Test Contract. Anqi Li. February Abstract. I examine a dynamic agency model with imperfect public monitoring where

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1 est Contract Anqi Li February 204 Abstract I examine a dynamic agency model with imperfect public monitoring where the outcome process depends arbitrarily on past actions and exhibits moderate serial correlation. he analysis establishes the first affirmative result concerning the possibility of attaining approximately efficient payoffs in this complex contracting environment for any fixed discount factor. he proof exploits the concentration inequality of McDiarmid 989 and Kontorovich and Ramanan 2008 which yields uniform probability bounds across all action profiles and all outcome processes. he construction overcomes many challenges faced by conventional incentive theories and yields a robust profit guarantee for practical contract designers whose knowledge about the outcome process is incomplete. Department of Economics, Washington University in St. Louis. angellianqi@gmail.com. I thank Ilya Segal for inspiration, guidance and support. I also thank George-Levi Gayle, SangMok Lee and Yiqing Xing for helpful discussions. his work is very preliminary. Comments and suggestions are highly welcomed.

2 Introduction In many real-world situations, we need effective incentive schemes that induce people to take hard-to-measure actions with important future consequences. For example, in firms and nonprofit organizations, it is useful to engage newly hired employees in learning-by-doing that familiarizes them with basic obligations before they assume increasingly important task. In universities and research labs, it is crucial to keep scientists to goal chasing until they generate major breakthroughs. And when incumbent politicians are being evaluated for reappointments, an important consideration is whether they have pursued ambitious plans that convey significant long-term benefits. Unfortunately, while the question of how to design incentive contracts under persistent technologies is very important from both a theoretical and a practical viewpoint, it has rarely been examined by prior studies on dynamic agency theories, most of which assume that actions have only transient impacts on the concurrent outcome. Even in the very few exceptions that allow actions to have persistent effects, 2 the authors commonly make strong functional form assumptions that quickly become unrealistic when the true technological persistence can take arbitrary forms and or when the analyst s knowledge about the outcome process is incomplete. Consider, for example, the problem that is commonly faced by a university department chair or a research lab manager who wants scientists to pursue ambitious projects, some of which may prove to be fruitful only when it comes close to tenure or promotion decisions. In this situation, is there an incentive contract that guarantees the chair or the manager a satisfactory outcome? And will the performance of the contract be robust even if the chair or the manager is clueless about how fast it takes to generate For example, Radner 985, Spear and Srivastava 987, Abreu et al. 990 and Sannikov 2008 make use of this assumption to obtain recursive characterizations of players value functions. 2 Notably, Sannikov 203 characterizes the optimal contract in an environment where the impact of past actions depends only on the time lag and decays exponentially over time. 2

3 major breakthroughs? If the answer is affirmative, then how is the solution different from contracts that are commonly prescribed by standard incentive theories? his paper investigates these research questions in a finite-horizon principal-agent model with an arbitrary discount factor where in each period, a risk-neutral principal observes an imperfect public signal or revenue that takes bounded values and depends arbitrarily on the past actions of a risk-averse agent. In this complex contracting environment, I propose a simple yet powerful Average-Revenue est Contract that yields a near-optimal profit when the horizon is long even if the principal knows few details about the underlying revenue process. he average-revenue test contract is straightforward to construct. For illustrative purpose, consider the case where the horizon lasts for periods without discounting. In this situation, the average-revenue test contract depends only on two parameter values: the per-period expected revenue µ and the per-period expected cost c of the optimal action profile in the complete information benchmark or the target action profile. Over the -period horizon, the principal delivers a fixed consumption to the agent at the beginning of each period. At the end of period, she compares the arithmetic average of realized revenues with a revenue target µ 2 +ε where ε is chosen arbitrarily from 0, 2. If the outcome is above the target, then the agent passes and nothing happens. Otherwise he fails and experiences a large utility loss αc where α is an arbitrary number that is greater than one. Albeit simple, the average-revenue test contract yields a per-period expected profit that is O 2 +ε -less than the optimal expected profit in the complete information benchmark when is large. More strikingly, this powerful result is established by an elementary proof that makes use of only two basic observations. First, I invoke McDiarmid 989 s concentration inequality for independent random variables to derive a uniform lower bound for the probability of a regular event which, loosely 3

4 speaking, occurs when the arithmetic average of realized revenues differs from its expected value at the realized action profile by at most 2 +ε that holds for all bounded revenue processes and all action profiles. Moreover, since this lower bound converges to one exponentially fast as grows to infinity, it is indeed without loss to focus exclusively on the regular event when the horizon is long. Second, I exploit the optimality of the target action profile to show that when the horizon is long, if the agent reaches the revenue target µ 2 +ε at the regular event, then the per-period action cost is at least c O 2 +ε because otherwise we have a contradiction where there is a cheaper way of approximating the revenue target than by taking the target action profile. hese two observations lead to a straightforward lower bound for the loss that the agent experience in equilibrium: P fail σ c O 2 +ε +P fail σ }{{} αc }{{} Penalty Action Cost In particular, when is large, we must have P fail σ O 2 +ε because otherwise the agent will prefer to take the target action profile which costs only c but allows him to pass with certainty at the regular event. his is the key result that helps establish the equilibrium profit bound when is large. Indeed, the proof continues to hold for any discount factor δ 0, see Appendix B and allows the principal to approximate a wide range of payoff targets see Appendix C. he analysis conveys two important insights. First, it establishes the first affirmative result concerning the possibility of attaining approximately Pareto-efficient outcomes in a dynamic game with imperfect public monitoring with where the outcome process depends arbitrarily on past actions. Methodologically, it illustrates how 4

5 an appropriate choice of the convergence theorem helps overcome numerous challenges faced by conventional incentive theories, including listed in an increasing order of generality the need to rule out technological persistence when Radner 98 and Radner 985 examine a similar question in a dynamic agency model with imperfect public monitoring; the concern for multi-step deviations that has plagued the analysis of many interesting contract design problems, e.g., where actions have persistent effects on future outcomes, where there is a need to screen between agents of varying innate abilities or where parties need to experiment jointly to figure out the profitability of a project; and more fundamentally, the need to know counterfactuals i.e., outcome distributions at deviation action profiles that increases significantly the knowledge burden on the contract designer see Section for elaborate discussions. he flexibility of the current framework also yields several immediate but important extensions. In particular, Section 4. allows revenues to exhibit moderate serial correlation to accommodate trends and fads whereas Section 4.2 establishes the existence of a near-efficient and almost budget-balanced incentive scheme in large partnerships with persistent technologies. Second, the analysis suggests that one only needs to know a few parameter values to construct a contract that guarantees a satisfactory performance. his result has two normative implications for practical contract designers who face complex contracting environments. On the one hand, it prescribes an upper bound for the cost of ignoring technological details and thus allows us to conduct straightforward cost-benefit analyses when it is costly to acquire information about the underlying environment. On the other hand, it asserts that the principal will run the danger of misspecification if she insists on adding details to the contract while her knowledge about these details is incomplete. In Section 5 I formalize the second idea by investigating a max-min game between the principal and an adversary where the principal 5

6 proposes a contract before the adversary specifies an arbitrarily complex revenue process to minimize the principal s profit. Specifically, I make use of concentration inequalities again to show that if the principal knows only a revenue target and 2 the cost of an action profile that achieves such target in expectation at the contracting stage, then the average-revenue test contract is essentially max-min optimal among the class of Lipschitz est Contracts which, roughly speaking, penalizes the agent if the realization of finitely many Lipschitz test statistics lies outside an acceptance region see Section 3.2 for the formal definition. Intuitively, the max-min test contract should accept outcomes that indicate that the agent is obedient and reject outcomes that suggest otherwise. For the principal to distinguish between these two outcomes, she essentially needs to know the expected value of max-min test statistics at varying action profiles. If, on the contrary, that the principal is clueless about the expected value of a test statistic e.g., has no idea about the expected revenue over the first half horizon because it is unclear how long it takes the agent to generate breakthroughs, then such test is misleading if it is conducted alone and is redundant if it is performed combinedly with the average-revenue test. he remainder of this paper proceeds as follows: Section 2 lays down the statistical background; Section 3 examines the baseline model in particular, Section 3. introduces the model setup, Section establishes the main result concerning the performance of the average-revenue contract and Section highlights the key difference between the main theorem and prior results; Section 4 investigates several extensions of the baseline model; Section 5 examines the robustness of the averagerevenue test contract; Section 6 concludes; Appendix A contains omitted proofs; Appendix B and C detail issues such as how to allow for discounting and how to modify the contract for achieving varying objectives. 6

7 2 Background Lipschitz est Statistic Let Ω be the population space and x = x, x 2,, x be a sample of size for some N. A test statistic : Ω R is Lipschitz with respect to the normalized Hamming metric or simply Lipschitz if for some c > 0, x x x t x t c for all x, x Ω 2. For a Lipschitz test statistic, define = inf { c > 0 : x x x t x t c for all x, x Ω } 2.2 as the Lipschitz constant, and say that the test statistic is -Lipschitz. Many common test statistics, such as all finite sample moments of bounded random variables, are Lipschitz: Fact. Let Ω R be a bounded set. hen xk t is sup x Ω x k inf x Ω x k - Lipschitz for all k N. Fact 2. Let Ω R be a bounded set and take an arbitrary δ 0,. hen δ δ δt x k t is sup x Ω x k inf x Ω x k -Lipschitz for all k N. Concentration Inequality Concentration of measure is a fairly general phenomenon which, roughly speaking, asserts that under certain regularity conditions, a function defined on a probability space almost always takes values that are close to its average value on the probability space. In this paper I make extensive use of McDiarmid 989 s concentration inequality for independent random variables and Kontorovich and Ramanan 2008 s concentration inequality for Markov chains with bounded contraction coefficients: 7

8 Fact 3 McDiarmid 989. Let X = X, X 2,, X be independent random variables for some N. hen for any ε > 0 and any Lipschitz test statistic, 2 ε2 P { X E X ε} 2 exp Fact 4 Kontorovich and Ramanan Let X = X, X 2,, X be a possibly inhomogenous Markov chain where X t takes value in a countable set Ω for all t =, 2,,. Denote by P the Markov measure. Define the initial distribution p 0 and transition kernels {p t } by t P {X, X 2,, X t = x, x 2,, x t } = p 0 x p s x s+ x s 2.4 for every t and x, x 2,, x t Ω t, and let θ t be the t th contraction coefficient, 3 i.e., s= θ t. = sup p t x p t x V 2.5 x,x Ω hen for any ε > 0 and any Lipschitz test statistic, ε 2 P { X E X ε} 2 exp 2 2 M where M. = max t + θ t + θ t θ t+ + + θ t θ 2.7 In particular, if there exists 0 < θ < such that θ t θ for all t =, 2,,, then the right hand side of 2.6 is bounded from above by 2 exp θ2 ε he total variation distance between two probability measures ν, ν 2 defined on a measurable. space Ω, B B stands for the Borel σ-algebra is defined by V = supa B ν A ν 2 A. 8

9 3 Baseline Model In this section I examine a baseline model where the outcome process depends only on the agent s past actions. Specifically, I introduce the model setup in Section 3., present the main result concerning the performance of the average-revenue test contract in Section 3.2 and discuss the key implications of the main result in Section Setup Preference he horizon is finite, t =, 2,..., for some N. here is a riskneutral principal she and a risk-averse agent he who face zero outside options at the outset and do not discount the future. 4 In period t =, 2,,, the agent takes an action a t R + at a cost ca t and derives a utility uψ t from consuming ψ t ψ, ψ R units of the consumption good, where c 0 and c0 = 0 whereas u is increasing, concave and smooth with lim ψ ψ uψ = and lim ψ ψ uψ =. Production echnology In period t, after the agent takes a t, the production technology yields a random revenue Y t that takes values in a bounded set y, y ] R. For the time being, I allow Y t to depend only on the t-period action profile a t = a, a 2,, a t for all t =, 2,,. he principal observes the realized revenue but not the agent s action. arget Action Profile Use σ to denote a -period random action profile. For notational convenience, 5 assume throughout that there is a pure action profile a = a,, a 2,,, a, that maximizes the expected profit in the complete benchmark where it suffices to pay the agent a fixed consumption that is just enough to 4 he assumption of no discounting is meant solely for notationanl convenience. See Appendix B for the case where players share a common discount factor δ 0,. 5 his is because I will use σ to denote the agent s strategy in later sections. 9

10 compensate him for the action cost, i.e., ] a = arg max E Y t σ u E ca t σ R + σ ] 3. Denote by C and R the cost and the expected revenue of a, respectively, i.e., C = ca t, 3.2 ] R = E Y t a 3.3 and define Π as the expected profit of a in the complete information benchmark, i.e., Π = R u C 3.4 From now on, I will call a the target action profile. o make the analysis interesting, I assume that a incurs a non-trivial per-period cost to the agent and yields a nontrivial per-period profit to the principal in the complete information benchmark when is large: Assumption. C, Π Θ as. 6 he next lemma plays an important role in the upcoming discussion. It asserts that if a -period action profile σ yields more or less the same revenue as a, then the action cost at σ cannot be significantly lower than C because otherwise a cannot be the optimal action profile in the complete information benchmark. Formally, 6 Specifically, I write h Θ if h c, c] for some c, c > 0 that are independent of. 0

11 Lemma. ake an arbitrary N. hen for all σ R +, u C u E ca t σ ] { ]} R E Y t σ Proof. his follows immediately from the optimality of a. 3.2 est Contract A Lipschitz test contract or simply test contract,, n, R, ψ, ψ consists of n N Lipschitz test statistics,, n, a rejection region R R n and transfer payments ψ, ψ. ime evolves as follows. At the outset, the principal proposes the contract to the agent which becomes binding if the agent prefers to accept the contract than to consume his outside option. In period t =, 2,,, the contract delivers a fixed consumption ψ. At the end of period, it computes test statistics y,, n y based on the realized revenues y = y,, y. If y,, n y R c, then the agent passes and consumes ψ as before. Otherwise he fails and consumes ψ Average-Revenue est Contract For the time being, I focus on the Average-Revenue est Contract AR, R AR, ψ AR, ψ AR that is defined by AR y = R AR =, R y t 3.5 b 3.6 where b = 2 +ε for some arbitrary ε 0, 2, and u ψ AR u ψ AR = αc 3.7

12 for some arbitrary α >. In words, the contract tests the hypothesis that the agent is taking the target action profile and rejects the hypothesis if the arithmetic average of realized revenues falls short of the revenue target by a small amount. Notably, the rejection region depends only on the expected revenue at the target action profile, but not on specificities of the revenue process. his feature plays a crucial role in the upcoming discussion. he average-revenue test contract induces a dynamic game where the agent s { strategy σ = σ t, : R+ t y, y ] } t R+ is a collection of mappings from past actions and past revenues to the current action. he solution concept is Bayesian Nash equilibrium. he equilibrium strategy σ maximizes the agent s expected utility if he agrees to participate, i.e., σ arg max u ψ AR αc P AR Y R AR σ σ IC And the contract is individually rational if the agent prefers to accept the contract than to abstain, i.e., u ψ AR αc P AR Y R AR σ 0 IR he main result of this paper asserts that the average-revenue test contract is near-optimal when the horizon is sufficiently long: heorem. Under Assumption, when is sufficiently large, there is an averagerevenue test contract AR, R AR, ψ AR, ψ AR that induces a Bayesian Nash equilibrium σ where i he agent rarely fails: P AR Y R AR σ Ob ; ii he expected action cost is close to the cost of a : E ca t σ ] C 2

13 Ob ; iii he contract is individually rational: u ψ AR αc P AR Y R AR σ 0; iv he profit is near-optimal: E Y ] t σ ψ AR Π Ob Proof of heorem I now streamline the proof of heorem. Specifically, for any given -period pure action profile a, I say that the realized outputs y are b -regular or simply regular if the arithmetic average of y is bounded around its expected value at a by b, i.e., y t E ] Y t a < b 3.8 he regular event satisfies two simple yet important properties. First, since that for any given a, Y = Y,, Y are independent random variables that take value in a bounded set y, y ], and 2 y = y t is y-lipschitz where y. = y y, it follows by McDiarmid 989 s concentration inequality that the probability of y being b -regular is bounded from below by 2 exp 2 b2. Noticeably, y 2 this lower bound holds for all action profiles and all technologies that satisfy the descriptions in Section 3.. Moreover, as, it converges to one at a speed that is faster than Ob. Formally, Lemma 2. ake an arbitrary a as given. hen i For all N, { P y t E ] } Y t a < b a 2 exp 2 b2 y 2 3

14 ii As, { P y t E ] } Y t a b a ob Proof. Part i follows immediately from McDiarmid 989 s concentration inequality. Part ii is obvious. Second, I observe that at the event where y are b -regular and the agent passes the test, the cost of the realized action profile cannot be significantly lower than C because otherwise a cannot be the optimal action profile in the complete information benchmark. Formally, Lemma 3. Suppose that Assumption holds and that y are generated by some pure action profile a. If y are b -regular and AR y R c, then ca t C Ob when is large. Proof. By assumption, we have AR y R c, or y t R b and 2 y are b -regular, or ] y t E Y t a < b herefore, ] E Y t a R 2b 4

15 hen it follows by Lemma that u ca t u C 2b Now since that u is increasing and smooth, and that C Θ by Assumption, we have the following result when is large: ca t = u u ca t u u C Ob C Ob Noticeably, the lower bounds prescribed by previous two lemmas hold for all action profiles and all revenue processes. Based on this observation, I now prove heorem i the reader should consult the Appendix for proofs of ii - iv: Proof. For notational convenience, denote by E a the event that realized outputs are b -regular given the realized action profile a, and by F the event that the agent fails the test. Define r = P E a df a σ as the equilibrium probability a that outputs are b -regular, and π = P E a, F df a σ as the equilibrium probability that outputs are b -regular yet the agent fails the test. It follows a from Lemma 2 that r 2 exp b2 2 y 2 for all N 3.9 I now compare the agent s equilibrium payoff with his expected payoff at the target 5

16 action profile a. I first bound the equilibirum payoff from above by u ψ AR π αc }{{} +r π C O b + r }{{} }{{} where - 3 are lower bounds for the expected loss i.e., action cost and monetary penalty at varying outcomes. Specifically, is attained if the agent pays the penalty for failing the test yet incurs no action cost, 2 is attained if he passes the test when revenues are regular and thus spends at least C O b on action cost see Lemma 3, and 3 is attained if he takes the lowest action and pays no penalty when revenues are irregular. I next notice that if the agent takes the target action profile a instead, then he spends C on action cost but fails the test only when revenues are irregular. hus, his expected payoff is bounded from below by u ψ AR { C + 2 exp } b2 αc 2 y 2 Subtracting the second expression from the first one yields the following upper bound for the net benefit of the equilibrium strategy: C { α π + r + 2α exp } b2 + r 2 y 2 π O b C { α π + 2α + exp } b2 + O b 2 y 2 C { α π + ob } + O b In particular, if π Ob, then the last line is strictly negative when is sufficiently large. his means that the target action profile yields a strictly higher payoff 6

17 than the equilibrium strategy, a contradiction. Now since that π Ob, we have P F σ π + r Ob + ob Ob Parts ii - iv: Follow immediately from Part i. See Appendix A for details Discussion and Related Literature How should the result be interpreted? It should be emphasized that heorem after some normalization is true for any fixed discount factor δ 0, Appendix B. his is because the two pillars of the proof, namely Lemmas 2 and 3, continue to. hold when Y t is replaced with Ŷt = δ t Y t. hus, the theorem is the first affirmative result that establishes the possibility of attaining near Pareto efficient payoffs in dynamic games with imperfect public monitoring, arbitrarily persistent technologies and fixed discount factors. It is also noteworthy that with some modification, the average-revenue test contract can approximate any payoff pair that is attainable by action profiles that are sufficiently costly for the agent and sufficiently profitable for the principal in the complete information benchmark Appendix C. In other words, the above construction is flexible enough to accommodate varying considerations such as redistribution and or imperfect knowledge about R C. How does the concentration inequality help establish the near-optimality of the Average-Revenue est Contract under arbitrary technological Persistence? he idea of using the Law of the Large Numbers to relax incentive constraints has proven to be useful for solving several important classes of problems. 7 However, no prior study 7 For example, in multi-agent screening problems, Jackson and Sonnenschein 2007 and Escobar and oikka 202 establish that reporting as truthfully as possible is an ε-equilibrium of a finitehorizon game where players are restricted to make announcements whose empirical distribution should match the theretical distribution of true types. In repeated games of imperfect monitoring, 7

18 as far as the author is aware has used this approach to establish Pareto efficiency results in dynamic public monitoring games with arbitrarily persistent technologies and fixed discount factors. he closest to the current framework are Radner 98 and Radner 985. Specifically, Radner 98 considers a finite-horizon agency model with neither technological persistence or discounting where he constructs an ε-equilibrium of the dynamic game when the horizon is sufficiently long. In Radner 985, the author circumvents the challenge in the finite horizon model by investigating an infinite-horizon agency game where he establishes a Folk heorem for the case of high discounting factors. In particular, he uses recursive methods to derive equilibrium payoff upper bounds under the assumption that there is no technological persistence and 2 strategies are memoryless across review blocks. Indeed, the average-revenue test contract is reminiscent of key elements of these two papers, namely the review block and the review strategy. However, the techniques developed there cannot be easily extended to study public monitoring games with arbitrarily persistent technologies. Specifically, Radner 98 makes use of weak convergence theorems e.g., Law of the Iterated Logarithm which do not give probability bounds that hold uniformly across all action profiles and all technologies. hus, if one were to extend his approach to derive equilibrium probability and payoff bounds in the current context, a necessary step is to characterize explicitly the equilibrium strategy i.e., derive F a σ in the proof of heorem, a problem that quickly becomes intractable as the strategy space grows exponentially with the horizon length. Meanwhile, the recursive method of Radner 985 hinges on two crucial assumptions that there is no technological persistence and that strategies are memoryless across Abreu et al. 99 shows that hiding monitoring outcomes and delaying their release improves the overall efficiency because a it allows parties to inflict severe penalties at rare outcomes and b prevents early unraveling a la Green and Porter 984. his result has proven to be useful for studying repeated private monitoring games see Compte 998 and Kandori and Matsushima 998 for example. 8

19 review blocks. herefore, it is not suitable for studying applications where the technology is sufficiently persistent and 2 the deadline of performance evaluation cannot be postponed infinitely, 8 such as how to incentivize assistant professors to pursue ambitious projects whose results may only get published when it comes close to the tenure decision, or how to encourage politicians to execute long-term plans whose benefits may only culminate right before re-elections take place. he average-revenue test contract is fundamentally different than standard incentive contracts, where the incentive payment typically depends on the outcome of a likelihood ratio test that compares the revenue distribution at the target action profile with its counterpart at deviation action profiles. his method, while being useful in certain restrictive environments e.g., when the outcome process exhibits no history-dependence, becomes limited when actions have long-term impacts because then the principal needs to know the agent s past actions to form the relevant likelihood ratio test in the first place. hus, she must consider multi-step deviations when designing the incentive contract, a problem that quickly becomes intractable when the technological persistence is allowed to take arbitrary forms. In contrast, the test contract exploits the concentration inequality to obtain uniform probability bounds that hold for all production technologies at all action profiles. his feature makes the concern for multi-step deviations irrelevant and yields robust performance bounds without making any assumption on the outcome distribution at deviation action profiles. More broadly, this counterfactual-free approach is applicable to more complex contract design problems where standard analyses are plagued by the concern for multi-step deviations, e.g., when there is a need to screen between agents of different innate abilities or when parties learn jointly if the technology is profitable 8 echnically, one can trivialize technological persistence by making the review block arbitrarily long. However, this trick does not help solve many practical contract design problems. 9

20 or not. 9 What is the implication of the result that heorem holds for arbitrary discount factors? he idea of inflicting severe penalties at unlikely outcomes appears first in Mirrlees 974 which approximates the first-best outcome in a static agency model where every deviation leads to significant changes in probabilities tail events. Later contributions, such as Radner 98, illustrate that this assumption holds naturally in dynamic environments where we can pool information over time to form a precise estimate of the agent s long-term action. Nevertheless, most prior constructions that make use of the Law of the Large Numbers assume high discount factors. In other words, it is tempting to believe that for this approach to work, one needs to collect enough information before parties loose patience in order. Fortunately, heorem shows that this assumption is superfluous. hus, the method that is developed here is applicable to a much larger set of applications than one would previously imagine. How much does the principal need to know to write a good contract? Strikingly, one needs to know virtually nothing else than C and R to construct the average revenue test contract not even the action cost of the equilibrium strategy. What derives this result is a simple yet powerful observation, that the optimality of the target action profile per se yields a tight upper bound for the agent s equilibrium payoff. Specifically, notice that if revenues are b -regular and the agent passes the test, then action cost at the realized action profile cannot be O b -cheaper than C because otherwise we will violate the optimality of the target action profile. his observation, together with straightforward payoff bounds at other equilibrium out- 9 he concern for multi-step deviation is still very important especially if the objective is to derive explicit characterizations of the optimal Bayesian contract rather than to obtain probability and payoff bounds see, for example, Fong 2007 and Hörner and Samuelson 203 for significant progresses in this direction. hat being said, I show in Section 5 that it is difficult to improve upon the average-revenue test contract if the principal knows little about the outcome process besides the expected revenue at the target action profile. 20

21 Pass Fail Regular u ψ AR C + O b u ψ AR αc Irregular Ignorable Event Ignorable Event able : Payoff Bounds at Varying Equilibrium Outcomes comes see able for a summary, allows us to obtain equilibrium payoff bounds without characterizing explicitly the equilibrium strategy. his minimalist approach makes the contract a desirable candidate in many realworld situations where the principal s knowledge about the detailed production technology is limited. For example, in a workplace where employees have more or less the same long-term productivity yet differ significantly in the detailed working style, it is natural for the manager to set revenue targets only especially if she cannot identify the exact workstyle of each individual Section 5 formalizes this idea. As another example, imagine that a newly established firm wants to infer the monitoring technology from the outcome of nearby firms yet has only access to aggregate level data like the average output/worker pay rather than individual level data such as the detailed contract. In this situation, the firm can still use the average-revenue test contract as a profit guarantee before spending more time and resources to back out the entire monitoring technology. How should we think about the consumption variation? he consumption variation in the average-revenue test contract has many real-world counterparts. For newly hired employees, it means promotion opportunities; for assistant professors, it means tenureship; for politicians, it means the prospect of re-election. Certainly, there are many real-world considerations e.g., limited liability, lucrative outside options, etc. that limit the penalty that can be inflicted on the agent. When these constraints are binding, there are at least two things that we can try. First, we may increase the baseline wage first and then inflict a severe penalty relative to the 2

22 baseline wage without violating the binding constraint. Second, we may divide the horizon into several blocks and apply the average-revenue test in each individual block, because as the block length shrinks, the consumption variation that is required for disciplining the agent decreases, too. However, this trick is not problem-free. First, it requires the principal to know more about the revenue process, especially the expected value of the test statistic in each block. Second, as the number of observations per block decreases, the statistical test will eventually become imprecise, creating new issues such as the classical tension between ype I and ype II errors. It will be interesting to see if these challenges can be overcomed using non-standard methods. 4 Extension his section investigates two extensions of the baseline model. First, I enrich the baseline model to allow revenues to exhibit moderate serial correlation. Second, I establish the existence of a near-efficient and almost budget-balanced incentive scheme in a partnership game with persistent monitoring technologies. 4. Serial Correlation One may wish to capture real-world phenomena such as trends and fads by allowing revenues to exhibit serial correlation. o formalize this idea, suppose that for each t =, 2,,, Y t takes value in a countable subset of y, y ] and depends only on a t, Y t,, Y t d for some d N. As before, assume that there is a unique pure action profile a that maximizes the principal s expected profit in the complete information benchmark, and use C, R and Π to denote the cost, the expected revenue and the expected profit in the complete information benchmark of this action profile, respectively. 22

23 For any given a, write the period-t revenue as Y t a, and define X t a = Yt a, Y t d+ a and X a = X a,, X a. 0 Clearly, X a is a Markov chain where X t takes value in a countable subset of y, y ] d for all t =, 2,,. Let θ t a be the t th contraction coefficient of this Markov chain see Equation 2.5 for the formal definition and assume that θ t a s are uniformly bounded away from one for all t,, a : Assumption 2. sup θ t a θ for some 0 < θ <. t,a R +, N Notice that Assumption 2 holds when the revenue process is moderately persistent, e.g., if P Y t = y a t, y t,, y t d+ θ < for all y, t, a t, y t, y t d+. In the current setting, a straightforward extension of heorem establishes that the average-revenue test contract remains near-optimal when the horizon is long enough: Corollary. Under Assumptions and 2, when is large, there is an averagerevenue test contract that induces a Bayesian Nash equilibrium that satisfies all properties being listed in heorem. Proof. Replace McDiarmid 989 s concentration inequality with Kontorovich and Ramanan 2008 s concentration inequality in the proof of heorem. See Appendix A for details. 4.2 Partnership Game I now investigate a multi-agent dynamic game of imperfect public monitoring where the outcome process is allowed to depend arbitrarily on players past actions. Using the insight from the baseline model, I establish the existence of a near-efficient and almost budget-balanced incentive scheme. 0 ake obvious care when defining X a,, X d a. 23

24 here is a team of n risk-neutral players who interact over a -period horizon. In period t, player i takes a private action a i,t R + at a cost c i a i,t before two types of signals Y i,t, i =,, n and X t are publicly realized. Specifically, Y i,t is a measure of player i s contribution to the team revenue which takes value in a bounded set y, y ] R and depends only on a t i = a i,,, a i,t, whereas X t is the period-t revenue that takes value in a bounded set x, x] R and depends only on a t = a t,, a t n. o make progress, I assume that over the -period horizon, the expected total revenue equals to the sum of expected individual contributions, i.e., Assumption 3. E X t ] a = n i= E Y i,t a i ] for all a. his assumption holds, for example, in large teams where individual contributions add up to the team output in a linear fashion. hat being said, notice that the assumption is a restriction on the first moment of the outcome process and thus is considerably weaker than the requirement that the revenue should equal to the sum of individual contributions at every realized outcome. Let a = a,, a n be the unique pure action profile that maximizes the expected surplus, i.e., a = arg max E X t σ R n + n c i a i,t i= σ ] 4. Define R i,, C i, as individual i s expected contribution and action cost at a i for Holmstrom 982 formalizes the tension between efficiency and budget-balanceness when there is moral hazard in teams. A decade later, Fudenberg, Levine and Maskin 994 proposes sufficient conditions for achieving these two objectives simultaneously under which they establish a Folk heorem for infinitely repeated games of imperfect public monitoring where players actions have transient effects on the concurrent outcome. In the spirit of Fudenberg, Levine and Maskin 994, I allow individual deviations to be uniquely identified by changes in the distribution of individual performance measures. Nevertheless, I allow actions to have persistent effects on future outcomes and do not require players to hold common knowledge in the detailed outcome process. 24

25 i =,, n, i.e., R i, = E C i, = Y i,t a i ] 4.2 c i a i,t 4.3 and use R, Π to denote the team s expected revenue and expected surplus at a : R = n R i, 4.4 i= Π = R n C i, 4.5 As before, assume that a is sufficiently profitable for the team and sufficiently costly for each individual when the horizon is long: i= Assumption 4. Π, C i,, i =, 2,, n Θ as. Under this assumption, it is straightforward to verify that there exists a collection of reals α = α,, α n such that 0 < α i < and α i R > C i, for all i. { } n Consider a modified average-revenue test contract i AR, R AR i, ψi AR, ψ AR i i= that consists of n individual average-revenue test contracts. Since that players are risk-neutral and do not discount the future, it is without loss to postpone the payment to the end of period, where the team computes the arithmetic average of each individual i s contribution given realized outcomes x = x,, x and y i = y i,,, y i,, AR i y i = y i,t

26 If the result lies in the rejection region: R AR i =, R i, α ib 4.7 where b = 2 +ε for some arbitrary ε 0, 2, then i fails the test and earns ψ AR i = Otherwise he passes the test and earns ψ AR i = α i R d 4.9 where d = 2 +δ for some arbitrary ε < δ <. For notational convenience, I use 2 ψ AR i, to denote i s transfer payment at the end of period. he team runs a budget surplus if x t n i= ψ i,, runs a deficit if x t < n i= ψ i, and balances the budget if x t = n i= ψ i,. For convenience, I assume that there is a disinterested third party who absorbs the surplus/makes up the deficit whenever the team fails to balance the budget. Clearly, the modified average-revenue test contract fails to satisfy ex-post budget-balanceness. But as I will demonstrate very soon, it is almost budget-balanced when the horizon is long in the sense that it runs a negligible per-period expected budget surplus and runs a deficit with a vanishing probability. he modified average-revenue test contract induces a dynamic game. Since that individual payoffs depend only on individual performance measures, I restrict attention to strategy profiles σ = σ,,, σ n, where σ i, = σ i,,t : R t + y, y ] } { t R+ maps i s past actions and his past performance measures to his current action choice. Given this restriction, a Bayesian Nash equilibrium σ = σ,,, σ n, is a pro- 26

27 file of strategies where each player maximizes his expected payoff over the -period horizon, i.e., for all i, σi, arg max E σ i, ] i, c i a i,t σ i, ψ AR 4.0 In the current setting, a straightforward extension of heorem establishes that when the horizon is long, there is a Bayesian Nash equilibrium of the game induced by the modified average-revenue test contract where each player attains a nearoptimal performance, 2 the team produces a near-optimal revenue, and 3 the team runs a negligible budget surplus almost all the time and a budget-deficit with a small probability. Formally, Corollary 2. Under Assumptions 3 and 4, when is large, there is a Bayesian Nash } n equilibrium σ { of the modified average-revenue test contract i AR, R AR i, ψi AR, ψ AR i such that i Each player attains a near-optimal performance: E Y ] i, σ R i, Ob ; ii he expected revenue is near-optimal: E X ] t σ R Ob ; iii he per-period expected budget-surplus is negligible: E X t ] ] + n i= ψ i, σ Od ; iv he probability of running a deficit is negligible: P X t < n Ob. i= ψ i, σ Proof. Due to limitations of space, I offer only a proof sketch here the reader should consult Appendix A for details. Part i is obvious. 27 i=

28 Part ii: when is large, the probability of the event where x, y,, y n are b - regular at the realized action profile is close to one McDiarmid 989. hus, the equilbrium expected total revenue equals approximately to the equilibrium expected total revenue at the regular event, which is in turn bounded around the equilbrium sum of individual contributions at the regular event Assumption 3 and the restriction on the strategy space. By Part i, the last term is closely bounded around the sum of individual contributions at the target action profile, or the revenue target. Part iii-iv: it follows from Part i and the definition of the incentive scheme that the team runs a budget surplus at the event where x, y,, y n are b -regular at the realized action profile and all players pass the test. his event occurs with a probability close to one when is large. Furthermore, at this event, the per-period budget-surplus is negligible because the sum of realized revenues is closely bounded around the revenue target. 5 Robustness Admittedly, one may improve upon the average-revenue test contract if she knows all details about the outcome process. For example, if the principal is confident about how much revenue that the agent is capable of generating over the first half horizon, then she can perform an additional test about whether the average revenue within this time frame is close to its theoretical mean. Yet in reality, this extra piece of information may not be available if the principal cannot distinguish between agents of varying working styles, some accumulate revenues fast at the outset whereas others take long to generate major breakthroughs. If we keep repeating this logic, we will eventually reach an extreme situation where the principal knows very little about the 28

29 revenue process besides the per-period expected revenue at the target action profile. 2 In this case, if the principal is aware that the revenue process can be arbitrarily complex, then what contract will she offer to guarantee herself a satisfactory profit? Perhaps not surprisingly, the answer to this question is the average-revenue test contract. In fact, I will go one step further and argue that if the principal insists on performing a Lipschitz test that is not a straightforward transformation of the average-revenue test, then asymptotically, such test is misleading if it is performed alone and is redundant if it is conducted together with the average-revenue test. o formalize this idea, I need to specify the penalty that the principal faces when performing a Lipschitz test whose expected value she is clueless about. Specifically, let us begin with a concrete example where the principal is tempted to test the arithmetic average of revenues over the first half horizon. However, all she knows is that the production technology is chosen by an adversary who wants to minimize her profit from a feasible set whereby for each x 0, 00], here is a production technology such that at the target action profile, the per-period expected revenue in the first 2 periods is x; here is a production technology and an action profile of the form at which the per-period expected revenue in the first 2 a,, a, 0,, 0 2 is x, 2 the expected revenue is 0 for all t = +,,, and 3 the total action cost is significantly 2 lower than the cost of the target action profile. In this situation, the principal faces a dilemma when testing the arithmetic average of revenues in the first 2 periods when is large. On the one hand, if this test is 2 Recall the discussion in Section that the average-revenue test contract can approximate any payoff target that is attainable by action profiles that are sufficiently costly for the agent and sufficiently profitable for the principal in the complete information benchmark see Appendix C for the formal result. hus, the maintained assumption here should be interpreted as the principal knows some feasible revenue-cost target rather than the principal knows exactly the expected revenue and cost at the optimal action profile. 29

30 performed alone, then loosely speaking, it has to pass the agent whenever the realized test statistic is between 0 and 00 so as not to destroy the agent s incentive. But then the agent will deviate to cheap actions that yield low revenues which allow him to pass with a high probability. In this case, the test is misleading. On the other hand, if this test is performed together with the average-revenue test, then regardless of the realization of the test statistic, the principal will pass the agent as long as the arithmetic average of revenues over the -period horizon is centered around its expected value at the target action profile. In this case, the test is redundant. he remainder of this section is devoted to making the above intuition precise. For any given horizon length, denote by { Y t : t =,, a } to denote the production technology at an action profile a and by φ = {{ Y t : t =,, a } } : a R + the entire production technology. Let a φ be the unique optimal action profile in the complete information benchmark, i.e., ] a φ = arg max E Y t σ ; φ u E ca t σ R + σ ] 5. and define C φ, R φ and Π φ as the cost, the expected revenue and the expected profit in the complete information benchmark of a φ, respectively, i.e., C φ = c a t, φ 5.2 ] R φ = E Y t a φ ; φ Π φ = R u C φ o make the analysis interesting, I assume that φ satisfies the following criteria: 30

31 Assumption 5. For all N, i Y t takes value in y, y ] and depends only on a t for all and a ; ii R φ iii C φ = µ for some µ Θ; = c for some c Θ; iv hose action profiles that are significantly cheaper than a φ are also considerably less profitable: there exist c 0, c, γ 0, and κ, c/c such that for all a such that ca t < κc, we have ] E Y t a ; φ u ca t < γπ φ 5.5 For any given, denote by A = { a : } ca t < c the set of all -period action profiles that cost less than c, and define Φ = {φ that satisfies Assumption 5} as the set of all production technologies that satisfy Assumption 5. At the outset, it is common knowledge that the principal knows only the horizon length, 2 φ Φ, 3 µ and c, and 4 u and c, which are assumed to satisfy descriptions in the baseline model, whereas the agent knows everything about the underlying environment. Under this assumption, I argue that it is difficult to improve upon the average-revenue test contract if the principal is concerned about the worst-case scenario where she commits to a test contract before Nature specifies details about the production technology i.e., chooses φ Φ to minimize her profit. Specifically, I demonstrate that if the principal has no clue about the expected value of a test statistc, then such test is misleading if it is conducted alone and is redundant if it is performed together with the average-revenue test. Misleading est Suppose that at the outset, the principal proposes a Lipschitz 3

32 test contract, R, ψ, ψ where a R c R consists of open intervals and some.= of their boundary points and uψ u ψ B Θ before Nature chooses φ Φ to minimize her expected profit, i.e., max,r,ψ,ψ satisfying a-b ] min E Y t ψ + Y R ψ ψ φ Φ σ φ ; φ, s.t. φ Φ, σ φ arg max uψ E Y R B σ ; φ ] ; IC σ ] uψ E Y R B σ φ ; φ 0 IR Denote the solution to this problem, i.e., the max-min test contract, by m, R m, ψ m, ψ m. o characterize the max-min test contract, notice that when is large, the average-revenue test contract yields a near-optimal profit for all φ Φ. Simple as it is, this observation yields a sharp characterization of the rejection region of the max-min test contract. Formally, denote by bd S and int S the boundary and the interior of an arbitrary set S R, respectively, by dx, y the Eclidean distance between two arbitrary points x, y R and by B ε x the ε-ball that is centered at x. hen Proposition. Under Assumptions and 5, there exist ε 0, 2 and N such that for all, i ii φ Φ. B δ E m Y φ, a φ ] a A,φ Φ x R c m B 2δ x where δ. = 2 +ε ; { E m Y a ; φ ]} { } x int R c m : inf y bdr c m dx, y δ = Proposition highlights the tension that the principal faces when is large. On the one hand, Part i asserts that a slight extension of the max-min acceptance 32