of agents working over a finite horizon, on a joint project whose true prospects are unknown. Each agent’s effort level is unobservable by the others. Over time, the agents receive interim public feedback about team performance; then they can adapt to new information by adjusting their effort levels. This feedback is noisy but informative about the agents’ efforts and the project’s potential: both high effort and good prospects statistically improve feedback. When the project ends, agents share the total output in a prespecified way. Our main finding is a mutual encouragement effect among team members, associated with learning about the project’s potential. In environments with uncertainty, team members’ optimism plays an important role in how hard they work. Optimistic agents exert more effort as they expect higher returns for it. At the same time, interim feedback about performance affects the members’ optimism. Given that a team member’s effort can affect interim feedback, each member has incentives to work harder to preclude setbacks and, thus, keep her teammates optimistic, encouraging them to exert effort. By doing this, she encourages higher effort on their part. The possibility of such mutual encouragement, which we call encouragement via belief manipulation, leads to an increased equilibrium effort level that counters the free-riding problem. Importantly, the encouragement effect in our model is present only when there is uncertainty about the prospects of the project. This suggests that introducing uncertainty into team production can be welfare improving.3 Similar insights regarding the role of uncertainty in incentivizing agents to take desirable actions have appeared elsewhere, notably in the career concerns literature pioneered by Hölmstrom (1999). Ours is a novel application of such an encouragement mechanism in a team context, which is further distinguished because it features mutual (as opposed to one-sided) encouragement among team members. Our analysis not only demonstrates that encouragement via belief manipulation can counter free-riding incentives, but also identifies a force that limits its benefits. When an agent engages in this type of encouragement, she creates optimism among her teammates, which in turn translates into high expectations of her own future effort. However, the agent does not share her teammates’ optimism and, therefore, chooses not to meet 2With few notable exceptions, the existing literature considers either static environments or dynamic environments with complete information, which explains why this channel has not previously appeared. 3This result is related to Hermalin (1998), which shows that providing only one member (leader) superior information could lead to a better outcome compared with complete information. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1025 their expectations. Therefore, optimism generated via belief manipulation is inherently short-lived. This “ratchet” effect limits the agents’ encouragement incentives, placing upper bounds on the equilibrium effort levels.4 We consider agents with heterogeneous costs of effort and show that agents with higher costs may work harder than the socially efficient level. Intuitively, the agents who are teamed with low-cost partners have a greater incentive to encourage them, since a low-cost agent’s effort is more sensitive to his beliefs. Therefore, the higher-cost agents, whose socially efficient effort levels are relatively low, may overwork. However, we derive an output sharing rule that eliminates effort overprovision and makes bounds on equilibrium effort coincide with their socially efficient levels. Furthermore, as the feedback becomes more responsive to manipulation, the equilibrium effort choices converge to socially efficient levels, provided that the right sharing rule is used. Our theory has implications for important questions concerning team design, namely optimal timing of feedback as well as if and when it is desirable to use joint performance measures. To study this, we introduce a principal who controls the release of feedback to the agents and seeks to maximize total output. We show that the principal may find it optimal to restrict agents’ access to feedback. Doing this allows the principal to leverage the encouragement incentives by controlling the magnitude of the ratchet effect. For instance, suppose that the principal fully blocks access to feedback during some final phase. Then, during this phase, the ratchet effect will be eliminated as agents’ underperformance does not affect others’ beliefs. Therefore, any optimism created early on will be long-lived, strengthening the encouragement incentives. We provide conditions under which “one-time feedback” maximizes team output. We also consider the problem of a principal who can observe individual outputs of team members. We show that such a principal may nevertheless choose to reward agents on joint output. Doing so introduces free-riding incentives, but it also activates the encouragement effect. We show that under certain conditions, the latter can be sufficiently strengthened via restricted feedback to overcome the former, leading the principal to prefer joint performance measures. Another contribution of this paper is to provide a framework that is tractable yet rich enough to study our economic question. Analyzing dynamic team incentives with uncertainty requires a model in which learning interacts with unobservable actions. In such models, however, characterization of behavior off the equilibrium path is severely complicated, as an agent’s deviation may cause her private belief to diverge from the public belief. Our setup circumvents this problem by separating the feedback from the production process, which greatly simplifies belief updating off the equilibrium path. Moreover, the speed of learning in our model does not depend on agents’ action, eliminating their incentives for experimentation. This distinguishes our model from the strategic experimentation literature (Bolton and Harris 1999,Keller et al. 2005, 4The ratchet effect—the effect of potentially causing high expectations of the agent’s future action—is extensively analyzed in the literature on dynamic agency models with asymmetric information (Weitzman 1980,Freixas et al. 1985) and dynamic moral hazard with learning and symmetric uncertainty (Bhaskar 2014,Prat and Jovanovic 2014,Cisternas 2018a,Bhaskar and Mailath 2019). 1026 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Bonatti and Hörner 2011). We believe that our framework opens further possibilities to analyze various aspects of team and feedback design.5 Our work connects to multiple strands of the literature. First, as we have noted, a large literature on teams analyzes moral hazard in groups (Olson 1965,Alchian and Demsetz 1972,Hölmstrom 1982). The literature mostly suggests that cooperation can be sustained by “punishments” of past behavior in the form of either lower monetary transfers or future non-cooperation by teammates.6Our paper demonstrates that encouragement incentives in the presence of uncertainty could alleviate free-riding when contractual remedies are not available. In a paper closely related to ours, Bonatti and Hörner (2011)considerdynamic moral hazard in teams in the presence of uncertainty. They utilize an exponential bandit framework in which the interaction ends when the common project has a “breakthrough.” The probability of a breakthrough depends on the agents’ instant effort level and the type of the project. By contrast, in our model, the total output of the project increases in agents’ cumulative effort. To the best of our knowledge, ours is the first paper to tractably incorporate learning in a team production model where agents’ payoffs depend on the whole history of effort. Second, as noted earlier, incentives to manipulate others’ beliefs by attempting to influence realizations of noisy signals have been investigated in various contexts. Since Hölmstrom (1999), the literature on career concerns has analyzed the “signal-jamming” incentives of a manager who attempts to affect the market belief about his innate ability. Riordan (1985) (oligopoly) and Fudenberg and Tirole (1986) (entrant–incumbent game) examine a firm’s incentive to make the competing firm more pessimistic about future profitability. Recently, Cisternas (2018a) substantially expands the career concerns model by allowing general (nonlinear) payoffs for the long-run player in a stationary environment. He shows how the ratchet effect shapes the player’s equilibrium incentives.7 Our paper complements Cisternas (2018a) by analyzing the evolution of the ratchet effect in a nonstationary environment. In general, we contribute to the signal-jamming literature by analyzing such incentives in a team production environment. Third, the economic literature has identified various forms of “encouragement effect” that manifest themselves via mechanisms that are different from ours. In games with complete information, the literature on dynamic contribution games (Admati and Perry 1991,Marx and Matthews 2000,Yildirim 2006) shows that a public project can be completed by agents who contribute small amounts from time to time. 5Our model can trivially accommodate heterogeneity among agents with respect to their productivity or ability to influence informative feedback and, less trivially, heterogeneity with respect to information. This, for instance, would open the door to addressing questions on allocation of heterogeneous agents into teams operating within an organization. 6In the literature on contracts with many agents, a group contract based on total output can mitigate moral hazard in teams (Hölmstrom 1982,Legros and Matthews 1993); in repeated partnership games, the threat of future non-cooperation following a deviation sustains various equilibrium dynamics (Radner et al. 1986). 7Cisternas (2018b) generalizes the career concerns model along another dimension by allowing investment in human capital. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1027 Georgiadis (2015) provides a continuous-time framework and derives insights for various team design issues. These papers assume that the payoff is realized only when the project’s state reaches a prespecified threshold. Therefore, the effort choices at different points in time are strategic complements, which is the channel through which the encouragement effect operates.8 Several papers analyze the encouragement effect in games with incomplete information, but differ from ours in one or more of these crucial aspects. We contribute to this literature by identifying a novel channel by which encouragement can operate. The encouragement mechanism in our paper is one of signal-jamming, and, thus, its existence crucially depends on (i) the presence of symmetric uncertainty, (ii) unobservable actions, and (iii) payoff externalities. Bolton and Harris (1999) consider a multi-agent experimentation model in which agents’ effort choices are publicly observable and payoff is not shared. They show that the possibility of eliciting future experimentation by others encourages current experimentation. Dong (2018) analyzes how an encouragement effect can result from asymmetric information in a canonical exponential bandit model with observable effort choices. She shows that the better-informed player increases his effort to “signal” his optimism to the uninformed player, leading to an increase in both players’ efforts.9 Campbell et al. (2014) consider a public good provision problem in which players decide whether to directly and credibly disclose their private information about production successes. They find that when the deadline is close, players hide successes to increase others’ effort incentives. Fourth, our result regarding the benefit of joint performance measures relates to several papers. Che and Yoo (2001) demonstrate that it may be desirable to use joint performance measures in dynamic environments when team members have an advantage in monitoring each other. Dai and Toikka (2018) reach an analogous conclusion in a static environment where there is large ambiguity about the production technology and the principal maximizes his payoff guarantee. In a contest model, Halac et al. (2017) show that changing both the reward and the information structure could improve the outcome. Our result that the use of joint performance measures can be optimal only when used in conjunction with information control is reminiscent of their conclusion. However their mechanism is distinct from ours: in their model, since the budget is fixed, agents who succeed early wish to discourage others’ effort under the shared prize scheme and restricted information helps eliminate this adverse incentive. Finally, there is a large literature in management regarding the effect of team potency—collective belief regarding the team’s ability to be successful—on team performance (Mathieu et al. 2008). The literature finds that team potency has a positive impact on performance through their respective effects on the actions of the team members (Gully et al. 2002). Our paper contributes to the literature by suggesting the novel hypothesis that the team members’ ability to affect team potency via feedback may have a positive effect on performance. 8See also Georgiadis (2017) for the effects of deadlines and monitoring frequencies on free-riding incentives. 9Klein and Wagner (2018) demonstrate that the encouragement effect can arise in a strategic investment model with experimentation due to private information. 1028 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) The remainder of this paper is organized as follows. Section 2 describes the model. Section 3 characterizes the equilibrium and discusses its dynamics. Section 4 conducts comparative statics. Section 5 analyzes the effect of the feedback timing and performance measures on team incentives. Section 6 concludes. The Appendix contains all the omitted proofs. 2. Model A team of Nagents undertakes a common project over a fixed time period [0T],where T<∞. At the beginning of the game, nature draws a persistent state θfrom a Gaussian distribution N(μ01/ν0). At every time t, agents simultaneously choose private effort levels ai(t) (i=1N). Agent i’s effort has a quadratic flow cost of ciai(t)2/2.Agents do not discount the payoffs. Agents observe neither the state nor the others’ effort choices, but make inferences based on public feedback Y(t). The feedback evolves according to a stochastic process dY(t) =θ+κ N  i=1 ai(t)dt +1 √ηdW (t) (1) where W(t)is a standard Brownian motion. Note that the drift of the above process is affected by both the unknown state (θ) and the agents’ unobservable actions (ai(t)). Here, κmeasures the responsiveness of feedback on the agents’ efforts and ηcontrols the feedback precision. We make a restriction on the feedback timing that the agents observe Y(t) at intervals of time >0. Formally, assume (without loss of generality) that T/ is an integer and that the agents observe Y(t) only at t=k (k=1T/). Note that with no discounting, the agents do not change their effort levels until the new feedback is observed. This effectively makes our model a discrete-time model. In Sections 3and 4, we mostly focus on the limit case of continuously observable feedback (→0). Our hybrid model—discrete feedback with continuous effort choices—is especially useful in analyzing the optimal feedback timing, which we address in Section 5.10 At the end of the project—that is, at time T—an output Pis realized. The amount of output depends on both the project state and the total effort: P=θT 0 N  i=1 ai(t) dt The output is shared according to an exogenous sharing rule s=(s1s2sN),with si≥0and N i=1si=1. 10Another advantage of our model, compared to a full continuous-time version, is that it admits a unique Nash equilibrium (Theorem 1). In Appendix B, we show that the unique Nash equilibrium converges to the unique linear Markov perfect equilibrium of the continuous-time version of our model, providing justification of the Markov perfection criteria. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1029 Givenaneffortprofilea={(a1(t)     aN(t))}t∈[0T]such that ai(t) is measurable with respect to the information available at time t,agenti’s expected payoff is EaθsiP−T 0 ci ai(t)2 2dt=EaθT 0siθ N  i=1 ai(t) −ci ai(t)2 2dt A public history of length t=k is a sequence Yk={Y(k)}{k=1k}.Agenti’s private history of length t,hi(t), is the combination of public history and his own past actions up to time t. Formally, hi(t) ={Yˆ k(t){ai(t)}t∈[0t)},where ˆ k(t) =max{k: k ≤t}. A pure strategy of agent iis a mapping from his private histories into R.Wefocus on pure strategy Nash equilibria. Note that because of the full support assumption on the feedback process, the only deviations that are detectable by agent iare his own. Consequently, all Nash equilibria are perfect Bayesian. Remarks about the model Two aspects of our model deserve further elaboration. First, note that the output Pexhibits complementarity between effort ai(t) and the state θ. Such complementarity ensures that the expected marginal product of an agent’s effort is higher when he is more optimistic and thereby generates encouragement incentives via belief manipulation. Second, feedback Y(t)is additively separable in agents’ actions and the state. This specification is not necessary for the presence of encouragement incentives, but it renders our dynamic model tractable. In particular, as Section 2.1 clarifies, this assumption implies that the speed of learning is independent of agents’ actions and, thus, eliminates considerations of experimentation motives. This feature distinguishes our mechanism from those in the literature on strategic experimentation (Bonatti and Hörner 2011,Keller et al. 2005). 2.1 Belief updating In this subsection, we analyze the agents’ belief updating process on and off the equilibrium path. As a benchmark case, suppose that the agents continuously observe feedback Y(t)at every instant. Let a∗(t) ={a∗ i(τ)}iτ∈[0t) be the agents’ common conjecture about their effort paths. Define Z(t) =1 tY(t)−κt 0 N  i=1 a∗ itdt By (1), for any t>0,Z(t) is distributed normally with mean θand precision ηt as long as the agents follow the conjectured effort path. Define public belief as the common posterior belief under the assumption that all agents follow a∗(t). Then public belief at time tis Gaussian with mean μ(t) ≡μY(t)a∗(t)=ν0μ0+ηtZ(t) ν0+ηt (2) and precision ν(t) =ν0+ηt.11 11Strictly speaking, the public posterior mean and other posteriors are functions of relevant histories and not just time. To save on notation, we drop references to the specific history and simply index the beliefs by time whenever this leads to no confusion. 1030 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) If an agent deviates from the conjectured effort path, however, his posterior belief differs from the public belief. Define agent i’s private belief at time tas a function of his private history hi(t) and his conjecture of others’ effort paths a∗ −i(t) ={a∗ j(τ)}j=iτ∈[0t). Define ˆ Zi(t) =1 tY(t)−κt 0ait+ j=i a∗ jtdt Then agent i’s private belief at time tis Gaussian with mean ˆμi(t) =ν0μ0+ηt ˆ Zi(t) ν0+ηt (3) and precision ˆνi(t) =ν(t) =ν0+ηt. Note that if all agents follow a∗(t),privatebelief coincides with public belief (that is, ˆμi(t) =μ(t))foranyt. For future use, we define ρ(t) =η/(ν0+ηt), and express (2)and(3)as μ(t) =1−ρ(t)tμ0+ρ(t)tZ(t) (4) ˆμi(t) =1−ρ(t)tμ0+ρ(t)t ˆ Zi(t) (5) Note that while agent i’s private belief is not affected by his own deviation from the conjectured path, the mean of public belief is. The parameter ρ(t) captures the rate that an agent’s effort at time taffects the future public belief. In our model, feedback is not observed continuously, but is observed at intervals of time >0. In this case, the belief updating processes are identical to (2)and(3), but beliefs are updated only at the moments when feedback is revealed. Formally, the public (private) belief path μ(t) (ˆμ i(t))isgivenby ˆμ i(t) =ˆμi(k) (μ(t) =μ(k))for t∈[k(k +1)). 3. Equilibrium Our model admits a unique Nash equilibrium. This equilibrium has a remarkably simple structure: After any history, the equilibrium action of each agent is linear in the mean of his private posterior belief. Our main result, Theorem 1, establishes the uniqueness of equilibrium and characterizes the equilibrium in the continuous-feedback limit (as →0) as solutions of a system of ordinary differential equations. We relegate all formal proofs to the Appendix. Theorem 1. For any >0, there exists a unique Nash equilibrium. In equilibrium, agent i’s action is given by a∗ ihi(t);=ξ i(t) ˆμ i(t) where the coefficient ξ i(t) is a deterministic function of time. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1037 Figure 3. Asymmetric effort costs c1=1and c2=06: left panel, s=(1/21/2);rightpanel, s=(3/85/8). the socially efficient levels 1/ci.21 In the left panel, where the agents’ shares are equal, the high-cost agent’s effort exceeds its socially efficient level (1/c1), while the low-cost agent’s effort level is bounded away from 1/c2.Proposition 4 states that it is always possible to fine-tune the sharing rule such that the upper bound of ξi(t) coincides with the socially efficient levels. The right panel of Figure 3 demonstrates this result by describing the equilibrium ξi(t) under s∗ i=(3/85/8). Proposition 4. If the sharing rule s=(s1s2sN)satisfies si=s∗ i≡ 1 ci N  j=1 1 cj  the upper bound of ξi(t) coincides with its socially efficient level, that is, ¯ ξi=1/ci. Note that s∗ iin Proposition 4 is not necessarily the share structure that maximizes the expected payoff, as ξi(t) is generally bounded away from ¯ ξi.Infact,ξi(t) may not reach its upper bound for any t,asshowninProposition 2. However, the following corollary, which is a straightforward implication of Propositions 3and 4, identifies a limit case under which the equilibrium ξi(t) converges to the socially efficient level. Corollary 1 (Convergence to socially efficient outcome). If the sharing rule satisfies si=s∗ ifor all i, the belief sensitivity ξi(t) converges pointwise to its socially efficient level for t∈[0T)as κ→∞. 21Given the uncertainty about θand the absence of learning considerations, the socially optimal level of effort for each agent iwould be μ(t)/ci,whereμ(t) is the unbiased mean belief about θgiven the realizations of the feedback. Since learning happens exogenously in our model, there are no intertemporal trade-offs between current and future surplus created by varying learning speeds. Therefore, the socially optimal efforts simply maximize the current surplus given the current information. Therefore, the socially optimal outcome would prescribe a constant belief sensitivity of ξ∗ i=1/ci. 1038 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) To understand this seemingly unexpected result, consider the marginal return to agent i’s effort when all agents’ belief sensitivities are close to their upper bounds (that is, ξj(t) =¯ ξjfor all j)andκis arbitrarily large. Equation (9) implies that if agent imarginally increases her effort, then the belief manipulation effect provides an initial boost to the others’ effort levels, the size of which is approximately ρ(t)κj=i¯ ξj. Then, due to the ratchet effect, the initial boost decays at an approximate rate of ρ(t)κ¯ ξi.22 Therefore, the total present value of the increase in the others’ efforts is approximately (j=i¯ ξj)/¯ ξi. This is simply the ratio of the sum of others’ belief sensitivities (which determine the size of the initial boost) and own belief sensitivity (which acts as a discount rate). Adding the direct marginal product of agent i’s effort (1) and multiplying by agent i’s share (si) yields the private marginal return of agent i’s effort: si⎛ ⎜ ⎜ ⎜ ⎝ 1+ j=i¯ ξj ¯ ξi ⎞ ⎟ ⎟ ⎟ ⎠=si N  j=1 ¯ ξj ¯ ξi  Social efficiency requires the private marginal return to equal the social marginal return, which implies that the sharing rule must satisfy si N  j=1 ¯ ξj ¯ ξi=1=⇒ si=¯ ξi N  j=1 ¯ ξj  This sharing rule is clearly feasible and, indeed, the shares characterized in Proposition 4 are found by plugging ¯ ξi=1 ciinto the above expression for si. 5. Restricted feedback and joint performance measures This section has two messages. First, a principal who can control the timing of the feedback can increase the team’s output relative to its level in our unique equilibrium. Second, this improvement can be so drastic that an output-maximizing principal may be better off choosing to employ such a measure and reward the agents on joint output even when he is able to observe the individual outputs of the team members, in spite of the fact that the latter would eliminate all free-riding concerns. 22By the time x>t, this boost is discounted for two reasons: (i) exogenous learning, by e−x tρ(l)dl, and (ii) ratcheting, by e−x tκρ(l) ¯ ξidl.Whenκis large, the variation in ρ(l) is negligible compared to κand, therefore, a linear approximation of the integral x tρ(l)dl by ρ(t)(x −t) becomes appropriate, leading to this assertion. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1039 5.1 Restricted feedback Consider a principal whose payoff is an increasing function of the team’s total output and who privately observes Y(t). Before production begins he can commit to a “feedback schedule,” which is a subset T⊂[0T], with the understanding that at each t∈T, the principal publicly and credibly reveals Y(t). In all the variations we consider below, agents’ equilibrium efforts are linear in their mean beliefs so that their equilibrium strategies, as usual, are characterized by their deterministic belief sensitivities. Let ξT i(t) represent the belief sensitivity of effort under feedback schedule T. Then since agent i’s time-teffort level on the equilibrium path is ξT i(t)μ(t), the expected output of a team is given by P(T)=Et=0T 0 θμ(t) N  i=1 ξT i(t) dt =T 0μ2 0+γT(t) ν0N  i=1 ξT i(t) dt (10) where γT(t) is a measure of the precision of information that the agents have at time t. Specifically, γT(t) =ητT(t) ν0+ητT(t) where τT(t) =sup{t∈T|t≤t}is the most recent date of feedback preceding time t. The form of expected output reveals that, all else being equal, the principal prefers to give as much information as possible (i.e., the output is increasing in γT(t)). Then the only potential reason he would delay release of feedback is its strategic impact on ξT i(·). Note that the expected output is also increasing in ξT i.Nextweconsideranexample illustrating the channel by which such a restriction may help. A numerical example To fix ideas, consider the simplest possible case: a team consisting of two symmetric agents, with the equal sharing rule. Furthermore, assume that ci=η=ν0=μ0=1. First, under “continuous feedback” (T=[0T]), the equilibrium belief sensitivities are calculated from (6)as ξ1(t) =ξ2(t) =1 1+1+t 1+Tκ We immediately verify that belief sensitivities are increasing in κbut are bounded. To illustrate how holding back feedback may boost belief sensitivities, consider a very crude scheme that reveals feedback at a unique instant, say ˜ t; i.e., consider T={˜ t}. The belief sensitivities under this feedback schedule are given by ξi(t) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 21+1 2(T −˜ t)ρ(˜ t)κif t<˜ t 1 2if t≥˜ t 1040 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Since there is no further feedback after ˜ t, the belief sensitivities are equal to the direct marginal benefit of effort (in this case, 1/2). The expression for ξi(t) for t<˜ talso accounts for this while additionally capturing the familiar benefit from belief manipulation. The expression for the latter can be understood as follows. The rate at which an agent’s effort at t∈[0˜ t) boosts his teammate’s time ˜ tbelief is κρ( ˜ t). Since the belief sensitivity of effort for t>˜ tis 1/2, the initial boost in the teammate’s effort is κρ( ˜ t)/2. Importantly, lack of further feedback eliminates both further learning and any ratchet behavior, allowing this boost to remain constant over t∈[˜ tT]. This leads to a total effort boost of (T −˜ t)κρ(˜ t)/2, accounting for the second additive term in the expression. The only gain from restricting the feedback in this manner is the boost in belief sensitivities prior to ˜ t.23 This gain operates through elimination of the ratchet effect over [0˜ t). Recall that in our baseline model, the ratchet effect dampens earlier effort incentives because it leads to rapid decay of the optimism generated by boosts to feedback. By shutting down learning after ˜ t, the one-time feedback schedule completely eliminates the ratchet effect. Importantly, in our baseline model, the ratchet effect is most severe when κis large and is the force that bounds belief sensitivities. Not surprisingly, and as is clear by inspecting the expression for ξi(t), when the ratchet effect is shut down, as κgrows large, the earlier belief sensitivities grow without bound. This advantage of one-time feedback eventually becomes sufficient so that it is preferred by the principal when κis large. Optimality of one-time feedback schedules The following proposition shows that the forces at work in the above example exist under a general environment, so that one-time feedback induces a greater output than continuous feedback if κis large. The proposition also characterizes the optimal timing of one-time feedback schedules. Proposition 5. Among all one-time feedback schedules, expected output is maximized when feedback is given at t∗=ην0T+ν2 0−ν0 η Moreover, there exists ¯κsuch that if κ>¯κ, then the expected output under T∗≡{t∗}exceeds expected output under continuous feedback T∗∗ ≡[0T]. As described in the above example, the negative ratchet effect does not exist under one-time feedback, since there is no subsequent feedback after belief divergence. Proposition 5 implies that when κis large enough, cost from the ratchet effect also becomes large so that the one-time feedback dominates the continuous feedback schedule. The optimal timing t∗of feedback balances the following trade-off. While delaying the feedback means a longer period for agents to work harder because of the belief manipulation incentives, the incentives become weaker as there exists a shorter period during which manipulated beliefs influence behavior. 23When feedback is restricted in this manner, γT(t) is weakly smaller for all tand, for t>˜ t,thebelief sensitivities are also smaller than under continuous (unrestricted) feedback. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1041 Next we show that the one-time feedback schedule is optimal for the principal among the class of “coarse” feedback schemes under sufficiently large κ.Let(T)represent the “coarseness” of a feedback schedule Tdefined by (T)=inf!""t−t""|tt∈T# Also, define T() as a set of feedback schedules Twith (T)>. Proposition 6. Fix >0. Then there exists ¯κsuch that for any κ>¯κ, the one-time feedback schedule T∗defined in Proposition 5 induces the largest expected output among the feedback schedules in T(). When away from the continuous-feedback limit, i.e., if period length >0,the coarseness of available feedback schemes is bounded below by .ThenProposition 6 implies that for fixed period length ,asκbecomes sufficiently large, T∗is optimal among all feasible feedback schemes.24 5.2 Joint versus individual performance measures Now suppose that the principal observes the individual outputs of team members. By paying each agent his/her own production, the principal can completely eliminate freeriding. However, doing so also eliminates the belief manipulation incentives as agents do not gain from others’ hard work. Under this “individual performance measure” (IPM), the belief sensitivity of each agent is simply 1/ciand, in particular, is independent of the feedback schedule. Therefore, the following result immediately follows by inspecting (10). Proposition 7. Under the IPM, the output-maximizing feedback schedule is the continuous feedback T∗∗. In light of Proposition 7, to argue that the principal may prefer joint performance measures (JPM, i.e., splitting the total output among agents) over IPM, it suffices to show that JPM together with the best one-time feedback schedule T∗is superior to IPM together with the continuous-feedback schedule T∗∗.Figure 4 illustrates the equilibrium belief sensitivities under each of these combinations as well as the benchmark case of JPM with T∗∗ for a symmetric team of two agents. Note that under full-information feedback, IPM generates larger belief sensitivities and, therefore, larger expected output than JPM.25 However, since under a one-time feedback schedule, the belief sensitivities over [0t∗)can be arbitrarily large, the inequality can be reversed. The following proposition provides sufficient conditions for this reversal. 24In the limit when →0, “coarseness” rules out continuous feedback over any positive-length interval. 25In general, this follows because under joint performance measures, the belief sensitivities are bounded above by ¯ ξi<1/ci. 1042 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Figure 4. Equilibrium belief sensitivity under different feedback schemes and performance measures. (T=10,N=2,ci=1,si=1/2.) Proposition 8. There exists ¯κsuch that if κ>¯κ, then the joint performance measure combined with one-time feedback generates more expected output than the individual performance measure with continuous feedback. The comparison of the two schemes (JPM, T∗) versus (IPM, T∗∗) boils down to the relative impact on output of the additional early effort generated by the former versus the increased precision generated by the latter. This result is intuitive: By earlier arguments, the larger is the κ, the more advantageous is restricted feedback along with JPM, and this advantage grows unboundedly, whereas the advantage generated by eliminating free-riding is bounded.26 6. Concluding remarks This paper contributes to the recent literature that investigates the effect of uncertainty on team behavior. While our model is simple, it highlights several interesting trade-offs that occur when joint production takes place over time and under uncertainty. Nevertheless, it may be of interest to understand how these trade-offs would interact with others that may appear in more general environments. Our model can trivially accommodate further heterogeneity (e.g., with respect to agent productivity, ability to impact feedback) and discounting as long as the linearquadratic-Gaussian structure is preserved. Moving away from the linear-quadratic- Gaussian framework appears to be a worthy path for future research even though it is more challenging. Allowing for more general production functions to accommodate interaction between agents’ effort choices or a different relationship between the state and marginal product of effort may lead to strengthening or weakening of the encouragement effect that we are highlighting. Investigating these relationships can lead to 26Che and Yoo (2001) and Dai and Toikka (2018) reach conclusions that are similar in spirit. Che and Yoo (2001) demonstrate that it may be desirable to use joint performance measures in dynamic environments when team members have an advantage in monitoring each other, while Dai and Toikka (2018)reachan analogous conclusion in a static environment where there is large ambiguity about the production technology and the principal maximizes his payoff guarantee. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1043 fruitful conclusions about the optimality of various feedback schemes and performance measures in different technological environments. A different type of heterogeneity that may be of interest is with respect to the agents’ information. Specifically, our linear-quadratic-Gaussian model can be extended to accommodate the asymmetric information among the agents: some members (experts) may have more precise information about the state than others (novices), leading essentially to a dynamic version of Hermalin (1998). This extension may provide insights into optimal dynamic sharing rules between experts and novices. The encouragement effect we highlight in this paper is likely to be present under different specifications. Here we assume that agents observe public feedback while their actions are private, and that the encouragement effect manifests itself as a form of signal jamming. Under the alternative specification where agents receive private feedback but take observable actions, effort incentives are likely to be boosted in an analogous fashion due to the agents’ incentives to signal their information to others.27 More broadly, analysis of the optimal contract combined with information design remains an interesting path for future research. Appendix The appendix consists of four parts. Appendix A contains all omitted proofs. Appendix B characterizes the linear Markov equilibrium in the continuous-time version of our model and shows the convergence result of the equilibrium in Theorem 1 as →0. Appendix C discusses potential non-monotonicity of the equilibrium belief sensitivity when is bounded away from 0.Appendix D describes an exercise that illustrates the trade-offs associated with determining the optimal level of uncertainty. Appendix A: Omitted proofs Proof of Theorem 1 We prove Theorem 1 in two steps. First, we show that there exists a unique Nash equilibrium in our model for any >0of the feedback interval (Proposition 9). Not only do we show the existence and uniqueness, but we also characterize the belief sensitivities ξiin recursive form (12).Second,wetakethelimitof→0and show that the recursive forms (12) converge to the system of differential equations (6)inTheorem 1. Given any pure strategy profile, the Gaussian belief updating process (4) implies that the agents’ beliefs after every length-thistory can be summarized by μ(t)ˆμ1(t)     ˆμN(t) where μ(t) and ˆμi(t) are the mean of the public belief and agent i’s private belief at time t, respectively. These beliefs are constant over intervals [k (k+1)),k=0K−1= (T −)/. Moreover, over such intervals, the marginal contribution of each agent’s effort 27See Cetemen (2018) for an analysis of dynamic team behavior when members have different information about the project quality. 1044 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) to feedback Y((k+1)) and to final output Pare constant. Since, in addition, effort cost is convex, we conclude that in any equilibrium, the effort choices are constant over such intervals. Thus, the setup is equivalent to a discrete time game with K≡T/ periods. Accordingly, we refer to the time interval [k(k+1)) as period k. For ease of notation, let μk,ˆμik,andaik (k=0K −1) stand for the values of μ(t),ˆμi(t),andai(t) for t∈[k(k +1)). It is convenient to define ˜ Zk=Z(k +1)−Z(k) and ρk=η νk  Then the evolution of the public belief can be expressed recursively as μk+1=(1−ρk+1)μk+ρk+1˜ Zk(11) The following proposition characterizes the unique equilibrium when feedback is observed at discrete intervals. Proposition 9. Fix >0. There exists a unique Nash equilibrium of the model. In equilibrium, agent i’s effort level for t∈[k(k +1)) (k=0K−1)is a∗ ik =ξik ˆμik where ξiK−1=si/ci,and ξik =si ci1+κ K−1  l=k+1 j=i ξjlρl l−1 $ m=k+1 (1−κξimρm)(12) for k=0K−1. Proof. We employ backward induction to prove the proposition. In the last period (k=K−1), each agent solves the problem a∗ iK−1=arg max a Esiθa+ j=i a∗ jK−1−ci a2 2 =argmax asiˆμiK−1a+ j=i a∗ jK−1−ci a2 2 The first-order condition yields agent i’s unique equilibrium effort a∗ iK−1=(si/ci)ˆμiK−1, which is linear in the mean of the private belief with coefficient ξiK−1=si/ci. Now suppose that the claim of the proposition holds for period k+1onward, that is, in an equilibrium of the game, agent iplays a∗ il =ξil ˆμil for l=k+1K−1,where ξil is defined in (12). Fix a public history of length k, and suppose that there exists an equilibrium in which ¯ aik is agent i’s equilibrium effort choice in period kfollowing the public history, Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1045 provided that he has not deviated in the past. Thus, all other agents anticipate that agent ichooses ¯ aik. Then it suffices to show that ¯ aik =ξik ˆμik. We compute agent i’s payoff from choosing arbitrary ain period k. First, using (11) we have μk+1−ˆμik+1=(1−ρk+1)(μk−ˆμik)+κρk+1(a −¯ aik) =ρk+1 ρk (μk−ˆμik)+κρk+1(a −¯ aik) (13) To calculate the belief divergence in period k+2onward, we use the induction hypothesis that in period l=k+1K−1agent iplays ail =ξil ˆμil, while the others expect him to play ¯ ail =ξilμl. Then, for all l=k+1K−1, μl+1−ˆμil+1=(1−ρl+1)(μl−ˆμil)−κρl+1ξil(μl−ˆμil)=ρl+1 ρl[1−κρlξil](μl−ˆμil) from which we obtain μl+1−ˆμil+1=(μk+1−ˆμik+1) l $ m=k+1 ρm+1 ρm[1−κρmξm] =(μk+1−ˆμik+1)ρl+1 ρk+1 l $ m=k+1[1−κρmξm](14) Substituting (13) into (14) and rearranging, we obtain μl+1=ˆμil+1+1 κρk (μk−ˆμik)+(a −¯ aik)ρl+1κ l $ m=k+1[1−κρmξm](15) Now agent i’s optimal effort a∗solves a∗=argmax asiˆμika+ j=i a∗ jk−ci a2 2 +Eksi K−1  l=k+1ˆμilξil ˆμil + j=i ξjlμl−ci ξ2 il 2ˆμ2 il Note that ˆμil for l=k+1K−1is independent of aand has expectation ˆμik.Substituting μlwith (13)and(15), eliminating additive terms that are independent of a, and replacing ˆμil with its expectation whenever appropriate, agent i’s problem can be rewritten as a∗=argmax asiˆμik1+κ K−1  l=k+1 j=i ξjlρl l−1 $ m=k+1 (1−κξimρm)a−ci a2 2 It is clear that the problem is concave in aand has a unique solution. The first-order condition immediately yields the desired result. 1046 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Now define ξ i(·)as a step function such that for t∈[k (k +1)), ξ i(t) =ξk Then, by Proposition 9,ξ i(·)constitutes the unique equilibrium of the game where the feedback is observed at intervals of length . It remains to show that as →0, the limit of ξ i(·)satisfies (6). Rewriting (12) in the recursive form yields ξik =si ci+κsi ci j=i ξjk+1ρk+1+(1−κξik+1ρk+1)ξik+1−si ci Rearranging and substituting for ρkand ξ i(·),weobtain ξ i(t) −ξ i(t +) =κρ(t +)si ci j=i ξ j(t +) −ξ i(t +)ξ i(t +) −si ci Then dividing by and letting →0, we obtain the desired result. Proof of Proposition 1 To establish monotonicity, we first observe that ˙ ξi(T) < 0. Suppose, for a contradiction, that there exist iand ˜ t∈[0T)such that ˙ ξi(˜ t) > 0. By the continuity of ˙ ξi(t),thereexists ˆ i>0such that ˙ ξi(t) < 0for t∈(T −ˆ iT]and ˙ ξi(T −ˆ i)=0. Without loss of generality, assume that i=1attains min{ˆ i|i=1N}, with the convention that ˆ j=∞if ˙ ξj(t) < 0for all t. This in particular implies that ˙ ξj(T −ˆ 1)≤0and ˙ ξi(t) < 0for t>T−ˆ 1for all j=1. Step 1. Suppose that for all j=1N,˙ ξj(T −ˆ 1)=0. In this case, manipulation of (6) reveals that each ξj(T −ˆ 1)must be equal to its upper bound given in (1). Since they are all nonincreasing over the interval (T −ˆ 1T], they must indeed be constant. In particular, it must be true that ξj(T) is equal to its upper bound, which violates the terminal condition ξj(T) =sj/cj. Therefore, we conclude that there exists j=1such that ˙ ξj(T −ˆ 1)<0. Step 2. Next we claim that ¨ ξ1(T −ˆ 1)>0. By taking derivatives of both sides of (6), and using ˙ ξ1(T −ˆ 1)=0and (6), we obtain ¨ ξ1(T −ˆ 1)=− ηκ ν0+ηt s1 c1 N  j=1 ˙ ξj(T −ˆ 1) Then, since ˙ ξj(T −ˆ 1)≤0with at least one strict inequality, we conclude that ¨ ξ1(T −ˆ 1)>0. Now, since ˙ ξ1is continuous, ˙ ξ1(T −ˆ 1)=0,and¨ ξ1(T −ˆ 1)>0,there exists >0such that ˙ ξ1(t) > 0whenever t∈(T −ˆ 1T −ˆ 1+), a contradiction, establishing that ˙ ξt≤0for all t. Monotonicity immediately implies the lower bound on ξi(t). Again, ˙ ξi(t) ≤0im- plies, by (6), that the term in parentheses on the right-hand side must be nonnegative. Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1053 ˙ δi(t) =−η νt δt+η νt κat−ξi(t)δi(t) +ˆμi(t)dt ˙ν(t) =η where Zi(t) =√η(Y(t)−κt 0(ai(t)+j=ia∗ j(t)) dt). Since the flow payoff is quadratic, it is natural to guess a quadratic value function of the form v1i(t) ˆμ2+v2i(t) ˆμδ +v3i(t) for the HJB equation. Then by the first-order condition with respect to ai(t),wehavethe equality ai(t) =si ciˆμ+ηκ ν(t)ci v2i(t) ˆμ⇒v2i(t) =ξi(t) −si ciν(t)ci ηκ  Taking the time derivative of both sides and rearranging gives ˙ v2i(t) =1 ηκ˙ ξi(t)ν(t)ci+˙ν(t)ξi(t)ci−˙ν(t)si=˙ ξi(t)ν(t)ci ηκ +ξi(t) 1 κci−si 1 κ Then applying the envelope theorem to the HJB equation (evaluated at δi=0) and plugging into the equations above, we reach29 ξi(t) −si ciν(t)ci ηκ η ν(t)κξi(t) +1=si N  j=i ξj(t) +˙ ξi(t)ν(t)ci ηκ +ξi(t) 1 κci−si 1 κ Rearranging yields ˙ ξi(t) =− ηκ ν0+ηt si ci j=i ξj(t) −ξi(t)ξi(t) −si ci which is identical to the unique Nash equilibrium in the continuous-feedback limit of our main model. The individual action (ai(t)) follows an Ito process: dai(t) =μ(t)−ηκ ν(t)si ci N  j ξj(t) −ξi(t)2dt +η ν(t)ξi(t) dZi(t) Note that the drift is negative if and only if μ(t) > 0, since the term within the parentheses on the right-hand side is always negative.30 The volatility term ( η ν(t)ξi(t))isdecreasing over time, since ν(t) is increasing, and ξ(t) is decreasing. The volatility term converges to 0as T→∞, given that ξ(t) →si ciand ν(t) →∞. Therefore, if the time horizon is sufficiently long, the equilibrium converges to the equilibrium of the static game with complete information. 29We evaluate the equation at δi=0since the public belief coincides with the private belief on the equilibrium path. 30Precisely, Zi(t) is a Brownian motion from the perspective of agent i. 1054 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Verification We can evaluate the parameters v1i(t),v2i(t),andv3i(t) as follows. By matching the coefficients, we obtain ˙ v1i(t) =−si N  j ξj(t) −ci 1 2ξi(t)2 ˙ v2i(t) =v2i(t)η νtκξi(t) +1−si j=i ξj(t) ˙ v3i(t) =−1 2 η2 ν2 t v1i(t) The parameter Vis twice continuously differentiable in δand ˆμ, and continuously differentiable in t. Since the form is quadratic, it satisfies the polynomial growth condition. Then, by Theorem 3.1 (Fleming and Soner (2006), Chapter IV), we conclude that the conjectured form solves the agent’s problem. Convergence from the discrete-feedback model It remains to show that for all i,a i(t) converges in distribution to ai(t). Recall that a i(t) =ξ i(n)μ i(n),wherenis such that t∈[n(n +1)]. The parameter ξ i(t) converges pointwise to ξi(t) and by the Donsker invariance principle, μ i(t) converges in distribution to μ i(t).ThenbyWhitt (1980) multiplication, ξ i(t)μ i(t) converges in distribution to ξi(t)μi(t). Appendix C: Non-monotonicity of belief sensitivity As discussed in the main text, when the period length is away from 0, the belief sensitivity ξit may be non-monotonic over time. To see this, consider a special case with two symmetric agents (i.e., c1=c2≡c/2,ands1=s2=1/2) and a horizon of three periods. Furthermore, let η0=η=κ=1. Note that in this case, ρt=κ/(t +2),t=012. Letting ξ1t=ξ2t≡ξt, it is easy to calculate ξ0,ξ1,andξ2as ξ2=1 c ξ1=1 c1+κ 1+31 c ξ0=1 c1+κ 1+21 c1+κ 1+31 c+1−κ 1+21 c1+κ 1+31 c κ 1+31 c   last period return  It is apparent by observation that ξ1is necessarily larger than ξ2, while ξ0can be larger or smaller than ξ1, depending on the values of ,κ,andc. In particular, for a fixed level of ,ifκis very large or cis very small, then ξ0falls below ξ1, since the term marked “last period return” becomes unboundedly negative. Intuitively, this is because in those cases, the ratchet effect in the middle period is large. To see this, notice that under the said conditions, ξ1can be very large. This means that the effort choice in the middle Theoretical Economics 15 (2020) Uncertainty-driven cooperation 1055 period is very sensitive to beliefs, either because the marginal cost of effort is very small (i.e., small c) or the feedback is very sensitive to effort choice (i.e., large κ). This implies that the impact of a divergence between public and private beliefs on effort choices at the beginning of this period is greatly amplified in this period. In particular, after an upwarddeviationinperiod0,playeriin period 1, being less optimistic than his teammate, chooses an effort level that is far below the teammate’s expectations. This in turn biases the period 2 belief of his teammate downward by a large amount and, thus, greatly reduces teammate’s period 2 effort—in fact to a level below what it would have been without the period 0 deviation of player i. The impact of this reduction overcomes the impact of the increase in the teammate’s effort in the middle period, rendering the marginal product of effort in the initial period small and possibly even negative. It is worth noting that this non-monotonicity is an artifact of our discrete-time specification. In fact, when actions can be adjusted frequently and feedback is received frequently, this type of behavior disappears. This is easily observed by inspecting the expressions for ξ0,ξ1,andξ2above. Intuitively, this is because the belief divergence after a deviation causes a ratchet effect that is the source of the non-monotonicity. The strength of the ratchet effect is positively related to the amount of information released in each period, which is proportional to the period length . Appendix D: Role of project uncertainty As discussed in Section 4, as the project uncertainty increases, there exists a trade-off between benefit from the belief manipulation incentives and the standard cost due to uninformed effort choices. In this appendix, we analyze the effect of project uncertainty on the agents’ expected payoff, which is used to plot Figure 2. Suppose that a manager of a team faces a choice of projects with varying uncertainty. The team manager tries to maximize the ex ante total payoff to the team. To clarify our analysis of the trade-off, we consider the case in which all projects have the same ex ante value under complete information. Recall that if the project state θis perfectly observed at the beginning, then the equilibrium action is a∗ i(t) =θ/N for all t∈[0T]. Since the state θis normally distributed with mean μ0and precision ν0, the agent’s ex ante expected payoff before the realization of θis E0T 0θ·a∗ i(t) −a∗ i(t)2 2dt=T N1−1 2NE0θ2 =T N1−1 2Nμ2 0+1 ν0 Note that the payoff structure of our model implies that the value of the project is convex in θ. Therefore, choosing a risky project (one with a small ν0) is always beneficial under complete information. 1056 Cetemen, Hwang, and Kaya Theoretical Economics 15 (2020) Now consider the original model where θis unknown. We consider the optimal choice of uncertainty ν0subject to a constraint μ2 0+1 ν0=kfor some k>0.Thisconstraint requires that the mean of the project decreases as its level of uncertainty increases, offsetting the inherent benefit of risk-taking described above. Given the constraint, the ex ante expected equilibrium payoff is given by E0T 0θ·ai(t) −ai(t)2 2dt=T 0 ξ(t)1−ξ(t) 2E0μ(t)2dt =T 0 ξ(t)1−ξ(t) 2k−1 ν(t)dt Note that as the project uncertainty becomes larger, the cost of uncertainty (captured by the term 1/ν(t)) increases, while the free-riding problem is alleviated since ξ(t) uniformly increases in ν0for all t∈(0T).Figure 2, which plots the above formula as a function of ν0, shows that there exists an optimal level of project uncertainty that balances the trade-off between the cost of uncertainty and the benefit of belief manipulation. References Admati, Anat R. and Motty Perry (1991), “Joint projects without commitment.” Review of Economic Studies, 58, 259–276. [1026] Alchian, Armen A. and Harold Demsetz (1972), “Production, information costs, and economic organization.” American Economic Review, 62, 777–795. 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