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Identification Design
JEL classification: D81, D83, C14, C21
Keywords: identification design, partial identification, prior-free, treatment effects
Frequentist data analysis is standard in regulatory evaluations of medical interventions and in scientific publishing. At the same time, existing models of strategic communication typically study Bayesian decision makers who filter evidence through an exogenous prior. Empirical researchers are themselves economic agents who face private incentives and may therefore be tempted to selectively disclose information, even when what they do disclose must accurately reflect their data. How should an objective decision maker respond?
In our model, the decision maker observes the population distribution of signals generated by an information structure. He understands the stochastic relationship between states and signals and views any admissible distribution of states that rationalizes the observed distribution of signals as plausible. His payoffs are state-dependent, he ranks actions by their worst-case payoff over the partially identified set of plausible state distributions, and an information structure implements an action if that action is worst-case optimal under that structure.
This framework is directly applicable to causal inference. Consider the classical treatment-effects environment with binary $Y$, binary treatment $T$, covariate $X$, and potential outcomes $(Y_{0},Y_{1})$. Treatment is unconfounded and the assignment mechanism $P(T\mid X)$ assigns each treatment to every covariate group with positive probability. If the policymaker observes the joint distribution of $(Y,X,T)$ then the average counterfactual outcomes $\mathbb{E}[Y_{0}]$ and $\mathbb{E}[Y_{1}]$ are exactly identified. In contrast, if the policymaker observes only the marginal distribution of $(Y,T)$ then uncertainty about the distribution of outcomes and treatments across groups translates into uncertainty about the consequences of extending each treatment to the untreated segment of the population. As we demonstrate in Section (ref), there are situations in which a fully informed policymaker chooses one treatment while a partially informed policymaker chooses another.
We develop three strands of results. First, we call the environment manipulable if for every action $\alpha$ and every admissible state distribution $\mu$ there exists an information structure that implements $\alpha$ when the true state distribution is $\mu$. We show that this condition holds if and only if the worst-case payoff for each action over the set of admissible state distributions is the same.
Next, we apply our characterization of manipulable environments to the general causal-inference framework with many treatments, many covariates, and many outcomes, and confirm that all such environments are indeed manipulable. We show more strongly that all such treatment-effects models are strongly manipulable via information structures that (i) are almost fully informative, in the sense that they disclose at least $n-1$ linearly independent statistics to the decision maker in a model with $n$ states; (ii) are generically robust to the policymaker's tiebreaking procedure; and (iii) limit the scope for ex post revelation of manipulation by point identifying the policymaker's payoff under the implemented action.
Drawing motivation from real-world disclosure policies, we conclude our study of our microeconometrics application by restricting attention to marginal information structures that reveal the joint distribution of the outcome variable, the treatment variable, and a selection of covariates. We develop tight identification results and sharp payoff bounds for these information structures, and apply them to a range of settings. While the policymaker's payoffs are exactly identified under the disclosure of a sufficiently rich set of covariates to verify the assignment mechanism in an experiment, observational studies remain vulnerable to partial manipulation via covariate selection. Our positive results for experiments extend to intermediate cases in which the assignment mechanism is unknown to the decision maker but is known to depend only on a particular set of covariates.
Third, and finally, we return to the general model and characterize the set of implementable actions in a class of non-manipulable environments that satisfy a pair of mild regularity criteria. There, action $\alpha$ is implementable under true state distribution $\mu$ if and only if there exists an admissible prior $\nu$ such that (i) the decision maker's payoff for $\alpha$ is no higher under $\nu$ than under $\mu$ and (ii) $\alpha$ is optimal in the counterfactual environment in which the decision maker knows that the state is distributed according to $\nu$. In all such cases, implementation is feasible via almost fully informative information structures.
\paragraph*{Related literature}
This paper endogenizes partial identification in the same way that Bayesian persuasion endogenizes Blackwell experiments. In doing so, we contribute to the literatures on experimentation and information design; strategic communication under ambiguity; robust decision making in economic theory; and partial identification in econometrics.
Blackwell's seminal papers Blackwell1951,Blackwell1953 introduce a model of statistical experimentation in which a Bayesian decision maker observes a signal generated by the experiment and updates his beliefs before acting. KamenicaGentzkow2011 endogenize the choice of experiment in a sender-receiver environment, characterize the sender-optimal experiment, and thereby launch the Bayesian persuasion literature. In turn, BergemannMorris2016 introduce the Bayes correlated equilibrium solution concept as a general framework for information design in Bayesian games with multiple agents, and survey the literature in BergemannMorris2019. In contrast, our decision maker responds to the entire distribution of signals rather than to individual realizations, has no prior, and evaluates actions by their worst-case payoff over the identified set. While one recent study (LinLiu2024) does provide a role for the unconditional message distribution in enforcing the commitment assumption in Bayesian persuasion, the receiver in that paper responds to individual signal realizations rather than the entire distribution, as in the standard formulation of the persuasion problem.
A growing literature studies persuasion and communication with ambiguity-averse agents. \citet*{BeaucheneLiLi2019} study a sender who strategically deploys ambiguous communication devices to an ambiguity-averse receiver who starts from a unique prior; \citet*{HedlundKauffeldtLammert2021} study persuasion of a receiver with an exogenous interval of priors and $\alpha$-maxmin preferences. In both papers, the receiver observes and responds to individual signal realizations. In a study more closely related to our own, KoesslerPahlke2025 design the coarseness of aggregate feedback about opponents' play in games with ambiguity-averse players, with beliefs restricted to be consistent with the observed aggregate data. Their decision makers respond to distributional information rather than to individual signals, as ours does; we differ in that our uncertainty is about the payoff-relevant state rather than about opponents' strategies, and in our consideration of a single-agent decision problem rather than a game.
Along those lines, our model of individual decision making was originally developed in earlier work by the author (Rosenthal2026PFB). There, we say that information structure $(\Sigma,E)$ is robustly more informative than $(\Sigma',E')$ if, for every decision problem, the decision maker's guaranteed payoff under the former matches or exceeds his guaranteed payoff under the latter. This order is implied by, but does not imply, Blackwell's classical order. Outside this framework, there is a small but growing literature on non-Bayesian decision making in Blackwell experiments and sender-receiver games. Whitmeyer2026 studies the information-monotonicity of non-Bayesian updating rules, and \citet*{YangYoderZentefis2025} study the value of finite-dimensional explanations of complex models, showing that no explanation can improve the payoff of a worst-case decision maker. An earlier stream of papers extends Blackwell\textquoteright s classical framework to maxmin expected-utility maximizers who make Bayesian updates to exogenously specified sets of priors (Celen2012MEU,HeyenWiesenfarth2015MEU,LiZhou2016Blackwell).
Finally, the partial-identification literature in econometrics initiated in Manski1990,Manski1997,Manski2003,Manski2007,Manski2013 and surveyed by Tamer2010,Molinari2020,KlineTamer2023 studies decision problems in which the distribution of the state is set-identified and actions are evaluated according to worst-case criteria. We differ from this literature by our endogenization of the identified set via our interpretation of the information structure as a choice variable. This perspective facilitates our characterization of implementable actions and our identification of maximally informative structures.
\paragraph*{Roadmap}
The paper is organized as follows. We motivate our application to causal inference in Section (ref), lay out the model in Section (ref), and provide our characterization of manipulable environments in Section (ref). We apply this framework to treatment-effects models in Section (ref); confirm that those models are manipulable in Section (ref); analyze marginal information structures in Section (ref); and discuss the real-world context for our application in Section (ref). Finally, we return to the general model and provide a characterization of implementability in non-manipulable environments in Section (ref), discuss and conclude in Section (ref), and apply the minimax regret decision making criterion to our treatment-effects model in Appendix (ref). Proofs and omitted supporting results are organized into Appendices (ref)--(ref).
A researcher discloses data from an observational study of treatment outcomes to a decision maker. The treatment $T$, untreated outcome $Y_{0}$, treated outcome $Y_{1}$, and covariate $X$ are each binary. The researcher observes the true distribution $\mu$ of $(Y_{T},X,T)$ shown in Table (ref) and the counterfactual outcomes $(Y_{0},Y_{1})$ are conditionally independent of $T$ given $X$.
The decision maker's problem is to choose a treatment $a\in\{0,1\}$ to maximize the worst-case expected outcome $\mathbb{E}[Y_{a}]$. If the researcher discloses the full joint distribution $\mu$, then the counterfactual means
are exactly identified and the decision maker declines to treat. If instead the researcher discloses only the marginal distribution of $(Y,T)$, then the decision maker evaluates treatment $a$ by its worst-case payoff $\mathbb{E}_{\nu}[Y_{a}]$ with respect to all joint distributions $\nu$ on $(Y,X,T)$ consistent with unconfoundedness $(Y_{0},Y_{1})\perp T\mid X$, strict overlap $\nu(t,x)>0$, and the disclosed marginal in Table (ref).
As the inverse-probability weighting formulae ((ref))--((ref)) make clear, the worst case for action $a$ is that treatment $T=a$ was assigned as frequently as possible to $X$-groups that benefit most from it. While the disclosed marginal distribution of $(Y,T)$ constrains these quantities, it does not identify them. The joint distribution $\nu$ in Table (ref) is consistent with the disclosed marginal and yields worst-case payoffs
for both treatments. Although treatment is suboptimal when the decision maker observes the full joint distribution of $(Y,X,T)$, it is robustly optimal when he observes only the marginal of $(Y,T)$. Uncertainty about observation-level treatment propensities reverses the full-information optimal policy.
We write $\mathbb{R}^{n}$ for the set of real vectors of length $n$ equipped with the standard metric and identify real valued functions on finite sets $X$ with vectors in $\mathbb{R}^{\vert X\vert}$. Given a finite set $X$, we write $\Delta(X)$ for the set of all probabilities on $X$ and interpret the elements of $\Delta(X)$ as real vectors. More broadly, given a metric space $X$, we give the set of Borel distributions $\Delta(X)$ the topology of weak convergence and write $\text{supp}(\nu)$ for the support of $\nu\in\Delta(X)$.
This paper makes use of standard results from linear algebra. Given a set of real vectors $S\subset\mathbb{R}^{n}$, we write $\text{span}(S)$ for its span. If $S$ is singleton we write $\text{span}\{d\}$; if $S$ is a vector space we write $\text{dim}(S)$ for its dimension. Finally, given a linear map $f:\mathbb{R}^{n}\to\mathbb{R}^{m}$, we write $\ker f\equiv\{v\in\mathbb{R}^{n}\mid f(v)=0\}$ for its null space.
\paragraph{States, actions, and beliefs}
The set $\Omega$ of states is a non-empty and finite; the set $A$ of actions is a non-empty compact metric space. The set $\mathcal{P}\subset\Delta(\Omega)$ of admissible state distributions\footnote{Despite our non-Bayesian framework, we follow the literature on decision making under uncertainty and occasionally refer to elements of $\mathcal{P}$ as priors.} contains the true distribution of the state of the world $\mu$ and the decision maker's utility $U:\Delta(A)\times\Delta(\Omega)\to\mathbb{R}$ is continuous. We write $\omega$ for generic elements of $\Omega$, $a$ for generic elements of $A$, $\nu$ for generic elements of $\mathcal{P}$, and $\alpha$ for generic elements of $\Delta(A)$. Tuple $(\Omega,\mathcal{P},A,U)$ is the environment.
\paragraph{Information structures}
The decision maker acts on objective information about the true distribution of the state of the world $\mu$. He observes the distribution of messages generated by information structure $(\Sigma,E)$, where $\Sigma$ is a finite set of messages and $E:\Omega\to\Delta(\Sigma)$ assigns message distributions $E(\cdot\mid\omega)$ to states $\omega$. Admissible state distribution $\nu$ is observationally equivalent to $\mu$ if and only if $E\nu=E\mu$, and we write \[ \mathcal{P}_{\mu}(\Sigma,E)\equiv\{\nu\in\mathcal{P}\mid E\nu=E\mu\}=\{\nu\in\mathcal{P}\mid(\nu-\mu)\in\ker E\} \] for the identified set of all such distributions. Information structure $(\Sigma,E)$ is fully informative if $\ker E$ has dimension $0$; uninformative if $\ker E$ has dimension $\vert\Omega\vert-1$; and almost fully informative if $\ker E$ has dimension at most $1$.
\paragraph{The decision maker's problem}
The decision maker's problem
is to maximize his worst-case payoff over the identified set. We say that information structure $(\Sigma,E)$ implements action $\alpha$ if $\alpha$ is a solution to problem ((ref)) under $(\Sigma,E)$, and strictly so if that solution is unique.
The identified set $\mathcal{P_{\mu}}(\Sigma,E)$ is jointly characterized by the exogenous set of admissible priors $\mathcal{P}$, the exogenous true distribution $\mu$, and the endogenous information structure $(\Sigma,E)$. Admissible prior $\nu$ is observationally equivalent to the true distribution $\mu$ if and only if $\nu-\mu\in\ker E$; from that point of view, our problem reduces to one in which the independent variable of interest is the null space of the stochastic map $E$ rather than $(\Sigma,E)$ itself.
Proposition (ref) reduces the implementation problem to a search over kernels. We study environments in which that search succeeds for every action, regardless of the true distribution of the state variable.
Manipulability is a strong criterion. As we show, it holds if and only if the worst-case payoff for every action is the same.
Let $(\Omega,\mathcal{P},A,U)$ be any environment. If $\inf_{\nu\in\mathcal{P}}U(\alpha,\nu)=\inf_{\nu\in\mathcal{P}}U(\beta,\nu)$ for all actions $\alpha,\beta$ then any uninformative information structure implements every action. Conversely, if $\inf_{\nu\in\mathcal{P}}U(\alpha,\nu)>\inf_{\nu\in\mathcal{P}}U(\beta,\nu)$ then no information structure implements $\beta$ under any true state distribution $\mu$ satisfying $U(\beta,\mu)<\inf_{\nu\in\mathcal{P}}U(\alpha,\nu)$. Accordingly, $(\Omega,\mathcal{P},A,U)$ is manipulable if and only if the decision maker's worst-case payoff over the set of admissible priors is constant across actions. This situation arises naturally in applications in which each action yields the same range of potential outcomes.
There are two or more treatments $T_{1},...,T_{k}$ in finite set $\mathcal{T}$; two or more outcomes $Y$ in finite set $\mathcal{Y}\subset\mathbb{R}$; and finitely many covariates $X\equiv(X_{1},...,X_{\ell})$. Each variable $X_{j}$ takes values in non-empty and finite set $\mathcal{X}_{j}$, and $X$ takes values in $\mathcal{X}\equiv\mathcal{X}_{1}\times...\times\mathcal{X}_{\ell}$.\footnote{Models with only an outcome variable $Y$ and treatment variable $T$ are equivalent to specifications of our model with $\vert\mathcal{X}\vert=1$. } While the latent state $(Y_{1},...,Y_{k},T,X)$ specifies the outcome $Y_{t}$ under each treatment $t$, the observed state $\omega\equiv(Y_{T},T,X)$ records only the outcome $Y_{T}$ of the treatment actually received $T$. Unconfoundedness $(Y_{1},...,Y_{k})\perp T\mid X$ identifies the counterfactual mean $\mathbb{E}_{\nu}[Y_{a}]$ from the distribution of observable variables $\nu$, and the set of actions $A$, the state space $\Omega$, and the decision maker's utility $U:\Delta(A)\times\Delta(\Omega)\to\mathbb{R}$ are respectively
The assignment mechanism $P(T\mid X)$ belongs to a non-empty set $\mathscr{P}$ of maps from $\mathcal{X}$ into $\Delta(\mathcal{T})$ satisfying the strict overlap assumption
and the set of admissible distributions
is the set of all distributions $\nu$ consistent with the structural assumptions embedded in $\mathscr{P}$. We call environment $(\Omega,\mathcal{P},A,U)$ a treatment-effects model if it satisfies ((ref))--((ref)), and highlight the two privileged special cases.
In an experiment $(\Omega,\mathcal{P},A,U)$, the decision maker is certain of the assignment mechanism $P$, which is intentionally designed, executed, and disclosed.
In an observational study $(\Omega,\mathcal{P},A,U)$, the decision maker has no specific exogenous knowledge of the treatment mix assigned to each group in the data. Instead, he knows only that the assignment mechanism $P$ satisfies strict overlap and that treatment is unconfounded. This is the natural setting for the analysis of non-experimental data.
In all treatment-effects models $(\Omega,\mathcal{P},A,U)$, the decision maker's payoffs have a cubical structure under which manipulability is evident. Prior $\nu$ is admissible if and only if there exists a family of conditional outcome distributions $\Phi:\mathcal{T}\times\mathcal{X}\to\Delta(\mathcal{Y})$, an admissible assignment mechanism $P$, and a distribution of weights $\lambda\in\Delta(\mathcal{X})$ such that
Direct substitution into ((ref)) yields \[ U(a,\nu)=\sum_{y,x}y\Phi(y\mid a,x)\lambda(x), \] the conditional distributions $\Phi$ and the weights $\lambda$ are free parameters, the group-specific conditional mean $\sum_{y}y\Phi(y\mid a,x)$ lies anywhere in the interval $[\min\mathcal{Y},\max\mathcal{Y}]$, and the set of admissible payoff vectors is
We prove Proposition (ref) by constructively verifying that the worst-case payoff for every action $\alpha$ is $\min\mathcal{Y}$ and applying Theorem (ref). In doing so, we establish that treatment-effects models are manipulable via uninformative information structures under which the agent's payoff for every action coincides with the global minimum.
This approach to establishing manipulability raises three questions. First, are these environments manipulable by information structures that disclose more information to the decision maker? Second, are they manipulable via information structures that do not rely on discretionary tiebreaking? Third, and related to both of the first two points, are they manipulable by information structures that do not immiserate the decision maker?
In order to formally address these questions, it will be helpful to call action--prior pair $(\alpha,\mu)$ eligible if (i) $\alpha$ is some degenerate action $a$ and (ii) $\mu$ is not a minimizer for $U(\alpha,\cdot)$ over the set of admissible priors $\mathcal{P}$. The first criterion is practically relevant; the second holds generically.
Proposition (ref) provides information structures with three features. First, they reveal at least $\vert\mathcal{Y}\vert\cdot\vert\mathcal{T}\vert\cdot\vert\mathcal{X}\vert-1$ linearly independent statistics in a model with $\vert\mathcal{Y}\vert\cdot\vert\mathcal{T}\vert\cdot\vert\mathcal{X}\vert$ states. Second, in practically relevant cases in which the implemented action does not involve randomization over treatments, implementation is generically strict. Third, in those same cases, the true distribution of outcomes $\mu(y,a,x)$ under the implemented action $a$ is exactly identified, and there is no ex post revelation of manipulation. The proof is based around ((ref)), with some care to account for the nonlinearity of the payoff functional \[ \nu\mapsto\sum_{y,x}\frac{y}{\nu(a\mid x)}\nu(y,a,x) \] in non-experimental settings.
The almost fully informative information structures invoked in our manipulability results for treatment-effects models are both informationally rich and simple in the abstract. At the same time, it is difficult to give them a practical interpretation, because real-world researchers do not typically produce data sets that are generated by information structures of arbitrary form. Instead, they disclose the joint distribution of the outcome variable, the treatment variable, and a selection of other variables.
Information structure $(\Sigma,E)$ is a marginal information structure if there exists an index\footnote{At one extreme, if $J$ is empty then $\mathcal{Z}$ is trivial and disclosure of the $(Y,T,Z)$ marginal is equivalent to disclosure of the $(Y,T)$ marginal. At the other extreme, if $J=\{1,...,\ell\}$ then $\mathcal{W}$ is trivial and disclosure of the $(Y,T,Z)$ marginal is equivalent to disclosure of the $(Y,T,X)$.} $J\subset\{1,...,\ell\}$, set $\mathcal{Z}\equiv\prod_{j\in J}\mathcal{X}_{j}$ of disclosed covariates $Z$, and set $\mathcal{W}\equiv\prod_{j\notin J}\mathcal{X}_{j}$ of omitted variables $W$ such that
Marginal information structures $(\Sigma,E)$ reveal the marginal distribution of their disclosed covariates $(Y,T,Z)$, and the matching-marginals condition that delineates the identified set \[ \mathcal{P}_{\mu}(\Sigma,E)=\{\nu\in\mathcal{P}\mid\forall(y,t,z)\;\nu(y,t,z)=\mu(y,t,z)\} \] implies that treatment effects models are not manipulable via marginal information structures.
In Example (ref), there exists a marginal information structure $(\Sigma,E)$ that implements action $\alpha$ under true state distribution $\mu$ if and only if the support of $\alpha$ is contained in the set of maximizers for the outcome map $\Phi$ if and only if $\alpha$ is optimal under fully informative information structures. Per our choice of $\Phi$, not every action is optimal under full revelation of $\mu$. Accordingly, no treatment-effects models are manipulable via marginal information structures. While this is a positive result for the decision maker, the dependency on $\mu$ raises the question as to when such protection is available regardless of the true state distribution.
As we show, treatment-effects models are identified by a marginal information structure $(\Sigma,E)$ if and only if every admissible assignment mechanism is measurable with respect to the disclosed covariates.
Theorem (ref) has implications in a range of canonical treatment-effects models. As we discuss in Section (ref), best practices for real-world experiments include a variety of disclosure requirements. Our result clarifies that compliance with these requirements makes manipulation impossible if and only if the disclosed covariates are rich enough to pin down the assignment probabilities. This impossibility result extends more broadly to environments in which there is uncertainty about the assignment mechanism $P$ but $P$ is known to depend only on a specific set of disclosed covariates $Z$.
In fact, under our maintained unconfoundedness and strict overlap assumptions, the global measurability criterion in Theorem (ref) is equivalent to the hypothesis that all of the latent distributions in our model satisfy the familiar selection on observables criterion $(Y_{1},...,Y_{k})\perp T\mid Z$. To see why, recall that the sufficiency of $(Y_{1},...,Y_{k})\perp T\mid Z$ for pointwise identification is a textbook result. In turn, global selection on observables is sufficient for global identification, and the theorem shows that global identification implies the measurability criterion. Finally, the measurability criterion implies for all admissible distributions of latent variables \[ \nu(y_{1},...,y_{k},t\mid z)=\sum_{w}\nu(y_{1},...,y_{k}\mid z,w)\nu(t\mid z,w)\nu(w\mid z)=\nu(y_{1},...,y_{k}\mid z)\nu(t\mid z), \] where the first equality follows from our maintained assumption $(Y_{1},...,Y_{k})\perp T\mid(Z,W)$ and the second from measurability. Global selection on observables follows.
In contrast to treatment-effects models that satisfy the theorem's measurability criterion, observational studies with omitted covariates are never identified. At the same time, we have already established in Example (ref) that these models are not manipulable via marginal information structures. Accordingly, observational studies with disclosed marginals occupy a middle ground. In order to characterize the extent to which these studies are partially manipulable, we develop bounds on the decision maker's payoffs under true state distribution $\mu$ and marginal information structure $(\Sigma,E)$. Our bounds are tight when the set of omitted covariates is large.
Assumption (ref) implies that the set of omitted variables is sufficiently rich to encode all latent counterfactual vectors $(Y_{1},...,Y_{k})$, as is the case if the omitted variables are interpreted as uncertain objects rather than fixed variables that are known to be relevant but excluded from the disclosed marginal.
The decision maker's payoff is the sum of the observed covariate-weighted average outcome for the treated group and the unobserved covariate-weighted average outcome for the untreated groups. While marginal information structures are such that the former is directly disclosed, the latter can best be interpreted by returning to the latent state space $(Y_{1},...,Y_{k},T,Z,W)$ and the assumptions that justify the inverse probability weighting payoff formula ((ref)).
First, strict overlap implies that every represented group $(z,w)$ receives treatment $a$ under every admissible state distribution. Second, unconfoundedness implies that the conditional distribution of treated outcomes $Y_{a}$ given cell $(b,z,w)$ coincides with the distribution of treated outcomes $Y_{a}$ given cell $(a,z,w)$. The outcomes $Y_{a}$ for the untreated cell $(b,z,w)$ are constrained to the set of outcomes $\{y\mid\mu(y,a,z)>0\}$; the disclosure of additional covariates monotonically tightens that constraint; and the choice of what to disclose is an instrument for manipulation.\footnote{This lies in contrast to the classical Manski (1990) framework in which there are no structural assumptions linking counterfactual outcomes across treated and untreated groups, the untreated might have any counterfactual outcome in $\mathcal{Y}$, and the $(Y,T,Z)$ marginal provides no tighter bounds than the $(Y,T)$ marginal itself.}
Proposition (ref) implies that there exists a marginal information structure $(\Sigma,E)$ that implements action $\alpha$ under true distribution $\mu$ if and only if there exists a set of covariates $Z$ under which $\alpha$ is optimal according to the lower bound given in Theorem (ref). As in our motivating example in Section (ref), it is not in general the case that such actions coincide with full-disclosure optimal policies. Instead, the action is determined by what is disclosed and what is omitted.
The misalignment between researcher incentives and public interest is an acknowledged phenomenon in the conveyance of scientific evidence, as discussed in Spiess2025 and taken as the premise of both that paper and several other recent studies in econometrics and economic theory (\citet*{AndrewsShapiro2021,BanerjeeChassangMonteroSnowberg2020,KasySpiess2024}). While we are distinguished from these works by our emphasis on identification rather than finite sample issues, the available evidence suggests that the targeting of statistical significance is not the only mechanism for researcher manipulation. LenzSahn2021 provide a replication study in which over 30% of the articles published in two volumes of the American Journal of Political Science achieve statistical significance via undisclosed and unjustified covariate adjustment, in which the authors emphasize that these adjustments typically increased the magnitude of the estimated effects rather than the precision with which they were estimated. Elsewhere, \citet*{Dwan2014} report discrepancy rates of $46-82\%$ for adjusted-versus-unadjusted analyses across cohort studies of randomized clinical trials.
Aside from the direct evidence that researchers sometimes engage in selective reporting practices, there is an abundance of indirect evidence of this behavior in the form of institutional guidelines aimed at inhibiting it. The 2007 Food and Drug Administration Amendments Act (FDAAA2007) mandates the registration of clinical trials and pre-specifications of primary outcomes on ClinicalTrials.gov before enrollment. More recent FDA guidance on covariate adjustment (FDA2023) requires prospective specification of covariate adjustment procedures before unblinding of comparative data; earlier guidance from the International Council for Harmonisation (ICH1998) requires pre-specification of statistical analysis plans in pharmaceutical trials.
On the scientific side, the CONSORT protocol for randomized control trials (\citet*{Hopewell2025}) provides guidelines on the disclosure of randomization procedures, pre-analysis plans, and the disclosure of both covariate-adjusted and unadjusted estimates. Within economics, the American Economic Association requires registration of field experiments prior to submission to its society journals (\citet*{Olken2015}) and allows for the disclosure of pre-analysis plans; separately, the Journal of Development Economics offers an editorial process in which authors disclose their full statistical analysis plan and the journal commits to a publication outcome before the results are known (\citet*{BogdanoskiFosterKarlanMiguel2018}). As in regulatory guidelines, all of these practices constrain the researcher's choice of information structure.
As a baseline positive recommendation, Theorem (ref) suggests that disclosures other than the joint distribution of the outcome variable, the treatment variable, and a selection of covariates ought to be regarded with skepticism. To the extent that the typical information structures deployed in practice are indeed of the marginal form, our results suggest positive prescriptions for evaluating those types of disclosures in three classes of environments.
First, completely randomized experiments (ImbensRubin2015) in which $P(T\mid X)=P(T)$ for all groups $X$ are commonplace. In keeping with received wisdom, payoffs in these experiments are exactly identified regardless of how many covariates are omitted. One notable exception to this practice lies in the study of prioritized school-admissions lotteries, in which some students are given a higher probability of acceptance than others (\citet*{AbdulkadirogluAngristNaritaPathak2017,AbdulkadirogluAngristNaritaPathak2022}). In that context, Theorem (ref) confirms the importance of researcher disclosure of the joint distribution of the outcome variable of interest, the received treatment, and the priority group for each student; disclosure of the specific admissions probabilities themselves is unnecessary.
Second, in some natural experiments it might be the case that the researcher understands which variables are relevant for assignment probabilities but is unsure about what the probabilities themselves are. As Theorem (ref) clarifies, those studies are identified (and therefore fully non-manipulable) if and only if the data set includes all variables that assignment depends on.
Finally, while observational studies remain partially manipulable via covariate selection, we offer two positive takeaways. First, if the researcher is constrained to disclose a given set of covariates, then there is no scope for intentional manipulation. Second, as the number of disclosed covariates grows, the bounds in Theorem (ref) imply that the worst-case (and best-case) payoffs for each treatment collapse towards their true values.
In Example (ref), we identify a failure of manipulability in an otherwise-manipulable environment that results from constraints on the information structure. At the same time, Theorem (ref) shows that not all environments are manipulable even in the absence of such constraints. The set of implementable actions in non-manipulable environments $(\Omega,\mathcal{P},A,U)$ has a simple characterization as long as $(\Omega,\mathcal{P},A,U)$ satisfy some additional regularity criteria.
As we show in Lemma (ref) in the Appendix, the technical conditions in Assumption (ref) imply that $\alpha$ is a solution to the decision maker's problem under information structure $(\Sigma,E)$ and true distribution $\mu$ if and only if there exists a state distribution $\nu$ in the identified set such that $(\alpha,\nu)$ are a saddle point of that problem satisfying
In all such cases, $\alpha$ can be implemented by an almost fully informative information structure $(\Sigma^{\nu},E^{\nu})$ that “targets” prior $\nu$ via its null space $\ker E=\text{span}\{\nu-\mu\}$.
Proposition (ref) provides sufficient conditions under which implementation is equivalent to implementation via almost fully informative information structures. If the decision maker is an expected utility maximizer and it is possible for information structures to condition messages on payoff-irrelevant information, then the set of implementable actions has a straightforward characterization that is closely related to the saddle inequalities ((ref)).
Assumption (ref) does two things. First, it imposes linearity on the decision maker's utility $U$. Second, it enriches the state space $\Omega$ with redundant states. In doing so, it provides greater control over the payoffs associated with the extreme points of the identified set.
Provided that Assumption (ref) and Assumption (ref) hold, action $\alpha$ is implementable if and only if there exists an admissible prior $\nu$ such that (i) $\alpha$ is full-information optimal under $\nu$ and (ii) $\alpha$ is no better under $\nu$ than under the true distribution $\mu$.\footnote{The conditions (i) $\alpha\in\alpha^{*}(\nu)$ and (ii) $U(\alpha,\mu)\geq U(\alpha,\nu)$ that characterize implementability in Theorem (ref) are necessary but not sufficient for the pair $(\alpha,\nu)$ to be a saddle point of the decision maker's problem under true state distribution $\mu$ and any information structure $(\Sigma,E)$ under which $\nu$ is observationally equivalent to $\mu$. We are grateful to an anonymous referee for pointing out that Kuzmics2017 observes Wald's complete class theorem implies that a maxmin decision maker who can randomize over acts always chooses as if she were a subjective expected utility maximizer. In our model, the \textquotedbl as if\textquotedbl representation arises from the sufficiency of Assumption (ref) for the existence of a saddle point in the identified set, and the substance lies in (a) showing that Assumption (ref) closes the gap between necessity and characterization; and (b) implementation is achievable via almost fully informative information structures. More broadly, our general model is insufficiently well-structured to guarantee the existence of a saddle point and our other results therefore do not rely on strong duality. } Together, these assumptions reduce to three criteria. First, the set of admissible priors $\mathcal{P}$ is compact and convex. Second, the decision maker is an expected utility maximizer. Third, either (i) the natural specification of the state space includes variables that are pure signals and have no direct influence on payoffs; or (ii) a motivated designer deliberately conditions signals on payoff-irrelevant information. On the one hand, while the compactness, convexity, and linearity criteria are not appropriate for our application to robust causal inference, they are standard in economic theory. On the other hand, while redundant signals are typically excluded, this is at least in part because they have no value to designers. To the extent that it is possible for the designer to create redundant states by making message probabilities contingent on variables that do not affect the decision maker's payoffs, their inclusion in our model is without loss of generality.
This paper proposes a model of identification design in which the decision maker observes the entire population-level distribution of messages generated by an information structure, views any admissible state distribution that is consistent with the distribution of those messages as plausible, and ranks actions by their worst-case payoff over the set of all such distributions. We provide a general characterization of manipulable environments in which there exists an information structure that implements each action under each true distribution of the state of the world, and apply our framework to robust causal inference in microeconometrics. There, we show that all treatment-effects models are manipulable; characterize the extent to which these environments are partially manipulable under a practically-motivated restriction to marginal information structures that disclose the joint distribution of the outcome variable, the received treatment, and a selection of covariates; and develop tight identification results and payoff bounds for that restricted class. Finally, we provide a general characterization of implementability for non-manipulable environments.
Our formulation of the decision maker's problem as one in which he maximizes his worst-case payoff is one of many ambiguity averse preferences considered in the decision theory and statistical decision theory literatures. We make two comments about our approach.
First, in keeping with the nonlinearity of the decision maker's underlying payoff functional \[ U(a,\nu)=\mathbb{E}_{\nu}[Y_{a}]=\sum_{y,x}\frac{y}{\nu(a\mid x)}\nu(y,a,x) \] in non-experimental instances of our treatment-effects application, we allow for but do not require $U$ to be of the expected utility form. This lies in contrast to both the maxmin expected utility criterion (GilboaSchmeidler1989) and the standard formulation of its counterparts, including the smooth ambiguity aversion (KlibanoffMarinacciMukerji2005) and variational preferences models (MaccheroniMarinacciRustichini2006) that nest maxmin expected utility either directly or as a limiting case.
Second, regardless of whether or not $U$ represents an expected utility preference, the outer maxmin criterion is consistent with the major premise of this paper in the sense that its only inputs are (i) the structural restrictions to the set of admissible distributions $\mathcal{P}$ and (ii) the restrictions on $\mathcal{P}$ implied by the information revealed by the information structure $(\Sigma,E)$. By way of contrast, both of the aforementioned models take either second-order beliefs (the former) or penalty terms (the latter) as exogenous inputs that prioritize some admissible priors over others. At the same time, maxmin is not the only suitable criterion for purely data-driven analysis; in Appendix (ref), we show how a version of our manipulability result holds under the alternative minimax regret criterion for a decision maker who restricts attention to pure-strategy treatments. More broadly, in contrast to results that follow from the monotonicity of the decision maker's payoff in the size of the uncertainty set (\citet*{LiZhou2016Blackwell,LiZhou2020,Wang2024,Rosenthal2026PFB}), our results do not immediately extend to general ambiguity averse preferences.
Along with a richer treatment of alternative ambiguity attitudes, this paper suggests several avenues for future work. First, extensions of our framework to dynamic settings or to games with multiple players. Second, a treatment of the finite sample issues that we have deliberately abstracted from in order to maintain our focus on identification. Third, applications to ordinary least squares, instrumental variables, and other standard models in microeconometrics other than the potential-outcomes setting. More broadly, we view our model as a starting point for the analysis of strategic information disclosure in environments with data-driven prior-free decision makers.