The exact contents of citations.db main_text.text for this paper — one flattened LaTeX string, title through conclusion, appendix excluded, unmodified except for removing email addresses. This is what our citation measures are computed over.
59,180 characters
Geometric Control of Decisions' Affordability
{\maketitle
\vspace{-4.2em}
\thispagestyle{empty}
\begin{abstract} \textbf{Abstract.}
This paper studies the performance of data-driven decisions from a geometric perspective.
A policymaker learns from an innovated donor population to decide whether to innovate groups in a distinct target population, and must compensate for any mistake.
I introduce certification: an estimator yields certified decisions when it controls the probability of a mistake, whenever intervention effects are sufficiently large in magnitude.
First, I show that certification implies a bound on worst-case compensation.
Then, I study matching estimators with positive weights and show that, in a large-sample regime, affordability by certification becomes a purely geometric problem.
I prove that a Delaunay interpolant, whose properties are well-known from results in computational geometry, delivers the best affordability guarantee.
Finally, I show how this result can be leveraged to guide donor-data collection plans to bring worst-case compensation cost below a target level.
I illustrate the gains of adopting this geometric point of view in targeting and collection plans with a semi-synthetic empirical application in development economics. \\
\noindent\textbf{Keywords:} Statistical Decision Theory, Matching Estimators, Delaunay Triangulation. \vspace{8em}
\end{abstract}}
\maketitle
\section{Introduction}
Matching estimators are often valued for their flexibility and for the relatively weak structure they impose when transporting information across units or populations \citep[see, e.g.][]{abadtie_imbens_2006,abadie_2021_rev}.
Yet, that same flexibility can leave the choice of weights ambiguous, especially when several donor combinations fit a target equally well \citep[see][]{Abadie2021}.
This paper studies that longstanding tension from a new (statistical) decision theoretic angle.
I consider a policy choice problem in which decisions must be learned from an innovated donor population and applied to a distinct target population, and I ask which matching weights are justified by that objective.
In this setting, a criterion based on worst-case loss disciplines the choice of weights and, in the large-sample limit, reduces it to a purely geometric problem.
Not only this perspective allows for analytical guarantees on such weights choice, but it also provides guidance for defining data collection plans that are empirically shown to outperform random sampling in reducing worst-case loss.
Consider two counterfactual states: the status quo and one innovation.
A policymaker (PM) decides whether to innovate groups in a target population.
She bases this decision on a sample drawn from an innovated donor population.
The donor and target populations do not coincide.
She is accountable for her actions: any wrong decision must be compensated.
Decisions are made by a threshold rule that takes the value one if the estimated intervention effect for units in the target is strictly positive.
She has a finite compensation budget and she searches for an estimator of the intervention effect that guarantees an affordable worst-case compensation.
The PM needs to commit to a choice of estimator before the data realize.
Once the target population realizes, decisions are made and the compensation cost must be paid out.
The guarantees the PM is looking for need to hold uniformly over the possible target populations, but pointwise over the estimating-data population.
I introduce certification as a means to provide affordability conditions.
An estimator produces certified decisions if the probability of making a mistake is uniformly controlled, provided that the intervention's effects are \textit{large enough} in absolute value.
As a first result, I show that the bounds on the effects' magnitude and on the probability of making mistakes provide an affordability constraint.
Then, I study a sufficient condition for an estimator to provide certified decisions in general, and specialize this condition to the set of linearly precise matching estimators with positive weights.
I show that, for this class of estimators, in finite samples, we can derive affordability constraints that depend on (i) a geometric feature of the donor data, and (ii) on individual heterogeneity in treatment effects.
Moreover, I show that, if we let the size of the donor data grow and we observe the full target population, the stochastic components vanish and certification reduces to a purely geometric problem.
As a first core contribution, I connect this asymptotic geometric problem with well known results in computational geometry \citep{Delaunay_1934aa}.
I define the Delaunay Matching Estimator by barycentric interpolation on the Delaunay triangulation of the donor covariates.
I then show that, within the class of matching estimators with positive weights, Delaunay Matching solves optimally the certification geometric problem at every feasible target point.
This result is proven via a standard lifting argument.
As a consequence, Delaunay Matching also delivers the best worst-case asymptotic affordability guarantee.
As a second core contribution, I study how new donor data should be collected when the policymaker aims to bring worst-case compensation below a target level.
Although I do not characterize the exact minimax collection rule, I use the optimality of Delaunay Matching together with geometric results in \citet{waldron} to motivate a new finite-sample collection rule, which I call the Geometric Plan.
The rule directs new donor observations toward regions of the donor covariate support where the current geometric approximation is weakest.
This refinement is shown empirically to deliver sizeable gains relative to random collection.
I illustrate the applicability of Delaunay Matching with a semi-synthetic application in development economics built on the NREGS Smartcards experiment of \citet{muralidharan_building_2016}.
In the original experiment, the government randomized the rollout of biometric Smartcards to deliver cash transfers across $296$ mandals in rural Andhra Pradesh.
The study then surveyed households in $880$ Gram Panchayats (GPs) to measure how the new payment system affected leakage and service delivery.
In the targeting exercise, control GPs define the target population, treated GPs define the donor population, and the policymaker decides cell by cell whether to deploy Smartcards based on two baseline covariates, innovating whenever the estimated gain from lower leakage is positive.
Cells are defined over quintiles of GPs' baseline log annual consumption and NREGS exposure.
The exercise keeps the real GP-level donor and target covariate spaces, but imposes a fictitious treatment-effect frontier.
Delaunay Matching takes a decision for $21$ out of $25$ target cells, certifies $16$ cells asymptotically and $14$ in finite samples, and no certified cell receives the wrong decision.
I then pretend a policymaker wants to reduce the current worst-case compensation cost by half collecting new donor data points.
The Geometric Plan achieves the target level with only three additional donor points, whereas a random collection plan fails to reach it within twenty steps.
This finding suggests sizeable gains in adopting a geometric perspective in designing cost-efficient collection plans for policy choice problems.
\paragraph{Related Literature.}
First, this study contributes to a growing literature that applies statistical decision theory to the problem of targeting interventions, highlighting the potential of adopting a geometric perspective to control worst-case loss.
There are three main differences with standard approaches \citep[e.g.][]{manski_statistical_2004,stoye_minimax_2009, kitagawa_who_2018,athey_policy_2021,mbakop_model_2021}.
First, the target and donor populations do not coincide.
Second, the PM is only worst-case against the target population, and pointwise on the donor population.
These two departures are also considered in papers that study problems of partial identification in treatment choice \citep{stoye_covariates_2012, kido_distributionally_2022, adjaho_external_2022, yata_2025, christensen_payoffs_2025, Olea_2026}.
\cite{stoye_covariates_2012}, \cite{yata_2025}, and \cite{Olea_2026} study how the length of the identified set affects how data-dependent policy recommendations should be and show that, when the sets are large, it may be optimal to adopt fractional or even no-data rules.
\cite{yata_2025} and \cite{adjaho_external_2022} use different measures of Wasserstein distance between the estimating and target populations to study how optimal recommendations would change along that metric.
None of these papers leverage the geometry of the estimating data to control the worst-case loss.
Moreover, all of these papers study the performance of decisions rules for a fixed estimator, while this paper does the opposite.
\\
Third, the assignment mechanism of the intervention is deterministic: all individuals in the donor population have received it and all in the target population have not.
Therefore, unconfoundedness (i.e. the assignment of the intervention in the estimating data being conditionally independent to potential outcomes) and strict overlap (i.e. each unit having a strictly positive probability of receiving the intervention or staying in the status-quo) do not hold.\footnote{For a reference of these assumptions see e.g. Section 3.1 \cite{manski_statistical_2004}, Ass. 2.1 \cite{kitagawa_who_2018}, Ass. 2.1, 3.1 \cite{mbakop_model_2021}.}
Relaxing the constraints on the assignment mechanism, however, comes with two costs.
First, I impose a partially linear potential outcomes equation that sums an unknown twice-differentiable function of covariates (henceforth causal law) and a random individual component.
Both the function and the individual component are potential, and therefore allowed to vary across counterfactual states.
Second, while the distribution of covariates and of the individual component is allowed to vary across the target and donor populations, the causal law is fixed between the two and the individual component is assumed to have conditional mean zero.
This study also contributes to the literature on matching and synthetic control estimators.
Synthetic control methods are usually studied in panel data settings where each treated unit can be approximated by a convex combination of donor units with nonnegative weights that sum to one, a feature that yields sparse and interpretable counterfactual comparisons \citep[e.g.][among others]{Abadie01062010,Ben-Michael02102021, abadie_2021_rev, synth_did_2021, Chernozhukov02102021, Kellogg02102021}.
The interest in the geometric properties of such estimators draws from results in \citet{Abadie2021}, who study synthetic controls for disaggregated data and show that, when treated units lie in the convex hull of the donor pool, the synthetic control problem may admit multiple solutions. Their penalized estimator restores uniqueness and sparsity, and they provide a Delaunay characterization of the donor units that receive positive weight.
I build on this geometric insight to study the decision theoretic properties of a similar class of estimators.
To the best of my knowledge, this paper is the first to derive optimal weights for a matching estimator that solves a policy choice decision problem.
Overall, this paper contributes to the broader literatures on statistical decision theory and causal inference highlighting the value of studying the performance of data-driven decisions from a geometric perspective.
The rest of the paper is organized as follows.
Section \ref{sec:PM_obj} describes the decision problem and lays out the main assumptions.
Section \ref{sec:one_solution} solves the decision problem in general and then specializes the result to matching estimators.
Section \ref{sec:delaunay} introduces Delaunay Matching and provides the optimality result.
Section \ref{sec:data_collection} describes how to leverage Delaunay Mathcing's optimality to define geometric data collection plans.
Section \ref{sec:semi_synth} illustrates the semi-synthetic empirical application.
Section \ref{sec:conclusions} concludes.
\section{Formal Description of the PM's Objective}\label{sec:PM_obj}
Denote by $(Y_i(0),Y_i(1))$ the random potential outcomes of unit $i$ under the status quo and the innovation. Let
\begin{equation}
\{(X_i,U_i(0),U_i(1))\}_{i=1}^N\sim_{\mathrm{i.i.d.}} P^N,
\end{equation}
with $P\in\mathcal P$. The PM observes the donor sample
\begin{equation}
S^N:=\{(X_i,Y_i(1))\}_{i=1}^N,
\end{equation}
with $(x,y)\in\mathcal X\times\mathcal Y$.
\begin{assumption}[Distributions Family]\label{ass:distr}
Define $\mathcal{P}=\{P\ \text{over}\ \mathcal{X}\times \mathcal{U}^2: \mathbb{E}_{P}[U_i(d)|X_i]=0,\ |U_i(d)|\leq \bar{U}_d< \infty,\ \text{for}\ d \in \{0,1\}\}$.
\end{assumption}
\begin{assumption}[Outcome Function]\label{ass:model}
Assume the potential outcomes functions have the following form:
\begin{align}
Y_i(D_i) & = f_{D_i}(X_i) + U_i(D_i)
\end{align}
where $f_{d}$ belongs to the admissible functions set $\mathcal{F}_c$ defined as:
\begin{equation}
f_0,f_1 \in \mathcal{F}_c:=\left\{f: \mathcal{X} \rightarrow \mathcal{Y}: \sup_{x\in\mathcal{X}}||\nabla^2 f(x) || \leq c<\infty\right\}
\end{equation}
for any $x\in\mathcal{X}$, and $|Y_i(1)- Y_i(0)|\leq \bar{\tau}$.
\end{assumption}
Define the individual treatment effect and conditional average treatment effect for a target unit $j$ as:
\begin{equation}
\tau_j:=Y_j(1)-Y_j(0),
\qquad
\tau(x):=\mathbb E_Q[\tau_j\mid X_j=x].
\end{equation}
\begin{definition}[Sample-specific treatment effect estimator]
Let $\mathcal M$ denote a class of estimators of $\tau_j$. For donor sample $S^N$, define
\begin{equation}
\hat\tau_j=m(X_j,Y_j(0),S^N).
\end{equation}
\end{definition}
\begin{remark}
The PM commits to a specific estimator $m(\cdot)$ to be applied to the donor sample for any draw $x$.
This condition can be rationalized by delegating the choice of $m(\cdot)$ to another agent, say the researcher, who does not observe the target sample.
We adopt the single-agent framework for expositional convenience.
\end{remark}
After the choice of $m$ is made,
\begin{equation}
\{(X_j,U_j(0),U_j(1))\}_{j=1}^J \sim Q^J,
\end{equation}
where $Q\in\mathcal P$. From this draw, the PM observes:
\begin{equation}
\{(X_j,Y_j(0))\}_{j=1}^J.
\end{equation}
The PM uses the donor sample to decide whether to innovate group $x$. Let
\begin{equation}
N_x:=\sum_{j=1}^J \mathbf 1\{X_j=x\}.
\end{equation}
For any $x$ such that $N_x\ge 1$, define the empirical treatment effect estimate
\begin{equation}\label{eq:tauhat_x_def}
\hat\tau_m(x)
:=
\frac{1}{N_x}
\sum_{j=1}^J
m(X_j,Y_j(0),S^N)\mathbf 1\{X_j=x\},
\end{equation}
and the empirical decision rule
\begin{equation}
\hat{d}_m(x):=\mathbf 1\{\hat\tau_m(x)\ge 0\}.
\end{equation}
Define the oracle decision rule as
\begin{equation} \label{eq:oracle}
d^*(x):=\mathbf 1\{\tau(x)\ge 0\}.
\end{equation}
\begin{definition}[PM's Objective] \label{def:affordable_decisions}
Assume the PM needs to compensate for any wrong decision. In particular, define the compensation loss at group $x$ as:
\begin{equation}\label{eq:loss}
\ell(x):=|\tau(x)| \cdot \mathbf 1\{d^*(x)\neq \hat{d}_m(x)\}.
\end{equation}
Given a compensation budget $B_0$, the PM aims at setting ex-ante an estimator $m(\cdot)$ such that:
\begin{equation}
\sup_{Q\in \mathcal{P}, f_0,f_1 \in \mathcal{F}_c} \mathbb{E}_{Q^J \times P^N}[\ell(X_j)] \leq B_0.
\end{equation}
That is, the PM wants to commit to an estimator that provides affordable decisions.
\end{definition}
\begin{remark}
Notice that the loss function defined in \eqref{eq:loss} is a group-level analogue of regret. Indeed,
\begin{align}
R(m) & :=\mathbb{E}_{Q^J \times P^N}[Y_j(d^*(X_j))] - \mathbb{E}_{Q^J \times P^N}[Y_j(\hat{d}_m(X_j))] \\
& =
\mathbb{E}_{Q^J \times P^N}[\tau_j \cdot (d^*(X_j)-\hat{d}_m(X_j))] \\
& = \mathbb{E}_{Q^J \times P^N}[\tau_j \cdot \mathbf{1}\{d^*(X_j)\neq\hat{d}_m(X_j)\}] \\
& = \mathbb{E}_{Q^J \times P^N}[\tau(X_j) \cdot \mathbf{1}\{d^*(X_j)\neq\hat{d}_m(X_j)\}]
\end{align}
The criterion $\ell(X_j)=|\tau(X_j)|\mathbf 1\{d^*(X_j)\neq\hat d_m(X_j)\}$ replaces the realized individual effect $\tau_j$ by its conditional mean magnitude $|\tau(X_j)|$.
\end{remark}
Solving the problem defined in Def. \ref{def:affordable_decisions} is challenging because the population of the estimating and the target samples do not coincide, and the assignment mechanism is deterministic.
As a result, the assumptions of unconfoundedness and strict overlap, which are common in the policy learning literature \citep[see, e.g. ][]{manski_statistical_2004,kitagawa_who_2018,athey_policy_2021,mbakop_model_2021}, fail.
I introduce certified decisions as an alternative building block for solving the PM's objective in this setting.
\begin{definition}[Certified decision]\label{def:PM_objective}
Fix a target error rate $\alpha\in(0,1)$. An estimator $m \in \mathcal M$ provides a certified decision if there exists a set $\mathcal C_\alpha(m):=\{x: |\tau(x)|> \gamma_\alpha(m)\}$ satisfying
\begin{equation} \label{eq:PM_objective}
\sup_{Q\in \mathcal{P}}\sup_{f_0,f_1\in\mathcal F_c} \mathbb P_{Q^J\times P^N} \bigl(d^*(X_j)\neq \hat{d}_m(X_j) \mid X_j \in \mathcal{C}_\alpha(m) \bigr)
\le \alpha.
\end{equation}
\end{definition}
\begin{figure}
\centering
\caption{Certified decisions: graphical illustration.}
\scalebox{1.15}{
\begin{tikzpicture}
\begin{axis}[
width=10.5cm, height=5cm,
axis lines=center,
axis line style={->,thick},
xlabel={$x$}, ylabel={$\tau(x)$},
xmin=0, xmax=10.5,
ymin=-2.0, ymax=2.0,
xtick=\empty, ytick={\empty},
clip=false,
]
\fill[orange!15] (axis cs:0,-0.6) rectangle (axis cs:10,0.6);
\draw[orange!65,dashed,thick]
(axis cs:0, 0.6)--(axis cs:10.15, 0.6)
node[right,font=\small,orange!80!black] {$\gamma_\alpha(m)$};
\draw[orange!65,dashed,thick]
(axis cs:0,-0.6)--(axis cs:10.15,-0.6)
node[right,font=\small,orange!80!black] {$-\gamma_\alpha(m)$};
\draw[gray!45,dashed] (axis cs:0,0)--(axis cs:10,0);
\node[orange!75!black,font=\small,align=center]
at (axis cs:2.5,0) {not\\certified};
\addplot[RoyalBlue,very thick,domain=0:10,
restrict y to domain=-10:-0.6,samples=300]
{1.5*sin(deg(0.9*x - 1.0))};
\addplot[RoyalBlue,very thick,domain=0:10,
restrict y to domain=0.6:10,samples=300]
{1.5*sin(deg(0.9*x - 1.0))};
\addplot[RoyalBlue,very thick,dashed,domain=0:10,
restrict y to domain=-0.6:0.6,samples=300]
{1.5*sin(deg(0.9*x - 1.0))};
\draw[<->,green!55!black,thick]
(axis cs:3.5,0.6)--(axis cs:3.5,1.85);
\node[green!55!black,font=\small,right]
at (axis cs:3.55,1.22) {$\mathcal{C}_\alpha(m)$};
\draw[<->,green!55!black,thick]
(axis cs:0.25,-0.6)--(axis cs:0.25,-1.85);
\node[green!55!black,font=\small,right]
at (axis cs:0.3,-1.22) {$\mathcal{C}_\alpha(m)$};
\node[black,font=\small,left]
at (axis cs:0,0) {0};
\end{axis}
\end{tikzpicture}}
\label{fig:certification}
\smallskip
\begin{minipage}{0.88\linewidth}
\footnotesize\textit{\textbf{Notes:}} The blue curve depicts a treatment effect function $\tau(x)$.
The orange shaded band is the non-certified region $\{x:|\tau(x)|\le\gamma_\alpha(m)\}$.
Solid blue segments outside the band form the certified set $\mathcal{C}_\alpha(m)$, marked by the green brackets.
The dashed blue segments correspond to portions of the curve that fall inside the non-certified band.
\end{minipage}
\end{figure}
Figure \ref{fig:certification} illustrates the definition.
The treatment effect function $\tau(x)$ (in blue) crosses the threshold $\gamma_\alpha(m)$ at several points.
The certified set $\mathcal{C}_\alpha(m)$ consists of those $x$ where $|\tau(x)|$ exceeds $\gamma_\alpha(m)$. Conditional on the event that the realized target draw falls in this set, the probability of making a mistake with $\hat{d}_m(X_j)$ is controlled at $\alpha$.
At points inside the orange band, $|\tau(x)|\le\gamma_\alpha(m)$, and no such certification guarantee is imposed.
The next Lemma notes that certification's parameters $\gamma_\alpha(m)$ and $\alpha$ imply an affordability constraint.
\begin{lemma}[Affordability of certified decisions] \label{lem:compensation}
Under Assumption \ref{ass:model}, and for a choice of $m(\cdot)$ that satisfies \eqref{eq:PM_objective},
\begin{equation}
\sup_{Q \in \mathcal{P}}\mathbb E_{Q^J \times P^N}[\ell(X_j)] \leq \max\{\gamma_\alpha(m),\alpha \bar\tau\}.
\end{equation}
As a consequence, $m(\cdot)$ achieves the PM's objective if:
\begin{equation}
\max\{\gamma_\alpha(m), \alpha \bar{\tau}\} \leq B_0.
\end{equation}
\end{lemma}
\hyperref[proof:lem:compensation]{The formal proof is in Appendix \ref{app:formal_proofs}.}
Lemma \ref{lem:compensation} shows that the worst-case compensation cost is bounded above by the maximum between $\gamma_\alpha(m)$ and $\alpha\bar{\tau}$.
The bound is sharp under either of the following sufficient conditions. First, there exists $(Q^\star,x^\star)\in \mathcal P \times \mathcal C_\alpha(m)$ such that $Q_X^\star=\delta_{x^\star}$, $|\tau(x^\star)|=\bar\tau$, and
\begin{equation}
\mathbb P_{(Q^\star)^J \times P^N}\bigl(d^*(X_j)\neq \hat d_m(X_j)\mid X_j\in \mathcal C_\alpha(m)\bigr)=\alpha.
\end{equation}
Second, there exists $(Q^\dagger,x^\dagger)\in \mathcal P \times \mathcal C_\alpha(m)^c$ such that $Q_X^\dagger=\delta_{x^\dagger}$, $|\tau(x^\dagger)|=\gamma_\alpha(m)$, and
\begin{equation}
\mathbb P_{(Q^\dagger)^J \times P^N}\bigl(d^*(X_j)\neq \hat d_m(X_j)\mid X_j\notin \mathcal C_\alpha(m)\bigr)=1.
\end{equation}
Under either condition, the upper bound in Lemma \ref{lem:compensation} is attained with equality.
The intuition is that, because the PM commits to a choice of $m$ \textit{before} $Q$ realizes and $\mathcal P$ does not constrain the target distribution, an adversary may concentrate all probability mass on the covariate value at which the corresponding local upper bound is attained.
\section{One Solution to the Decision Problem} \label{sec:one_solution}
This section proceeds in three steps.
I first state a general sufficient condition for certification, then I introduce a class of estimators $\mathcal M_w$ that make this condition operational in finite samples, and finally I characterize the asymptotic certification conditions and its affordability implication.
The following Lemma introduces a sufficient condition for a general estimator $m(\cdot)$ to provide certified decisions.
\begin{lemma}[Sufficient condition for certification]\label{lem:suff_cert}
Let $\mathcal C_\alpha(m):=\{x\in\mathcal X:|\tau(x)|>\gamma_\alpha(m)\}$.
Any estimator $m \in \mathcal{M}$ that satisfies
\begin{equation}\label{eq:suff_cert_error_bound}
\sup_{Q \in \mathcal{P}}\sup_{f_0,f_1 \in \mathcal{F}_c}
\mathbb{P}_{Q^J \times P^N}\left(|\hat{\tau}_m(X_j)-\tau(X_j)|\geq \gamma_\alpha(m) \mid X_j \in \mathcal{C}_\alpha(m)\right)\leq \alpha,
\end{equation}
also satisfies
\begin{equation}
\sup_{Q \in \mathcal{P}}\sup_{f_0,f_1 \in \mathcal{F}_c}
\mathbb{P}_{Q^J \times P^N}(d^*(X_j)\neq \hat d_m(X_j) \mid X_j \in \mathcal{C}_\alpha(m))\leq \alpha.
\end{equation}
\end{lemma}
\hyperref[proof:lem:suff_cert]{The formal proof is in Appendix \ref{app:formal_proofs}.}
Lemma \ref{lem:suff_cert} states that, if the probability of the absolute estimation error being larger than the certification boundary $\gamma_\alpha(m)$ is controlled by $\alpha$ conditional on the realized target draw falling in $\mathcal C_\alpha(m)$, then the decision estimated through $m$ is certified.
\begin{remark}
The sufficient condition in Lemma \ref{lem:suff_cert} is conceptually simple but, in general, difficult to verify. It requires a uniform high-probability bound on the unobservable estimation error $|\hat{\tau}_m(x)-\tau(x)|$.
For a generic estimator $m(\cdot)$, such a bound is not available in finite samples without additional structure linking the estimator to observable quantities and to the smoothness restrictions imposed by $\mathcal F_c$.
\end{remark}
This technical challenge motivates the class $\mathcal M_w$ defined below, which is designed precisely to make this condition operational.
For estimators in $\mathcal M_w$, the estimation error admits a decomposition into a geometric approximation term and two bounded stochastic terms.
Each component can be controlled uniformly under Assumptions \ref{ass:distr} and \ref{ass:model}, yielding an explicit and computable certification and affordability condition.
\begin{definition}[Matching estimators with positive weights]\label{def:matching_positive}
Fix a block size $n$. Let $\mathcal M_w\subseteq\mathcal M$ denote the class of block-level estimators defined by
\begin{equation}
\hat\tau_{j,w}
=
m_w(X_j,Y_j(0),S^n)
=
\sum_{i=1}^n w(X_j,X_i)Y_i(1)-Y_j(0),
\end{equation}
where, for each $x\in\mathcal X$, the weights satisfy
\begin{equation}
w(x,X_i)\ge 0,
\qquad
\sum_{i=1}^n w(x,X_i)=1,
\qquad
\sum_{i=1}^n w(x,X_i)X_i=x.
\end{equation}
\end{definition}
\begin{remark}[Domain of $\mathcal M_w$]\label{rem:domain_mw}
The constraints in Definition \ref{def:matching_positive} imply that positive affine-exact weights exist only if
\begin{equation}
x\in \mathrm{conv}(\{X_i\}_{i=1}^n).
\end{equation}
Accordingly, a block-level estimator $m_w\in\mathcal M_w$ is well defined only on the convex hull of the donor block. In the finite-sample construction based on a partition of the full donor sample into $K_N$ blocks, the aggregate estimator $\hat\tau_w(x)$ is therefore defined on the random feasible set
\begin{equation}
\mathcal X:=\bigcap_{k=1}^{K_N} \mathrm{conv}(\{X_{i,k}\}_{i=1}^n).
\end{equation}
All subsequent statements for $\mathcal M_w$ and for Delaunay matching are understood on this feasible region.
\end{remark}
To obtain explicit finite-sample certification bounds, I now return to the full donor sample $S^N$ and treat the block construction as part of the estimator rather than as part of the data-generating process. Suppose that $N$ is a multiple of $n$, and partition the donor sample into $K_N:=N/n$ mutually exclusive blocks of size $n$:
\begin{equation}
S_{k,N}^n:=\{(X_{i,k},Y_{i,k}(1))\}_{i=1}^n,
\qquad k=1,\dots,K_N.
\end{equation}
Because the original donor observations are i.i.d. under $P^N$, the resulting blocks are mutually independent and each has law $P^n$. Let
\begin{equation}
\mathbf X_N^n:=\{X_{i,k}\}_{i\le n,\ k\le K_N}.
\end{equation}
For each donor block $k$, let
\begin{equation}
\hat\tau_{j,k,w}:=m_w(X_j,Y_j(0),S_{k,N}^n).
\end{equation}
For $m_w\in\mathcal M_w$, define the aggregate estimator
\begin{equation}
\hat\tau_w(x)
:=
\frac{1}{K_N N_x}
\sum_{j=1}^J\sum_{k=1}^{K_N}
\hat\tau_{j,k,w}\mathbf 1\{X_j=x\},
\qquad N_x\ge 1.
\end{equation}
Equivalently,
\begin{equation}\label{eq:tauhat_w_simplified}
\hat\tau_w(x)
=
\frac{1}{K_N}\sum_{k=1}^{K_N}\sum_{i=1}^n \hat w_{i,k}(x)Y_{i,k}(1)
-
\frac{1}{N_x}\sum_{j:X_j=x}Y_j(0),
\end{equation}
where $\hat w_{i,k}(x):=w(x,X_{i,k})$.
The following Theorem leverages the property of the class $\mathcal{M}_w$ to derive finite-sample certification conditions.
\begin{theorem}[Finite-sample certified decisions]\label{thm:finite_dec}
Under Assumptions \ref{ass:distr} and \ref{ass:model}, for any $m_w\in\mathcal M_w$, and any $\alpha\in(0,1)$, define
\begin{equation}
\mathcal{C}_\alpha(m_w):= \left\{x :|\tau(x)|>\sup_{x\in \mathcal{X}}r_\alpha(x,m_w)\right\},
\end{equation}
then,
\begin{equation}
\sup_{Q \in \mathcal{P}, f_0, f_1 \in \mathcal{F}_c}\mathbb P_{Q^J\times P^N}\bigl(\hat{d}_m(X_j)\neq d^*(X_j) \mid X_j \in \mathcal{C}_{\alpha}(m_w),\mathbf X_N^n \bigr)\le \alpha,
\end{equation}
almost surely,
where,
\begin{equation}\label{eq:main_radius_bound}
r_\alpha(x,m_w)
=
\frac{c}{2K_N}\sum_{k=1}^{K_N} \sum_{i=1}^n \hat w_{i,k}(x)\|X_{i,k}-x\|^2
+
\frac{\bar U_1}{K_N}\sqrt{2\log(4/\alpha)\sum_{k=1}^{K_N}\sum_{i=1}^n \hat w_{i,k}(x)^2}
+
\bar U_0\sqrt{2\log(4/\alpha)}.
\end{equation}
\end{theorem}
\hyperref[proof:thm:finite_dec]{The formal proof is in Appendix \ref{app:formal_proofs}.} Theorem \ref{thm:finite_dec} delivers an ex-ante certification guarantee, conditional on the donor covariates, based on the scalar boundary $\sup_{x\in\mathcal X} r_\alpha(x,m_w)$.
The radius $r_\alpha(x,m_w)$ itself is composed of three terms.
First, a geometric term that measures the weighted distance between the target point and the donor points, averaged over the $K_N$ donor blocks.
Second, an idiosyncratic term due to the individual component in Ass. \ref{ass:model} of donor units, which is bounded using a standard concentration inequality \citep[see][]{hoeffding_probability_1963}.
Third, a second idiosyncratic term due to the individual component of target units, bounded at its worst-case ex-ante value corresponding to the smallest feasible count $N_x=1$.
The following corollary turns the certification guarantee of Theorem \ref{thm:finite_dec} into an affordability constraint.
\begin{corollary}[Worst-case compensation cost from finite sample decisions]\label{cor:aff_finite}
By Theorem \ref{thm:finite_dec} and Lemma \ref{lem:compensation}, applied conditionally on $\mathbf X_N^n$,
\begin{equation}
\sup_{Q \in \mathcal{P}, f_0,f_1 \in \mathcal{F}_c} \mathbb{E}_{Q^J \times P^N}[\ell(X_j)\mid\mathbf X_N^n] \leq \max\left\{\sup_{x\in\mathcal{X}} r_\alpha(x,m_w), \alpha \bar{\tau}\right\}.
\end{equation}
almost surely.
\end{corollary}
Because the policymaker commits to $m_w$ after observing the donor design but before the target distribution $Q$ realizes, a least-favorable $Q_X$ concentrates its mass on the covariate values where the donor-design-conditional radius $r_\alpha(x,m_w)$ is largest.
Hence, conditional on the donor design, the worst-case expected compensation is governed by the maximum between the supremum over $x$ of the certification boundary and $\alpha\bar{\tau}$.
The finite-sample radius in Theorem \ref{thm:finite_dec} isolates one geometric component and two stochastic components.
In the ex-ante finite guarantee, the target-side term is evaluated at its worst-case value. In the large-sample regime, the underlying stochastic terms vanish as $N$ and $N_x$ grow, so the certification problem becomes purely geometric.
The next theorem formalizes this asymptotic certification result and defines the limiting certification radius $r_\infty(x,m_w)$.
\begin{theorem}[Asymptotic certified decisions]\label{thm:asymp_dec}
Under Assumptions \ref{ass:distr} and \ref{ass:model}, let the donor sample size satisfy $N\to\infty$ with fixed block size $n$, so that $K_N=N/n\to\infty$, and suppose also that $N_x \to\infty$. Define
\begin{equation}
r_\infty(x,m_w):=
\frac{c}{2}\mathbb E_{P^n}\!\left[\sum_{i=1}^n \hat w_i(x)\|X_i-x\|^2\right].
\end{equation}
Then, for
\begin{equation}
\mathcal{C}(m):= \left\{x \in \mathcal{X}:|\tau(x)|> \sup_{x\in\mathcal{X}} r_\infty(x,m_w)\right\},
\end{equation}
and every $Q\in\mathcal P$ and $f_0,f_1\in\mathcal F_c$, it follows that
\begin{equation}
\mathbb P_{Q^J\times P^N}\bigl(\hat d_m(X_j)\neq d^*(X_j) \mid X_j \in \mathcal{C}(m)\bigr)\to 0.
\end{equation}
\end{theorem}
\hyperref[proof:thm:asymp_dec]{The formal proof is in Appendix \ref{app:formal_proofs}.}
Theorem \ref{thm:asymp_dec} shows that, if we observed a large donor sample and partitioned it into infinitely many mutually exclusive blocks of fixed size $n$, and if we could perfectly estimate the conditional expectation in the status quo in the target population, certification would depend only on how well the donor covariates geometrically span the target point $x$.
\begin{remark}[Fixed target groups]\label{rem:fixed_groups}
An alternative interpretation of the condition $N_x\to\infty$ is that the policymaker partitions the target population ex ante through a measurable map $\pi:\mathcal X\to\mathcal G$ and takes decisions at the group level, while still forming the matching prediction pointwise in $x$. Then the relevant replication condition is $N_g\to\infty$ for each group $g\in\mathcal G$, which makes the target-side control noise vanish within each group. This modification does not change the worst-case geometric criterion. Indeed, the asymptotic certification boundary for a group $g$ is
\begin{equation}
\sup_{x\in \pi^{-1}(g)} r_\infty(x,m_w),
\end{equation}
so that
\begin{equation}
\sup_{g\in\mathcal G}\sup_{x\in \pi^{-1}(g)} r_\infty(x,m_w)
=
\sup_{x\in\mathcal X} r_\infty(x,m_w).
\end{equation}
Because the grouping rule is fixed before the target law realizes, a least-favorable $Q$ can still concentrate its mass on covariate values with the worst geometry. Appendix \ref{app:formal_proofs} gives a formal proof of this observation.
\end{remark}
The following corollary translates this limiting certification result into an affordability bound.
As in Corollary \ref{cor:aff_finite}, because the policymaker commits to $m_w$ after observing the donor design but before the target distribution realizes, a least-favorable $Q_X$ concentrates its mass on the covariate values where the asymptotic certification radius is largest.
Unlike Corollary \ref{cor:aff_finite}, there is no maximum with $\alpha \bar{\tau}$ because the certification error probability vanishes asymptotically.
\begin{corollary}[Worst-case compensation cost from large sample decisions]\label{cor:aff_asympt}
By Theorem \ref{thm:asymp_dec} and the same decomposition used in Lemma \ref{lem:compensation},
\begin{equation}
\sup_{Q \in \mathcal{P}, f_0,f_1 \in \mathcal{F}_c} \limsup_{N, N_x \to \infty} \mathbb{E}_{Q^J \times P^N}[\ell(X_j)] \leq \sup_{x\in\mathcal{X}} r_\infty(x,m_w).
\end{equation}
\end{corollary}
\section{Delaunay Matching Optimality}\label{sec:delaunay}
In Theorem \ref{thm:asymp_dec} and Corollary \ref{cor:aff_asympt}, the estimator enters only through the geometric term
\begin{equation}
\sum_{i=1}^n \hat w_i(x)\|X_i-x\|^2.
\end{equation}
This section identifies the $m_w \in \mathcal{M}_w$ that minimizes that quantity.
\begin{definition}[Delaunay triangulation]\label{def:delaunay_triangulation}
For a donor block $k$, a triangulation $T_k$ of $\mathrm{conv}(\{X_{i,k}\}_{i=1}^n)$ is a Delaunay triangulation if every simplex $\sigma\in T_k$ has a circumsphere whose interior contains no donor location $X_{i,k}$. In two dimensions, circumspheres reduce to circumcircles, yielding the empty circumcircle property illustrated in Figure \ref{fig:delaunay_empty_circumcircle}.
\end{definition}
\begin{figure}[h]
\centering
\caption{Delaunay Triangulation}
\label{fig:delaunay_empty_circumcircle}
\includegraphics[width=0.62\linewidth]{Figures/delaunay_empty_circumcircle.pdf}
\smallskip
\begin{minipage}{0.88\linewidth}
\footnotesize\textit{\textbf{Notes:}} The gray edges depict a Delaunay triangulation of donor covariate locations in $\mathbb R^2$. For the highlighted simplex $\sigma$, the dashed circumcircle has center $c$ and contains no donor point in its interior. This is the two-dimensional version of Definition \ref{def:delaunay_triangulation}.
\end{minipage}
\end{figure}
Throughout, donor covariate locations are assumed distinct and in general position, so the Delaunay triangulation is simplicial \citep{Delaunay_1934aa}.
For each donor block $k$, let $T_{\mathrm{del},k}$ denote the Delaunay triangulation of $\{X_{i,k}\}_{i=1}^n$. For any $x\in \mathrm{conv}(\{X_{i,k}\}_{i=1}^n)$, let
\begin{equation}
\sigma_k^{\mathrm{del}}(x)=\mathrm{conv}(V_{1,k}(x),\dots,V_{d+1,k}(x))\in T_{\mathrm{del},k}
\end{equation}
be any Delaunay simplex containing $x$, and let $\hat w_{i,k}^{\mathrm{del}}(x)$ denote the corresponding barycentric weights, with zero weight assigned to donor points outside the selected simplex.\footnote{Because the triangulation is simplicial, if $x$ lies on a shared face, the resulting weight vector is independent of which containing simplex is selected.}
Let $m_{\mathrm{del}}\in\mathcal M_w$ denote the Delaunay Matching Estimator (DME), that is the matching estimator induced by these Delaunay weights.
\begin{theorem}[Pointwise optimality of Delaunay weights]\label{thm:delaunay_budget}
For each donor block $k$ and each feasible target point $x\in \mathrm{conv}(\{X_{i,k}\}_{i=1}^n)$,
\begin{equation}
\{\hat w_{i,k}^{\mathrm{del}}(x)\}_{i=1}^n
\in
\operatorname*{argmin}_{w\in\mathbb R^n}
\left\{
\sum_{i=1}^n w_i\|X_{i,k}-x\|^2:
w_i\ge 0,\;
\sum_{i=1}^n w_i=1,\;
\sum_{i=1}^n w_iX_{i,k}=x
\right\}.
\end{equation}
\end{theorem}
\hyperref[proof:thm:delaunay_budget]{The formal proof is in Appendix \ref{app:formal_proofs}.}
Theorem \ref{thm:delaunay_budget} is a pointwise statement: at every feasible target point, Delaunay weights solve the local geometric approximation problem over the full class of positive affine-exact weights.
\begin{figure}[h]
\centering
\caption{Geometric illustration of the Delaunay optimality argument.}
\label{fig:delaunay_pointwise_argument}
\includegraphics[width=0.78\linewidth]{Figures/delaunay_pointwise_argument.pdf}
\smallskip
\begin{minipage}{0.92\linewidth}
\footnotesize\textit{\textbf{Notes:}} The horizontal axes $(u_1,u_2)$ index covariate space, while the vertical axis records the lifted height $z=\|u\|^2$. The donor points are lifted to the paraboloid, the blue simplex is the lift of the Delaunay simplex containing $x$, and the green plane is the affine interpolant through its lifted vertices. The crimson curve is the lifted circumcircle, namely the intersection of the paraboloid with that affine plane. Because the Delaunay circumsphere is empty, this plane lies weakly below every lifted donor point. At the target covariate value $x$, the plane has height $\ell(x)=\sum_i \hat w_i^{\mathrm{del}}(x)\|X_i\|^2$, while the paraboloid has height $\|x\|^2$. Their vertical gap is therefore $\sum_i \hat w_i^{\mathrm{del}}(x)\|X_i-x\|^2$. Any other feasible weights averaging to $x$ correspond to a convex combination of lifted donor points and so must produce a height weakly above the same plane.
\end{minipage}
\end{figure}
\begin{remark}[Geometric proof sketch]
Figure \ref{fig:delaunay_pointwise_argument} summarizes the argument. Fix a target point $x$. Every feasible set of weights $w$ determines a lifted point
\begin{equation}
\left(x,\sum_{i=1}^n w_i\|X_i\|^2\right)
\end{equation}
in the convex hull of the lifted donor points $(X_i,\|X_i\|^2)$. By the identity
\begin{equation}
\sum_{i=1}^n w_i\|X_i-x\|^2
=
\sum_{i=1}^n w_i\|X_i\|^2-\|x\|^2,
\end{equation}
the geometric criterion is exactly the vertical distance between that lifted point and the paraboloid height $\|x\|^2$.
In the figure, the Delaunay simplex containing $x$ determines an affine plane whose intersection with the paraboloid is the lifted circumcircle.
Because the circumcircle is empty (the defining condition for a triangulation to be Delaunay), that plane lies weakly below every lifted donor point.
Therefore the Delaunay barycentric weights attain the lowest feasible lifted point above $x$, which proves the pointwise inequality of Theorem \ref{thm:delaunay_budget}.
Averaging this pointwise dominance over donor samples yields the bound for $r_\infty(x,m_w)$, and taking the supremum over $x$ gives Corollary \ref{cor:delaunay_affordability}.
\end{remark}
Because the dominance result in Theorem \ref{thm:delaunay_budget} holds sample by sample and point by point, it carries over directly to the asymptotic certification radius.
\begin{corollary}[Implication for asymptotic affordability]\label{cor:delaunay_affordability}
For every $m_w\in\mathcal M_w$ and every $x\in\mathcal X$,
\begin{equation}
r_\infty(x,m_{\mathrm{del}})
\le
r_\infty(x,m_w).
\end{equation}
Consequently,
\begin{equation}
\sup_{x\in\mathcal X} r_\infty(x,m_{\mathrm{del}})
\le
\sup_{x\in\mathcal X} r_\infty(x,m_w).
\end{equation}
Therefore, by Corollary \ref{cor:aff_asympt},
\begin{equation}
\sup_{Q \in \mathcal{P}, f_0,f_1 \in \mathcal{F}_c}\limsup_{N,N_x \to \infty} \mathbb{E}_{Q^J \times P^N}[\ell(X_j)]
\le
\sup_{x\in\mathcal X} r_\infty(x,m_{\mathrm{del}})
\le
\sup_{x\in\mathcal X} r_\infty(x,m_w)
\end{equation}
for every $m_w\in\mathcal M_w$.
\end{corollary}
\hyperref[proof:cor:delaunay_affordability]{The formal proof is in Appendix \ref{app:formal_proofs}.}
Corollary \ref{cor:delaunay_affordability} shows that Delaunay Matching Estimator provides the best asymptotic affordability guarantee in the set of matching estimators with positive weights.
\section{Geometric Data Collection Plans}\label{sec:data_collection}
Corollary \ref{cor:aff_asympt} suggests a natural donor-design problem.
Suppose the policymaker has committed to Delaunay Matching.
Then, the relevant object for controlling asymptotic worst-case compensation is the geometric part of the certification radius.
Conditioning on the realized donor covariates $\mathbf X_N^n$, define the design-conditional geometric criterion
\begin{equation}\label{eq:geo_design_criterion}
\hat r(x)
:=
\frac{c}{2K_N}
\sum_{k=1}^{K_N}\sum_{i=1}^n
\hat w_{i,k}^{\mathrm{del}}(x)\|X_{i,k}-x\|^2.
\end{equation}
Eq. \ref{eq:geo_design_criterion} provides the sample analogue of the asymptotic affordability bound in Corollary \ref{cor:aff_asympt}.
I focus on \textit{local refinement} plans that preserve the current feasible region $\mathcal X$.
Accordingly, an additional donor covariate assigned to block $k$ must lie in $\mathrm{conv}(\{X_{i,k}\}_{i=1}^n)$.
\begin{definition}[One-step collection plan]\label{def:collection_plan}
A one-step local collection plan is a measurable rule
\begin{equation}
\pi(\mathbf X_N^n)=(k,z),
\qquad
k\in\{1,\dots,K_N\},
\qquad
z\in \mathrm{conv}(\{X_{i,k}\}_{i=1}^n).
\end{equation}
After augmenting donor block $k$ with the new covariate point $z$ and recomputing the Delaunay triangulation on that block, let $\hat r_{(k,z)}(x)$ denote the updated version of \eqref{eq:geo_design_criterion}.
The exact one-step refinement problem is:
\begin{equation}\label{eq:exact_collection_problem}
(k_N^\star,z_N^\star)
\in
\operatorname*{argmin}_{k,z}
\sup_{x\in \mathcal X}
\hat r_{(k,z)}(x).
\end{equation}
\end{definition}
Unfortunately, this problem does not admit a closed-form solution in general and may be computationally infeasible to solve numerically.
However, we can use some standard results in computational geometry to find an approximate solution.
For block $k$, define
\begin{equation}
g_k(x):=
\sum_{i=1}^n \hat w_{i,k}^{\mathrm{del}}(x)\|X_{i,k}-x\|^2,
\qquad
\hat r(x)=\frac{c}{2K_N}\sum_{k=1}^{K_N} g_k(x).
\end{equation}
For a simplex $\sigma$, let $r_{\mathrm{mc}}(\sigma)$ denote the radius of the smallest Euclidean ball containing it.
Then Waldron's geometric bound \citep{waldron} implies that, for every donor block $k$ and every $x\in\mathcal X$,
\begin{equation}
g_k(x)\le \max_{\sigma\in T_{\mathrm{del},k}} r_{\mathrm{mc}}(\sigma)^2.
\end{equation}
Moreover, if $\sigma\in T_{\mathrm{del},k}$ and $x\in \sigma$, the local loss on that simplex is maximized at the center of its minimum enclosing ball.
For each donor block $k$, let
\begin{equation}
\sigma_k^\star
\in
\operatorname*{argmax}_{\sigma\in T_{\mathrm{del},k}} r_{\mathrm{mc}}(\sigma),
\qquad
\rho_k^\star:=r_{\mathrm{mc}}(\sigma_k^\star)^2.
\end{equation}
Then, by \eqref{eq:geo_design_criterion},
\begin{equation}\label{eq:max_block_proxy}
\sup_{x\in\mathcal X}\hat r(x)
\le
\frac{c}{2}\max_{1\le k\le K_N}\rho_k^\star.
\end{equation}
To relate \eqref{eq:max_block_proxy} to the exact criterion in \eqref{eq:exact_collection_problem}, let $\rho_\ell^{(k,z)}$ denote the updated analogue of $\rho_\ell^\star$ after adding the new donor point $z$ to block $k$.
Then
\begin{equation}
\sup_{x\in \mathcal X}\hat r_{(k,z)}(x)
\le
\frac{c}{2K_N}\sum_{\ell=1}^{K_N}\rho_\ell^{(k,z)}
\le
\frac{c}{2}\max_{1\le \ell \le K_N}\rho_\ell^{(k,z)}.
\end{equation}
Therefore, minimizing the largest updated worst-simplex radius is a conservative approximation to the exact refinement problem.
It is conservative because the exact criterion averages block-specific geometric losses before taking the supremum over $x$, whereas the proxy first takes the supremum within each block and only then across blocks.
\begin{definition}[Geometric Plan]\label{def:worst_simplex_center}
Let
\begin{equation}
\hat{k}
\in
\operatorname*{argmax}_{1\le k\le K_N}\rho_k^\star,
\end{equation}
and let $c^*_{\hat{k}}$ denote the center of the minimum enclosing ball of the simplex $\sigma_{\hat{k}}^*$.
The Geometric Plan places the next donor covariate at
\begin{equation}
\hat{z}:={c}^*_{\hat{k}}.
\end{equation}
\end{definition}
Definition \ref{def:worst_simplex_center} is a greedy approximation to the exact refinement problem in Definition \ref{def:collection_plan}.
It selects the block that binds the conservative upper bound \eqref{eq:max_block_proxy} and, within that block, places the new donor point at the location where the current simplex-level geometric loss is largest.
\section{Semi-Synthetic Application}\label{sec:semi_synth}
This section builds on the experimental setting of \citet{muralidharan_building_2016}, who study the rollout of biometric Smartcards for NREGS and Social Security Pension payments in rural Andhra Pradesh.
In the original experiment, the government randomized the order of Smartcard conversion across $296$ eligible mandals in eight districts, assigning $112$ mandals to treatment, $139$ to a buffer group, and $45$ to control.
The buffer group was introduced to preserve a gap between treated and control mandals long enough to field endline surveys after rollout in treated areas but before rollout in control areas.
The survey sample covered $880$ Gram Panchayats (GPs), with ten households per GP: six drawn from the NREGS jobcard frame and four from the pension beneficiary frame.
The endline sample contains $8{,}114$ households.
\subsection{Targeting Smartcards with Delaunay Matching}
The targeting exercise asks where a policymaker should deploy the Smartcard payment system in subgroups of the target population.
I therefore treat GPs in control mandals as targets, GPs in treated mandals as donors, and aggregate the target population into $25$ empirical cells obtained by a $5\times 5$ quantile partition of two normalized baseline covariates: baseline log annual consumption and a GP-level baseline NREGS payment measure.
I partition the original treatment group into $40$ mutually exclusive donor blocks that are balanced across the 25 target subgroups.
Each donor block yields a Delaunay triangulation, and predictions are averaged across blocks.
In Figure \ref{fig:semiemp_donor_triangulations} I show four examples of such blocks and plot the Delaunay triangulation of each block.
\begin{figure}[h]
\centering
\caption{Donor Blocks and Delaunay Triangulations}
\label{fig:semiemp_donor_triangulations}
\includegraphics[width=0.88\linewidth]{code/synth_appl/figures/semiemp_graph0_donor_triangulations.pdf}
\smallskip
\begin{minipage}{0.92\linewidth}
\footnotesize\textit{\textbf{Notes:}} Each panel reports one donor block from the pooled-support partition used in the semi-synthetic application. Blue points are treated donor GPs and gray segments are Delaunay edges. The four blocks are selected to span the empirical distribution of the donor worst-simplex radius, so the figure visualizes how local geometric coverage varies across donor designs. In every panel, both covariates are rescaled to $[0,1]$.
\end{minipage}
\end{figure}
I impose a known synthetic treatment effect $\tau(x)$, scaled to the empirical dispersion of gains from lower leakage.
In the main specification I consider a linear $\tau(x)$ whose frontier lies on the first principal-component direction of the target support to ensure that there is mass on both sides of the frontier and to rule out trivial decisions.
In the left-hand panel of Figure \ref{fig:semiemp_tau_decisions} I plot the oracle decision frontier, highlighting in yellow the 25 target points, scaled by their relative size.
Note that this synthetic $\tau(x)$ creates a transparent frontier with some cells far from the decision boundary and others close to it, making it possible to see whether the certification rule expands where geometry is favorable and contracts where the problem is intrinsically harder.
Within each target point, I estimate $\tau(x)$ using DME as defined in Section \ref{sec:delaunay} and assign each cell to the innovation if the estimated intervention's effect is positive.
In the right-hand panel of Figure \ref{fig:semiemp_tau_decisions} I plot the donor and target covariate space, where the latter is color-coded according to the decision made.
Blue target points are assigned to the status quo, while red target points are assigned to the innovation.
Black target points lie outside the convex hull of at least one donor block, and gray target points are not asymptotically certified.
Note that gray points lie close to the oracle frontier, and black points lie at the corners of the covariate space.
\begin{figure}[h]
\centering
\caption{Treatment-effect surface and Covariate Space}
\label{fig:semiemp_tau_decisions}
\includegraphics[width=0.8\linewidth]{code/synth_appl/figures/semiemp_graphAB_tau_decisions.pdf}
\smallskip
\begin{minipage}{0.96\linewidth}
\footnotesize\textit{\textbf{Notes:}} The left panel plots the true synthetic treatment-effect surface $\tau(x)$ over the empirical covariate support. The solid black line is the zero-effect frontier $\tau(x)=0$. Faint white points mark treated donor GPs, while gold circles mark target-cell centroids; circle size is proportional to the number of target GPs in the cell, $N_x$. The right panel plots the asymptotic Delaunay decision map for the same cells under the pooled-donor design. Red cells are certified for innovation, blue cells are certified for the status quo, gray cells are covered by the donor triangulations but not certified, and black cells are not covered. By construction, higher values of $\tau(x)$ correspond to larger gains from lower leakage.
\end{minipage}
\end{figure}
Figure \ref{fig:semiemp_certification} plots the 25 target points sorted by true $\tau(x)$ (in black).
The DME estimate is plotted in green for correct decisions and red for wrong decisions.
The dark band denotes the asymptotic certification radius (see Theorem \ref{thm:asymp_dec} for a definition) and the lighter band denotes the finite sample radius (see Theorem \ref{thm:finite_dec} for a definition).
$21$ of $25$ target cells are covered by the donor triangulations, $16$ are asymptotically certified (see Theorem \ref{thm:asymp_dec} for a definition), and $14$ remain certified after adding the finite-sample stochastic terms (see Theorem \ref{thm:finite_dec} for a definition).
Moreover, all cells that are actually certified are assigned the oracle decision, while the few mistakes occur only among uncertified cells.
\begin{figure}[h!]
\centering
\caption{Decisions and Certification Sets}
\label{fig:semiemp_certification}
\includegraphics[width=0.8\linewidth]{code/synth_appl/figures/semiemp_graphC_certification.pdf}
\smallskip
\begin{minipage}{0.96\linewidth}
\footnotesize\textit{\textbf{Notes:}} Covered target cells are ordered on the horizontal axis by their true synthetic effect $\tau(x)$. Black points plot the true cell effect, while colored points plot the estimated effect $\hat\tau(x)$. Green points indicate that the induced decision agrees with the oracle policy and red points indicate a mistake. The gray ribbon is the asymptotic certification band $\pm r_\infty(x,m_{\mathrm{del}})$ and the pink ribbon is the finite-sample band obtained by adding the stochastic terms from Theorem \ref{thm:finite_dec}. Cells with $|\tau(x)|$ outside the relevant band are certified. The figure shows that finite-sample certification is more conservative, especially in cells near the zero-effect frontier and in cells with weaker geometric coverage.
\end{minipage}
\end{figure}
\subsection{Optimal Collection Plans}
In this section I illustrate the geometric collection plan in the context of \citet{muralidharan_building_2016} and evaluate its performance in terms of worst-case compensation cost against a random collection plan.
The Geometric Plan targets the blocks with the largest minimum-enclosing-ball radius across simplices and places a new donor unit at the center of the worst simplex's minimum enclosing ball.
By contrast, a random plan places new donor units at randomly selected locations (within the convex hull) in randomly selected subsamples.
The empirical exercise considers a target loss equal to half the current one and aims to define a collection plan that achieves that target.
\begin{figure}[h!]
\centering
\caption{Blocks' Refinement - Geometric Plan vs Random Plan}
\label{fig:semiemp_collection_paths}
\includegraphics[width=0.96\linewidth]{code/synth_appl/figures/semiemp_graph0_donor_triangulations_collection_5step_global_compare.pdf}
\smallskip
\begin{minipage}{0.92\linewidth}
\footnotesize\textit{\textbf{Notes:}} Each column fixes one of the four showcased donor blocks from Figure \ref{fig:semiemp_donor_triangulations}. In both rows, the planner is restricted to collect new donor points only within these same four blocks. Blue points are the original treated donor GPs and gray segments are Delaunay edges after five sequential additions under the displayed rule. Gold triangles mark newly collected donor points, labeled by their global insertion order. In the top row, the Geometric Plan selects, at each step, the showcased block with the largest current worst-simplex radius and inserts the center of that simplex. In the bottom row, the random benchmark draws one showcased block uniformly at random and then draws a point uniformly inside that block's current worst simplex. In every panel, both covariates are rescaled to $[0,1]$.
\end{minipage}
\end{figure}
In Figure \ref{fig:semiemp_collection_paths}, I illustrate the difference between the Geometric Plan (see Definition \ref{def:worst_simplex_center}) and a random plan.
The figure restricts attention to four random blocks and five donor additions, purely for illustrative purposes.
The first row shows the Geometric Plan, and the second row shows the random plan.
For each block (each column), we can see the Delaunay triangulation in the covariate space and the five points added by the plan, sorted by their order of addition.
The Geometric Plan adds new donor points in blocks $23$ and $39$, where the worst triangles are visually the coarsest across blocks, and turns a few large simplices into smaller and more regular ones.
The random plan behaves differently: it spreads the same number of additions across blocks and places them at generic interior locations, so the largest simplices often remain essentially unchanged after five additions.
\begin{figure}
\centering
\caption{Worst Case Loss - Geometric Plan vs Random Plan}
\label{fig:semiemp_budget_paths}
\includegraphics[width=0.88\linewidth]{code/synth_appl/figures/semiemp_budget_paths.pdf}
\smallskip
\begin{minipage}{0.92\linewidth}
\footnotesize\textit{\textbf{Notes:}} The figure reports the stopping-budget experiment based on the full pooled-donor design of the semi-synthetic application. Step $0$ is the baseline design, and the horizontal axis counts the total number of donor points added thereafter. The vertical axis reports the worst-case asymptotic compensation budget across covered target GPs, using the known curvature bound $c$ and the geometric approximation term from Corollary \ref{cor:aff_asympt}. The red line is the deterministic Geometric Plan path, as defined in Definition \ref{def:worst_simplex_center}. The blue line is the median path across seeded random-block-plus-random-point refinements, and the shaded band is the 10th--90th percentile range across those random replications. The dashed horizontal line marks the target budget used in the experiment.
\end{minipage}
\end{figure}
Figure \ref{fig:semiemp_budget_paths} shows that this geometric difference maps directly into the policymaker's objective.
Starting from a baseline worst-case asymptotic budget of about $52.2$, the geometric path lowers the bound to $41.6$ after one addition, to $30.4$ after two, and to $25.9$ after three, thereby crossing the target budget line with only three extra donor points.
The decline continues up to step $8$, where the path stabilizes around $15.9$.
One interesting finding is the plateau after step $8$.
That happens because, after the large triangles that drive most of the worst-case loss are regularized by the plan, worst-case triangles across blocks look more and more similar.
As a result, most additions improve the supremum only marginally.
The random benchmark, instead, delivers little systematic progress.
Its median path remains at the baseline level over the full $20$-step horizon, and even its lower decile stays above the target budget.
This happens because it is unlikely that, by placing a random point within and across blocks, we can catch the original worst triangle.
This result motivates the theory developed in this paper and the interest in geometry.
Once we can control worst-case loss geometrically, we also know how to design cost-efficient collection plans that achieve sizeable gains against the random collection benchmark.
\section{Conclusions}\label{sec:conclusions}
This paper studied how a policymaker can learn policy decisions from an innovated donor population and apply them on a distinct target population.
I introduced certification as a criterion linking uniform control of mistake probabilities, away from the decision frontier, to worst-case loss.
I then showed that, for positive affine-exact matching estimators, the finite-sample certification problem decomposes into a geometric approximation term and stochastic terms, while in large samples the stochastic components vanish and the problem becomes purely geometric.
I next connected this asymptotic geometric problem to Delaunay triangulations and proved that Delaunay Matching solves the relevant pointwise approximation problem optimally within the class considered.
I then used that result to motivate a geometric donor-data collection rule aimed at reducing worst-case compensation below a target level.
Finally, in the semi-synthetic application based on the Smartcards experiment \citep{muralidharan_building_2016}, I illustrated that this geometric perspective was informative both for targeting decisions and for designing collection plans, and showed that geometric collections deliver substantial gains relative to random collection.
\bibliography{bib}