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Identification Design
\maketitle
\begin{abstract}
This paper develops a model of \textit{identification design} and applies it to robust causal inference in microeconometrics. The decision maker observes the population distribution of signals generated by an information structure and ranks actions by their worst-case payoff over the set of admissible state distributions consistent with those signals. We call an environment \textit{manipulable} if every action is implementable under all true distributions of the state variable, and show this holds if and only if all actions share the same worst-case payoff.
We confirm in application that all treatment-effects models are manipulable, and moreover that manipulation is feasible via \textit{almost fully informative} information structures that conceal at most one dimension of information from the decision maker. As in practice, we consider a restriction to \textit{marginal information structures} that disclose the joint distribution of the outcome variable, treatment variable, and a selection of covariates. In that context, we provide necessary and sufficient conditions for exact identification and sharp payoff bounds for disclosures that do not satisfy those conditions. In doing so, we show that the disclosure of a sufficiently rich set of covariates to verify faithful execution of the assignment mechanism eliminates all scope for manipulation in experiments, while observational studies remain partially manipulable via covariate selection.
\end{abstract}
\medskip
\noindent
\textit{JEL classification}: D81, D83, C14, C21
\medskip
\noindent \textit{Keywords}: identification design, partial identification, prior-free, treatment effects
\section{Introduction}
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 \emph{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{Section: motivating example},
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
\emph{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 \emph{strongly manipulable} via information structures
that (i) are \emph{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 \emph{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 \citeyearpar{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. \citet{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, \citet{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 \citet{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
(\citet{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, \citet{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 (\citet{Rosenthal2026PFB}).
There, we say that information structure $(\Sigma,E)$ is \emph{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.
\citet{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 (\citet{Celen2012MEU,HeyenWiesenfarth2015MEU,LiZhou2016Blackwell}).
Finally, the partial-identification literature in econometrics initiated
in \citet{Manski1990,Manski1997,Manski2003,Manski2007,Manski2013}
and surveyed by \citet{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{Section: motivating example}, lay
out the model in Section \ref{SECTION: MODEL}, and provide our characterization
of manipulable environments in Section \ref{Section: Manip}. We apply
this framework to treatment-effects models in Section \ref{Section: TE};
confirm that those models are manipulable in Section \ref{subsec: universal manip};
analyze marginal information structures in Section \ref{partial manipulability};
and discuss the real-world context for our application in Section
\ref{section: discussion}. Finally, we return to the general model
and provide a characterization of implementability in non-manipulable
environments in Section \ref{sec:Non-manipulable-environments}, discuss
and conclude in Section \ref{section: conclusion}, and apply the
minimax regret decision making criterion to our treatment-effects
model in Appendix \ref{AppendX: regret}. Proofs and omitted supporting
results are organized into Appendices \ref{appendix: manip}--\ref{appendix: non manip}.
\section{\label{Section: motivating example}Motivating Example}
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{t1} and the counterfactual outcomes $(Y_{0},Y_{1})$
are conditionally independent of $T$ given $X$.
\begin{table}[h!]
\centering
\caption{Observed distribution $\mu(Y,X,T)$}
\setlength{\tabcolsep}{10pt}
\label{t1}
\begin{tabular}{c|cccc}
& $(X{=}0,T{=}0)$ & $(X{=}1,T{=}0)$ & $(X{=}0,T{=}1)$ & $(X{=}1,T{=}1)$ \\
\hline
$Y=0$ & 0.40 & 0.05 & 0.10 & 0.30 \\
$Y=1$ & 0.00 & 0.05 & 0.00 & 0.10 \\
\end{tabular}
\end{table}
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
\begin{align}
\mathbb{E}[Y_{0}] & =\sum_{y,t,x}\frac{\mathbf{1}\{t=0\}y}{\mu(t=0\mid X=x)}\mu(y,t,x)=0.25,\label{eq1}\\
\mathbb{E}[Y_{1}] & =\sum_{y,t,x}\frac{\mathbf{1}\{t=1\}y}{\mu(t=1\mid X=x)}\mu(y,t,x)=0.125\label{eq2}
\end{align}
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{t2}.
\begin{table}[h!]
\centering
\caption{Disclosed marginal distribution $\mu(Y,T)$}
\setlength{\tabcolsep}{15pt}
\label{t2}
\begin{tabular}{c|cc}
& $T=0$ & $T=1$ \\
\hline
$Y=0$ & 0.45 & 0.40 \\
$Y=1$ & 0.05 & 0.10 \\
\end{tabular}
\end{table}
As the inverse-probability weighting formulae (\ref{eq1})--(\ref{eq2})
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{t3} is consistent with the disclosed
marginal and yields worst-case payoffs
\begin{align*}
\mathbb{E}_{\nu^ {}}[Y_{0}] & =0.05, & \mathbb{E}_{\nu^ {}}[Y_{1}] & =0.10
\end{align*}
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.
\begin{table}[h!]
\centering
\caption{Worst-case distribution $\nu$ of $(Y, X, T)$ under partial disclosure}
\vspace{0.5em}
\setlength{\tabcolsep}{10pt}
\label{t3}
\begin{tabular}{c|cccc}
& $(X{=}0,T{=}0)$ & $(X{=}1,T{=}0)$ & $(X{=}0,T{=}1)$ & $(X{=}1,T{=}1)$ \\
\hline
$Y=0$ & 0.45 & Negligible & Negligible & 0.40 \\
$Y=1$ & 0.05 & Negligible & Negligible & 0.10 \\
\end{tabular}
\end{table}
\section{\label{SECTION: MODEL}Model}
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 \emph{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 \emph{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 \emph{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 \emph{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 \emph{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 \emph{identified set }of all such distributions. Information
structure $(\Sigma,E)$ is \emph{fully informative }if $\ker E$ has
dimension $0$; \emph{uninformative }if $\ker E$ has dimension $\vert\Omega\vert-1$;
and \emph{almost fully informative} if $\ker E$ has dimension at
most $1$.
\paragraph{The decision maker's problem}
The \emph{decision maker's problem }
\begin{equation}
\max_{\alpha\in\Delta(A)}\;\inf_{\nu\in\mathcal{P_{\mu}}(\Sigma,E)}\;U(\alpha,\nu).\label{DM}
\end{equation}
is to maximize his worst-case payoff over the identified set. We say
that information structure $(\Sigma,E)$ \emph{implements }action
$\alpha$ if $\alpha$ is a solution to problem (\ref{DM}) under
$(\Sigma,E)$, and strictly so if that solution is unique.
\section{\label{Section: Manip}Manipulability}
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.
\begin{prop}
\label{Prop which null}There exists an information structure $(\Sigma,E)$
with $\ker E=D$ if and only if $D$ is a linear subspace of $\mathbb{R}^{\vert\Omega\vert}$
satisfying $\sum_{\omega\in\Omega}d(\omega)=0$ for all directions
$d\in D$.
\end{prop}
Proposition \ref{Prop which null} 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.
\begin{defn}
Environment $(\Omega,\mathcal{P},A,U)$ is \emph{manipulable }if for
all actions $\alpha\in\Delta(A)$ and for all admissible state distributions
$\mu$ there exists an information structure $(\Sigma,E)$ that implements
action $\alpha$ when the true state distribution is $\mu$.
\end{defn}
Manipulability is a strong criterion. As we show, it holds if and
only if the worst-case payoff for every action is the same.
\begin{thm}
\label{thm Manipulability}Environment $(\Omega,\mathcal{P},A,U)$
is manipulable if and only if $\inf_{\nu\in\mathcal{P}}U(\alpha,\nu)=\inf_{\nu\in\mathcal{P}}U(\beta,\nu)$
for all actions $\alpha,\beta$.
\end{thm}
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.
\section{\label{Section: TE}Treatment effects}
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
\begin{align}
A & \equiv\mathcal{T}, & \Omega & \equiv\mathcal{Y}\times\mathcal{X}\times\mathcal{T}, & U(\alpha,\nu)\equiv\int_{A}\mathbb{E}_{\nu}[Y_{a}]\;\mathrm{d}\alpha & =\int_{A}\sum_{y,x}\frac{y}{\nu(a\mid x)}\nu(y,a,x)\;\mathrm{d}\alpha.\label{TE 1}
\end{align}
The assignment mechanism\emph{ }$P(T\mid X)$ belongs to a non-empty
set $\mathscr{P}$ of maps from $\mathcal{X}$ into $\Delta(\mathcal{T})$
satisfying the strict overlap assumption
\begin{equation}
\forall P\in\mathscr{P}\;\forall t\in\mathcal{T}\;\forall x\in\mathcal{X}\;0<P(T=t\mid X=x)<1,\label{TE 2}
\end{equation}
and the set of admissible distributions
\begin{equation}
\mathcal{P}\equiv\bigcup_{P\in\mathscr{P}}\{\nu\in\Delta(\Omega)\mid\forall t\in\mathcal{T}\;\forall x\in\mathcal{X}\;\nu(T=t\mid X=x)=P(T=t\mid X=x)\}\label{TE 3}
\end{equation}
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 \emph{treatment-effects model }if it satisfies (\ref{TE 1})--(\ref{TE 3}),
and highlight the two privileged special cases.
\begin{defn}
Treatment-effects model $(\Omega,\mathcal{P},A,U)$ is an \emph{experiment
}if $\mathscr{P}=\{P\}$.
\end{defn}
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.
\begin{defn}
Treatment-effects model $(\Omega,\mathcal{P},A,U)$ is an \emph{observational
study }if $\mathscr{P}$ is the set of all assignment mechanisms $P$
satisfying (\ref{TE 2}).
\end{defn}
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.
\subsection{\label{subsec: universal manip}Manipulability}
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
\begin{align*}
\nu(y,t,x) & =\Phi(y\mid t,x)P(t\mid x)\lambda(x).
\end{align*}
Direct substitution into (\ref{TE 1}) 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
\begin{equation}
\{(U(a,\nu))_{a\in A}\mid\nu\in\mathcal{P}\}=[\min\mathcal{Y},\max\mathcal{Y}]^{A}.\label{displasy: te cube}
\end{equation}
\begin{prop}
\label{prop: manip}Treatment-effects models $(\Omega,\mathcal{P},A,U)$
are manipulable.
\end{prop}
We prove Proposition \ref{prop: manip} by constructively verifying
that the worst-case payoff for every action $\alpha$ is $\min\mathcal{Y}$
and applying Theorem \ref{thm Manipulability}. 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 \emph{$(\alpha,\mu)$ 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.
\begin{defn}
Manipulable environment $(\Omega,\mathcal{P},A,U)$ is \emph{strongly
manipulable }if
\begin{enumerate}
\item[(i)] for all action--prior pairs $(\alpha,\mu)$ there exists an almost
fully informative information structure $(\Sigma,E)$ that implements
action $\alpha$ when the true state distribution is $\mu$;
\item[(ii)] for all eligible action--prior pairs $(\alpha,\mu)$ there exists
an almost fully informative information structure $(\Sigma,E)$ that
(a) strictly implements action $\alpha$ when the true state distribution
is $\mu$ and (b) satisfies $U(\alpha,\nu)=U(\alpha,\mu)$ for all
admissible priors $\nu$ observationally equivalent to $\mu$.
\end{enumerate}
Treatment-effects models satisfy all of these criteria.
\end{defn}
\begin{prop}
\label{prop: strict manip}Treatment-effects models $(\Omega,\mathcal{P},A,U)$
are strongly manipulable.
\end{prop}
Proposition \ref{prop: strict manip} 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{displasy: te cube}), 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.
\subsection{\label{partial manipulability}Marginal information structures}
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 \emph{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 \emph{disclosed }covariates $Z$, and set $\mathcal{W}\equiv\prod_{j\notin J}\mathcal{X}_{j}$
of \emph{omitted }variables $W$ such that
\begin{align*}
\Sigma^ {} & =\mathcal{Y}\times\mathcal{T}\times\mathcal{Z}, & E^ {}((y,t,z)\mid(y',t',z',w')) & =\mathbf{1}\{(y,t,z)=(y',t',z')\}.
\end{align*}
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.
\begin{example}
\label{Example 1}Let $(\Omega,\mathcal{P},A,U)$ be any treatment-effects
model, $P$ any admissible assignment mechanism, $\lambda$ any distribution
of covariate groups $(z,w)$, and $\Phi:\mathcal{T}\to\mathcal{Y}$
any non-constant degenerate assignment of outcomes to treatments.
Under \emph{any }marginal information structure $(\Sigma,E)$ with
disclosed covariates $Z$ ---\emph{ }including cases in which every
covariate is omitted ---\emph{ }true state distribution
\[
\mu(y,t,z,w)\equiv\mathbf{1}\{y=\Phi(t)\}P(t\mid z,w)\lambda(z,w)
\]
is such that the decision maker's payoffs under every observationally
equivalent prior $\nu$ satisfy $U(\alpha,\nu)=\sum_{a}\Phi(a)\alpha(a)=U(\alpha,\mu)$
for all actions $\alpha$.
\end{example}
In Example \ref{Example 1}, 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.
\begin{defn}
Information structure $(\Sigma,E)$ \emph{identifies }$(\Omega,\mathcal{P},A,U)$
if $U(\alpha,\nu)=U(\alpha,\mu)$ for all actions $\alpha$, all $\nu\in\mathcal{P}_{\mu}(\Sigma,E)$,
and all $\mu\in\mathcal{P}$.
\end{defn}
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.
\begin{thm}
\label{theorem: global identification}Consider treatment-effects
model $(\Omega,\mathcal{P},A,U)$. Marginal information structure
$(\Sigma,E)$ with disclosed covariates $Z$ identifies treatment-effects
model $(\Omega,\mathcal{P},A,U)$ if and only if $P(t\mid z,w)=P(t\mid z)$
for all admissible assignment mechanisms $P$ and all $(t,z,w)$.
\end{thm}
Theorem \ref{theorem: global identification} has implications in
a range of canonical treatment-effects models. As we discuss in Section
\ref{section: discussion}, 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{theorem: global identification}
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{Example 1} 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.
\begin{assumption}
\label{Assumption: richness}(Richness) The set of omitted covariates
$\mathcal{W}$ satisfies $\vert\mathcal{W}\vert\geq\vert\mathcal{Y}\vert^{\vert A\vert}$.
\end{assumption}
Assumption \ref{Assumption: richness} 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.
\begin{thm}
\label{theorem 3 sharp bounds}Consider treatment-effects model $(\Omega,\mathcal{P},A,U)$,
admissible prior $\mu$, marginal information structure $(\Sigma,E)$
with disclosed covariates $Z$, and define
\begin{align}
L(a) & \equiv\sum_{y,z}y\mu(y,a,z)+\sum_{z}\min\{y\mid\mu(y,a,z)>0\}\,\sum_{t\neq a}\sum_{y}\mu(y,t,z),\label{display: L(a)}\\
H(a) & \equiv\sum_{y,z}y\mu(y,a,z)+\sum_{z}\max\{y\mid\mu(y,a,z)>0\}\sum_{t\neq a}\sum_{y}\mu(y,t,z).\label{display: H(A)}
\end{align}
If $L(a)=H(a)$ define $V(a)\equiv\{L(a)\}=\{H(a)\}$; otherwise,
define $V(a)\equiv(L(a),H(a))$.
\begin{enumerate}
\item[(i)] If admissible prior $\nu$ satisfies $\nu(y,t,z)=\mu(y,t,z)$ for
all $(y,t,z)$ then $U(a,\nu)\in V(a)$ for all actions $a\in A$.
\item[(ii)] Conversely, if $(\Omega,\mathcal{P},A,U)$ is an observational study,
Assumption \ref{Assumption: richness} holds, and payoff map $\Phi:A\to\mathbb{R}$
satisfies $\phi(a)\in V(a)$ for all $a\in A$, then there exists
an admissible distribution $\nu$ with $\nu(y,t,z)=\mu(y,t,z)$ for
all $(y,t,z)$ and $U(a,\nu)=\Phi(a)$ for all $a\in A$.
\end{enumerate}
\end{thm}
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{TE 1}).
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.}
\begin{prop}
\label{prop: obs 1}Consider observational study $(\Omega,\mathcal{P},A,U)$,
admissible prior $\mu$, marginal information structure $(\Sigma,E)$
with disclosed covariates $Z$, and suppose Assumption~\ref{Assumption: richness}
holds. Information structure $(\Sigma,E)$ implements action $\alpha$
if and only if the lower bound $L$ defined in (\ref{display: L(a)})
satisfies $L(a)\geq L(b)$ for all actions $a\in A$ in the support
of $\alpha$ and all actions $b\in A$.
\end{prop}
Proposition \ref{prop: obs 1} implies that there exists a\emph{ }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{theorem 3 sharp bounds}. As in
our motivating example in Section \ref{Section: motivating example},
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.
\subsection{\label{section: discussion}Practical context}
The misalignment between researcher incentives and public interest
is an acknowledged phenomenon in the conveyance of scientific evidence,
as discussed in \citet{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. \citet{LenzSahn2021} provide a replication
study in which over 30\% of the articles published in two volumes
of the \emph{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
(\citet{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 (\citet{FDA2023})
requires prospective specification of covariate adjustment procedures
before unblinding of comparative data; earlier guidance from the International
Council for Harmonisation (\citet{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
\emph{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{thm Manipulability}
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 (\citet{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{theorem: global identification} 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{theorem: global identification} 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 \emph{intentional} manipulation. Second,
as the number of disclosed covariates grows, the bounds in Theorem
\ref{theorem 3 sharp bounds} imply that the worst-case (and best-case)
payoffs for each treatment collapse towards their true values.
\section{\label{sec:Non-manipulable-environments}Non-manipulable environments}
In Example \ref{Example 1}, 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{thm Manipulability}
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.
\begin{assumption}
\label{Assumption: regularity}(Regularity) The set of admissible
priors $\mathcal{P}$ is compact and convex, the map $\alpha\mapsto U(\alpha,\nu)$
is quasiconcave for each admissible prior $\nu$, and the map $\nu\mapsto U(\alpha,\nu)$
is quasiconvex for each action $\alpha$.
\end{assumption}
As we show in Lemma \ref{Lemma: maxmin to saddle} in the Appendix,
the technical conditions in Assumption \ref{Assumption: regularity}
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 \emph{saddle point} of that problem
satisfying
\begin{equation}
\forall\alpha'\in\Delta(A)\;\forall\nu'\in\mathcal{P}_{\mu}(\Sigma,E)\;U(\alpha,\nu')\geq U(\alpha,\nu)\geq U(\alpha',\nu).\label{display: saddle}
\end{equation}
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 $\ker E=\text{span}\{\nu-\mu\}$.
\begin{prop}
\label{prop: almost fully informative-1}Suppose Assumption \ref{Assumption: regularity}
holds. There exists an information structure $(\Sigma,E)$ that implements
action $\alpha$ if and only if there exists an almost fully informative
information structure that implements $\alpha$.
\end{prop}
Proposition \ref{prop: almost fully informative-1} 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{display: saddle}).
\begin{assumption}
\label{Assumption: irrel}(Payoff-irrelevant signals) The decision
maker is an expected utility maximizer with Bernoulli utility function
$u:A\times\Omega\to\mathbb{R}$ and for each state $\omega$ there
exists a distinct state $\omega'$ such that
\begin{enumerate}
\item[(i)] $u(a,\omega)=u(a,\omega')$ for all actions $a\in A$; and
\item[(ii)] if $\nu(s)=\nu'(s)$ for all $s\in\Omega\setminus\{\omega,\omega'\}$
then $\nu\in\mathcal{P}$ if and only if $\nu'\in\mathcal{P}$.
\end{enumerate}
\end{assumption}
Assumption \ref{Assumption: irrel} 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.
\begin{thm}
\label{Theorem : main}Consider environment $(\Omega,\mathcal{P},A,U)$.
\begin{enumerate}
\item[(i)] Suppose Assumption \ref{Assumption: regularity} holds. If there
exists an information structure $(\Sigma,E)$ that implements action
$\alpha$ under true state distribution $\mu$ then there exists an
admissible prior $\nu\in\mathcal{P}$ such that $\alpha\in\alpha^{*}(\nu)$
and $U(\alpha,\mu)\geq U(\alpha,\nu)$.
\item[(ii)] Suppose Assumptions \ref{Assumption: regularity}--\ref{Assumption: irrel}
hold. If there exists an admissible prior $\nu\in\mathcal{P}$ such
that $\alpha\in\alpha^{*}(\nu)$ and $U(\alpha,\mu)\geq U(\alpha,\nu)$
then there exists an almost fully informative information structure
$(\Sigma,E)$ that implements action $\alpha$ under true state distribution
$\mu$.
\end{enumerate}
\end{thm}
Provided that Assumption \ref{Assumption: regularity} and Assumption
\ref{Assumption: irrel} 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{Theorem : main}
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 \citet{Kuzmics2017} observes Wald's
complete class theorem implies that a maxmin decision maker who can
randomize over acts always chooses \emph{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{Assumption: regularity}
for the existence of a saddle point in the identified set, and the
substance lies in (a) showing that Assumption \ref{Assumption: irrel}
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 \emph{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.
\section{\label{section: conclusion}Discussion and conclusions}
This paper proposes a model of \emph{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\emph{ 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 \emph{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
(\citet{GilboaSchmeidler1989}) and the standard formulation of its
counterparts, including the smooth ambiguity aversion (\citet{KlibanoffMarinacciMukerji2005})
and variational preferences models (\citet{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{AppendX: regret}, 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.
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