Data and incentives

Liang, Annie; Madsen, Erik Article Data and incentives Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Liang, Annie; Madsen, Erik (2024) : Data and incentives, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 1, pp. 407-448, https://doi.org/10.3982/TE5289 This Version is available at: https://hdl.handle.net/10419/296463 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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 19 (2024), 407–448 1555-7561/20240407 Data and incentives Annie Liang Department of Economics, Northwestern University Erik Madsen Department of Economics, New York University “Big data” gives markets access to previously unmeasured characteristics of individual agents. Policymakers must decide whether and how to regulate the use of this data. We study how new data affects incentives for agents to exert effort in settings such as the labor market, where an agent’s quality is initially unknown but is forecast from an observable outcome. We show that measurement of a new covariate has a systematic effect on the average effort exerted by agents, with the direction of the effect determined by whether the covariate is informative about long-run quality versus a shock to short-run outcomes. For a class of covariates satisfying a statistical property that we call strong homoskedasticity,thiseffectis uniform across agents. More generally, new measurements can impact agents unequally, and we show that these distributional effects have a first-order impact on social welfare. Keywords. Big data, forecasting, effort incentives, career concerns. JEL classification. C72, D83, L51. 1. Introduction Online platforms and data brokers extensively track, record, and aggregate consumer activities, producing measurements of everything from the size of an individual’s social network,1to how often they move residences,2to the amount of time they spend playing video games.3These new measurements are increasingly available to firms and organizations, who may find them useful as predictors of economic outcomes, such as Annie Liang: [email protected] Erik Madsen: [email protected] We are grateful to Eduardo Azevedo, Cuimin Ba, Dirk Bergemann, Alessandro Bonatti, Sylvain Chassang, Yash Deshpande, Ben Golub, Matt Jackson, Yizhou Jin, Navin Kartik, Rishabh Kirpalani, Alessandro Lizzeri, Steven Matthews, Xiaosheng Mu, Larry Samuelson, Andrzej Skrzypacz, Juuso Toikka, and Weijie Zhong for useful conversations, and to National Science Foundation Grant SES-1851629 for financial support. We thank Changhwa Lee for valuable research assistance on this project. 1The finance startup Lenddo evaluated borrowers on the basis of factors such as “how many friends or followers they have on their social networks” (Gage,2012). 2The alternative credit scoring company ZestFinance used borrowers’ frequency of residence changes to predict their creditworthiness (Lippert,2014). 3China’s widely-publicized social credit scoring system reportedly plans to incorporate data on how many video games a consumer purchases and how much time they spend playing them (Canales and Mok, 2022). ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5289 408 Liang and Madsen Theoretical Economics 19 (2024) a worker’s future productivity in a new job.4Regulating such uses of personal data has emerged as an important policy issue,5but our understanding of when and how to do so remains preliminary. The use of new data in such settings may have far-ranging social impacts beyond direct privacy concerns, reshaping the creation and distribution of economic surplus. In this paper, we study the impact of new data on markets in which moral hazard is an important concern.6A motivating application is the labor market, where wages and job opportunities are commonly tied to market forecasts of a worker’s productivity based on past output. A well-recognized consequence of this practice is that workers are incentivized to work hard to improve the market’s forecast. Since output is typically socially valuable, new data which impacts workers’ effort incentives may negatively affect labor market productivity absent regulation. We propose a simple model of reputational incentives that isolates the effect of new data on moral hazard. Our model builds on the classic “career concerns” framework of Holmström (1999), in which an agent exerts effort to improve an outcome used by a market to forecast his type. Different from Holmström (1999), we suppose that the market additionally bases its forecast on auxiliary data consisting of covariates describing the agent, which are observed prior to his choice of effort. We separate covariates into two categories: Some covariates, which we call attributes, describe the agent’s type; while others, which we call circumstances, are informative about a transient shock to his outcome. For example, a worker’s creativity is an attribute, while an illness or injury is a circumstance. We model the acquisition of new data as an expansion of the set of covariates that are measured and may be used for forecasting. Measurement of new covariates updates the market’s beliefs about a given agent’s type and shock, reshaping incentives for effort. Our main results characterize how measurement of a new covariate impacts the population distribution of effort and aggregate welfare. Our basic positive result is that incorporating a new covariate into the market’s type forecast leads to both a systematic reduction in uncertainty across the population and a redistribution of uncertainty between agents. While the systematic effect moves the effort of all agents in the same direction, the redistributionary effect leads to heterogeneous effort responses, which may differ even directionally across agents. Each of these effects has a first-order effect on aggregate welfare, and we find that the redistributionary effect can oppose and even overturn the welfare impact of reducing uncertainty. 4Employers already widely use similar information collected from internet searches to screen potential hires on the basis of factors such as social media activity (CareerBuilder,2018). 5For instance, proposed European Union rules for artificial intelligence have flagged automated employment screening systems as “high risk” applications subject to strict regulation, in particular regarding the data sets they rely on (European Commission,2021). 6Our approach complements recent work focusing on how data collection impacts markets shaped by asymmetric information. See, for instance, Bergemann, Bonatti, and Gan (2022), Elliott, Galeotti, and Koh (2022), Yang (2022) for price discrimination; Ichihashi (2019), Hidir and Vellodi (2021), Gomes and Pavan (2022) for matching on platforms; and Braverman and Chassang (2022), Brunnermeier, Lamba, and Segura- Rodriguez (2021) for insurance pricing. Theoretical Economics 19 (2024) Data and incentives 409 We formalize our positive findings through a pair of theorems. Theorem 1shows that quite generally, measurement of a new attribute reduces average effort in the population, while measurement of a new circumstance increases average effort. Importantly, this result does not guarantee that all agents change their effort by the same amount, or even in the same direction. Theorem 2shows that any heterogeneity in the effort responses of different agents is entirely attributable to a redistribution of uncertainty across the population. Our normative findings are summarized in Theorem 3. It establishes that, in the absence of redistribution of uncertainty, the directional effect on welfare of a newly measured covariate is jointly determined by its classification as an attribute or circumstance along with the weight that agents place on their future reputations. It further shows that greater redistribution of uncertainty leads to reduced welfare gains from measurement of a covariate, and that this reduction can be so extreme that the measurement of some covariates is never welfare-enhancing, regardless of the magnitude of agents’ reputational concerns. Our work contributes to an emerging literature studying the use of personal data for forecasting. Existing work has highlighted incentives for agents to game forecasts by distorting (Ball (2022), Bonatti and Cisternas (2020), Frankel and Kartik (2022), Haghtalab, Immorlica, Lucier, and Wang (2020), Hu, Immorlica, and Vaughan (2019)) or misreporting (Eliaz and Spiegler (2019,2022)) their covariates. Such incentives are especially important when a small number of covariates shape forecasts in a well-understood way. We study the complementary question of how data usage impacts incentives for agents to directly improve outcomes. These incentives are particularly relevant when outcomes are a primary forecasting input, making them a natural target for manipulation; or when algorithms used to incorporate additional covariates into forecasts are opaque, obscuring effective strategies for gaming them. Additionally, our model builds on the career concerns literature. Compared to the original Holmström (1999) model, we focus on a two-period model which incorporates auxiliary signals and non-Gaussian information structures. We share these modeling features with the closely related work of Dewatripont, Jewitt, and Tirole (1999)andRodina (2018).7These papers show that under certain conditions, signals about a single agent’s type lower his effort while signals about the shock raise it.8In our model with a population of heterogeneous agents, a similar result holds on average across agents 7An adjacent literature on relative performance comparisons, e.g., Meyer and Vickers (1997), considers settings in which the additional signal is not exogenous, but is instead generated by the outcome of another agent with correlated unobservables. See also Tirole (2021), in which the additional signal is an outcome in another domain in which the agent exerts effort. 8Rodina (2018) additionally studies whether garbling the outcome signal can improve agent incentives, a question also examined by Hörner and Lambert (2021) and Smolin (2021) in related contexts. By contrast, we assume that the agent’s outcome is perfectly observable and focus on regulation of access to additional covariates. 410 Liang and Madsen Theoretical Economics 19 (2024) (Theorem 1). But additionally, introduction of new data potentially changes the crosssectional dispersion of effort in the population. As we show in Section 3.3, this distributional effect has a first-order effect on welfare, and in some cases it can override the welfare effect of an average change in effort. Our results therefore highlight the importance of explicitly modeling the heterogeneity across agents that is present in applications. The remainder of this paper proceeds as follows. Section 2describes our model; Section 3establishes our main results about the impact of new measurements on effort and welfare; Section 4discusses extensions; and Section 5concludes. Supporting analyses and all proofs are collected in the Appendix. 2. Model In Section 2.1, we describe our basic model of reputational incentives for effort, which is a 2-period version of the Holmström (1999) career concerns model with general information structures. In Section 2.2, we augment the model by introducing auxiliary data. 2.1 Effort and welfare An agent participates in a market across two periods t=1, 2. He possesses a quality type θ∼Fθ, which is persistent across time and unknown to himself and the market. In period 1, the agent privately chooses an effort level e∈R+at cost C(e)=1 2e2.(We extend our results to general cost functions in Section 4.3.) The agent’s effort choice, along with his quality θand a transient shock ε∼Fε, determine the realization of an observable outcome Y=e+θ+ε. We assume that E(θ)=μ>0 while E(ε)=0. The agent’s reward from his period-1 interaction is independent of Yandnormalizedtobe0. 9His period-1 payoff is therefore U1=−1 2e2. In period 2, the agent receives a reputational payoff standing in for returns from future participation in the market. This payoff is equal to the market’s expectation of his quality conditional on the outcome variable Y.10 Since the agent’s effort choice is private, the market’s forecast is based on a conjectured level of effort ˆ e. Letting Yˆ e≡ ˆ e+θ+εbe the outcome supposing that the market’s effort conjecture is correct, the agent’s second-period payoff conditional on the realized outcome Y=yis U2=Eˆ e(θ|Y=y), 9This normalization does not rule out wage payments which depend on the market’s forecast of period-1 effort, as in Holmström (1999). Such effects do not impact equilibrium effort or social surplus, and so we do not explicitly model them. 10None of our results would change if the agent’s reputational payoff were instead the market’s expectation of any strictly increasing function of θ. Our model therefore accommodates a variety of interpretations for the source of reputational returns from effort. Theoretical Economics 19 (2024) Data and incentives 411 where Eˆ e(θ|Y=y)denotes the market’s (potentially misspecified) expectation of θ, updated based on the realized outcome assuming that Y=Yˆ e. The agent’s ex post payoff from participating in the market in both periods is a weighted sum of payoffs across the two periods: U=(1−β)·U1+β·U2, where β∈(0, 1)is the reputation weight, which denotes the importance to the agent of future reputational rewards versus current effort costs. The agent’s expected payoff under effort level eis therefore Ee(U)=β·EeEˆ e(θ|Y)−(1−β)·e2 2, where Eedenotes the expectation operator given the true effort level e. In equilibrium, the agent must have no incentive to deviate from the market’s conjectured level of effort. Let e∗denote equilibrium effort. Then in equilibrium the marginal value of effort (i.e., the equilibrium marginal impact of effort on the expected reputational reward), discounted by its relative weight β/(1−β), equals the marginal cost of effort: β 1−β·∂ ∂eEeEe∗(θ|Y)e=e∗ =e∗. Because effort impacts the outcome additively, the marginal value of effort appearing in this first-order condition is independent of e∗and may be written as MV ≡E0∂ ∂Y E0(θ|Y),(1) where E0denotes the expectation operator assuming that the agent does not exert effort to distort the outcome (see Appendix D.1 for details). As (1) does not depend on e∗,the unique effort level satisfying the first-order condition is then e∗=β 1−β·MV . Throughout this paper, we will assume that the first-order approach is valid, so that e∗constitutes the unique equilibrium effort choice. We measure welfare using a standard criterion that treats both the outcome variable Yand the agent’s effort cost C(e)as welfare-relevant.11 An agent whose type is θand effort choice is ethus generates welfare w(θ,e)≡Ee(Y|θ)−C(e)=θ+e−1 2e2.(2) This function is strictly concave in effort and maximized at the “first-best” effort level eFB =1 no matter the agent’s type. Appendix Aextends our analysis to alternative welfare specifications incorporating learning-by-doing and unproductive gaming. 11The assumption that Ycontributes to social welfare is appropriate for settings such as the labor market, in which the outcome captures productive output or some other socially valuable activity. We do not include the agent’s equilibrium reputational payoff in the welfare calculation because on average that payoff is fixed at Ee∗(Ee∗(θ|Y)) =μ, independent of the equilibrium effort level. (This property continues to hold when the model is augmented with data.). 412 Liang and Madsen Theoretical Economics 19 (2024) 2.2 Data and beliefs We now augment the basic model by supposing that θand εare predictable from underlying (and potentially measurable) covariates, with data revealing a subset of these covariates. We refer to those covariates which predict θas attributes, denoted by the random variables (a1,,aJ), and those covariates which predict εas circumstances, denoted by the random variables (c1,,cK). Specifically, the type θand shock εsatisfy θ=f1(a1)+···+fJ(aJ)+uθ ε=g1(c1)+···+gK(cK)+uε where each fj,j∈{1, ,J},andgk,k∈{1, ,K}, is a deterministic and one-to-one effect size function. (This specification nests the standard linear regression model as a special case when all effect size functions are affine.) For convenience, we will define the type components θj≡fj(aj)for each j=1, ,Jand shock components εk≡gk(ck) for each k=1, ,K.12 The idiosyncratic noise terms uθand uεare independent of one another and of all covariates, and have full support on the reals.13 We allow for correlation between attributes and between circumstances, but assume that the vector of attributes is independent of the vector of circumstances, i.e., (a1,,aJ)⊥⊥ (c1,,cK), implying in particular that θ⊥⊥ ε. (We consider covariates which are correlated with both the type and shockinSection4.1.) Some covariates are measured, making them observable to the agent and the market. We use J⊆{1, ,J}to denote the set of measured attributes and K⊆{1, ,K}to denote the set of measured circumstances. All measured covariates are observed at the outset of the interaction, leading the agent and market to share a common belief that the agent’s type and shock follow their distributions conditional on the agent’s measured covariate values. We view symmetric uncertainty as a natural conceptual benchmark that allows us to cleanly disentangle moral hazard from issues of selection. In Section 2.3,we discuss how our results would change if either side had additional private information. The interaction then proceeds as described in Section 2.1, with appropriate adjustments to the calculation of equilibrium effort. Conditioning on the measured covariates, the agent’s marginal value of effort changes from (1) to the quantity MVJ,K≡E0∂ ∂Y E0(θ|Y,aJ,cK)|aJ,cK(3) and equilibrium effort becomes e∗ J,K=β 1−β·MVJ,K.(4) 12Invertibility of the effect size functions implies that observation of a new covariate ajor ckis equivalent to observation of the corresponding type or shock component θjor εk. Some of our results, particularly those involving strong homoskedasticity, do not depend on invertibility. 13The full support assumption ensures that the distributions of θand εconditional on any family of measured covariates have full support, simplifying our proofs. All of our results continue to hold in the absence of full support, and so we will make free use of examples which do not feature full-support idiosyncratic noise terms. Theoretical Economics 19 (2024) Data and incentives 413 Note that both the marginal value of effort and equilibrium effort may vary with the values of the agent’s measured covariates, and they are therefore both random quantities. We interpret this randomness from a population perspective, by supposing that the market interacts with a continuum of agents possessing varying attributes and circumstances. From this perspective, random variation in e∗ J,Kcorresponds to a distribution of effort across the population of agents. Aggregate welfare given measured covariates (J,K)is the expectation of realized welfare w(θ,e∗ J,K)as defined in (2), averaging over variation in the type θand effort e∗ J,Kacross the population: W(J,K)≡Ewθ,e∗ J,K=μ+Ee∗ J,K−1 2e∗ J,K2. (Recall that μ≡E(θ)is the unconditional average quality in the population.) Aggregate welfare is maximized when all agents exert the first-best level eFB =1. Our main results compare effort and welfare when the set of measured covariates changes from some baseline family (J,K)to an expanded family (J∪{j},K)or (J,K∪{k})containing one additional covariate. To simplify exposition, throughout the main text we develop our results assuming that the baseline family is (J,K)= (∅,∅), while the expanded family is either ({1},∅)or (∅,{1}), corresponding to measurement of attribute 1 or circumstance 1. (Our results extend straightforwardly to general baselines—see Appendix Bfor details.) In this context, we define η≡j>1fj(aj)+uθ and δ≡k>1gk(ck)+uεand decompose the type and shock as θ=θ1+η,ε=ε1+δ, so that the type component θ1and shock component ε1summarize the information revealed by a new measurement, while ηand δare the residual unknowns. We impose a set of standard regularity conditions on the distributions of these variables. Assumption 1 (Admissibility). The random variables θand εhave log-concave density functions, and the conditional random variables η|a1and δ|c1have log-concave density functions for every realization of a1and c1. Assumption 2(Differentiability). For every effort level e, the derivative ∂ ∂Y Ee(θ|Y)exists and is uniformly bounded across all realizations of Y. Additionally, for every effort level eand realization of (a1,c1), the derivatives ∂ ∂Y Ee(θ|Y,a1)and ∂ ∂Y Ee(θ|Y,c1)exist and are uniformly bounded across all realizations of Y. In additive statistical inference models, log-concavity is a canonical assumption ensuring that better outcomes correspond to improved inferences about latent variables.14 14Specifically, if an analyst observes an outcome Zwhich is decomposable as Z=X+Y,whereboth Xand Yare unobserved, and if Xand Yare statistically independent and have log-concave density functions, then upon observing Zhis posterior beliefs about Xand Yare higher (in the first-order stochastic dominance order) for larger realizations of Z(Milgrom (1981)).. 414 Liang and Madsen Theoretical Economics 19 (2024) Assumption 1ensures that in both the baseline and the expanded environments, better (worse) realizations of Ylead to higher (lower) posterior beliefs about both the type and shock. Assumption 2ensures that conditional expectations are sufficiently smooth that we can take derivatives and exchange derivatives and expectations where required. 2.3 Discussion of modeling choices Private information on the side of the market Our model assumes that the market does not know more about an agent’s type or shock than the agent does himself. In some applications, private information on the side of the market is possible if the market has access to data on past outcomes for other agents with similar covariates. (This view of big data is embedded in, for instance, the “inverse selection” model of Brunnermeier, Lamba, and Segura-Rodriguez (2021).) We show in Section 4.2 that our results continue to hold under this sort of informational asymmetry, so long as measurement of a new covariate leads the agent to believe that the market has gained new information about his type or shock. Private information on the side of the agent Another possibility is that the agent knows more about his type and shock than the market. This asymmetry implies not only that the agent knows more about his own covariates, but also that he can discern how these covariates impact his type and shock distributions (via the effect size functions fjand gk), a demanding assumption in many applications. Nevertheless, our results continue to hold with private information on the agent’s side whenever the market’s posterior type expectation is linear in the outcome signal (for instance, whenever (θ1,,θJ)and (ε1,,εK)follow elliptical distributions). Beyond these settings, the agent’s perceived marginal value of effort may vary with private information about his type or shock, complicating the analysis, but we conjecture that our main results extend more broadly. Exogeneity of covariates Our model contrasts covariates, which are fixed characteristics of the agent (at least in the short run); and the outcome Y, which is susceptible to manipulation by effort. We view this dichotomy as a useful one for several reasons. First, θ and εmay be determined by the aggregation of many covariates, each of which individually plays only a small role. In such contexts, our exercise can be viewed as focusing on the agent’s incentives to influence the relatively informative outcome signal Y, while abstracting from any costly distortion of less-informative individual covariates. Second, to compute the value of manipulating a covariate, the agent must know the precise shape of the effect size function describing how that covariate impacts the outcome, which is more demanding than the knowledge requirements that we impose. 3. Main results Our main results characterize the impact of measuring a new covariate on population effort and welfare. We first show that for a broad class of covariates, measuring a new attribute decreases population effort on average, while measuring a new circumstance increases it (Section 3.1). However, outside a narrower class of covariates, new measurements may yield effort responses of heterogeneous magnitude and even direction Theoretical Economics 19 (2024) Data and incentives 421 (a1>0.05)falls by roughly 50%, to e∗∗ H≈0.077 ·β 1−β, while effort for all remaining workers rises fivefold to e∗∗ L≈0.82 ·β 1−β. Intuitively, although residential stability has only a small direct impact on job performance, it has a large impact on the market’s uncertainty about worker reliability. The market’s uncertainty about workers who move very frequently (residential stability falls below the bottom 5% percentile) increases, while the market’s uncertainty about the type of all remaining workers decreases. Since, moreover, EMV 2 +A≈0.039 >MV2≈0.026, Theorem 3implies that aggregate welfare decreases upon measuring attribute 1 for any reputation weight β. 4. Extensions We now analyze several extensions of our framework. Section 4.1 studies the effect of covariates, which are correlated with both the type and shock. Section 4.2 establishes that our results are robust to agent uncertainty about the market’s beliefs. Section 4.3 relaxes the assumption that the agent’s effort costs are quadratic. 4.1 General covariates Our main results have assumed that individual covariates are informative about the agent’s type θor shock ε, but not both. In some applications, covariates may plausibly lie somewhere between these two extremes. We now show how our results can be adapted to accommodate such covariates. As in our main results, we focus on a baseline in which no covariates are observed and an expanded data set consisting of a single covariate, whose value we will denote by the random variable X. We allow this covariate to be correlated with both θand εin a very general way, which we summarize by its effect on the conditional mean of the outcome. Let Y0≡θ+εbe the baseline outcome ignoring the agent’s effort, and define the random variable ¯ Y0≡E(θ+ε|X)to be the conditional mean of the baseline outcome given the covariate. We maintain an invertibility assumption ensuring that measuring Xis equivalent to observing the conditional mean outcome ¯ Y0. Assumption 3 (General invertibility). E(θ+ε|X=x)is a one-to-one function of x. This condition is analogous to the invertibility assumptions imposed on the effect size functions fjand gkin the baseline model, and it serves the same purpose. We additionally impose admissibility and differentiability assumptions analogous to Assumptions 1and 2in the baseline analysis. Assumption 4 (General admissibility). (¯ Y0,Y0)are statistically affiliated. Assumption 5(Generaldifferentiability). For every effort level e, the derivative ∂ ∂Y Ee(θ| Y)exists and is uniformly bounded across all realizations of Y. For every effort level eand realization of X, the derivative ∂ ∂Y Ee(θ|Y,X)exists and is uniformly bounded across all realizations of Y. 422 Liang and Madsen Theoretical Economics 19 (2024) Assumption 4serves the same role as Assumption 1in ensuring that better outcomes correspond to improved inferences about latent variables. In particular, if the “net residual” Y0≡Y0−¯ Y0is independent of the realization of X, affiliation of (¯ Y0,Y0)reduces to log-concavity of the density function of Y0. Meanwhile, Assumption 5is a straightforward adaptation of Assumption 2. Finally, we impose an assumption ensuring that θand εare correlated only through the covariate X. We maintain it to focus on the simplest context in which correlation between the type and shock might arise. Assumption 6 (General independence). (θ,ε)are independent conditional on X. We now derive conditions under which measuring Xincreases or decreases effort, extending the results of Theorem 1to this setting. Proposition 1. Suppose that Assumptions 3–6hold. 1. If (¯ Y0,θ)and (¯ Y0,−ε)are each statistically affiliated, then measuring Xreduces average effort. 2. If (¯ Y0,−θ)and (¯ Y0,ε)are each statistically affiliated, then measuring Xincreases average effort. This result establishes that a covariate which is positively associated with one component of the outcome, and is simultaneously negatively associated with the remaining component, has an unambiguous impact on the expected marginal value of effort. The positive association condition here is a direct analog of the affiliation condition in Theorem 1, and is needed for the same reason. Meanwhile, the negative association condition rules out scenarios in which a good covariate realization implies both a high type and a high shock. Since these inferences have conflicting effects on the marginal value of effort, the net effect of measuring such a covariate is inherently ambiguous. By contrast, if a good covariate realization suggests a high type and a low shock, or vice versa, the two effects reinforce and the measurement has an unambiguous impact on average effort. This result can be strengthened to obtain a uniform effect on effort under homoskedasticity conditions similar to those imposed in Theorem 2. In particular, let θ ≡θ−E(θ|X)be the residual unobserved type component after measuring X.Define ε similarly with respect to the shock. Then if the joint distribution of (θ,ε) is independent of the realization of X, measuring Xaffects effort uniformly across all agents. To illustrate these forces concretely, we analyze the effect of measuring a new covariate in a multivariate Gaussian setting. Suppose that θand εare decomposable as θ=μ+b·X+Z,ε=d·X+W Theoretical Economics 19 (2024) Data and incentives 423 where X∼N(0, σ2 x),Z∼N(0, σ2 z),andW∼N(0, σ2 w)are mutually independent and μ,b,anddare known constants. The following lemma ensures that the regularity assumptions imposed in Proposition 1are satisfied in this setting whenever b+d= 0, a condition we will maintain going forward.26 Lemma 1. Assumptions 3–6are satisfied in a multivariate Gaussian setting with a general covariate whenever b+d= 0. We now check when the conditions identified in Proposition 1under which measuring Xincreases or decreases effort are satisfied. (¯ Y0,θ)and (¯ Y0,ε)are each jointly Gaussian, and (¯ Y0,±θ)are positively correlated if and only if sign(b+d)=±sign(b). Similarly, (¯ Y0,±ε)are positively correlated if and only if sign(b+d)=±sign(d).Then whenever b>0, Proposition 1implies that measuring Xreduces effort if b+d>0≥d, i.e., d∈(−b,0 ]. Similarly, whenever d>0, measuring Xincreases effort if b+d>0≥b, i.e., b∈(−d,0 ]. The bounds d≤0andb≤0 illustrate the general point made earlier: Measuring X has an unambiguous effect on effort only if its informativeness about one component of the outcome is reinforced rather than opposed by its informativeness about the remaining component. The remaining condition b+d>0 ensures that better outcomes correspond to improved inferences about the type or shock in the baseline, without which the expected directional effect of a measurement can reverse. We can verify these results by explicitly calculating MV and MV+, the marginal value of effort before and after measuring X. The following result summarizes the calculation. Proposition 2. sign(MV −MV+)=sign((b+d)( b σ2 z −d σ2 w)). If b>0andd∈(−b,0 ], this result implies that MV+<MV, in line with the prediction of Proposition 1. Similarly, if d>0andb∈(−d,0 ],thenMV+>MV.Conversely, if both band dare positive, the sign of MV −MV+is ambiguous. Depending on the sizes of these coefficients relative to the residual uncertainty about θand ε, measuring Xcould move the marginal value of effort in either direction. 4.2 Model uncertainty and misspecification Suppose that, contrary to our assumptions in the baseline model, the agent is subjectively uncertain about the market’s perceived distribution of (θ,ε)given his measured covariates. Such a situation may arise if he does not know which set of covariates the market observes, or if he does not know how the market maps his covariate values into perceived type and shock distributions. This subjective uncertainty can be modeled by supposing the agent possesses beliefs over possible joint distributions of (θ,ε)that the market might hold when forecasting the agent’s type. (It is not important that the market’s true model be contained in the 26If b+d=0, then Xdoes not impact Yand cannot be estimated by observing the outcome. As a result, measuring it has no impact on the marginal value of effort. 424 Liang and Madsen Theoretical Economics 19 (2024) support of the agent’s beliefs, so the agent may be misspecified.) We will continue to maintain the assumption that the agent is not asymmetrically informed about his type, and so his own subjective belief about the distribution of his type and outcome is the expectation of his belief about the market’s distribution. In this setting, all of our results extend in the following sense: If the agent becomes convinced that the market’s statistical model has become “better-informed” about the agent’s type or shock, his effort will move in the direction predicted by our results, so long as the corresponding statistical assumptions hold for each model in the support of the agent’s beliefs. More precisely, an agent believes the market has become “betterinformed” if he thinks that, regardless of what statistical model it is in fact using, the market has gained access to an additional attribute or additional covariate. In that case, the marginal value of effort moves in the same direction for every model in the support of the agent’s beliefs, and the expected marginal value of effort therefore moves in this direction as well. Our main results therefore continue to hold in this environment. 4.3 General convex cost functions We have established our main results under the assumption that effort costs take the form C(e)=1 2e2. Under this cost function, equilibrium effort is identical to the marginal value of effort, allowing us to characterize the former by analyzing the latter. More generally, when Cis a strictly convex cost function, equilibrium effort is a uniquely determined, strictly increasing function of the marginal value of effort: e∗ J,K=C−1β 1−β·MVJ,K, where MVJ,Kis as defined in (3). As a result, under such a cost function, a deterministic shift in the marginal value of effort implies a change in effort in the same direction. This implies in particular that the results of Theorem 2under strong homoskedasticity extend immediately. Theorem 1for affiliated covariates extends so long as all agents change their effort in the same direction, and more generally under a condition on the third derivative of the effort cost function.27 (In Appendix C, we present similar, but more restrictive, generalizations of the welfare results from Section 3.3.) Proposition 3. Suppose Assumptions 1–2hold. (a) If attribute 1is affiliated, then measuring it reduces average effort if C ≥0or all agents change their effort in the same direction. (b) If circumstance 1is affiliated, then measuring it increases average effort if C ≤0 or all agents change their effort in the same direction. 27The proof of this result is a straightforward application of the proof of Theorem 1,combinedwiththe logic of the discussion following the proposition statement. Theoretical Economics 19 (2024) Data and incentives 425 The new force which arises under general cost functions is that average effort may respond to mean-preserving spreads of the marginal value of effort. To illustrate this possibility, consider any cost function C(e)∝ek,wherek>1. If k>2, then under such a cost function the marginal cost of effort is convex, so equilibrium effort is a concave function of the marginal value of effort. Hence, any mean-preserving spread of the marginal value of effort reduces average effort. Conversely, if 2 >k>1, effort is a convex function of the marginal value of effort, and a mean-preserving spread of the marginal value of effort increases average effort. Measuring a new affiliated covariate has two effects: It shifts the average marginal value of effort, and (whenever strong homoskedasticity fails) it may additionally introduce a spread in the distribution of marginal values. If the marginal cost of effort is convex, this second effect tends to reduce equilibrium effort. Thus, when a new attribute is measured, these two forces work together to lower average effort, and the results of Theorem 1continue to hold. A similar outcome holds when the marginal cost of effort is concave and a new circumstance is measured. When the two forces conflict, the net effect on effort is ambiguous. In particular, if agents change their effort in different directions, average effort could move in the opposite direction from the average marginal value of effort. 5. Conclusion As firms and governments move toward collecting large consumer data sets as inputs to decision-making, the question of whether and how to regulate the usage of personal data has emerged as an important policy question. Recent regulations, such as the European Union’s General Data Protection Regulation, have focused on protecting consumer privacy and improving transparency regarding what kind of data is being collected. An important complementary consideration is how data impacts economic outcomes. In this paper, we have focused on one such factor—the effect that market access to novel covariates has on incentives for hidden effort. Our results indicate that forecasting from data on enduring personal attributes decreases average effort across the population, while conversely data reflecting short-lived circumstances boosts effort. It is therefore important to distinguish between these two classes of data when regulating data usage. Further, new data may lead to increased variation in effort across workers, an outcome which has a first-order impact on welfare. This finding suggests that regulators should also take into account the distributional effects of new data when deciding whether to permit its use in particular markets. One way to interpret the attributes and circumstances in our model is as standins for covariates with different levels of persistence in a dynamic model, where the agent exerts effort over multiple periods and his type evolves over time. Generalizing our results to a many-period setting is technically challenging under non-Gaussian information structures, since effort deviations today may distort future returns to effort. Nonetheless, doing so would permit a richer study of the welfare implications of forecasting from data with varying persistence, making it an important avenue for future research. 426 Liang and Madsen Theoretical Economics 19 (2024) Appendix A: Alternative welfare specifications In this appendix, we extend our welfare analysis to consider alternative environments in which effort improves future as well as current outcomes (“learning-by-doing”) or is partially dissipative (“gaming” effort). A.1 Learning-by-doing In some applications, effort may improve future as well as current outcomes, for instance in labor market settings featuring learning-by-doing. In that case, the agent’s type is not constant over time but instead improves with past effort, and effort has socially beneficial effects over multiple periods. Our model can be modified to accommodate this feature by allowing the agent’s type θ(t), which determines the average outcome in period t, to be time-dependent. Concretely, we will suppose that θ(2)=θ(1)+γ·e,whereγ>0 is a learning-by-doing parameter. Period-1 output is Y=e+θ(1)+ε, while the agent’s period-2 reputational reward is E(θ(2)|Y). The presence of learning by doing does not affect equilibrium effort, because the agent’s reputational reward is based on the market’s forecast of his effort (which is fixed) rather than his true effort. This expectation is Ee∗θ(2)|Y=(1+γ)·e∗+Ee∗θ(1)|Y. Exerting additional effort is therefore valuable to the agent only insofar as it improves the market’s forecast of θ(1), exactly as in our main model. Thus, equation (4) continues to characterize equilibrium effort. The socially optimal effort level, however, becomes eFB =1+γin this model. Equilibrium effort therefore falls below the first-best level for a broader range of reputation weights βas the learning-by-doing parameter γincreases. An analogue of Theorem 3 continues to hold, where the threshold reputation weights β∗and β∗are increasing in γ. In other words, increased learning-by-doing makes circumstances (which boost effort) more attractive and attributes (which reduce it) less so at any given reputational weight. A.2 “Gaming” effort In other applications, effort may be dissipative and serve to distort a signal of quality without producing social value. This possibility may arise, for instance, in labor market settings in which a worker can spend time performing “influence activities” to increase the visibility of his accomplishments (as in Milgrom and Roberts (1988)). It may also arise in educational settings where the outcome variable is a test score that can be improved by test prep with no further educational value (as in Frankel and Kartik (2022)). Theoretical Economics 19 (2024) Data and incentives 427 To accommodate this possibility, our welfare criterion can be modified to discount the welfare benefits of effort: w(θ,e)=θ+δ·e−1 2e2, where δ∈[0, 1)measures the proportion of effort which is socially beneficial. When δ=0, effort is totally unproductive, while δ∈(0, 1)captures situations in which some fraction of effort contributes social value. The dissipative nature of effort has no impact on equilibrium effort, but reduces the first-best effort level to eFB =δ. Equilibrium effort will therefore exceed the firstbest level for a broader range of reputation weights βas effort becomes increasingly dissipative. A result analogous to Theorem 3can be established in this setting, with the threshold reputation weights β∗and β∗increasing in δ. One interesting case is δ=0, in which effort is fully dissipative effort. In that case, measuring new attributes improves welfare while measuring new circumstances diminishes it, regardless of the reputation weight β. (The one exception is for an attribute with significant disparate impact, which may still be welfare-reducing for all β.) Appendix B: Results for a general baseline The results of Section 3can be straightforwardly generalized to accommodate settings in which some covariates are initially measured by the market. Given any sets J⊆{1, ,J}of measured attributes and K⊆{1, ,K}of measured circumstances, define ηJ≡ j/∈J θj+uθ,δK≡ k/∈K εk+uε to be the sums of all unmeasured components of the agent’s type and shock. Fix a baseline family (J,K)of measured covariates. Affiliation and strong homoskedasticity may be generalized to this environment as follows. Definition B.1 (Affiliation). The attribute j/∈Jis J-affiliated if (θj,ηJ∪{j})is affiliated conditional on aJ. The circumstance k/∈Kis K-affiliated if (εk,δK∪{k})is affiliated conditional on cK. Definition B.2 (Strong homoskedasticity). The attribute j/∈Jsatisfies J-strong homoskedasticity if ηJ∪{j}−E(ηJ∪{j}|aJ∪{j})is independent of ajconditional on aJ. The circumstance k/∈Ksatisfies K-strong homoskedasticity if δK∪{k}−E(δK∪{k}| cK∪{k})is independent of ckconditional on cK. As formulated, these definitions apply across all (J,K)-subpopulations of agents, where each subpopulation consists of all agents sharing a particular realization of (aJ,cK). They could alternatively be formulated more narrowly to apply only for a particular set of realized covariates, if the analyst is primarily interested in the impact of a new covariate on a particular subpopulation of agents. 428 Liang and Madsen Theoretical Economics 19 (2024) Assumptions 1and 2, which imposed log-concavity on latent variables and boundedness of derivatives of conditional expectations, must also be extended for a general set of baseline measured covariates. We split these conditions into an assumption we maintain in the baseline environment, and a condition imposed on newly measured covariates. Assumption B.1 (Baseline admissibility). The conditional distributions θ|aJand ε| cKhave log-concave density functions, and for every effort level eand realization of (aJ,cK), the derivative ∂ ∂Y Ee(θ|Y,aJ,cK)exists and is uniformly bounded across all realizations of Y. Definition B.3 (Admissible covariates). An attribute j/∈Jis J-admissible if: (1) ηJ∪{j}|aJ∪{j}has a log-concave density function for every realization of aJ∪{j};and(2) for every effort level eand realization of covariates (aJ∪{j},cK), the derivative ∂ ∂Y Ee(θ| Y,aJ∪{j},cK)exists and is uniformly bounded across all realizations of Y. A circumstance k/∈Kis K-admissible if: (1) δK∪{k}|cK∪{k}has a log-concave density function for every realization of cK∪{k}; and (2) for every every effort level eand realization of covariates (aJ,cK∪{k}), the derivative ∂ ∂Y Ee(θ|Y,aJ,cK∪{k})exists and is uniformly bounded across all realizations of Y. With these concepts, we can generalize Theorems 1and 2as follows. Theorem B.1. Suppose Assumption B.1 holds. (a) If attribute jis J-admissible and satisfies J-affiliation, then measuring it weakly reduces average effort within each (J,K)-subpopulation. (b) If circumstance kis K-admissible and satisfies K-affiliation, then measuring it weakly increases average effort within each (J,K)-subpopulation. Theorem B.2. Suppose Assumption B.1 holds. (a) If attribute jis J-admissible and satisfies J-strong homoskedasticity, then measuring it weakly reduces every agent’s effort. Further, the magnitude of the effort change is the same for every agent in each (J,K)-subpopulation. (b) If circumstance kis K-admissible and satisfies K-strong homoskedasticity, then measuring it weakly increases every agent’s effort. Further, the magnitude of the effort change is the same for every agent in each (J,K)-subpopulation. These results can be applied repeatedly to assess the impact of measuring multiple covariates, so long as admissibility and the corresponding statistical condition (affiliation or strong homoskedasticity) holds for each of the measured covariates relative to its respective baseline. Note in particular that the exponential and multivariate normal settings of Examples 2and 3satisfy affiliation and strong homoskedasticity, respectively, for any baseline and newly measured covariate. Our welfare results also hold under general baselines using appropriate notions of regularity and strict regularity. Theoretical Economics 19 (2024) Data and incentives 429 Definition B.4. Fix a baseline family of measured covariates (J,K).Then: •An attribute j/∈Jis (J,K)-regular if, conditional on any realization of (aJ,cK), measuring jweakly reduces the marginal value of effort on average. It is strictly (J,K)-regular if the reduction is strict for a positive fraction of realizations of (aJ,cK). •A circumstance k/∈Kis (J,K)-regular if, conditional on any realization of (aJ,cK), measuring kweakly increases the marginal value of effort on average. It is strictly (J,K)-regular if the increase is strict for a positive fraction of realizations of (aJ,cK). Weak regularity imposes monotonicity separately on each subpopulation of agents. Strict regularity imposes the stronger requirement of strict monotonicity for a positive fraction of agents. (In the special case of a baseline with no observed covariates, strict regularity trivially implies strict monotonicity for all agents, corresponding to Definition 3.) The following result generalizes Theorem 3to general baselines. Theorem B.3. Fix a baseline family of measured covariates (J,K). (a) For every (J,K)-regular attribute j/∈J, there exists a threshold reputation weight β∗∈(0, 1]such that measuring jis welfare-improving if and only if β>β ∗.Moreover, β∗<1if and only if EMV 2 J∪{j},K<EMV 2 J,K(B.1) where MVJ,Kis as defined in (3). (b) For every (J,K)-regular circumstance k/∈K, there exists a threshold reputation weight β∗∈[0, 1)such that measuring kis welfare-improving if and only if β<β ∗. Moreover, β∗>0if and only if kis strictly (J,K)-regular. Appendix C: Welfare under general convex costs Theorem 3, our main welfare result, can be extended to nonquadratic effort cost functions under the same conditions as Proposition 3, assuming that effort costs follow a power law. We state and prove this result for general baselines, as in the analysis of Appendix B. Proposition C.1. Suppose that C(e)=Aekfor some A>0and k>1. Fix a baseline family of measured covariates (J,K). (a) Suppose that the market measures the additional regular attribute j/∈J.Ifeither k≥2or else all agents change their effort in the same direction, then there exists a threshold reputation weight β∗∈(0, 1]such that the measurement is welfareimproving if and only if β>β ∗.Ifjis strictly regular and all agents change their effort in the same direction, then β∗<1. 430 Liang and Madsen Theoretical Economics 19 (2024) (b) Suppose that the market measures the additional regular circumstance k/∈K.If either k≤2or else all agents change their effort in the same direction, then there exists a threshold reputation weight β∗∈[0, 1)such that the measurement is welfareimproving if and only if β<β ∗.Ifkis strictly regular, then β∗>0. For general cost functions, fully characterizing how aggregate welfare changes with β becomes intractable. However, it can be shown that if the effort cost function is approximately quadratic near zero, then when βis small, newly measured regular attributes reduce aggregate welfare and newly measured regular circumstances increase them; while for large β, these effects reverse. Proof of Proposition C.1. If a newly measured covariate is regular but not strictly regular, then expected effort is unchanged while the distribution of effort in each subpopulation undergoes a mean-preserving spread under the measurement. Then since effort costs are strictly convex, aggregate welfare must at least weakly decrease no matter the value of β, corresponding to β∗=1 for an attribute and β∗=0 for a circumstance. For the remainder of the proof, we assume that the newly measured attribute is strictly regular. Let δ≡β/(1−β).Notethate∗ J,Kdepends on βonly through δ, and we will write e∗ J,K(δ)to make this dependence explicit. We first consider the case in which the market measures a new attribute j.Allnotation is as in the proof of Theorem B.3. Define W(δ)≡Ew0e∗ J,K(δ)−w0e∗ J,K(δ), where w0(e)≡e−C(e). Recall that given any family of measured covariates, equilibrium effort in a given subpopulation satisfies e∗=(C)−1(δ·MV ),whereMV is the corresponding subpopulation marginal value of effort. Thus, ∂e∗ ∂δ =MV CC−1(δ·MV ) and ∂ ∂δwe∗=1−Ce∗∂e∗ ∂δ =(1−δ·MV )·MV ·1 CC−1(δ·MV ). When C(e)=Aek,wehave 1 CC−1(δ·MV )=A·(δ·MV )1 k−1−1, where A≡1/((k−1)(Ak)1/(k−1))>0. Hence, ∂ ∂δwe∗=A·δ1 k−1−1·(1−δ·MV )·MV 1 k−1. Theoretical Economics 19 (2024) Data and incentives 437 Proof. This is established along very similar lines to the proof of Lemma D.2.Fixrealizations of (aJ,cK), condition all distributions on their values, and suppress explicit conditioning. Let ρη(u)be the density of ηJand ρδ|ε(x|z)be the conditional density of δ−K|εk. The conditions required for the steps of the proof of Lemma D.2 to go through are that ρη(u)is log-concave, ρδ|ε(x|z)is log-concave in xfor all z,and(δK,εk)are affiliated. The first two properties follow from Assumption B.1, while the final property holds by K-affiliation of circumstance k. We compare MVJ,Kwith MVJ,Kin a manner very similar to the case of an additional attribute. Fix realizations of (aJ,cK), and define Fε(z|y)≡Prεk≤z|Y0=y,aJ,cK to be the conditional CDF of εkgiven the outcome Y0. Decompose Y0as Y0=μ(J,K)+ηJ+δK. Taking expectations of each side conditional on (Y0,εk,aJ,cK)yields Y0=μ(J,K)+EηJ|Y0,εk,aJ,cK+EδK|Y0,εk,aJ,cK. Hence, dFε(z|y)EηJ|Y0=y,εk=z,aJ,cK =y−μ(J,K)−dFε(z|y)EδK|Y0=y,εk=z,aJ,cK.(D.2) Lemma D.3 directly implies that dFε(z|y)EδK|Y0=y,εk=z,aJ,cK is weakly increasing in y,so(D.2) is weakly decreasing in y. Following the same logic as in the attributes case, monotonicity of (D.2) implies that ∂ ∂Y0EηJ|Y0,aJ,cK≤E∂ ∂Y0EηJ|Y0,εk,aJ,cKY0,aJ,cK, and it follows that MVJ,K≤E[MVJ,K|aJ,cK]. Thus, the marginal value of effort in each subpopulation in the baseline is weakly lower than the expected marginal value of effort when the circumstance kis additionally measured. D.3 Proofs of Theorems 2and B.2 We prove Theorem B.2,fromwhichTheorem2follows immediately as a corollary. 438 Liang and Madsen Theoretical Economics 19 (2024) D.3.1 Part (a) Fix a baseline family of measured covariates (J,K). As established in Lemma D.1, the marginal value of effort is MV (J,K)=E∂ ∂Y0EηJ|Y0,aJ,cKaJ,cK, where Y0≡θ+εis the baseline value of the outcome after subtracting out the agent’s effort. Now suppose the market additionally observes the attribute j/∈J,andletJ≡J∪ {j}. Under the expanded family of measured covariates, the marginal value of effort becomes MVJ,K=E∂ ∂Y0EηJ|Y0,aJ,cKaJ,cK, where, conditional on (aJ,cK),MVJ,Kis a random variable whose value is a function of the realization of aj. The outcome Y0may be decomposed as Y0=μ(J,K)+ηJ+δK,(D.3) where μ(J,K)≡ j∈J θj+ k∈K εk is constant conditional on (aJ,cK). The residual type component ηJmay be further decomposed as ηJ=¯ηJ|j+ηJ, where ¯ηJ|j≡E[ηJ|aJ],ηJ≡θ−E[θ|aJ]. We may therefore rewrite (D.3)as Y0=μ(J,K)+¯ηJ|j+ηJ+δK. Now, note that ηJ−E[ηJ|aJ]= j∈J θj+ηJ−E j∈J θj+ηJ|aJ=ηJ. Hence, J-strong homoskedasticity of attribute jis equivalent to the assumption that ηJis independent of ajconditional on (aJ,cK). Therefore, under J-strong homoskedasticity, Y0depends on ajonly through ¯ηJ|j. It follows that under J-strong homoskedasticity, E[ηJ|Y0,aJ,cK]=E[ηJ|Y0,¯ηJ|j,aJ,cK], and the latter expectation depends on ajonly through ¯ηJ|j. Using this fact, we may write EηJ|Y0,aJ,cK=¯ηJ|j+EηJ|Y0,¯ηJ|j,aJ,cK Theoretical Economics 19 (2024) Data and incentives 439 and MVJ,K=E∂ ∂Y0EηJ|Y0,¯ηJ|j,aJ,cK¯ηJ|j,aJ,cK. The theorem holds if can we show that the conditional expectation of ηJis less responsive to the realization of the outcome Ythan the conditional expectation of the original residual ηJ.NotethatηJis the sum of the (conditionally) independent variables ¯ηJ|jand ηJ, so uncertainty about ηJis mechanically lower than uncertainty about ηJ. But this does not directly translate into a statement that the posterior expectation of ηJis less sensitive to the realization of Y. In general, we are not even guaranteed that higher realizations of Ylead to higher inferences about θJonce we have conditioned on the realization of ¯ηJ|j.29 We next prove a key technical lemma, which will imply an analogue of admissibility for our transformed environment. Lemma D.4. (ηJ,¯ηJ|j,Y0)are affiliated conditional on (aJ,cK). Proof. Fix a set of realizations of (aJ,cK), and condition all distributions on these values. To economize on notation, we suppress explicit conditioning on these covariates throughout this proof. Let ˜ρη,θ,Y(u,t,y)be the conditional joint density of (ηJ,¯ηJ|j,Y0). We will show that ˜ρη,θ,Yis log-supermodular. Use ˜ρθ(t)to denote the density of ¯ηJ|j,˜ρη|θ(u|t)to denote the conditional density of ηJ|¯ηJ|j,and ˜ρY|η(y|u)to denote the conditional density of Y0|ηJ.NotethatY0 is independent of ¯ηJ|jconditional on ηJ.So, ˜ρη,θ,Ymay be decomposed as ˜ρη,θ,Y(u,t,y)=˜ρθ(t)˜ρη|θ(u|t)˜ρY|η(y|u). It is therefore sufficient to show that ˜ρY|ηand ˜ρη|θare log-supermodular. First, consider ˜ρY|η. Decompose Y0as Y0=μ(J,K)+ηJ+δK. Let ρδ(z)be the density of δK.Then ˜ρY|η(y|u)=ρδ(y−μ(J,K)−u). Under Assumption B.1,ρδis log-concave, meaning ˜ρY|ηis log-supermodular. As for ˜ρη|θ,let ˜ρη(w)be the density of ηJ. Decompose ηJas ηJ=¯ηJ|j+ηJ, and recall that if jis J-strongly homoskedastic, then ηJis independent of ajand hence ¯ηJ|j. It follows that ˜ρη|θ(u|t)=˜ρη(u−t), 29Recall that our admissibility assumptions are imposed on the original type component θj, and not on the constructed ¯ηJ|j. 440 Liang and Madsen Theoretical Economics 19 (2024) and hence, ∂2 ∂u∂t log ˜ρη|θ(u|t)=− ∂2 ∂w2log ˜ρη(w)w=u−t =− ∂2 ∂u2log ˜ρη|θ(u|t). Now, let ρη|a(u|α)denote the conditional density of ηJ|aj. Define ζ(α)≡fj(α)+E[ηJ|aj=α], so that ηJ=ζ(aj)+ηJ. Strong homoskedasticity implies that ρη|a(u|α)=˜ρηu−ζ(α)=˜ρη|θu|ζ(α). Let ˜ ≡{t:ζ(α)=tfor some α∈Aj}denote the support of ¯ηJ|j.Fixanyt∈˜ .Then for all uand every α∈Ajsuch that ζ(α)=t, ∂2 ∂u2log ˜ρη|θ(u|t)=∂2 ∂u2logρη|a(u|α). Let ρη|adenote the conditional density of ηJ|aj.Thenρη|a(u|α)=ρη|a(u−fj(α)| α),sothat ∂2 ∂u2log ˜ρη|θ(u|t)=∂2 ∂w2log ρη|a(w|α)w=u−fj(α) . Under Assumption B.1,ρη|ais log-concave and this final derivative is nonpositive, meaning ∂2 ∂u∂t log ˜ρη|θ(u|t)=− ∂2 ∂u2log ˜ρη|θ(u|t)≥0 for every uand t∈˜ .Hence, ˜ρη|θis log-supermodular, as desired. Following arguments identical to those used for the proof of Theorem B.1 (with ηJ and ¯ηJ|jplaying the roles of ηJand θj, respectively), Lemma D.4 implies MVJ,K≥E[MVJ,K|aJ,cK]. Thus, the marginal value of effort in each subpopulation in the baseline setting is weakly higher than the expected marginal value once attribute jis additionally measured, for any realizations of (aJ,cK). To complete the proof, we must establish that monotonicity holds uniformly across realizations of aj, and not just on average. This follows immediately from the fact that MVJ,Kis independent of ajconditional on (aJ,cK). To see this, decompose Yas Y=e+μ(J,K)+¯ηJ|j+ηJ+δK. Theoretical Economics 19 (2024) Data and incentives 441 Strong homoskedasticity implies that ηJis independent of ajconditional on (aJ, cK).Hence,ajenters the market’s inference problem only via a known additive shift ¯ηJ|jto the agent’s type distribution and, its value therefore does not impact incentives for effort. So, these incentives must be independent of aj,asclaimed. D.3.2 Part (b) Suppose the market observes the additional circumstance k/∈K.Let K≡K∪{k}. Under the expanded family of measured covariates, the marginal value of effort is MVJ,K=E∂ ∂Y0EηJ|Y0,aJ,cKaJ,cK. Define ¯ δK,k≡E[δK|cK],δK≡ε−E[ε|cK]. Then Y0may be decomposed as Y0=μ(J,K)+ηJ+¯ δK,k+δK. Under strong homoskedasticity, δKis independent of ckconditional on (aJ,cK),and so the distribution of Y0depends on ckonly through ¯ δK,k.Thus, EηJ|Y0,aJ,cK=EηJ|Y0,¯ δK,k,aJ,cK, and the latter random variable depends on ckonly through ¯ δK,k. Therefore, in a manner analogous to the attribute case, the marginal value of effort after measuring jdepends on ckonly through ¯ δK,kand may be written MVJ,K=E∂ ∂Y0EηJ|Y0,¯ δK,k,aJ,cK¯ δK,k,aJ,cK. Lemma D.5. (¯ δK,k,δK,Y0)are affiliated conditional on (aJ,cK). Proof. This proof follows along very similar lines to the proof of Lemma D.4.Fix realizations of (aJ,cK), condition all distributions on their values, and suppress explicit conditioning on these covariates. Let ρη(u)be the conditional density of ηJand ρδ|c(x|z)be the conditional density of δK|ck. The conditions required for the steps of the proof of Lemma D.4 to go through are that ρη(u)is log-concave, ρδ|c(x|z)is log-concave in xfor all z,andδKis independent of ¯ δK|k. The first two properties are ensured by Assumption B.1, while the final property holds under strong homoskedasticity. We compare MVJ,Kand MVJ,Kin a manner very similar to the attribute case. Fix realizations of (aJ,cK). Define  Fε(z|y)≡Pr¯ δK,k≤z|Y0=y,aJ,cK 442 Liang and Madsen Theoretical Economics 19 (2024) to be the distribution function of ¯ δK,kconditional on the outcome Y0. Decompose Y0 as Y0=μ(J,K)+ηJ+δK. Taking expectations of each side conditional on (Y0,¯ δK,k,aJ,cK)yield Y0=μ(J,K)+EηJ|Y0,¯ δK,k,aJ,cK+EδK|Y0,¯ δK,k,aJ,cK. Hence, d Fεz|yEηJ|Y0=y,¯ δK,k=z,aJ,cK =y−μ(J,K)−d Fεz|yEδK|Y0=y,¯ δK,k=z,aJ,cK.(D.4) Lemma D.5 directly implies that d Fε(z|y)EδK|Y0=y,¯ δK,k=z,aJ,cK is nondecreasing in y,so(D.4) is nonincreasing in y. Following the same logic as in the attributes case, monotonicity of (D.4) implies that ∂ ∂Y0EηJ|Y0,aJ,cK≤E∂ ∂Y0EηJ|Y0,¯ δK,k,aJ,cKY0,aJ,cK, and it follows that MVJ,K≤E[MVJ,K|aJ,cK]. Thus, the marginal value of effort in each subpopulation in the baseline is weakly lower than the expected marginal value of effort when the circumstance kis additionally measured. The final step in the proof is to establish that monotonicity holds uniformly across realizations of the additional circumstance, and not just on average. This follows from arguments nearly identical to those used for the attributes case. D.4 Proof of Theorem 3 Define δ≡β/(1−β). Then given any set of measured covariates, equilibrium effort can be written as e∗ J,K=δ·MVJ,K. We will characterize the aggregate welfare change from measuring a new covariate as a function of δ, which can be mapped onto an equivalent characterization in terms of β. Consider first the case in which the market observes the new attribute j.LetJ≡ J∪{j}. The aggregate change in welfare from measuring j, as a function of δ,is W(δ)=Ew0(δ·MVJ,K)−w0(δ·MVJ,K), Theoretical Economics 19 (2024) Data and incentives 443 where w0(e)≡e−1 2e2. The aggregate change in welfare is a quadratic function of δ, whose first derivative is d dδW(δ)=E[MVJ,K−MVJ,K]−δ·EMV 2 J,K−MV 2 J,K. Observe that W(0)=0and d dδW(δ)|δ=0=E[MVJ,K−MVJ,K≤0 with the inequality implied by regularity of j.So,W (δ)is a quadratic function that vanishes at δ=0 and is nonincreasing there. Suppose first that jis strictly regular, so that Wis strictly decreasing at 0. Then its shape must then satisfy either of: •W(δ)is strictly convex and intersects zero exactly once for δ>0, •W(δ)is weakly concave and does not intersect zero for any δ>0. Define δ∗≡inf{δ>0:W (δ)≥0}. This threshold lies in (0, ∞), and has the property that W(δ)>0forδ>δ ∗,andW (δ)≤0forδ∈(0, δ∗].Further,δ∗<∞if and only if Wis strictly convex. Convexity is determined by the sign of d2 dδ2W (δ)=EMV 2 J,K−MV 2 J,K. Hence, δ∗<∞if and only if E[MV 2 J,K]>E[MV 2 J,K]. Letting β∗≡δ∗/(1+δ∗)therefore yields a reputation weight threshold with the desired properties. Suppose instead that jis weakly but not strictly regular. Write EMV 2 J,K|aJ,cK =EMVJ,K−E[MVJ,K|aJ,cK]2|aJ,cK+E[MVJ,K|aJ,cK]2 =Var(MVJ,K|aJ,cK)+E[MVJ,K|aJ,cK]2. Hence, EMV 2 J,K|aJ,cK−MV 2 J,K =E[MVJ,K|aJ,cK]2−MV 2 J,K+Var(MVJ,K|aJ,cK). Weak but not strict regularity implies that MVJ,K=E[MVJ,K|aJ,cK]with probability 1, so that EMV 2 J,K|aJ,cK−MV 2 J,K=Var(MVJ,K|aJ,cK)≥0 with probability 1. Therefore, Wis at least weakly concave. Weak but not strict regularity additionally implies that Whas zero slope at δ=0. Hence, in this case 444 Liang and Madsen Theoretical Economics 19 (2024) W (δ)≤0forallδ>0, and we set δ∗=∞and β∗=∞. We may summarize the work for the strictly and weakly regular cases with the conclusion that β∗<∞if and only if E[MV 2 J,K]>E[MV 2 J,K]. Indeed, in the strictly regular case, this equivalence was directly established, while in the weakly regular case, it is true both that β∗=∞and that the latter inequality cannot hold. Now suppose the market observes the new circumstance k. Calculations very similar to those for the attribute case show that Wis a quadratic function of δwhich vanishes at δ=0 and is weakly increasing there, and strictly increasing if kis strictly regular. It is concave if and only if E[MV 2 J,K]≥[MV 2 J,K], with the concavity strict if and only if the inequality is. If kis only weakly regular, then the calculations for the attribute case show that E[MV 2 J,K]≥E[MV 2 J,K], meaning that Wmust be at least weakly concave, in which case W(δ)≤0forallδ>0. In this case, we set δ∗=0andβ∗=δ∗/(1+δ∗)=0. Going forward, suppose that kis strictly regular. Defining δ∗≡inf{δ>0:W (δ)≤ 0}yields a threshold in (0, ∞]with the property that W(δ)<0forallδ>δ ∗and W(δ)>0forallδ∈(0, δ∗). This threshold is finite if and only if Wis a strictly concave function. Calculations very similar to the attribute case imply that EMV 2 J,K|aJ,cK−MV 2 J,K =E[MVJ,K|aJ,cK]2−MV 2 J,K+Var(MVJ,K|aJ,cK). Strict regularity implies that the difference between the first two terms on the rhs is nonnegative for every realization of (aJ,cK), and positive with positive probability. Further, the conditional variance of MVJ,Kmust be nonnegative. Therefore, EMV 2 J,K|aJ,cK≥MV 2 J,K, with the inequality strict with positive probability. Taking the unconditional expectation of both sides yields the desired strict concavity of W, implying δ∗<∞. Letting β∗≡ δ∗/(1+δ∗)yields a reputation weight with the desired properties. D.5 Proof of Proposition 1 We prove the first part of the result, with the second following from nearly identical arguments. To streamline notation, throughout this proof we will drop superscripts on Y0 and ¯ Y0. Under Assumption 3, conditioning on the value of Xis equivalent to conditioning on the value of ¯ Y. The desired result can therefore be established using a technique very similar to the one employed in the proof of Theorem B.1(a)subsequenttoLemma D.2,with ¯ Yplaying the role of θj. That argument requires two conditions: (A) (¯ Y,Y) are statistically affiliated, and (B) (¯ Y,θ)are statistically affiliated conditional on Y.We maintain condition A by hypothesis, so it remains only to establish condition B. Let ρY|¯ Y,θ(y|y,t)be the conditional density of Ygiven (¯ Y,θ),ρ¯ Y,θbe the joint density of (¯ Y,θ),andρYbe the marginal density of Y. Then by Bayes’ rule, ρ¯ Y,θ|Yy,t|y=ρY|¯ Y,θy|y,tρ¯ Y,θy,t ρY(y). Theoretical Economics 19 (2024) Data and incentives 445 Because (¯ Y,θ)are affiliated, the density ρ¯ Y,θis log-supermodular. To establish condition B, it is therefore sufficient to show that ρY|¯ Y,θ(y|y,t)is log-supermodular in (y,t), holding yfixed. Let ρε|¯ Y,θ(e|y,t)and ρε|¯ Y(e|y)be the conditional densities of εgiven (¯ Y,θ)and ¯ Y, respectively. Given the conditional independence of θand εgiven Xand the invertibility of E(¯ Y|X), it follows that θand εare independent conditional on ¯ Y. Therefore, ρε|¯ Y,θe|y,t=ρε|¯ Ye|y for all t, allowing us to write ρY|¯ Y,θy|y,t=ρε|¯ Y,θy−t|y,t=ρε|¯ Yy−t|y. Let ρε,¯ Y(e,y)be the joint density of (ε,¯ Y)and ρ¯ Y(y)be the marginal density of ¯ Y. Then using Bayes’ rule, we have ρY|¯ Y,θy|y,t=ρε,¯ Yy−t,y ρ¯ Yy. Condition B therefore follows if ρε,¯ Yis log-submodular. Equivalently, ρ−ε,¯ Ymust be logsupermodular, where ρ−ε,¯ Yis the joint density of (−ε,¯ Y). This condition is equivalent to affiliation of (−ε,¯ Y),ashypothesized. D.6 Proof of Lemma 1 In a multivariate Gaussian environment, the conditional mean of output is ¯ Y0=μ+(b+d)X, satisfying Assumption 3whenever b+d= 0. Since (X,Z,W)are jointly Gaussian and ¯ Y0and Y0are each linear combinations of (X,Z,W),thepair(¯ Y0,Y0)are also jointly Gaussian. Additionally, Y0=¯ Y0+Z+W, and since ¯ Y0is independent of Z+Wit follows that Y0and ¯ Y0are positively correlated. Therefore, (¯ Y0,Y0)are affiliated, satisfying Assumption 4. Moreover, conditional expectations in multivariate Gaussian environments are linear in the conditioning variable, ensuring that Assumption 5is satisfied. Finally, mutual independence of X,Z,and Wimplies that (θ,ε)are independent conditional on X, satisfying Assumption 6. D.7 Proof of Proposition 2 Prior to the measurement, (θ,Y0)have joint distribution θ Y0∼Nμ μ,σ2 θσθ,Y σθ,Yσ2 Y 446 Liang and Madsen Theoretical Economics 19 (2024) where σ2 θ=b2σ2 x+σ2 z,σθ,Y=b(b+d)σ2 x+σ2 z,σ2 Y=(b+d)2σ2 x+σ2 z+σ2 w. The market’s type forecast given equilibrium effort e∗is therefore Ee∗(θ|Y)=Eθ|Y0=Y−e∗=μ+σθ,Y σ2 Y ·Y−μ−e∗, implying that the baseline marginal value of effort is MV =σθ,Y σ2 Y =b(b+d)σ2 x+σ2 z (b+d)2σ2 x+σ2 z+σ2 w . Meanwhile, after measuring X, the conditional joint distribution of (θ,Y0)is θ Y0X∼N μ+bX μ+(b+d)X,σ2 zσ2 z σ2 zσ2 z+σ2 w. Given equilibrium effort e∗∗, the market’s posterior type forecast after measuring Xis Ee∗∗ (θ|X,Y)=Eθ|X,Y0=Y−e∗∗=μ+bX +σ2 z σ2 z+σ2 wY−e∗∗ −μ−(b+d)X. The agent’s marginal value of effort under the expanded data set is therefore MV+=σ2 z σ2 z+σ2 w . Comparing the expressions for MV and MV+just derived yields the identity in the proposition statement. References Ball, Ian (2022), “Scoring strategic agents.” Working Paper. [409] Bergemann, Dirk, Alessandro Bonatti, and Tan Gan (2022), “The economics of social data.” RAND Journal of Economics, 53, 263–296. [408] Bonatti, Alessandro and Gonzalo Cisternas (2020), “Consumer scores and price discrimination.” The Review of Economic Studies, 87, 750–791. [409] Braverman, Mark and Sylvain Chassang (2022), “Data-driven incentive alignment in capitation schemes.” Journal of Public Economics, 207, 104584. 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