$\sum\limits_{\mathbb{w}_{n^{*}}}h^{\prime\prime}(\mathbb{w}_{n^{*}})\mathbb{f}^{G_{n^{*}}}_{red}(\mathbb{w}_{n^{*}})$
if one only sums over elements $\mathbb{w}_{n^{*}}$ with positive probability
under $G_{n^{*}}$. When $\Lambda$ is bijective for $V_{i}$, then there is only
a single composition $\mathbb{u}_{n^{*}}$ with positive probability, given
$\mathbb{v}_{n^{*}}$. Thus, when $Y_{i}$ is binary, the number of terms with
positive probability is equal to
$\Big{(}^{n^{*}+2|\mathcal{V}|-1}_{n^{*}}\Big{)}$. For example, in the real
data analysis of Section 6, inference for
$\mathbb{P}_{n^{*}=20}^{F}\Big{(}\mathbb{Y}_{n^{*}}^{G^{\Lambda_{2}}_{n^{*}=20}}(1)=x\Big{)}$
via parametric compositional g-formula estimation involved summing over
$\Big{(}^{n^{*}+2|\mathcal{V}|-1}_{n^{*}}\Big{)}=\Big{(}^{20+2\times
3-1}_{20}\Big{)}=53130$ terms. We have provided code for these computations at
GitHub.

###  A.5 Sub-cluster regimes

Let $\mathbf{W}_{n}$ be an unmeasured random vector denoting the membership of
each individual $i$ to one of $m_{n}<n$ sub-clusters that partition the
(super) cluster of size $n$. Let
$\boldsymbol{\kappa}_{m_{n}}\equiv\\{\kappa_{n,1},\dots,\kappa_{n,m_{n}}\\}$
denote a partition of the $\kappa_{n}$ treatment units across the $m_{n}$ sub-
clusters. In a slight abuse of notation, we let $\boldsymbol{\kappa}_{m_{n}}$
both denote a random variable, and an arbitrary realization.

Now we define a general class of sub-cluster regimes:

######  Definition S6 (Sub-cluster regimes).

Let $\delta$ be partitioned into
$\delta_{\mathbf{R}}\equiv\\{\delta_{\mathbf{R},1},\dots,\delta_{\mathbf{R},m_{n}}\\}$,
and
$\delta^{\prime}_{\mathbf{A}}\equiv\\{\delta^{\prime}_{\mathbf{A},1},\dots,\delta^{\prime}_{\mathbf{A},m_{n}}\\}$.
Let $\mathbf{G}_{m}$ denote an $m\times m$ matrix of generalized resource-
limited regimes, whose entry in its $r^{th}$ row and $p^{th}$ column, denoted
by $[\mathbf{G}_{m}]_{r,p}$, is some element in $\Pi_{r}^{F}(p)$, i.e. a
generalized regime for a cluster of size $r$ with $p$ treatment units. Then,
define the class of sub-cluster regimes $\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$
for which the following properties hold:

  * 1\.

Within each sub-cluster characterized by $W_{i}=c$, each individual index is
renumbered from $1,\dots,\mathbb{W}_{n}(c)$, according to the rank ordering of
the original indices among such individuals.

  * 2\.

Within each such sub-cluster characterized by $W_{i}=c$, individuals are
assigned treatment according to
$[\mathbf{G}_{m}]_{\mathbb{W}_{n}(c),\kappa_{n,c}}$ using the set of
randomizers $\\{\delta_{\mathbf{R},c},\delta^{\prime}_{\mathbf{A},c}\\}$.

  * 3\.

The sets of randomizers
$\Big{\\{}\\{\delta_{\mathbf{R},c},\delta^{\prime}_{\mathbf{A},c}\\}\mid
c\in\\{1,\dots,m_{n}\\}\Big{\\}}$ are mutually independent.

In words, sub-cluster dynamic regimes determine treatment assignment
independently by sub-cluster, where the regime implemented in each sub-cluster
$c$ is some arbitrary CR in the general class defined by the number of
individuals in that sub-cluster, $\mathbb{w}_{n}(c)$ and its corresponding
treatment supply $\kappa_{n,c}$. Note that when their exists only a single
sub-cluster, $m_{n}=1$, then
$\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})\equiv\Pi^{F}_{n}(\kappa_{n}).$

We leverage sub-cluster regimes in defining an alternative positivity
condition in Appendix B, which, together with weak positivity condition $C0$,
implies the strong positivity condition $C1$ articulated in Section 5.

$\Pi_{n}^{F}(\kappa_{n})$$\Pi_{n}(\kappa_{n})$$\Pi^{*}_{n}(\kappa_{n},L_{i})$$\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda_{\mathbb{l}_{n}}})$$\Pi^{d*}_{n}(\kappa_{n},L_{i})$$\Pi^{d\epsilon}_{n}(\kappa_{n},\Lambda,Q_{L})$$\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$$\Pi^{D}_{n}(\kappa_{n})$$\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$
Figure S1: Diagram depicting relations between classes of cluster dynamic
treatment regimes (C-DTR). $\Pi_{n}^{F}(\kappa_{n})$: generalized constrained
C-DTRs; $\Pi_{n}(\kappa_{n})$: constrained C-DTRs with exactly $\kappa_{n}$
treated units; $\Pi^{*}_{n}(\kappa_{n},L_{i})$: stochastic $L$-equitable
C-DTRs; $\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda_{\mathbb{l}_{n}}})$:
mixed-deterministic C-DTRs; $\Pi^{d*}_{n}(\kappa_{n},L_{i})$: deterministic
C-DTRs; $\Pi^{d\epsilon}_{n}(\kappa_{n},\Lambda,Q_{L})$: $\epsilon-L$-rank-
preserving C-DTRs; $\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$: $L$-rank-
preserving DTRs; $\Pi^{D}_{n}(\kappa_{n})$: fully-deterministic C-DTRs;
$\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$ generalized sub-cluster C-DTRs.

##  Appendix B Inference

This appendix elaborates on several inference-related topics deferred from the
main text. Section B.1 motivates a set of inverse-probability weighted (IPW)
estimators of individual-level g-formula parameters, and notes important
distinctions from conventional IPW estimators. Section B.2 provides an example
demonstrating the insufficiency of weak positivity condition $C0$ for the
general existence of a consistent estimator defined with respect to an
asymptotic law $\mathbb{P}_{0}$ following condition $D$, and thus motivating
the strong positivity condition $C1$ presented in Section 5. Section B.3
provides an alternative to positivity condition $C1$ that is stronger but that
may be easier to justify by subject matter experts. Section B.4 discusses
inference for large-cluster parameters, which faces challenges due to the
irregularity of these parameters when they are defined by a general CR.
Therein, we develop conditions and results for parametric, semiparametric, and
semiparametric efficient estimation of large-cluster parameters defined by
candidate and optimal CRs, adapting results from luedtke2016optimal. Notably,
the conditions considered in this section ensure the regularity of the large-
cluster parameters, in part through restrictions on the class of regimes
considered. Section B.5 presents strategies for consistent estimation of
large-cluster parameters under conditions that do not ensure their regularity,
adapted from online-learning-based estimators developed by
luedtke2016statistical. These strategies are important whenever investigators
are interested in clinically implementing regimes defined by low-dimensional
covariates, which often enjoy practical advantages over their high-dimensional
counterparts.

###  B.1 IPW

To motive a class of inverse-probability weighted estimators for cluster
parameters, define the weight variable

$\tilde{W}_{i}^{G_{n^{*}},j}\coloneqq\frac{\tilde{q}^{*}_{j}(A_{i}\mid
L_{i})}{\tilde{q}_{n}(A_{i}\mid L_{i})}.$

Then, using simple algebra, we can show that

$\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}\tilde{W}_{i}^{G_{n^{*}},j}=\sum\limits_{o}y\tilde{f}_{j}^{G_{n}}(o).$

As such, this appropriately-weighted empirical mean analogue of $Y_{i}$ will
also be asymptotically consistent for its corresponding causal cluster
parameter, $\mathbb{E}_{\mathbb{P}_{n^{*}}^{F}}[Y_{j}^{G_{n^{*}}}]$ under the
conditions of Theorem 3. These relations hold despite the fact that a
classical IPW representation of the individual-level g-formula
$f_{j}^{G_{n^{*}}}(o)$ would suggest a different expression. Motivated by the
density ratio
$\frac{f_{j}^{G_{n^{*}}}(o)}{\mathbb{P}_{n^{*}}(O_{i}=o)}=\frac{q^{*}_{j}(a\mid
l)}{q_{i}(a\mid l)}$ (under the conditions of Theorem 3), define the weight
variable

$W^{G_{n^{*}},j}_{i}\coloneqq\frac{q^{*}_{j}(A_{i}\mid L_{i})}{q_{i}(A_{i}\mid
L_{i})}.$

We thus have an IPW representation of the individual-level g-formula of
Theorem 1: by the Radon-Nikodym Theorem, as in richardson2013single, we have
that

$\mathbb{E}_{\mathbb{P}^{F}_{n^{*}}}[Y_{j}^{G_{n^{*}}}]=\mathbb{E}_{\mathbb{P}_{n^{*}}}[Y_{i}W^{G_{n^{*}},j}_{i}]$

under the conditions of Theorem 1. We emphasize that the weights
$W^{G_{n^{*}},j}_{i}$ are constructed with the density functions $q_{i}$
whereas $\tilde{W}_{i}^{G_{n^{*}},j}$ are constructed with a density function
$\tilde{q}_{n}$. While we have that $\tilde{q}_{n}$ converges to
$\overline{q}_{0}$ (under strong finite-cluster positivity $C1$), we do not
have any guarantees that $\overline{q}_{0}=q_{i}$. The following corollary may
be illustrative, where we note the algebraic relation

$\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}=\sum\limits_{o\in\mathcal{O}_{i}}y\tilde{Q}_{Y}(y\mid
a,l)\tilde{q}_{n}(a\mid l)\tilde{Q}_{L}(l).$

######  Corollary S2.

Consider an asymptotic law $\mathbb{P}_{0}^{F}$ following conditions $D$ and
$\mathbf{f}_{0}$ such that $\lim\tilde{q}_{n}(a\mid l)=\overline{q}_{0}(a\mid
l)$ for all $a,l$. Then,
$\lim\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}=\mathbb{E}_{\mathbb{P}_{0}}[\overline{Y}_{0}]=\mathbb{E}_{\mathbb{P}_{0}}[Y_{j}\frac{\overline{q}_{0}(A_{j}\mid
L_{j})}{q_{j}(A_{j}\mid L_{j})}].$

Thus, we see that the probability limit of the empirical mean analogue of
$Y_{i}$ is equivalent (under stated conditions) to the expectation of a
weighted outcome $Y_{j}$ whose likelihood-ratio weight has a denominator and
numerator defined by the measures $q_{j}$ and $\overline{q}_{0}$,
respectively. Thus, we see that the probability limit of
$\frac{1}{n}\sum\limits_{i=1}^{n}Y_{i}\tilde{W}_{i}^{G_{n},j}$ will result in
the appropriate cancellations of measures, and that it agrees with the
classical IPW representation of the individual-level g-formula
$\mathbb{E}_{\mathbb{P}_{0}}[Y_{i}\frac{\overline{q}_{0}(A_{i}\mid
L_{i})}{q_{i}(A_{i}\mid L_{i})}\frac{q^{*}_{j}(A_{i}\mid
L_{i})}{\overline{q}_{0}(A_{i}\mid L_{i})}]$.

The preceding results focused on IPW representation of finite cluster
parameter $f_{j}^{G_{n^{*}}}(o)$. When interest is in the large cluster
parameter $f^{\mathbf{G}_{0}}(o)$, analogous estimators may be obtained using
weights

$\tilde{W}_{i}^{\mathbf{G}_{0}}\coloneqq\frac{\tilde{q}^{*}_{0}(A_{i}\mid
L_{i})}{\tilde{q}_{n}(A_{i}\mid L_{i})},$

provided that $f^{\mathbf{G}_{0}}(o)$ is indeed a regular parameter, for
example under the conditions of Lemma S2 in Appendix Section B.4.

###  B.2 Counterexample to the existence of a consistent plug-in estimator of
$f_{i}^{G_{n}}$ under conditions of Theorem 1.

Theorem 1 provides a functional of individual-level observed data parameters
$Q_{L}$ and $Q_{Y}$ that equals a finite cluster counterfactual parameter,
provided that functional is well-defined. In line with conventional approaches
richardson2013single, weak cluster positivity condition $C0$ guarantees that
functional is well-defined. In section 5, we articulated an a notion of
asymptotics defined by an asymptotic law $\mathbb{P}_{0}^{F}$. Here we
demonstrate by example that the asymptotic analogues of the conditions of
Theorem 1 (i.e. condition $D$) and an asymptotic analogue of condition $C0$
are not sufficient for the consistency result of Theorem 3.

Let condition $D^{*}$ denote the intersection of condition $D$ and the
following condition, where we remind the reader that we have defined
$q_{0}(a\mid l)\coloneqq\underset{n\to\infty}{\lim}\overline{q}_{n}(a\mid l)$:

  * $D3.$

${q}_{0}$ is well-defined and if $\overline{q}^{*}_{n}(a\mid l)Q_{L}(l)>0$
then ${q}_{0}(a\mid l)>0$, for all $a,l$.

In other words, we will show by counterexample that the following proposition
is false.

######  Proposition.

Consider $\mathbb{P}_{0}^{F}$ and
$G_{n^{*}}\in\Pi_{n^{\ast}}(\kappa_{n^{\ast}})$ following condition $D^{*}$.
Then,

$\displaystyle\boldsymbol{\lim}\sum\limits_{o}h^{\prime}(o)\tilde{f}_{i}^{G_{n^{*}}}(o)=\mathbb{E}_{\mathbb{P}^{F}_{n^{*}}}[h^{\prime}(O_{i}^{G_{n^{*}}+})].$
(S10)

We do so by finding a law $\mathbb{P}_{0}^{F}$ following condition $D^{*}$ in
which $\tilde{Q}_{Y,n}$ does not converge and thus
$\tilde{f}_{i}^{G_{n^{*}}}(o)$ does not converge.

Consider a setting with binary covariate $L\in\mathcal{L}\equiv\\{0,1\\}$, and
consider an asymptotic law $\mathbb{P}_{0}^{F}$ following conditions $D$,
characterized by $Q_{L}(1)=0.5$ and $\lim\frac{\kappa_{n}}{n}=0.5$:
asymptotically, half of the cluster can receive treatment. Furthermore,
suppose that for each $n$, $\mathbb{P}_{n}$ is induced in part by
$\mathbf{f}_{n,\mathbf{A}}\coloneqq\\{f_{A_{i}},\dots,f_{A_{n}}\\}$ that is
equivalent to some mixed-deterministic CR
$G_{n}\in\Pi^{d}_{n}(\kappa_{n},L_{i},q^{*}_{\Lambda_{\mathbb{l}_{n}}})$ such
that for each $n$, for all $\mathbb{l}_{n}$ $q^{*}_{\Lambda_{\mathbb{l}_{n}}}$
puts equal mass on a $\Lambda$ such that $\Lambda(1)>\Lambda(0)$ and on a
$\Lambda^{*}$ such that $\Lambda^{*}(0)>\Lambda^{*}(1)$. Thus, $q_{i}(1\mid
l)=0.5$ for each $l$ and each $i$ and all $n$. It follows then that
$\overline{q}_{0}(1\mid l)=0.5$, and so $\mathbb{P}_{0}^{F}$ will follow
condition $D^{*}$ fro any $G_{n^{*}}$.

Under such an asymptotic law $\mathbb{P}_{0}^{F}$ we will have that
$\lim\limits_{n\to\infty}\mathbb{P}_{n}\Big{(}\mathbb{B}_{n}(1,l)=0\Big{)}=0.5$
for each $l$. In words, the probability that a given cluster will have exactly
0 treated individuals with covariate history $l$ is exactly 0.5, for each $l$.
We thus have that

$\lim\limits_{n\to\infty}\mathbb{P}_{n}\Bigg{(}\Big{|}\frac{\mathbb{O}_{n}(y,a,l)}{\mathbb{B}_{n}(a,l)}-Q_{Y}(y\mid
a,l)\Big{|}>\epsilon^{\prime}\Bigg{)}=0.5,$

for all $\epsilon^{\prime}>0$, which violates the convergence condition for
consistent estimation of the parameters described by $Q_{Y}$ under the
asymoptotic regime in this case.

###  B.3 Alternative conditions for asymptotic consistency: sub-cluster
positivity

In the preceding Appendix Section B.2, we illustrated that an asymptotic law
following condition $D$ could not provide any guarantees for the convergence
of relevant estimators of finite-cluster parameters (e.g.
$\tilde{f}^{G_{n^{*}}}_{i}(o)$), because no such guarantees can be made about
the convergence of $\tilde{{q}}_{n}$, and by consequence the convergence of
$\tilde{Q}_{Y,n}$. Strong cluster positivity condition $C1$ provides direct
guarantees for such convergence, by asserting that the probability limit of
$\tilde{{q}}_{n}$ is well-defined. However, this limiting condition is
arguably abstract; subject matter experts may find it difficult to reason
about its truth-value for a given setting. Therefore, in this Appendix Section
we provide a stronger condition on the treatment assignment mechanisms
$\mathbf{f}_{0,\mathbf{A}}$ in $\mathbb{S}_{0}$ (for a given asymptotic law
$\mathbb{P}_{0}$), which, together with $C0$ implies the strong cluster
positivity condition $C1$, and which may recognizably capture observed data
generating mechanisms in many settings.

Let
$P^{\kappa}_{0}\equiv\\{P^{\kappa}_{1},P^{\kappa}_{2},P^{\kappa}_{3},\dots,P^{\kappa}_{k},\dots,P^{\kappa}_{m},\dots\\}$
denote some sequence of distributions, whereby, for each $k$, $P^{\kappa}_{k}$
is some probability mass function over the space $\\{0,\dots,k\\}$.
Furthermore define the alpha-mixing coefficient introduced by
rosenblatt1956central, commonly used in interference settings (see for
example, savje2021average),

$\alpha_{\mathbb{P}_{n}}(V_{1},V_{2})=\sup\limits_{v_{1}\in\sigma(V_{1}),v_{2}\in\sigma(V_{2})}\Big{|}\mathbb{P}_{n}(V_{1}\in
v_{1},V_{2}\in v_{2})-\mathbb{P}_{n}(V_{1}\in v_{1})\mathbb{P}_{n}(V_{2}\in
v_{2})\Big{|},$

where $\sigma(X)$ is the sub-sigma algebra generated by $X$.

Now we consider the following alternative conditions for an asymptotic law
$\mathbb{P}_{0}^{F}$:

  * $C2.$

Sub-cluster positivity $\mathbb{P}_{0}^{F}$, is such that for all $a,l$, the
following conditions hold:

    * 1\.

For all $n$, $\mathbb{P}_{n}(\mathbb{W}_{n}(c)\leq m)=1$ for some fixed
$m\in\mathbb{N}^{+}$ for all $c$.

    * 2\.

For some fixed random variable $V^{\dagger}$ with support on
$\\{0,\dots,m\\}$, we have that
$(\mathbb{W}_{n}(c)-V^{\dagger})=o_{\mathbb{P}_{0}}(1)$ for all
$c\in\mathcal{C}_{n}\subset\\{1,\dots,m_{n}\\}$ and
$\frac{m_{n}-|\mathcal{C}_{n}|}{m_{n}}=o(1)$.

    * 3\.

For $c_{1}\neq c_{2}$ two sub-cluster indices,
$\alpha_{\mathbb{P}_{n}^{F}}(\mathbb{W}_{n}(c_{1}),\mathbb{W}_{n}(c_{2}))=o(1)$.

    * 4\.

For all $n$, $\mathbb{P}_{n}\in\mathbb{P}_{0}$ is characterized by an
$\mathbf{f}_{n,\mathbf{A}}$ equivalent to a regime
$G^{*}_{n}\in\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$, with
$\delta\equiv\epsilon_{\mathbf{A}_{n}}$.

    * 5\.

$\kappa_{n,c}$ is distributed according $P^{\kappa}_{\mathbb{W}_{n}(c)}\in
P^{\kappa}_{0}$, with the elements of $\boldsymbol{\kappa}_{m_{n}}$ mutually
independent and independent of $\boldsymbol{\epsilon}_{n}$.

    * 6\.

$\epsilon_{\mathbf{W}_{n}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\boldsymbol{\epsilon}_{n}$.

We direct the reader to Appendix A.5 for explicit definitions of the class of
sub-cluster regimes $\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$. Conditions $C2.1$ to
$C2.3$ place mild but important conditions on the sub-cluster asymptotic
regime. Considered jointly, they ensure that the number of clusters grows
sufficiently fast, that the number of individuals per sub-cluster converges to
some common distribution for almost all sub-clusters, and that jointly, the
sub-cluster sizes are merely weakly dependent. Condition $C2.4$ ensures that
any sub-cluster with fixed size $n_{c}$ will have a common distribution for
the number of treatment units available to it, and condition $C2.5$ ensures
that, once the number of treatment units available for that sub cluster is
additionally fixed to $\kappa_{n,c}$, any such cluster will follow a common
regime $[\mathbf{G}_{m}]_{n_{c},\kappa_{n,c}}$. A consequence of $C2.1$ to
$C2.5$ considered jointly is that $\tilde{q}_{n}$ converges sufficiently fast
to an empirical average of weakly dependent and identically distributed random
variables with a number of terms that grows proportional to $n$. Condition
$C2.6$ ensures that sub-cluster membership may be excluded as a covariate from
identification functionals.

Sub-cluster positivity condition $C2$ describes a data-generating mechanism
that may be justifiable via subject matter reasoning in many cases. For
example, a setting in which investigators receive data in trenches from an
increasing number of intensive care units of bounded size, i.e., that of our
observed data setting, would arguably be characterized by condition $C2$. In
the following results, we assert that condition $C2$ does indeed imply a
stronger version of convergence of $\tilde{q}_{n}$, and indeed when considered
in conjunction with weak positivity condition $C0$, does imply strong
positivity condition $C1$.

######  Lemma S1.

Consider an asymptotic law $\mathbb{P}_{0}^{F}$ following conditions $D$ and
sub-cluster positivity condition $C2$. Then $\tilde{q}_{n}$ is
$L^{2}(\mathbb{P}_{0})$ consistent for $\overline{q}_{0}$, i.e.
$\mathbb{E}[\lVert\tilde{q}_{n}-\overline{q}_{0}\rVert_{2}^{2}]=o(1).$

######  Corollary S3.

Consider a finite cluster regime of interest $G_{n^{*}}$. Under the conditions
of Lemma S1, Weak finite cluster positivity condition $C0$ implies strong
finite cluster positivity condition $C1$.

###  B.4 Estimation of regular large cluster parameters

In this appendix, we study estimators for the expected average potential
outcome in a large-cluster under an arbitrary asymptotic regime
$\mathbf{G}_{0}$, and also under the optimal such regime
$\mathbf{G}_{0}^{\mathbf{opt}}$. We do so by deriving their efficient
influence functions, and establish the regularity, asymptotic linearity (RAL),
and efficiency of their corresponding estimators among this class (of RAL
estimators) under a nonparametric model and appropriate regularity conditions.
Our results build on those of luedtke2016optimal and related work
van2014targeted, luedtke2016statistical, qiu2020optimal. Interestingly, the
conditions, estimators, and their resulting properties are largely isomorphic
to those in this literature, despite the substantially different nature of the
causal parameters we consider, the joint distributions in which they are
defined, and the model restrictions assumed for identification.

The crux of our arguments resolves around the following proposition: that our
parameter of interest – the expected average outcome over all individuals in a
large-cluster under a cluster- dynamic regime – is equivalent at all laws in
our model, to the expected outcome for a single solitary individual under a
particular individualized dynamic regime. Insodoing, we establish that the
statistical (observed data) parameter of interest is identical to that studied
by luedtke2016optimal. As such, under isomorphic conditions to those in
luedtke2016optimal, we inherit the properties of these estimators while
benefiting from the interpretive properties of our causal estimands and the
realistic models under which inference is done.

In Section B.4.1 we establish the linkages between our large-cluster
parameters and previously studied individual level parameters; In Section
B.4.2, we re-express the efficient influence functions (EIF) and resulting
EIF-based estimators of luedtke2016optimal; and in Section B.4.3, we adapt
conditions of luedtke2016optimal in terms of our notation and extend their
results.

####  B.4.1 Re-expressing large-cluster CRs as stochastic IRs

Note that $Y_{i}^{\mathbf{a}_{n}}$, $i=1,..n$ are marginally mutually
independent under $\mathcal{M}^{AB}_{n}$, and that
$Y_{i}^{\mathbf{a}_{n}}=Y_{i}^{a_{i}}$ for the $a_{i}\subset\mathbf{a}_{n}$
(See Proposition S6 in Appendix C). By consequence, the expectation of a given
individual’s outcome under any CR in $\Pi^{F}(\kappa_{n})$ can be
characterized by the probability with which that individual is assigned
treatment under that regime, conditional on their covariates, i.e. it is
characterized by $q^{*}_{i}$. This is easily appreciated upon inspection of
the individual-level g-formula function $f_{i}^{G_{n}}$, given in (1): by
Theorem 1, $\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{G_{n}+})]$ is
only a function of $i$ via $q^{*}_{i}$. Therefore, any two regimes $G_{n}$ and
$G_{n}^{\dagger}$ that result in the same $q^{*}_{i}$ will be equivalent in
the sense that
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{G_{n}+})]=\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{G^{\dagger}_{n}+})]$.
As a special case, we might consider a particular pair of regimes: 1) an
arbitrary stochastic CR $G_{n}$; and 2) a stochastic IR $g_{i}$ in which
treatment is assigned as a random draw from the treatment assignment density
$q^{*}_{n}$ under $G_{n}$.

######  Proposition S6.

Consider an IR $g_{i}$ that only intervenes on treatment for individual $i$,
assigning $A^{g_{i}+}_{i}$ as a random draw from $q^{*}_{i}$, the conditional
intervention density for an arbitrary $G_{n}\in\Pi_{n}(\kappa_{n})$. Under
$\mathcal{M}^{AB}_{n}$,

$\displaystyle\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{g_{i}})]=$
$\displaystyle\sum\limits_{\mathcal{O}_{i}}h^{\prime}(o)f_{i}^{G_{n}}(o).$
(S11)

###### Proof.

The result follows by showing that $\mathbb{P}_{n}^{F}(Y^{g_{i}}_{i}=y\mid
A^{g_{i}+}_{i}=a,L_{i}=l)=Q_{Y}(y\mid a,l)$ via identical arguments to the
proof of Lemma S4 and applying Proposition 1 in Appendix C. ∎

The following corollary extends Proposition S6 to large-cluster regimes.

######  Corollary S4.

Consider an asymptotic regime $\mathbf{G}_{0}$ following conditions $E$ under
a law $\mathbb{P}_{0}^{F}$ following conditions $D$, and a law
$\mathbb{P}_{n}^{F}\in\mathbb{P}_{0}^{F}$. Furthermore, consider an IR
$g^{*}_{i}$ that only intervenes on treatment for individual $i$, assigning
$A^{g^{*}_{i}+}_{i}$ as a random draw from $q^{*}_{0}$, the conditional
intervention density for the asymptotic regime $\mathbf{G}_{0}$ under
$\mathbb{P}_{0}^{F}$. Then the following identities hold, for all $i$ and all
$n$:

$\mathbb{E}_{\mathbb{P}_{n}^{F}}[Y_{i}^{g^{*}_{i}}]=\mathbb{E}_{\mathbb{P}_{0}^{F}}[Y_{i}^{g^{*}_{i}}]=\mathbb{E}_{\mathbb{P}_{0}^{F}}[\overline{Y}_{0}^{\mathbf{G}_{0}}].$

Let $g^{\mathbf{opt}}_{i}$ be such a regime assigning
$A^{g^{\mathbf{opt}}_{i}+}_{i}$ as a random draw from $q^{\mathbf{opt}}_{0}$,
the conditional intervention density under the optimal asymptotic regime
$\mathbf{G}^{\mathbf{opt}}_{0}$. Likewise, the following identities hold, for
all $i$ and all $n$:

$\mathbb{E}_{\mathbb{P}_{n}^{F}}[Y_{i}^{g^{\mathbf{opt}}_{i}}]=\mathbb{E}_{\mathbb{P}_{0}^{F}}[Y_{i}^{g^{\mathbf{opt}}_{i}}]=\mathbb{E}_{\mathbb{P}_{0}^{F}}[\overline{Y}_{0}^{\mathbf{G}^{\mathbf{opt}}_{0}}].$

In the following derivations, it will be useful to express the single
parameter $\mathbb{E}_{\mathbb{P}_{n}^{F}}[Y_{i}^{g^{\mathbf{opt}}_{i}}]$ –
the value under a stochastic regime dependent on $\delta$ and $L_{i}$ – as a
particular average over a set of parameters that each correspond to the value
under a deterministic regime dependent on $L_{i}$ alone. To that end, note
that any stochastic regime $g^{*}_{i}$ corresponding to an asymptotic regime
$\mathbf{G}_{0}$ can be formulated as a deterministic function with respect to
$L_{i}$ and a randomizer term $\delta\sim\text{Unif}(0,1)$, specifically:

$\displaystyle g^{*}_{i}:\\{\delta,l\\}\mapsto I\Big{(}\delta<q^{*}_{0}(1\mid
l)\Big{)}.$ (S12)

Thus conditional on a specific value of $\delta$, $g^{*}_{i}$ is equivalent to
a specific deterministic regime that is a function of $L_{i}$ alone,
$g^{*}_{i,\delta}:l\mapsto I\Big{(}\delta<q^{*}_{0}(1\mid l)\Big{)}$. Let
$P^{F}\equiv\mathbb{P}^{F}_{1}\in\mathbb{P}_{0}^{F}$ following conditions $D$,
and let $P\equiv P(P^{F})$. Then, we have the following equivalence:

$\displaystyle\mathbb{E}_{P^{F}}[Y_{i}^{g^{*}_{i}}]=\mathbb{E}_{\delta}\Big{[}\mathbb{E}_{P^{F}}[Y_{i}^{g^{*}_{i,\delta}}]\Big{]}.$
(S13)

####  B.4.2 Constructing a one-step estimator for the value of
$G^{\mathbf{opt}}_{n}$

In the preceding Section, we’ve argued that when we are interested in the
value of an asymptotic regime $\mathbf{G}_{0}$ for a large cluster following
an asymptotic law $\mathbb{P}_{0}^{F}$ (under conditions $D$ and $E$), we can
do so by instead targeting the value under an appropriately-equivalent class
of individual-level regimes $g_{i,\delta}$, deterministic with respect to
$L_{i}$, applied to a single individual in a cluster entirely of their own,
$\mathbb{E}_{P^{F}}[Y_{i}^{g^{*}_{i,\delta}}]$. Thus, such parameters are
identical to those classically studied by the causal inference literature,
under iid settings, where investigators target the value under some
deterministic regime $g$, i.e., $\mathbb{E}_{P^{F}}[Y^{g}]$, where $Y\equiv
Y_{i}$ when the observed data $\mathbf{O}_{n}\equiv\\{O_{1},\dots,O_{n}\\}$
are viewed as $n$ independent draws from the common law $P$.

We leverage this formulation to motivate articulation of the efficient
influence function for
$\mathbb{E}_{\mathbb{P}_{0}^{F}}[\overline{Y}_{0}^{\mathbf{G}_{0}}]$, via the
efficient influence function of
$\mathbb{E}_{P^{F}}[Y_{i}^{g^{*}_{i,\delta}}]$.

Let
$\Psi^{\mathbf{G}_{0}}\coloneqq\sum\limits_{o\in\mathcal{O}_{i}}yf^{\mathbf{G}_{0}}(o)$
denote the indentifying functional for the expected average outcome under the
asymptotic regime $\mathbf{G}_{0}$, and let
$\Psi^{\mathbf{opt}}\coloneqq\sum\limits_{o\in\mathcal{O}_{i}}yf^{\mathbf{G}_{0}^{\mathbf{opt}}}(o)$
denote the identifying functional for that parameter under the asymptotic
optimal regime $\mathbf{G}_{0}^{\mathbf{opt}}$. Note that
$\Psi^{\mathbf{G}_{0}}$ is simply the well-studied individual-level g-formula
with respect to $P$, an arbitrary margin of the more elaborated law
$\mathbb{P}_{n}$.

Let $\overline{P}$ be the observed margins of a distribution in
$\mathcal{M}_{1}$, which factorizes according to $Q_{Y}$, $\overline{q}_{0}$,
and $Q_{L}$, with respect to the asymptotic law $\mathbb{P}_{0}$. We consider
this law $\overline{P}$ because, under condition set $E$ and the strong
cluster positivity conditions $C1$, it represents the limiting distribution of
the empirical law
$\tilde{P}_{n}\equiv\\{\tilde{Q}_{Y,n},\tilde{q}_{n},\tilde{Q}_{L,n}\\}$. Then
we define the following two random variables $\Phi_{1}(\overline{P},g)(O_{i})$
and $\Phi_{2}(g,\eta)(O_{i})$, with respect to a deterministic regime $g$:

$\displaystyle\Phi_{1}(\overline{P},g)(o)\coloneqq$
$\displaystyle\frac{I\big{(}a=g(l)\big{)}}{\overline{q}_{0}(a\mid
l)}\Big{(}y-\mathbb{E}_{\overline{P}}[Y_{i}\mid a,l]\Big{)}$
$\displaystyle+\mathbb{E}_{\overline{P}}[Y_{i}\mid
g(l),l]-\mathbb{E}_{\overline{P}}\big{[}\mathbb{E}_{\overline{P}}[Y_{i}\mid
g(L_{i}),L_{i}]\big{]}.$ (S14) $\displaystyle\Phi_{2}(g,\eta)(o)\coloneqq$
$\displaystyle\eta(g(l)-\kappa^{*}).$ (S15)

$\Phi_{1}(\overline{P},g)$ is recognizable as the EIF for the g-formula
parameter under a fixed deterministic regime $g$ and $\Phi_{2}(g,\eta)$ has
been shown by luedtke2016optimal to be an additional additive term for the EIF
when the regime $g$ is the $\kappa^{*}$ resource-constrained optimal regime
that assigns treatment to all individuals with a CATE greater than $\eta$.

Let $g^{*}$ be an arbitrary stochastic regime that depends on $L_{i}$ and
$\delta$, and $g^{*}_{\delta}$ a corresponding deterministic regime with
$\delta$ fixed. We further define $\Phi_{0}(\overline{P},g^{*},\eta)(O_{i})$
to be the expectation of the sum of
$\Phi_{1}(\overline{P},g^{*}_{\delta})(O_{i})$ and
$\Phi_{2}(g^{*}_{\delta},\eta)(O_{i})$, taken with respect to the distribution
of $\delta$, and we let $\sigma^{2}(\overline{P},g^{*},\eta)$ denote the
variance of $\Phi_{0}(\overline{P},g^{*},\eta)(O_{i})$ under $\overline{P}$:

$\displaystyle\Phi_{0}(\overline{P},g^{*},\eta)(o)$
$\displaystyle\coloneqq\mathbb{E}_{\delta}\Big{[}\Phi_{1}(\overline{P},g^{*}_{\delta})(o)+\Phi_{2}(g^{*}_{\delta},\eta)(o)\Big{]}.$
(S16) $\displaystyle\sigma^{2}(\overline{P},g^{*},\eta)$
$\displaystyle\coloneqq\text{Var}_{\overline{P}}\Big{(}\Phi_{0}(\overline{P},g^{*},\eta)(O_{i})\Big{)}.$
(S17)

luedtke2016optimal showed that, under conditions,
$\Phi_{0}(\overline{P},g^{*},\eta)(O_{i})$ represents the EIF when the regime
$g^{*}$ is the $\kappa^{*}$ resource-constrained optimal regime that assigns
treatment to all individuals with a CATE greater than $\eta$, and randomly
assigns treatment to individuals with CATE equal to $\eta$ with precisely the
probability for which the marginal probability of assigned treatment is
exactly $\kappa^{*}$.

In a slight abuse of notation, let $\tilde{Q}_{Y,n}$, $\tilde{q}_{n}$,
$\tilde{Q}_{L,n}$ denote functionals of $\mathbb{O}_{n}$ designed to
approximate $Q_{Y}$, $\overline{q}_{0}$, $Q_{L}$, respectively, possibly via
flexible regression or machine learning procedures. Then define
$\tilde{g}^{*}_{i}$ and $\tilde{g}^{\mathbf{opt}}_{i}$ to be the appropriately
equivalent individual regimes, learned from the data $\tilde{P}_{n}$,
corresponding to $\mathbf{G}_{0}$ and $\mathbf{G}_{0}^{\mathbf{opt}}$,
respectively:

$\displaystyle\tilde{g}^{*}_{i}$ $\displaystyle:\delta\times l\mapsto
I\Big{(}\delta<\tilde{q}^{*}_{0}(1\mid l)\Big{)},$ (S18)
$\displaystyle\tilde{g}^{\mathbf{opt}}_{i}$ $\displaystyle:\delta\times
l\mapsto I\Big{(}\delta<\tilde{q}^{\mathbf{opt}}_{0}(1\mid l)\Big{)}.$ (S19)

With these objects defined, let $\tilde{\Psi}^{\mathbf{G}_{0}}_{n}$ and
$\tilde{\Psi}^{\mathbf{opt}}_{n}$ denote one-step estimators for
$\Psi^{\mathbf{G}_{0}}$ and $\Psi^{\mathbf{opt}}$, and let
$\tilde{\sigma}^{2,\mathbf{G}_{0}}_{n}$ and
$\tilde{\sigma}^{2,\mathbf{opt}}_{n}$ denote estimators for their asymptotic
variances, respectively:

$\displaystyle\tilde{\Psi}^{\mathbf{G}_{0}}_{n}$
$\displaystyle\coloneqq\sum\limits_{o\in\mathcal{O}_{i}}y\tilde{f}^{\mathbf{G}_{0}}(o)+\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{*}_{i},0)(O_{i}),$
(S20) $\displaystyle\tilde{\Psi}^{\mathbf{opt}}_{n}$
$\displaystyle\coloneqq\sum\limits_{o\in\mathcal{O}_{i}}y\tilde{f}^{\mathbf{G}^{\mathbf{opt}}_{0}}(o)+\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{\mathbf{opt}}_{i},\tilde{\eta}_{0})(O_{i}),$
(S21) $\displaystyle\tilde{\sigma}^{2,\mathbf{G}_{0}}_{n}$
$\displaystyle\coloneqq\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{*}_{i},0)(O_{i})^{2},$
(S22) $\displaystyle\tilde{\sigma}^{2,\mathbf{opt}}_{n}$
$\displaystyle\coloneqq\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{\mathbf{opt}}_{i},\tilde{\eta}_{0})(O_{i})^{2}.$
(S23)

We emphasize that the argument for $\eta$ with respect to the function
$\Phi_{0}$ in the definition of $\tilde{\Psi}^{\mathbf{G}_{0}}_{n}$ is 0,
indicating that the empirical analogue of the term deriving from $\Phi_{2}$
(the additional term in the influence function for the resource-constrained
optimal regime) is omitted from the estimator for the value under a fixed
asymptotic regime $\mathbf{G}_{0}$.

####  B.4.3 Results for a one-step estimator

In the following, we re-state regularity conditions from luedtke2016optimal.
To do so, we define for convenience some additional objects. First, let
$S_{\Lambda,P}$ be the survival function of $\Lambda(L_{i})$, i.e.,
$S_{\Lambda,P}:c\mapsto P(\Lambda(L_{i})<c)$. Furthermore define the following
second-order remainder terms:

$\displaystyle R_{1,0}(g^{*},\tilde{P}_{n})$
$\displaystyle=\mathbb{E}_{\delta}\Bigg{[}\mathbb{E}_{P}\Big{[}\Big{(}1-\frac{\overline{q}_{0}(g^{*}_{\delta}(L_{i})\mid
L_{i})}{\tilde{q}_{n}(g^{*}_{\delta}(L_{i})\mid
L_{i})}\Big{)}\Big{(}\mathbb{E}_{\tilde{P}_{n}}[Y_{i}\mid
g^{*}_{\delta}(L_{i}),L_{i}]-\mathbb{E}_{P}[Y_{i}\mid
g^{*}_{\delta}(L_{i}),L_{i}]\Big{)}\Big{]}\Bigg{]},$ (S24) $\displaystyle
R_{2,0}(g^{*},g^{\dagger},\Lambda,\omega)$
$\displaystyle=\mathbb{E}_{\delta}\Bigg{[}\mathbb{E}_{P}\Big{[}\big{\\{}g^{*}_{\delta}(L_{i})-g^{\dagger}_{\delta}(L_{i})\big{\\}}\big{\\{}\Lambda(L_{i})-\omega\big{\\}}\Big{]}\Bigg{]}.$
(S25)

The following conditions are articulated with respect to a general large-
cluster parameter under asymptotic regime $\mathbf{G}_{0}$, characterized by a
rank-function $\Lambda$, with equivalent IR $g^{*}_{i}$ under
$P\in\mathbb{P}_{0}.$

  * F1.

$\Lambda(L_{i})$ has continuous density $f_{0}$ at $\omega_{0}$ under $P$ and
$0<f_{0}(\omega_{0})<\infty$.

  * F2.

$S_{\Lambda,P}$ is continuous in a neighborhood of $\omega_{0}$.

  * F3.

$P\big{(}\Lambda(L_{i})=\omega\big{)}=0$ for all $\omega$ in a neighborhood of
$\omega_{0}$.

  * F4.

$P\big{(}\tau<\overline{q}_{0}(1\mid L_{i})<1-\tau\big{)}=1$ for some
$\tau>0$.

  * F5.

$P\big{(}\tau<\tilde{q}_{n}(1\mid L_{i})<1-\tau\big{)}=1$ with probability
approaching 1 for some $\tau>0$.

  * F6.

$R_{1,0}(\tilde{g}^{*}_{i},\tilde{P}_{n})=o_{P}(n^{1/2})$.

  * F7.

$\mathbb{E}_{\overline{P}}\Big{[}\Big{(}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{*}_{i},0)(O_{i})-\Phi_{0}(P,g^{*}_{i},0)(O_{i})\Big{)}^{2}\Big{]}=o_{P}(1)$,
and $\mathbb{E}_{\tilde{P}_{n}}[Y_{i}\mid A_{i}=a,L_{i}=l]$ belongs to a
$P$-Donsker class and is an $L^{2}(P)$-consistent estimator of
$\mathbb{E}_{P}[Y_{i}\mid A_{i}=a,L_{i}=l]$.

  * F8.

$\Phi_{0}(\tilde{P}_{n},\tilde{g}^{*}_{i},0)$ belongs to a $P-$Donsker class
$\mathcal{D}$ with probability approaching 1.

  * F9.

$\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{*}_{i},0)(O_{i})=o_{P}(n^{1/2})$.

Let Condition $F$ denote the intersection of conditions $F1$-$F9$. For
inference on the optimal asymptotic regime $\mathbf{G}^{\mathbf{opt}}_{0}$,
consider the following modified conditions:

  * $F1^{*}.$

$\Delta_{0}(L_{i})$ has continuous density $f^{*}_{0}$ at $\eta_{0}$ under $P$
and $0<f^{*}_{0}(\eta_{0})<\infty$.

  * $F2^{*}.$

$S_{\Delta_{0},P}$ is continuous in a neighborhood of $\eta_{0}$.

  * $F3^{*}.$

$P\big{(}\Delta_{0}(L_{i})=\eta\big{)}=0$ for all $\eta$ in a neighborhood of
$\eta_{0}$.

  * $F4^{*}.$

$P\big{(}\tau^{*}<\overline{q}_{0}(1\mid L_{i})<1-\tau^{*}\big{)}=1$ for some
$\tau^{*}>0$.

  * $F5^{*}.$

$P\big{(}\tau^{*}<\tilde{q}_{n}(1\mid L_{i})<1-\tau^{*}\big{)}=1$ with
probability approaching 1 for some $\tau^{*}>0$.

  * $F6^{*}.$

$R_{1,0}(\tilde{g}^{\mathbf{opt}}_{i},\tilde{P}_{n})=o_{P}(n^{1/2})$ and
$R_{2,0}(\tilde{g}^{\mathbf{opt}}_{i},{g}^{\mathbf{opt}}_{i},\Delta_{0},\eta_{0})=o_{P}(n^{1/2})$.

  * $F7^{*}.$

$\mathbb{E}_{\overline{P}}\Big{[}\Big{(}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{\mathbf{opt}}_{i},{\eta}_{0})(O_{i})-\Phi_{0}(\overline{P},g^{\mathbf{opt}}_{i},\eta_{0})(O_{i})\Big{)}^{2}\Big{]}=o_{P}(1)$,
and $\mathbb{E}_{\tilde{P}_{n}}[Y_{i}\mid A_{i}=a,L_{i}=l]$ belongs to a
$P$-Donsker class and is an $L^{2}(P)$-consistent estimator of
$\mathbb{E}_{P}[Y_{i}\mid A_{i}=a,L_{i}=l]$.

  * $F8^{*}.$

$\Phi_{0}(\tilde{P}_{n},\tilde{g}^{\mathbf{opt}}_{i},\eta_{0})$ belongs to a
$P-$Donsker class $\mathcal{D}$ with probability approaching 1.

  * $F9^{*}.$

$\frac{1}{n}\sum\limits_{i=1}^{n}\Phi_{0}(\tilde{P}_{n},\tilde{g}^{\mathbf{opt}}_{i},\eta_{0})(O_{i})=o_{P}(n^{1/2})$.

  * $F10^{*}$

$\tilde{\Delta}_{n}$ is consistent for $\Delta_{0}$ in $L^{1}(P)$ and
$\tilde{\Delta}_{n}$ belongs to a Glivenko-Cantelli class with probability
approaching 1.

Let Condition $F^{*}$ denote the intersection of these adapted conditions. The
following Theorem consolidates results from luedtke2016optimal.

######  Theorem S1.

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$ and an
asymptotic optimal regime $\mathbf{G}_{0}^{\mathbf{opt}}$ following condition
$E$. Suppose further that condition $F^{*}$ holds. Then the following
properties also hold:

  * (1.)

$\Psi^{\mathbf{opt}}$ is a pathwise differentiable parameter with efficient
non-parametric influence function
$\Phi_{0}(\overline{P}_{0},g^{\mathbf{opt}}_{i},\eta_{0})$.

  * (2.)

$\tilde{\Psi}^{\mathbf{opt}}_{n}$ is the asymptotically-efficient, regular and
asymptotically linear estimator of $\Psi^{\mathbf{opt}}$.

  * (3.)

An asymptotically valid two-sided $1-\alpha$ confidence interval is given by

$\tilde{\Psi}^{\mathbf{opt}}_{n}\pm
z_{1-\alpha/2}\frac{\tilde{\sigma}^{2,\mathbf{opt}}_{n}}{\sqrt{n}}.$

The subsequent proposition extends these results to estimators for
$\Psi^{\mathbf{G}_{0}}$.

######  Proposition S7.

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$ and an
asymptotic regime $\mathbf{G}_{0}$ following condition $E$. Suppose further
that condition $F$ holds. Then the following properties also hold:

  * (1.)

$\Psi^{\mathbf{G}_{0}}$ is a pathwise differentiable parameter with efficient
non-parametric influence function $\Phi_{0}(\overline{P}_{0},g^{*}_{i},0)$.

  * (2.)

$\tilde{\Psi}^{\mathbf{G}_{0}}_{n}$ is the asymptotically-efficient RAL
estimator of $\Psi^{\mathbf{G}_{0}}$.

  * (3.)

An asymptotically valid two-sided $1-\alpha$ confidence interval is given by

$\tilde{\Psi}^{\mathbf{G}_{0}}_{n}\pm
z_{1-\alpha/2}\frac{\tilde{\sigma}^{2,\mathbf{G}_{0}}_{n}}{\sqrt{n}}.$

The following results establish consistency of $\tilde{\omega}_{n}$, thereby
providing a sufficient condition for the regularity of
$\Psi^{\mathbf{G}_{0}}$.

######  Lemma S2.

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$ and an
asymptotic regime $\mathbf{G}_{0}$ following condition $E$. Suppose further
that Conditions $F2$ and $F3$ hold. Then, $\tilde{\omega}_{n}$ converges to
$\omega_{0}$ in probability.

Lemma S2 can be proved in the same manner as Lemma 5 in luedtke2016optimal.

Now, consider a large-cluster positivity condition analogous to strong finite-
cluster positivity condition $C1$.

  * C1∗.

Strong large-cluster positivity Consider an asymptotic regime $\mathbf{G}_{0}$
following condition $E$. $\mathbb{P}_{0}^{F}$ is such that $\overline{q}_{0}$
is well-defined and, for all $a,l$, if ${q}^{*}_{0}(a\mid l)Q_{L}(l)>0$ then
$\overline{q}_{0}(a\mid l)>0.$

The following corollary extends the results of Theorem 3 to large cluster
parameters:

######  Corollary S5.

Consider an asymptotic law $\mathbb{P}_{0}$ following conditions $D$ and an
asymptotic regime $\mathbf{G}_{0}$ following conditions $E$. Suppose that
$\tilde{\omega}_{n}$ is consistent for $\omega_{0}$, possibly via the
conditions of Lemma S2, and that the strong large-cluster positivity condition
$C1^{*}$ holds (possibly via the conjunction of sub-cluster positivity
condition $C2$ and weak large-cluster positivity condition $C0^{*})$. Then we
have that $\sum\limits_{o\in\mathcal{O}_{i}}y\tilde{f}^{\mathbf{G}_{0}}(o)$ is
an asymptotically consistent estimator for $\Psi^{\mathbf{G}_{0}}$. Under sub-
cluster positivity condition $C2$ we additional have that
$\sum\limits_{o\in\mathcal{O}_{i}}y\tilde{f}^{\mathbf{G}_{0}}(o)$ is a RAL
estimator for $\Psi^{\mathbf{G}_{0}}$.

###  B.5 Estimation of irregular large-cluster parameters

Following luedtke2016optimal, let $\\{l_{n}\\}_{n\in\mathbb{N}}$ be the
sequence of sizes of the smallest subsample of $\mathbf{O}_{n}$ on which the
optimal rule is learned. For $j=l_{n},\ldots,n$, let $\tilde{P}_{n,j}$ be the
empirical distribution of $\\{O_{1},\ldots,O_{j}\\}$, and let
$\tilde{q}_{n,j}$, $\tilde{g}_{i,j}^{*}$ and $\tilde{\eta}_{n,j}$ be the
estimates of $\bar{q}_{0}$, $g_{i}^{*}$, and ${\eta}_{0}$ based on
$\\{O_{1},\ldots,O_{j-1}\\}$.

Conditions $F^{*}$ are reasonable if $L_{i}$ is continuous, but if the
covariates are discrete then will not hold be verified: they would contradict
conditions $F1^{*}$ and $F2^{*}$. The following results will therefore
consider a set of conditions that exclude $F1^{*}$ and $F2^{*}$. In
particular, we let $F^{\dagger}$ denote the intersection of $F4^{*}$,
$F7^{*}$, and the following boundedness condition for the outcomes $Y_{i}$:

  * $F3^{\dagger}$.

$\overline{P}(|Y_{i}|<M)=1$ for some $M<\infty$.

luedtke2016statistical consider a slightly weaker version of condition
$F7^{*}$, that we do not restate here to focus on relations between conditions
for the estimators of this section compared to estimators considered in
Appendix B.4.3.

Then, luedtke2016statistical develop an online estimator. Let $\ell_{n}$ be
the size of the smallest sample on which we train our models and learn the
optimal rule.

For $j=1,\ldots,n$, let $(\tilde{q}_{n,j},\tilde{Q}_{n,j},\tilde{g}_{i,j})$ be
estimators of $(\bar{q}_{0},\mathbb{E}_{\overline{P}},g^{\textbf{opt}})$
respectively, based on $(O_{1},\ldots,O_{j-1})$.

Let

$\tilde{\sigma}^{2}_{0,n,j}:=\text{Var}_{\overline{P}}(\Phi_{0}(\tilde{P}_{n,j},\tilde{g}_{i,j},\tilde{\eta}_{n,j})\mid
O_{1},\ldots,O_{j-1}),$

and $\tilde{\sigma}^{2}_{n,j}$ be an estimate of $\tilde{\sigma}^{2}_{0,n,j}$
based on $(O_{1},\ldots,O_{j-1})$.

Furthermore, let

$\Gamma_{n}:=\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\tilde{\sigma}_{n,j}^{-1}.$

Let
$\tilde{\Phi}_{n,j}:=\Phi_{0}(\tilde{P}_{n,j},\tilde{g}_{i,j},\tilde{\eta}_{n,j})$.
Then, luedtke2016statistical define the following estimator.

$\hat{\Psi}_{n}^{\textbf{opt}}:=\Gamma_{n}^{-1}\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\tilde{\sigma}_{n,j}^{-1}\tilde{\Phi}_{n,j}(O_{j}).$

We further consider the following remainder terms:

$\displaystyle
R_{1n}:=\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\frac{\mathbb{E}_{\overline{P}}\left[\left(1-\frac{\bar{q}_{0}(\tilde{g}_{i,j}(L_{i})\mid
L_{i})}{\tilde{q}_{n,j}(\tilde{g}_{i,j}(L_{i})\mid
L_{i})}\right)(\tilde{Q}_{n,j}(\tilde{g}_{i,j}(L_{i}),L_{i})-\mathbb{E}_{\overline{P}}[Y_{i}\mid\tilde{g}_{i,j}(L_{i}),L_{i}])\right]}{\tilde{\sigma}_{n,j}},$
$\displaystyle
R_{2n}:=\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\frac{\mathbb{E}_{\bar{P}}[\mathbb{E}_{\overline{P}}[Y_{i}\mid\tilde{g}_{i,j}(L_{i}),L_{i}]]-\Psi^{\textbf{opt}}}{\tilde{\sigma}_{n,j}}.$

The following assumptions from luedtke2016statistical are required for
$\hat{\Psi}_{n}^{\textbf{opt}}$ to be asymptotically normal.

  * G1.

$n-\ell_{n}\to\infty$ as $n\to\infty$.

  * G2.

(Lindeberg-like condition) $\forall\epsilon>0$,

$\displaystyle\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\mathbb{E}_{\overline{P}}\left[\left(\frac{\tilde{\Phi}_{n,j}(O_{j})}{\tilde{\sigma}_{n,j}}\right)^{2}I\left[\frac{|\tilde{\Phi}_{n,j}(O_{j})|}{\tilde{\sigma}_{n,j}}>\epsilon\sqrt{n-\ell_{n}}\right]\Bigg{|}O_{1},\ldots,O_{j-1}\right]=o_{\overline{P}}(1).$

  * G3.

$\frac{1}{n-\ell_{n}}\sum\limits_{j=\ell_{n}+1}^{n}\frac{\tilde{\sigma}^{2}_{0,n,j}}{\tilde{\sigma}^{2}_{n,j}}\to
1$ in probability.

  * G4.

$R_{1n}=o_{\overline{P}}(n^{-1/2})$.

  * G5.

$R_{2n}=o_{\overline{P}}(n^{-1/2})$.

The following theorem is an immediate consequence of Theorem 2 in
luedtke2016statistical.

######  Theorem S2 (luedtke2016statistical).

Under Conditions $F^{\dagger}$ and G,

$\Gamma_{n}\sqrt{n-\ell_{n}}(\hat{\Psi}_{n}^{\textbf{{opt}}}-\Psi^{\textbf{{opt}}})\overset{d}{\to}\mathcal{N}(0,1),$

as $n\to\infty$.

Theorem S2 implies the following interval is a valid $1-\alpha$ CI for
$\Psi(P_{0})$,

$\hat{\Psi}_{n}^{\textbf{opt}}\pm
z_{1-\alpha/2}\frac{\Gamma_{n}^{-1}}{\sqrt{n-\ell_{n}}},$

where $z_{1-\alpha/2}$ denotes the $1-\alpha/2$ quantile of a standard normal
random variable.

As the asymptotic normality of $\hat{\Psi}_{n}^{\textbf{opt}}$ is proved in
luedtke2016statistical using a martingale central limit theorem, the proof of
Theorem 2 in luedtke2016statistical is also valid in our interference setting,
as it does not rely on the i.i.d. assumption.

##  Appendix C Proofs of main results

Here, we re-state and prove formal results, which we organize in two main
sections. Appendix C.1 provides proofs for identification-related results
presented inSections 3 and 4 of the main text, and Appendices A and D.
Appendix C.2 provides proofs for estimation-related results in the main text
Section 5, and Appendix B.

###  C.1 Identification results

####  C.1.1 Finite-cluster parameters

We begin by establishing the following lemma.

######  Lemma S3.

Under $\mathcal{M}_{n}^{AB}$, the following conditions hold for all $l,a$, and
for all $i\neq j$:

$\displaystyle\\{L_{i}\in\mathbf{L}_{n}\\}\text{ are mutually independent.}$
(S26) $\displaystyle\\{Y_{i}\in\mathbf{Y}_{n}\mid L_{i}=l,A_{i}=a\\}\text{ are
mutually independent.}$ (S27)
$\displaystyle\mathbb{P}_{n}(L_{i}=l)=\mathbb{P}_{n}(L_{j}=l).$ (S28)
$\displaystyle\mathbb{P}_{n}(Y_{i}=y\mid
L_{i}=l,A_{i}=a)=\mathbb{P}_{n}(Y_{j}=y\mid L_{j}=l,A_{j}=a).$ (S29)

###### Proof.

We prove the proposition in (S26) by showing that
$\mathbb{P}_{n}(\mathbf{L}_{n}=\mathbf{l}_{n})=\prod\limits_{i=1}^{n}\mathbb{P}_{n}(L_{i}=l_{i})$.

$\displaystyle\mathbb{P}_{n}(\mathbf{L}_{n}=\mathbf{l}_{n})=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(L_{i}=l_{i}\mid
L_{1}=l_{1},\dots,L_{i-1}=l_{i-1})$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(f_{L_{i}}(\epsilon_{L_{i}})=l_{i}\mid
f_{L_{1}}(\epsilon_{L_{1}})=l_{1},\dots,f_{L_{i-1}}(\epsilon_{L_{i-1}})=l_{i-1})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(f_{L_{i}}(\epsilon_{L_{i}})=l_{i})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(L_{i}=l_{i}).$

The first equality follows by laws of probability. The second equality follows
by definition of the structural equation model and the restrictions on the
parents of $L_{i}$ in Condition A1 (Conditional noninterference). The third
equality follows by the mutual independence of the error terms for each
$L_{i}$. The final equality is by the definition of the structural equation
model. We prove the proposition in (S27) by showing that
$\mathbb{P}_{n}(\mathbf{Y}_{n}=\mathbf{y}_{n}\mid\mathbf{B}_{n}=\mathbf{b}_{n})=\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}=y_{i}\mid\mathbf{B}_{n}=\mathbf{b}_{n})$,
where we let $\mathbf{V}_{-i}\coloneqq\mathbf{V}_{n}\setminus V_{i}$ and let
$\overline{L}_{i}\coloneqq\\{L_{1},\dots,L_{i}\\}$.

$\displaystyle\mathbb{P}_{n}(\mathbf{Y}_{n}=\mathbf{y}_{n}\mid\mathbf{B}_{n}=\mathbf{b}_{n})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}=y_{i}\mid
B_{i}=b_{i},\mathbf{B}_{-i}=\mathbf{b}_{-i},\overline{Y}_{i-1}=\overline{y}_{i-1})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(f^{\prime}_{Y_{i}}(\epsilon_{Y_{i}})=y_{i}\mid
B_{i}=b_{i},f^{\prime}_{\mathbf{B}_{-i}}(\epsilon_{\mathbf{A}_{-i}},\epsilon_{\mathbf{L}_{-i}})=\mathbf{b}_{-i},\overline{f}^{\prime}_{Y_{i-1}}(\overline{\epsilon}_{Y_{i-1}})=\overline{y}_{i-1}).$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(f^{\prime}_{Y_{i}}(\epsilon_{Y_{i}})=y_{i}\mid
B_{i}=b_{i})$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}=y_{i}\mid\mathbf{B}_{n}=\mathbf{b}_{n}).$

The first equality follows by laws of probability. The second equality follows
by definition of the structural equation model and restrictions on the parent
sets of $Y_{i}$ in Condition $A1$, where we have let
$f^{\prime}_{Y_{i}}(\cdot)\equiv f_{Y_{i}}(\cdot,B_{i}=b_{i})$ denote the
structural equation for $Y_{i}$ with $B_{i}$ fixed to $b_{i}$,
$f^{\prime}_{\mathbf{B}_{-i}}(\cdot)$ denote the vector valued structural
equation for $\mathbf{B}_{-i}$ with $B_{i}$ fixed to $b_{i}$, and
$\overline{f}^{\prime}_{\mathbf{Y}_{i-1}}(\cdot)$ denote the vector valued
structural equation for $\overline{Y}_{i-1}$ conditional on
$\mathbf{B}_{-i}=\mathbf{b}_{-i}$. The third equality follows by the error
independence structure for $\epsilon_{Y_{i}}$ in Condition A1 (Conditional
noninterference). The final equality is again by the definition of the
structural equation model, and by reversing the step between the second and
third equality.

We prove the propositions in (S28) and (S29) by showing that the elements of
$\mathbf{L}_{n}$ are identically distributed variables and likewise that
$\\{Y_{i}:B_{i}=b\\}$ are identically distributed. First, by condition $A3$
(conditional identically distributed errors) we have the error terms for each
$L_{i}$ have some common distribution, say that of $V_{L}$. Then we have that
each $L_{i}=f_{L_{i}}(\epsilon_{L_{i}})=f_{L}(\epsilon_{L_{i}})\sim
f_{L}(V_{L})$, that is, each $L_{i}$ is some fixed function $f_{L}$ of some
variables with a fixed distribution, where $f_{L_{i}}=f_{L}$ follows by
condition $A2$ (structural invariance). Likewise, let $f^{\prime}_{Y_{i}}$ be
defined as above. By condition $A3$ and $A1$ we have the error terms for each
$Y_{i}$ have some common distribution conditional on $B_{i}=b_{i}$, say that
of $V_{Y}$. Then we have that, given $B_{i}=b_{i}$, each
$Y_{i}=f^{\prime}_{Y_{i}}(\epsilon_{Y_{i}})=f^{\prime}_{Y}(\epsilon_{Y_{i}})\sim
f^{\prime}_{Y}(V_{Y})$, that is, each $Y_{i}$ is some fixed function
$f^{\prime}_{Y}$ of some variables with a fixed distribution, where
$f^{\prime}_{Y_{i}}=f_{Y}$ follows by condition $A2$ (structural invariance).
Thus, $\mathbb{P}_{n}(L_{i}=l)$ and $\mathbb{P}_{n}(Y_{i}=y\mid B_{i}=b)$ are
thus described by common measures $Q_{L}$ and $Q_{Y}$, respectively. ∎

######  Corollary S6.

Under $\mathcal{M}_{n}^{AB}$, the following conditions hold for all $l,a$:

$\displaystyle\\{Y_{i}^{G_{n}}\in\mathbf{Y}_{n}^{G_{n}}\mid
L_{i}=l,A^{G_{n}+}_{i}=a\\}\text{ are mutually independent.}$ (S30)
$\displaystyle\mathbb{P}_{n}^{F}(Y_{i}^{\mathbf{a}_{n}}=Y_{i}^{a_{i}})=1\text{
for each }i,a_{i}\in\mathbf{a}_{n}.$ (S31)
$\displaystyle\\{Y_{i}^{\mathbf{a}_{n}}\in\mathbf{Y}_{n}^{\mathbf{a}_{n}}\\}\text{
are mutually independent.}$ (S32)

Property (S30) of Corollary S6 follows immediately by recognizing that
$Y_{i}^{G_{n}}$ given $L_{i}=l,A^{G_{n}+}_{i}=a$ is equal to
$f^{*}_{Y}(\epsilon_{Y_{i}})$ with $f^{*}_{Y}$ equivalent to $f_{Y}$ with
$A_{i}$ and $L_{i}$ fixed to $a$ and $l$. Property (S31) follows directly by
inspection of the structural equation for each $Y_{i}$ and its arguments when
$A_{i}$ is fixed to $a_{i}$. Property (S32) follows if
$\mathbb{P}_{n}(\mathbf{Y}_{n}^{\mathbf{a}_{n}}=\mathbf{y}_{n})=\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y^{\mathbf{a}_{n}}_{i}=y_{i})$.

$\displaystyle\mathbb{P}_{n}(\mathbf{Y}_{n}^{\mathbf{a}_{n}}=\mathbf{y}_{n})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}^{\mathbf{a}_{n}}=y_{i}\mid\overline{Y}^{\mathbf{a}_{n}}_{i-1}=\overline{y}_{i-1})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}^{a_{i}}=y_{i}\mid
Y_{1}^{a_{1}}=y_{1},\dots,Y_{i-1}^{a_{i-1}}=y_{i-1})$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}^{a_{i}}=y_{i})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}_{n}(Y_{i}^{\mathbf{a}_{n}}=y_{i}).$

The first equality follows by laws of probability. The second follows by
Property (S31). The third follows by independencies evident on the DAG
$\mathcal{G}(\mathbf{a}_{n})$ in Figure S2b (See the Proof of Proposition S5).
The fourth follows again by Property (S31).

The subsequent corollary follows simply by recognizing that, under the
conditions of the previous Lemma, $\mathbb{L}_{n}$, $\mathbb{O}_{n}$ given
$\mathbb{B}_{n}=\mathbb{b}_{n}$, and $\mathbb{O}_{n}^{G_{n}+}$ given
$\mathbb{B}^{G_{n}+}_{n}=\mathbb{b}_{n}$ may be formulated as grouped Poisson-
multinomial random variables.

######  Corollary S7.

Consider a law $\mathbb{P}_{n}$ under $\mathcal{M}^{AB}_{n}$. Then,

$\displaystyle\mathbb{Q}_{L,n}(\mathbb{l}_{n})$
$\displaystyle=\frac{n!}{\prod\limits_{l\in\mathcal{L}}\mathbb{l}_{n}(l)!}\prod\limits_{l\in\mathcal{L}}Q_{L}(l)^{\mathbb{l}_{n}(l)},$
(S33) $\displaystyle\mathbb{Q}_{L,n\mid l}(\mathbb{l}_{n})$
$\displaystyle=\frac{(n-1)!}{(\mathbb{l}_{n}(l)-1)!\prod\limits_{l^{\prime}\in\mathcal{L}\setminus
l}\mathbb{l}_{n}(l^{\prime})!}Q_{L}(l)^{\mathbb{l}_{n}(l)-1}\prod\limits_{l^{\prime}\in\mathcal{L}\setminus
l}Q_{L}(l^{\prime})^{\mathbb{l}_{n}(l^{\prime})},$ (S34)
$\displaystyle\mathbb{Q}_{Y,n}(\mathbb{o}_{n}\mid\mathbb{b}_{n})$
$\displaystyle=\prod\limits_{a,l}\frac{\mathbb{b}_{n}(a,l)!}{\prod\limits_{y\in\mathcal{Y}}\mathbb{o}_{n}(y,a,l)!}\prod\limits_{y\in\mathcal{Y}}Q_{Y}(y\mid
a,l)^{\mathbb{o}_{n}(y,a,l)},$ (S35)
$\displaystyle\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})$
$\displaystyle=\prod\limits_{a,l}\frac{\mathbb{b}_{n}(a,l)!}{\prod\limits_{y\in\mathcal{Y}}\mathbb{o}_{n}(y,a,l)!}\prod\limits_{y\in\mathcal{Y}}\mathbb{P}_{n}^{F}(Y^{G_{n}}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)^{\mathbb{o}_{n}(y,a,l)}.$ (S36)

Note that for such laws, $\\{\mathbb{Q}_{L,n},\mathbb{Q}_{L,n\mid
l},\mathbb{Q}_{Y,n}\\}$ are defined entirely in terms of $\\{Q_{L},Q_{Y}\\}$
and known quantities, as is $\mathbb{q}^{*}_{n}$ and $q^{*}_{i}$ (see
Proposition S3 in Appendix A).

######  Lemma S4.

Consider a law $\mathbb{P}_{n}^{F}\in\mathcal{M}^{AB}_{n}$. Then:

$\displaystyle\mathbb{P}_{n}^{F}(Y^{G_{n}}_{i}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)=Q_{Y}(y\mid a,l),$ (S37)

and

$\displaystyle\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})=\mathbb{Q}_{Y,n}(\mathbb{o}_{n}\mid\mathbb{b}_{n}).$
(S38)

###### Proof.

First we prove the proposition of (S37).

$\displaystyle\mathbb{P}_{n}^{F}(Y^{G_{n}}_{i}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{n}^{F}(f_{Y}(\epsilon_{Y_{i}},a,l)=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{n}^{F}(f_{Y}(\epsilon_{Y_{i}},a,l)=y\mid
G^{*}_{n}(\epsilon_{\mathbf{L}},\delta)=a,L_{i}=l)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{n}^{F}(f_{Y}(\epsilon_{Y_{i}},a,l)=y\mid
G^{\prime}_{n}(\epsilon_{\mathbf{L}_{-i}},\delta)=a,L_{i}=l)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{n}^{F}(f_{Y}(\epsilon_{Y_{i}},a,l)=y\mid L_{i}=l)$
$\displaystyle=$ $\displaystyle\mathbb{P}_{n}^{F}(Y^{a}=y\mid L_{i}=l)$
$\displaystyle=$ $\displaystyle\mathbb{P}_{n}^{F}(Y^{a}=y\mid
A_{i},=a,L_{i}=l)$ $\displaystyle=$ $\displaystyle\mathbb{P}_{n}^{F}(Y=y\mid
A_{i},=a,L_{i}=l)=Q_{Y}(y\mid a,l).$

The first equality follows by definition of the structural equation model, and
by conditions $A1$ (conditional noninterference) and $A2$ (structural
invariance). The second equality follows by definition of the CR $G_{n}$, and
by defining a function $G_{n}^{*}$ that is a composition of $G_{n}$ and
$f_{\mathbf{L}_{n}}$ . The third follows by defining a function
$G_{n}^{\prime}$ that is equivalent to $G^{*}_{n}$ with $L_{i}$ fixed to $l$.
The fourth equality follows by condition $A1$ and the exogeneity of the
randomizer term $\delta$. The fifth equality follows by definition of the
structural equation model. The sixth follows by condition $B1$ (no individual-
level confounding of outcomes). The final equalities follow again by the
structural equation model and then by Lemma S3.

The proposition in (S38) follows immediately by applying (S37) to (S36) and
noting (S35) in Lemma S7.

∎

######  Theorem (Re-statement of Theorem 1 of Section 3).

Consider a law $\mathbb{P}_{n}\in\mathcal{M}^{AB}_{n}$ and a regime
$G_{n}\in\Pi_{n}(\kappa_{n})$. Then

$\mathbb{E}[h(\mathbb{O}_{n}^{G_{n}+})]=\sum\limits_{\mathbb{o}_{n}}h(\mathbb{o}_{n})\mathbb{f}^{G_{n}}(\mathbb{o}_{n})\
\text{and}\
\mathbb{E}[h^{\prime}(O_{i}^{G_{n}+})]=\sum\limits_{o}h^{\prime}(o)f_{i}^{G_{n}}(o),$

whenever the right hand sides of the equations are well-defined.

###### Proof.

We first show that
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h^{\prime}(O_{i}^{G_{n}+})]=\sum\limits_{o}h^{\prime}(o)f_{i}^{G_{n}}(o)$.

$\displaystyle\mathbb{P}_{n}^{F}(Y^{G_{n}}_{i}=y,A^{G_{n}+}_{i}=a,L_{i}=l)$
$\displaystyle=$ $\displaystyle\mathbb{P}_{n}^{F}(Y^{G_{n}}_{i}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)\mathbb{P}_{n}^{F}(A^{G_{n}+}_{i}=a\mid
L_{i}=l)\mathbb{P}_{n}^{F}(L_{i}=l)$ $\displaystyle=$
$\displaystyle\mathbb{P}_{n}(Y_{i}=y\mid
A_{i}=a,L_{i}=l)\mathbb{P}_{n}^{F}(A^{G_{n}+}_{i}=a\mid
L_{i}=l)\mathbb{P}_{n}(L_{i}=l)$ $\displaystyle=$ $\displaystyle Q_{y}(y\mid
a,l)q^{*}_{i}(a\mid l)Q_{L}(l)=f_{i}^{G_{n}}(o).$

The first equality follows by laws of probability. The second follows by Lemma
S4. The third follows by Lemma S3 and definitions, and the final follows by
definition. Then the result follows immediately by definition of the
expectation.

Now we prove
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h(\mathbb{O}_{n}^{G_{n}+})]=\sum\limits_{\mathbb{o}_{n}}h(\mathbb{o}_{n})\mathbb{f}^{G_{n}}(\mathbb{o}_{n})$.

$\displaystyle\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n})$
$\displaystyle=\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n},\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n},\mathbb{L}_{n}=\mathbb{l}_{n})$
$\displaystyle=\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})\mathbb{P}_{n}^{F}(\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n}\mid\mathbb{L}_{n}=\mathbb{l}_{n})\mathbb{P}_{n}^{F}(\mathbb{L}_{n}=\mathbb{l}_{n})$
$\displaystyle=\mathbb{P}_{n}(\mathbb{O}_{n}=\mathbb{o}_{n}\mid\mathbb{B}_{n}=\mathbb{b}_{n})\mathbb{P}_{n}^{F}(\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n}\mid\mathbb{L}_{n}=\mathbb{l}_{n})\mathbb{P}_{n}(\mathbb{L}_{n}=\mathbb{l}_{n})=\mathbb{f}^{G_{n}}(\mathbb{o}_{n})$

The first equality follows because $\mathbb{B}_{n}^{G_{n}+}$ and
$\mathbb{L}_{n}$ are coarsenings of $\mathbb{O}_{n}^{G_{n}+}$. The second
follows by laws of probability. The third follows by Lemma S4, and the final
follows by definition. Then the result follows immediately. ∎

In Appendix A, we prove Proposition S1. Proposition S1 is a stronger version
of Proposition 1 in Section 3 of the main text except with respect to the
larger class of stochastic $L_{i}$-equitable regimes
$\Pi_{n}^{*}(\kappa_{n},L_{i})$ defined in Appendix A for which the
$L_{i}$-rank-preserving regimes $\Pi_{n}^{d}(\kappa_{n},L_{i},\Lambda)$
considered in Proposition 1 are a special sub-class.

######  Proposition (Re-statement of Proposition S5 of Appendix A).

Consider a law $\mathbb{P}_{n}^{F}\in\mathcal{M}^{AB}_{n}$, a real-valued
function $h^{\prime\prime}$ and a regime $G_{n}\in\Pi_{n}(\kappa_{n})$, with
$G_{n}$ a function of at most $\\{\mathbf{V}_{n},\delta\\}$, i.e.,
$G_{n}:\\{\mathcal{V}\\}^{n}\times[0,1]\rightarrow\\{0,1\\}^{n}$, and with
$V_{i}\coloneqq c(L_{i})$ for all $i$. Then
$\mathbb{E}_{\mathbb{P}^{F}_{n}}[h(\mathbb{W}_{n}^{G_{n}+})]=\sum\limits_{\mathbb{w}_{n}}h^{\prime\prime}(\mathbb{w}_{n})\mathbb{f}^{G_{n}}_{red}(\mathbb{w}_{n})$
whenever the right hand side of the equation is well-defined, and also that
$q^{*}_{i}(1\mid l)=\sum\limits_{\mathbb{v}_{n}}q^{*}_{i,\mathbb{v}_{n}}(1\mid
c(l))\mathbb{Q}_{V,n\mid c(l)}(\mathbb{v}_{n}).$

###### Proof.

Our proof for the reduced compositional g-formula result hinges on the
proposition that
$\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})=\mathbb{Q}^{*}_{Y,n}(\mathbb{w}_{n}\mid\mathbb{u}_{n})$.
We first show that, under a
$G_{n}:\\{\mathcal{V}\\}^{n}\times[0,1]\rightarrow\\{0,1\\}^{n}$, we have that
$\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{w}_{n}\mid\mathbb{U}_{n}^{G_{n}+}=\mathbb{u}_{n})$
decomposes isomorphically to
$\mathbb{P}_{n}^{F}(\mathbb{O}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})$,
as in expression (S36) of Corollary S7. In the following, we use graphical
arguments for convenience. In particular we posit a particular DAG model for
the joint distribution of a subset of the variables in
$\mathbf{O}^{G_{n}+}_{n}$, where graphical rules of d-separation can be used
to infer conditional independencies at all distributions in the DAG model.
When $\mathcal{M}_{n}^{AB}$ (appropriately marginalized) is a submodel of the
DAG model, then a d-separation in the DAG can be used to infer a conditional
independence in a law $\mathbb{P}_{n}^{F}\in\mathcal{M}_{n}^{AB}$.

We present such a DAG $\mathcal{G}(G_{n})$ in Figure S2a, representing a model
for the joint distribution of $\\{O_{i}^{G+},\mathbf{L}_{-i}\\}$, where we use
bi-directed arrows between nodes to denote lack of independence between error
terms for those nodes. To show that this DAG model contains all distributions
for $\\{O_{i}^{G+},\mathbf{L}_{-i}\\}$ in $\mathcal{M}_{n}^{AB}$, consider a
supermodel of $\mathcal{M}_{n}^{AB}$ that is defined only by assumption $A1$
(Conditional noninterference), which we denote by $\mathcal{M}_{n}^{A1}$. By
$A1$, only $\\{A_{i}^{G_{n}+},L_{i}\\}$ can have arrows into $Y_{i}^{G_{n}}$ .
By definition of $A_{i}^{G_{n}+}$, only $\\{V_{i},\mathbf{V}_{-i}\\}$ may have
directed arrows into $A_{i}^{G_{n}+}$ in the DAG model. By definition of
$V_{i}\coloneqq c(L_{i})$, only $L_{i}$ may have an arrow into $V_{i}$ and
likewise only $\mathbf{L}_{-i}$ may have an arrow into $\mathbf{V}_{-i}$. The
assumed error independencies between
$\\{\epsilon_{L_{1}},\dots,\epsilon_{L_{n}}\\}$ preclude any bidirected arrows
between $L_{i}$ and $\mathbf{L}_{-i}\\}$. The assumed error independencies
between $\epsilon_{Y_{i}}$ and
$\\{\boldsymbol{\epsilon}_{n}\setminus\epsilon_{i}\\}$ preclude any bidirected
arrows between $Y_{i}$ and $\mathbf{L}_{-i}$. A second DAG
$\mathcal{G}(\mathbf{a}_{n})$ in Figure S2b represents a model for the joint
distribution of
$\\{\mathbf{O}_{n}^{\mathbf{a}_{n}}\setminus\mathbf{a}_{n}\\}$, which is a
supermodel for $\mathcal{M}_{n}^{A1}$ by identical arguments.

Now we proceed with the proof. To show that
$\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})$
decomposes analogous to (S36) of Corollary S7 , we argue that
$\\{Y^{G_{n}}_{i}\in\mathbf{Y}^{G_{n}}_{n}\mid V_{i}=v,A^{G_{n}+}_{i}=a\\}$
are mutually independent. We establish the latter by showing that
$\mathbb{P}^{F}_{n}(\mathbf{Y}^{G_{n}}_{n}=\mathbf{y}_{n}\mid\mathbf{U}^{G_{n}+}_{n}=\mathbf{u}_{n})=\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y^{G_{n}}_{i}=y_{i}\mid\mathbf{U}^{G_{n}+}_{n}=\mathbf{u}_{n})$.

$\displaystyle\mathbb{P}^{F}_{n}(\mathbf{Y}^{G_{n}}_{n}=\mathbf{y}_{n}\mid\mathbf{U}^{G_{n}+}_{n}=\mathbf{u}_{n})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y_{i}\mid\mathbf{A}^{G_{n}+}_{n}=\mathbf{a}_{n},\mathbf{V}_{n}=\mathbf{v}_{n},\overline{Y}^{G_{n}}_{i-1}=\overline{y}_{i-1})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y_{i}\mid\mathbf{A}^{G_{n}+}_{n}=\mathbf{a}_{n},\mathbf{V}_{n}=\mathbf{v}_{n},\overline{Y}^{G_{n}}_{i-1}=\overline{y}_{i-1},\delta)$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{a_{i}}=y_{i}\mid\mathbf{A}^{G_{n}+}_{n}=\mathbf{a}_{n},\mathbf{V}_{n}=\mathbf{v}_{n},\overline{Y}^{\overline{a}_{i-1}}_{i-1}=\overline{y}_{i-1},\delta)$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{a_{i}}=y_{i}\mid\mathbf{V}_{n}=\mathbf{v}_{n},\overline{Y}^{\overline{a}_{i-1}}_{i-1}=\overline{y}_{i-1})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{a_{i}}=y_{i}\mid\mathbf{V}_{n}=\mathbf{v}_{n})$
$\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{a_{i}}=y_{i}\mid
A^{G_{n}+}_{i}=a_{i},\mathbf{V}_{n}=\mathbf{v}_{n})$ $\displaystyle=$
$\displaystyle\prod\limits_{i=1}^{n}\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y_{i}\mid\mathbf{U}^{G_{n}+}_{n}=\mathbf{u}_{n}).$

$A^{G_{n}+}_{i}$$\mathbf{L}_{-i}$${L}_{i}$$\mathbf{V}_{-i}$${V}_{i}$$Y^{G_{n}}_{i}$
(a)

$\mathbf{L}_{-i}$${L}_{i}$$\mathbf{V}_{-i}$${V}_{i}$$\mathbf{Y}^{\mathbf{a}_{-i}}_{-i}$$Y^{a_{i}}_{i}$
(b)

Figure S2: (a) DAG $\mathcal{G}({G_{n}})$ representing a model for the joint
distribution of $\\{O_{i}^{G_{n}+},\mathbf{L}_{-i}\\}$ under condition $A1$.
(b) DAG $\mathcal{G}(\mathbf{a}_{n})$ representing a model for the joint
distribution of $\\{\mathbf{O}_{n}^{\mathbf{a}_{n}}\setminus\mathbf{a}_{n}\\}$
under condition $A1$.

The first equality follows by laws of probability. The second equality follows
by the strict exogeneity of the randomizer, $\delta$. The third follows by
definition of the structural equation model for $Y_{i}$. The fourth follows by
noting that $\mathbf{A}^{G_{n}+}_{n}$ is a constant conditional on
$\mathbf{V}_{n}=\mathbf{v}_{n}$ and $\delta$, and then using the strict
exogeneity of the randomizer, to remove $\delta$ from the conditioning set.
The fifth equality follows from DAG $\mathcal{G}(\mathbf{a}_{n})$, where we
can read that $Y_{i}^{a_{i}}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathbf{Y}_{-i}^{\mathbf{a}_{-i}}\mid\mathbf{V}_{n}$.
The sixth equality follows by reversing the steps in the first through third
equalities. The final equality follows from DAG $\mathcal{G}(G_{n})$, where we
can read that $Y_{i}^{G_{n}+}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathbf{V}_{-i}\mid A_{i}^{G_{n}+},V_{i}$, and
again reversing the steps in the first throught third equalities.

Now we have that
$\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{o}_{n}\mid\mathbb{B}_{n}^{G_{n}+}=\mathbb{b}_{n})=\mathbb{Q}^{*}_{Y,n}(\mathbb{w}_{n}\mid\mathbb{u}_{n})$
if $\mathbb{P}_{n}(Y_{i}^{G_{n}+}=y\mid U^{G_{n}+}_{i}=u)=Q^{*}_{Y}(y\mid
a,v)$, and so we establish the latter.

$\displaystyle\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y\mid U^{G_{n}+}_{i}=u)=$
$\displaystyle\sum\limits_{l:c(l)=v}\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)\mathbb{P}^{F}_{n}(L_{i}=l\mid
A^{G_{n}+}_{i}=a,V_{i}=v)$ $\displaystyle=$
$\displaystyle\sum\limits_{l:c(l)=v}\mathbb{P}^{F}_{n}(Y_{i}^{G_{n}}=y\mid
A^{G_{n}+}_{i}=a,L_{i}=l)Q_{L\mid v}(l)$ $\displaystyle=$
$\displaystyle\sum\limits_{l:c(l)=v}Q_{Y}(y\mid a,l)Q_{L\mid v}(l)$

The first follows by laws of probability, and noting that $V_{i}$ is
degenerate conditional on $L_{i}$. The second follows from DAG
$\mathcal{G}(G_{n})$, where we can read that $L_{i}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A_{i}^{G_{n}+}\mid V_{i}$. The third follows
by Lemma S4.

Following arguments isomorphic to the proof of Theorem 1, and using the
immediately preceding results, we have that:

$\displaystyle\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{w}_{n})$
$\displaystyle=\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{w}_{n},\mathbb{U}_{n}^{G_{n}+}=\mathbb{u}_{n},\mathbb{V}_{n}=\mathbb{v}_{n})$
$\displaystyle=\mathbb{P}_{n}^{F}(\mathbb{W}_{n}^{G_{n}+}=\mathbb{w}_{n}\mid\mathbb{U}_{n}^{G_{n}+}=\mathbb{u}_{n})\mathbb{P}_{n}^{F}(\mathbb{U}_{n}^{G_{n}+}=\mathbb{u}_{n}\mid\mathbb{V}_{n}=\mathbb{v}_{n})\mathbb{P}_{n}^{F}(\mathbb{V}_{n}=\mathbb{v}_{n})$
$\displaystyle=\mathbb{P}_{n}(\mathbb{W}_{n}=\mathbb{w}_{n}\mid\mathbb{u}_{n}=\mathbb{u}_{n})\mathbb{P}_{n}^{F}(\mathbb{U}_{n}^{G_{n}+}=\mathbb{u}_{n}\mid\mathbb{V}_{n}=\mathbb{v}_{n})\mathbb{P}_{n}(\mathbb{V}_{n}=\mathbb{v}_{n})=\mathbb{f}_{red}^{G_{n}}(\mathbb{w}_{n}).$

This concludes the proof for the reduced compositional g-formula identity of
Proposition S5. Now we show that $q^{*}_{i}(1\mid
l)=\sum\limits_{\mathbb{v}_{n}}q^{*}_{i,\mathbb{v}_{n}}(1\mid
c(l))\mathbb{Q}_{V,n\mid c(l)}(\mathbb{v}_{n})$. The result follows
immediately upon noting the following identities:

$\displaystyle\mathbb{P}^{F}_{n}(A^{G_{n}+}_{i}=a\mid L_{i}=l)=$
$\displaystyle\mathbb{P}^{F}_{n}(A^{G_{n}+}_{i}=a\mid L_{i}=l,V_{i}=c(l))$
$\displaystyle=$ $\displaystyle\mathbb{P}^{F}_{n}(A^{G_{n}+}_{i}=a\mid
V_{i}=c(l))$ $\displaystyle=$
$\displaystyle\sum\limits_{\mathbb{v}_{n}}q^{*}_{i,\mathbb{v}_{n}}(1\mid
c(l))\mathbb{Q}_{V,n\mid c(l)}(\mathbb{v}_{n}).$

The first follows by noting that $V_{i}$ is degenerate conditional on $L_{i}$.
The second follows from DAG $\mathcal{G}(G_{n})$, where we can read that
$L_{i}\mathchoice{\mathrel{\hbox to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}A_{i}^{G_{n}+}\mid V_{i}$. The third follows
by laws of probability. This concludes the proof.

∎

####  C.1.2 Large-cluster parameters

In this appendix, we provide a proof for Theorem 2 of Section 4 that provides
an identification functional for the large population parameter
$\mathbb{E}_{\mathbb{P}^{F}_{0}}[\overline{Y}_{0}^{\mathbf{G}_{0}}]$:

######  Theorem (Re-statement of Theorem 2 of Section 4).

Consider an asymptotic law $\mathbb{P}_{0}$ and an asymptotic regime
$\mathbf{G}_{0}$ following conditions $D$ and $E$, respectively. Then,
$\mathbb{E}[\overline{Y}^{\mathbf{G}_{0}}_{0}]=\sum\limits_{o\in\mathcal{O}_{i}}yf^{\mathbf{G}_{0}}(o)$
whenever the right-hand side is well-defined.

######  Proposition (Re-statement of Proposition 2 of Section 4).

Consider an asymptotic law $\mathbb{P}_{0}$ and an asymptotic regime
$\mathbf{G}_{0}$ following conditions $D$ and $E$, respectively. Then,

$\displaystyle q^{*}_{0}(1\mid l)$
$\displaystyle=\begin{cases}\frac{\kappa^{*}-\mathbb{P}_{0}(\Lambda(L_{i})>\omega_{0})}{\mathbb{P}_{0}(\Lambda(L_{i})=\omega_{0})}&:\Lambda(l)=\omega_{0},\\\
I\Big{(}\Lambda(l)>\omega_{0}\Big{)}&:\text{otherwise},\end{cases}$ (S39)

where
$\omega_{0}\coloneqq\textup{inf}\bigg{\\{}c:\mathbb{P}_{0}(\Lambda(L_{i})>c)\leq\kappa^{*}\bigg{\\}}.$

We defined
$\mathbb{E}_{\mathbb{P}^{F}_{0}}[\overline{Y}^{\mathbf{G}_{0}}_{0}]$ as the
limit of a sequence of expectations of cluster average potential outcomes
$\underset{n\to\infty}{\lim}\mathbb{E}_{\mathbb{P}^{F}_{n}}[\overline{Y}^{{G}_{n}}_{n}]$
corresponding to clusters indexed by $n$, where this sequence is defined by an
asymptotic law $\mathbb{P}^{F}_{0}$ and an asymptotic regime $\mathbf{G}_{0}$.
Identification of these limiting parameters follows via restrictions on
$\mathbb{P}^{F}_{0}$ and $\mathbf{G}_{0}$, which we restate here for
convenience.

  * $D1.$

$\mathbb{P}_{n}\in\mathcal{M}_{n}^{AB}$, for all $n\in\mathbb{N}^{+}.$

  * $D2.$

$\\{f_{L_{i}},\mathbb{P}_{n}^{\epsilon_{L_{i}}},f_{Y_{i}},\mathbb{P}_{n}^{\epsilon_{Y_{i}\mid
pa(y)}}\\}\equiv\\{f_{L_{i}},\mathbb{P}_{n^{\prime}}^{\epsilon_{L_{i}}},f_{Y_{i}},\mathbb{P}_{n^{\prime}}^{\epsilon_{Y_{i}\mid
pa(y)}}\\}$ for all $n,n^{\prime}\in\mathbb{N}^{+}.$

  * $E1.$

For each $n$, $G_{n}\in\Pi^{d}_{n}(\kappa_{n},L_{i},\Lambda)$.

  * $E2.$

For each $n$, $\kappa_{n}=\lfloor n\times\kappa^{*}\rfloor$, with
$\kappa^{*}\in[0,1].$

Then consider the following sequence of identities:

$\displaystyle\underset{n\to\infty}{\lim}\mathbb{E}_{\mathbb{P}^{F}_{n}}[\overline{Y}^{{G}_{n}}_{n}],$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{\lim}\mathbb{E}_{\mathbb{P}^{F}_{n}}[{Y}^{{G}_{n}}_{i}],$
$\displaystyle=$
$\displaystyle\underset{n\to\infty}{\lim}\sum\limits_{o}yQ_{Y}(y\mid
a,l)q^{*}_{i}(a\mid l)Q_{L}(l),$ $\displaystyle=$
$\displaystyle\sum\limits_{o}yQ_{Y}(y\mid
a,l)\big{\\{}\underset{n\to\infty}{\lim}q^{*}_{i}(a\mid l)\big{\\}}Q_{L}(l).$
(S40)

The first equality holds by the proof of Proposition S1, because by conditions
$D1$ and $E1$, $G_{n}$ and $\mathbb{P}_{n}$ satisfy the conditions of
Proposition S1 for each $n$. The second equality holds by Theorem 1 because
condition $D1$ ensures that $\mathbb{P}_{n}$ satisfies its conditions. The
third equality holds by condition $D2$, which ensures that the measures
$Q_{Y}$ and $Q_{L}$ are invariant in $n$ for $\mathbb{P}_{0}$.

It thus remains to prove the following Lemma:

######  Lemma S5.

Consider an asymptotic law $\mathbb{P}_{0}^{F}$ and an asymptotic regime
$\mathbf{G}_{0}$ following conditions $D$ and $E$, respectively. Then
$\underset{n\to\infty}{\lim}q^{*}_{i}(a\mid l)=q^{*}_{0}(a\mid l)$ for all
$a,l$.

###### Proof.

First, note that by Lemma S3 and condition $D$ that
$\boldsymbol{\lim}\frac{\mathbb{L}_{n}(l)}{n}=Q_{L}(l)$ for all $l$. Second,
note that, by laws of probability,

$q^{*}_{i}(1\mid l)=\sum\limits_{\mathbb{l}_{n}}q^{*}_{i,\mathbb{l}_{n}}(1\mid
l)\mathbb{P}_{n}(\mathbb{L}_{n}=\mathbb{l}_{n}\mid L_{i}=l).$

Then define the sequence
$\mathbb{l}_{0}\coloneqq\\{\mathbb{l}^{0}_{1},\mathbb{l}^{0}_{2},\mathbb{l}^{0}_{3},\dots\\}$
such that
$\lim\mathbb{P}_{n}(\lVert\frac{\mathbb{l}^{0}_{n}}{n}-Q_{L}\rVert_{1}<\epsilon)=1$
for all $\epsilon>0$ with respect to $\mathbb{l}_{0}$ and $\mathbb{P}_{0}$. We
thus have that $\lim q^{*}_{i}(1\mid l)=\lim
q^{*}_{i,\mathbb{l}^{0}_{n}}(1\mid l)$. Recall the following result (from
Proposition S4 of Appendix A) for $q^{*}_{i,\mathbb{l}_{n}}$ under a
$L_{i}$-rank-preserving regime, where we had defined
$S_{\Lambda}(m)\coloneqq\sum\limits_{\mathcal{L}}\mathbb{l}_{n}(l)I(\Lambda(l)\leq
m)$, and $S^{-}_{\Lambda}(m)$ to be the corresponding function denoting the
number of such individuals with coarsened rank group strictly greater than
$m$:

$\displaystyle q^{*}_{i,\mathbb{l}_{n}}(1\mid l)$
$\displaystyle=\begin{cases}\frac{\kappa_{n}-S_{\Lambda}^{-}(\omega_{\mathbb{l}_{n}})}{S_{\Lambda}(\omega_{\mathbb{l}_{n}})-S_{\Lambda}^{-}(\omega_{\mathbb{l}_{n}})}&:\Lambda(l)=\omega_{\mathbb{l}_{n}},\\\
I\Big{(}\Lambda(l)>\omega_{\mathbb{l}_{n}}\Big{)}&:\text{otherwise}.\end{cases}$

with

$\displaystyle\omega_{\mathbb{l}_{n}}\coloneqq\textup{inf}\bigg{\\{}c\in\mathbb{R}:S_{\Lambda}(c)\leq\kappa_{n}\bigg{\\}}.$

Note that $\omega_{\mathbb{l}_{n}}$ is alternatively expressed as
$\textup{inf}\bigg{\\{}c\in\mathbb{R}:\sum\limits_{\mathcal{L}}\frac{\mathbb{l}_{n}(l)}{n}I(\Lambda(l)>c)\leq\frac{\kappa_{n}}{n}\bigg{\\}}$

Thus we have that:

$\displaystyle\underset{n\to\infty}{\lim}\omega_{\mathbb{l}_{n}^{0}}=$
$\displaystyle\underset{n\to\infty}{\lim}\textup{inf}\bigg{\\{}c\in\mathbb{R}:\sum\limits_{\mathcal{L}}\frac{\mathbb{l}^{0}_{n}(l)}{n}I(\Lambda(l)>c)\leq\frac{\kappa_{n}}{n}\bigg{\\}}$
$\displaystyle=$
$\displaystyle\textup{inf}\bigg{\\{}c\in\mathbb{R}:\sum\limits_{\mathcal{L}}Q_{L}(l)I(\Lambda(l)>c)\leq\kappa^{*}\bigg{\\}}=\omega_{0},$

where we use $E2$, so that
$\underset{n\to\infty}{\lim}\frac{\kappa_{n}}{n}=\kappa^{*}$

We can also make the following re-formulation:

$\displaystyle\frac{\kappa_{n}-S_{\Lambda}^{-}(\omega_{\mathbb{l}_{n}})}{S_{\Lambda}(\omega_{\mathbb{l}_{n}})-S_{\Lambda}^{-}(\omega_{\mathbb{l}_{n}})}=\frac{\frac{\kappa_{n}}{n}-\sum\limits_{\mathcal{L}}\frac{\mathbb{l}_{n}(l)}{n}I(\Lambda(l)>\omega_{\mathbb{l}_{n}})}{\sum\limits_{\mathcal{L}}\frac{\mathbb{l}_{n}(l)}{n}I(\Lambda(l)=\omega_{\mathbb{l}_{n}})}$

Thus we have that:

$\displaystyle\underset{n\to\infty}{\lim}\frac{\kappa_{n}-S_{\Lambda}^{-}(\omega_{\mathbb{l}^{0}_{n}})}{S_{\Lambda}(\omega_{\mathbb{l}^{0}_{n}})-S_{\Lambda}^{-}(\omega_{\mathbb{l}^{0}_{n}})}=$
$\displaystyle\frac{\kappa^{*}-\sum\limits_{\mathcal{L}}Q_{L}(l)I(\Lambda(l)>\omega_{0})}{\sum\limits_{\mathcal{L}}Q_{L}(l)I(\Lambda(l)=\omega_{0})}.$

Because $\Lambda$ is fixed and invariant to $n$ in the asymptotic
$\mathbf{G}_{0}$ by condition $E1$, then we have the result $\lim
q^{*}_{i,\mathbb{l}^{0}_{n}}(1\mid l)=q_{0}^{*}(a\mid l)$, thus also proving
Proposition 2 from Section 4. The result in Theorem 2 follows by plugging in
$q_{0}^{*}(a\mid l)$ for $\underset{n\to\infty}{\lim}q^{*}_{i}(a\mid l)$ in
expression (S40).

∎

####  C.1.3 Optimal regimes

We restate the optimal regime identification result of Section 4.

######  Proposition (Re-statement of Proposition 3 of Section 4).

Consider a law $\mathbb{P}_{n}\in\mathcal{M}_{n}^{AB}$. The CR
$G^{\mathbf{opt}}_{n}\equiv G_{n}\in\Pi_{n}(\kappa_{n})$ that maximizes
$\mathbb{E}[\overline{Y}_{n}^{G_{n}}]$ is the $V$-rank-preserving CR
$G_{n}\in\Pi_{n}^{d}(\kappa_{n},V_{i},\Lambda)$ characterized by
$\Lambda\equiv\Delta$.

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$, and an
asymptotic regime $\mathbf{G}_{0}$ following condition $E$. If
$G_{n^{*}}\in\mathbf{G}_{0}$ is such an optimal regime for
$\mathbb{P}_{n^{*}}(\mathbb{P}_{0})$ for some $n^{*}>1$ then
$G_{n}\in\mathbf{G}_{0}$ is the optimal such regime for
$\mathbb{P}_{n}(\mathbb{P}_{0})$ for all $n$. Let $q^{\mathbf{opt}}_{0}$
denote the intervention density under the asymptotic regime
$\mathbf{G}^{\mathbf{opt}}_{0}\equiv\Big{(}G^{\mathbf{opt}}_{n=1},G^{\mathbf{opt}}_{n=2},G^{\mathbf{opt}}_{n=3},\dots\Big{)}$.
Then, $\mathbb{E}[\overline{Y}_{0}^{\mathbf{G}^{\mathbf{opt}}_{0}}]$ is
identified as in Theorem 2 and its intervention density $q^{\mathbf{opt}}_{0}$
is identified as in (4) of Proposition 2 where we take
$\Lambda=\Delta_{0}\coloneqq\Delta$ and
$\omega_{0}=\eta_{0}\coloneqq\inf\\{c\mid\mathbb{P}_{0}(\Delta_{0}(L_{i})>c)\leq\kappa^{*}\\}.$

The formulation of $G^{\mathbf{opt}}_{n}$ for a
$\mathbb{P}^{F}_{n}\in\mathcal{M}^{AB}_{n}$ follows from recognizing the
optimization problem as a special case of the fractional knapsack problem of
Dantzig (1957) where rewards for each individual $i$ are characterized by
$\Delta_{\mathbb{P}_{n}^{F}}(L_{i})$, thus corresponding to a regime in
$\Pi_{n}^{d}(\kappa_{n},V_{i},\Lambda)$ with
$\Lambda=\Delta_{\mathbb{P}_{n}^{F}}(L_{i})$. The large-cluster result follows
from the fact that for an asymptotic law $\mathbb{P}_{0}$ under conditions
$D$, $\Delta_{\mathbb{P}_{n}^{F}}(L_{i})$ is fixed to some
$\Delta_{\mathbb{P}_{0}^{F}}$ for all $n$. The identification result for
$q^{\mathbf{opt}}_{0}$ follows simply by substituting
$\Delta_{0}\coloneqq\Delta_{\mathbb{P}_{0}^{F}}$ for $\Lambda$ in $q_{0}^{*}$,
as formulated in Proposition 2.

Note that Proposition S8 follows simply by an extension of the fractional
knapsack problem that only allocates resources with non-negative rewards,
which is captured by a generalized rank-and-treat regime that only provides
treatment when $\Delta_{\mathbb{P}_{n}^{F}}(l_{i})$ is non-negative.

###  C.2 Inference results

####  C.2.1 Asymptotic consistency for finite-cluster parameters

We re-state the main asymptotic consistency result of Theorem 3 from Section
5.

######  Theorem (Re-statement of Theorem 3 of Section 5).

Consider $\mathbb{P}_{0}$ following conditions $D$ and
$G_{n^{*}}\in\Pi_{n^{\ast}}(\kappa_{n^{\ast}})$ following $C1$. Then,

$\displaystyle\boldsymbol{\lim}\sum\limits_{\mathbb{o}_{n^{\ast}}}h(\mathbb{o}_{n^{\ast}})\tilde{\mathbb{f}}^{G_{n^{\ast}}}(\mathbb{o}_{n^{*}})=\mathbb{E}[h(\mathbb{O}_{n^{\ast}}^{G_{n^{\ast}}+})],$
(S41)
$\displaystyle\boldsymbol{\lim}\sum\limits_{o}h^{\prime}(o)\tilde{f}_{i}^{G_{n^{*}}}(o)=\mathbb{E}[h^{\prime}(O_{i}^{G_{n^{*}}+})].$
(S42)

###### Proof.

Theorem 3 follows if
$\boldsymbol{\lim}\tilde{\mathbb{f}}^{G_{n^{\ast}}}(\mathbb{o}_{n^{*}})={\mathbb{f}}^{G_{n^{\ast}}}(\mathbb{o}_{n^{*}})$
and likewise
$\boldsymbol{\lim}\tilde{f}_{n^{*},i}^{G_{n^{*}}}(o)={f}_{i}^{G_{n^{*}}}(o)$.

First we remind the reader that ${f}_{i}^{G_{n^{*}}}$ and
${\mathbb{f}}^{G_{n^{\ast}}}$ are both smooth functionals of $Q_{L}$ and
$Q_{Y}$ that are fixed to $n^{*}$ across the index $n$ of the asymptotic law
$\mathbb{P}_{0}$. Furthermore, we remind the reader that
$\tilde{f}_{i}^{G_{n^{*}}}$ and $\tilde{\mathbb{f}}^{G_{n^{\ast}}}$ are simply
plug-in estimators of ${f}_{i}^{G_{n^{*}}}$ and ${\mathbb{f}}^{G_{n^{\ast}}}$,
with respect to $\tilde{Q}_{L}$ and $\tilde{Q}_{Y}$. Thus, our result follows
via Slutsky’s theorem if $\boldsymbol{\lim}\tilde{Q}_{L}(l)=Q_{L}(l)$ for all
$l$ and if $\boldsymbol{\lim}\tilde{Q}_{Y}(y\mid a,l)={Q}_{Y}(y\mid a,l)$ for
all $a,l$ such that $\overline{q}^{*}_{n}(a\mid
l)Q_{L}(l)>0\implies\overline{q}_{n}(a\mid l)>0.$

Under an asymptotic law $\mathbb{P}^{F}_{0}$ following condition $D$ we have
that $\boldsymbol{\lim}\frac{\mathbb{L}_{n}(l)}{n}=Q_{L}(l)$ for all $l$,
since
$\boldsymbol{\lim}\frac{\mathbb{L}_{n}(l)}{n}=\frac{1}{n}\sum\limits_{i=1}^{n}I(L_{i}=l)$
is an empirical average of iid random variables (see Lemma S3), and Condition
$D$ ensures that $Q_{L}$ remains fixed across laws in $\mathbb{P}^{F}_{0}$.
$\frac{\mathbb{O}_{n}(y,a,l)}{\mathbb{B}_{n}(a,l)}$ is also an empirical
average of a randomly varying number of iid random variables. The
counterexample of Appendix B illustrates that weak positivity condition $C0$
is not sufficient for consistency of
$\boldsymbol{\lim}\frac{\mathbb{O}_{n}(y,a,l)}{\mathbb{B}_{n}(a,l)}$ because
$\mathbb{B}_{n}(a,l)$ under these conditions will not necessarily grow with
some rate that is a function of $n$. Strong positivity condition $C1$ ensures
that $\mathbb{B}_{n}(a,l)$ indeed grows at such a rate so that the number of
individuals in this set will almost surely approach infinitely many as $n$
grows. Thus we have that
$\boldsymbol{\lim}\frac{\mathbb{O}_{n}(y,a,l)}{\mathbb{B}_{n}(a,l)}={Q}_{Y}(y\mid
a,l)$.

∎

####  C.2.2 Semiparametric efficiency for regular large-cluster parameters

######  Theorem (Re-statement of Theorem S1 from Appendix B).

Consider an asymptotic law $\mathbb{P}_{0}$ following condition $D$ and an
asymptotic optimal regime $\mathbf{G}_{0}^{\mathbf{opt}}$ following condition
$E$. Suppose further that condition $F^{*}$ holds. Then the following
properties also hold:

  * (1.)

$\Psi^{\mathbf{opt}}$ is a pathwise differentiable parameter with efficient
non-parametric influence function
$\Phi_{0}(\overline{P}_{0},g^{\mathbf{opt}}_{i},\eta_{0})$.

  * (2.)

$\tilde{\Psi}^{\mathbf{opt}}_{n}$ is the asymptotically-efficient, regular and
asymptotically linear estimator of $\Psi^{\mathbf{opt}}$.

  * (3.)

An asymptotically valid two-sided $1-\alpha$ confidence interval is given by

$\tilde{\Psi}^{\mathbf{opt}}_{n}\pm
z_{1-\alpha/2}\frac{\tilde{\sigma}^{2,\mathbf{opt}}_{n}}{\sqrt{n}}.$

###### Proof.

We adapt the proof from luedtke2016optimal. Let
$Pf:=\mathbb{E}_{P}[f(O_{i})]$. Asymptotic linearity and normality, will not
follow from standard arguments because the elements of $\mathbf{O}_{n}$ are
not iid. The rest of the proof follows from luedtke2016optimal, where in a
slight abuse of notation, we at times let $\Phi_{0}(\cdot)$ denote the random
variable $\Phi_{0}(\cdot)(O_{i})$, and likewise for $\Phi_{1}(\cdot)$ and
$\Phi_{2}(\cdot)$.

luedtke2016optimal show that

$\displaystyle\tilde{\Psi}_{n}^{\mathbf{opt}}-\Psi^{\mathbf{opt}}$
$\displaystyle=(\tilde{P}_{n}-\bar{P}_{0})\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta)+(\tilde{P}_{n}-\bar{P})(\Phi_{0}(\tilde{P}_{n},\tilde{g}_{i}^{\mathbf{opt}},\eta)-\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta))$
$\displaystyle+R_{0}(\tilde{g}_{i}^{\mathbf{opt}},\tilde{P}_{n})+o_{\bar{P}}(n^{-1/2}).$
(S43)

.

The second and third terms of Equation (S43) are $o_{\bar{P}}(n^{-1/2})$ by
the assumptions of Theorem 4 in luedtke2016optimal (a subset of $F^{*}$).
luedtke2016optimal use the central limit theorem (and hence the i.i.d.
assumption) to show asymptotic normality of
$\sqrt{n}(\tilde{P}_{n}-\bar{P})\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta)$.
As the elements of $\mathbf{O}_{n}$ are not iid, we cannot use the central
limit theorem to obtain the same conclusion; we need to show that
$\sqrt{n}(\tilde{P}_{n}-\bar{P})\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta)$
is asymptotically normal.

Recall the definition of $\Phi_{0}$:

$\displaystyle\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta)(o)$
$\displaystyle=\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})(o)+\Phi_{2}(g_{i}^{\mathbf{opt}},\eta)(o)$
$\displaystyle=\frac{I(a=g_{i}^{\mathbf{opt}}(l))}{\bar{q}_{0}(a\mid
l)}(y-\mathbb{E}_{\bar{P}}(Y_{i}\mid A_{i}=a,L_{i}=l))$
$\displaystyle+\\{\mathbb{E}_{\bar{P}}(Y_{i}\mid
A_{i}=g_{i}^{\mathbf{opt}}(l),L_{i}=l)-\mathbb{E}_{\bar{P}}[\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})]\\}$
$\displaystyle-\eta(g_{i}^{\mathbf{opt}}(l)-\kappa^{*}).$ (S44)

The first equality follows from the definition of $\Phi_{0}$ as
$g_{i}^{\mathbf{opt}}$ is deterministic by Condition $F3^{*}$ (and thus not a
function of $\delta)$ so we can omit the expectation over $\delta$ in
expression S17, and the second equality follows from the definitions of
$\Phi_{1}$ and $\Phi_{2}$.

By the law of total expectation and as $\tilde{P}_{n}$ has the empirical
distribution of $L_{i}$ marginally (given by $\mathbb{L}_{n}$),

$\displaystyle\tilde{P}_{n}\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})=\frac{1}{n}\sum_{i=1}^{n}\tilde{P}_{n}(\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})\mid
L_{i}).$

However,

$\displaystyle\tilde{P}_{n}(\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})\mid L_{i})$
$\displaystyle=\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}(Y_{i}-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))\ \Big{|}\ L_{i}\right]$
$\displaystyle=\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}Y_{i}\ \Big{|}\
L_{i}\right]-\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\mathbb{E}_{\bar{P}}(Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i})\
\Big{|}\ L_{i}\right].$ (S45)

The first equality comes from the definition of $\Phi_{1}$, and the second
equality follows from linearity of the expectation.

We focus on the first term of Equation (S45).

$\displaystyle\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}Y_{i}\ \Big{|}\ L_{i}\right]$
$\displaystyle=\frac{1}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\tilde{P}_{n}\left[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))Y_{i}\ \Big{|}\
L_{i}\right].$

The equality follows by linearity of the expectation as
$\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid L_{i})$ is a constant conditional
on $L_{i}$.

Then,

$\displaystyle\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}Y_{i}\ \Big{|}\ L_{i}\right]$
$\displaystyle=\frac{1}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\tilde{P}_{n}\left[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))Y_{i}\ \Big{|}\
L_{i}\right]$
$\displaystyle=\frac{1}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i}).$

The first equality is the same as the previous equation, and the second
inequality follows as

$\displaystyle\tilde{P}_{n}[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))Y_{i}\mid
L_{i}]$
$\displaystyle=\tilde{P}_{n}[\tilde{P}_{n}[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))Y_{i}\mid
A_{i},L_{i}]\mid L_{i}]$
$\displaystyle=\tilde{P}_{n}[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))\tilde{P}_{n}[Y_{i}\mid
A_{i},L_{i}]\mid L_{i}]$
$\displaystyle=\tilde{P}_{n}[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))\tilde{P}_{n}[Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}]\mid L_{i}]$
$\displaystyle=\tilde{P}_{n}[I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))\mid
L_{i}]\tilde{P}_{n}[Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i}]$
$\displaystyle=\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})\tilde{P}_{n}[Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i}].$

The first equality follows from the law of total expectation; the second
equality follows as $I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))$ is a constant
conditional on $A_{i}$ and $L_{i}$; the third equality follows as
$I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))\tilde{P}_{n}[Y_{i}\mid
A_{i},L_{i}]=I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))\tilde{P}_{n}[Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}]$ almost surely; the fourth equality follows
as $\tilde{P}_{n}[Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i}]$ is a constant;
the last equality follows by the definition of $\tilde{q}_{n}$.

However,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}Y_{i}\ \Big{|}\ L_{i}\right]$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}-1\right)\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})-\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\right)\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}).$

The first equality follows from the previous equation; the second equality
follows from adding and subtracting
$n^{-1}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$; the third equality follows from
replacing $1$ by $\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})/\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid L_{i})$.

Similarly,

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\tilde{P}_{n}\left[\frac{I(A_{i}=g_{i}^{\mathbf{opt}}(L_{i}))}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\mathbb{E}_{\bar{P}}(Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i})\
\Big{|}\ L_{i}\right]$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})-\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\right)\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}).$

This equality follows as $\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$ and
$\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid L_{i})$ are constants, by
definition of $\tilde{q}_{n}$, and by adding and subtracting
$n^{-1}\sum_{i=1}^{n}\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$.

Hence,

$\displaystyle\tilde{P}_{n}\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})$
$\displaystyle=\frac{1}{n}\sum_{i=1}^{n}\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})-\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})).$

This equality follows from our previous derivations.

Using similar derivations based on the law of total expectation, we can
replace empirical averages and expectations under $\tilde{P}_{n}$ with
expectations with respect to $\bar{P}$, and replace $\tilde{q}_{n}$ with
$\bar{q}_{0}$. Hence, we find
$\bar{P}\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})=0$. Thus,

$\displaystyle(\tilde{P}_{n}-\bar{P})\Phi_{0}(\bar{P},g_{i}^{\mathbf{opt}},\eta)$
$\displaystyle=(\tilde{P}_{n}-\bar{P})\Phi_{1}(\bar{P},g_{i}^{\mathbf{opt}})+(\tilde{P}_{n}-\bar{P})\Phi_{2}(g_{i}^{\mathbf{opt}},\eta)-(\tilde{P}_{n}-\bar{P})[\eta(g_{i}^{\mathbf{opt}}-\kappa^{*})]$
$\displaystyle=(\tilde{P}_{n}-\bar{P})(\Phi_{2}(g_{i}^{\mathbf{opt}},\eta)-\eta(g_{i}^{\mathbf{opt}}(L_{i})-\kappa^{*}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left(\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})-\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\right)(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$

The first equality follows from definitions of $\Phi_{0}$, $\Phi_{1}$, and
$\Phi_{2}$; the second equality follows from our previous derivations.

However,

$\displaystyle\sqrt{n}(\tilde{P}_{n}-\bar{P})(\Phi_{2}(g_{i}^{\mathbf{opt}},\eta)(L_{i})-\eta(g_{i}^{\mathbf{opt}}(L_{i})-\kappa^{*}))$

is asymptotically normal by the central limit theorem, as the $L_{i}$ are
i.i.d and $\Phi_{2}(g_{i}^{\mathbf{opt}},\eta)$ is only a function of $L_{i}$.
Furthermore,

$\displaystyle\sqrt{n}\left[\frac{1}{n}\sum_{i=1}^{n}\bigg{(}\frac{\tilde{q}_{n}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})-\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}{\bar{q}_{0}(g_{i}^{\mathbf{opt}}(L_{i})\mid
L_{i})}\right)(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))$
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}(\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i})-\mathbb{E}_{\bar{P}}(Y_{i}\mid
g_{i}^{\mathbf{opt}}(L_{i}),L_{i}))\bigg{]}=o_{\bar{P}}(n^{-1/2}),$

if $\mathbb{E}_{\tilde{P}_{n}}(Y_{i}\mid g_{i}^{\mathbf{opt}}(L_{i}),L_{i})$
verifies conditions analogous to Assumptions 4 and 5 in luedtke2016optimal,
that is, Condition $F7^{*}$, and $\tilde{q}_{n}$ is bounded. ∎

######  Remark.

The proof of Proposition S7 is a direct extension of this proof for a $g_{i}$
defined by a known $\Lambda$ function instead of $\Delta_{0}$.

####  C.2.3 Sub-cluster positivity

######  Lemma (Re-statment of Lemma S1 from Appendix B).

Consider an asymptotic law $\mathbb{P}_{0}^{F}$ following conditions $D$ and
sub-cluster positivity condition $C2$. Then $\tilde{q}_{n}$ is
$L^{2}(\mathbb{P}_{0})$ consistent for $\overline{q}_{0}$, i.e.
$\mathbb{E}[\lVert\tilde{q}_{n}-\overline{q}_{0}\rVert_{2}^{2}]=o(1).$

###### Proof.

Consider the following equations.

$\displaystyle\mathbb{E}_{\mathbb{P}_{0}^{F}}[(\tilde{q}_{n}(a\mid
l)-\bar{q}_{0}(a\mid l))^{2}]=o(1),$ (S46) $\displaystyle\tilde{q}_{n}(a\mid
l)-\bar{q}_{0}(a\mid l)=o_{\mathbb{P}_{0}}(1).$ (S47)

We aim to prove (S46), which implies (S47), the limit condition in the Strong
Positivity Conditions $C1$ and $C1^{*}$, which we consider in Appendix B.2.
For $L^{2}$-consistency, we need (S46).

We prove Equation (S46) in several steps. We show that $\tilde{q}_{n}(a|l)$
can be expressed as a sum of $m_{n}$ functions of sub-cluster variables in
Lemma S7, that is, the number of terms in the sum grows proportionately to
$n$. However, we cannot apply the usual central limit theorem, since sub-
cluster sizes $\mathbb{W}_{n}(c)$ are not i.i.d. Therefore, we show
$\tilde{q}_{n}(a|l)$ can be expressed as a sum of functions of
$\mathbb{W}_{n}(c)$ and independent error terms, and can be approximated with
a normal distribution in Lemma S6. Finally, we prove Equation (S46) holds
using this approximation.

Lemma S6 is used to approximate the mean and variance of $\tilde{q}_{n}(a|l)$
or any general sum of functions of the $\mathbb{W}_{n}(c)$ and independent
error terms. When Conditions $D$ and $C2$ hold, these approximations will be
sufficient as $n$ grows to prove that Equation (S46) holds as we show after
Lemma S7.

For a given $n$, under sub-cluster positivity $C2$, we have that
$\mathbb{W}_{n}(c)$ are only defined for $c\in\\{1,\dots,m_{n}<n\\}$. We want
to use the central limit theorem of [Theorem 27.4]billingsley_probability_1995
to find estimates of the first two moments of the sum of functions of the sub-
cluster sizes $\mathbb{W}_{n}(c)$. The CLT of [Theorem
27.4]billingsley_probability_1995 requires a stochastic process that is
defined for sub-clusters $c$ beyond the original $m_{n}$. Hence, we define for
each $n$, $\mathbb{W}_{n}(c)$ for $c\in\\{m_{n}+1,\dots,2n\\}$. In particular,
we let $\mathbb{W}_{n}(m_{n}+1),\ldots,\mathbb{W}_{n}(n)$ be defined as random
variables with degenerate distributions at $0$, and let
$\mathbb{W}_{n}(n+1),\ldots,\mathbb{W}_{n}(2n)$ be another independent set of
sub-cluster sizes of a cluster of size $n$ with the same joint distribution as
$\mathbb{W}_{n}(1),\ldots,\mathbb{W}_{n}(n)$. We define
$\mathbb{W}_{n}(jn+1),\ldots,\mathbb{W}_{n}(\\{j+1\\}n)$ analogously for
$j=2,3,4,\dots$. This construction is used only to give us approximations of
the mean and variance of a sum of functions of the sub-cluster sizes, but we
will need further argumentation to show that Equation (S46) holds.

######  Lemma S6.

Consider an asymptotic law $\mathbb{P}_{0}^{F}$ following conditions D and
sub-cluster positivity condition C2. Suppose that we have a function
$f_{n}^{\prime}$ of $\mathbb{W}_{n}(c)$ and the $\epsilon_{\mathbb{W}_{n},c}$,
where the elements of
$\\{{\epsilon}_{\mathbb{w}_{n},1},\dots,{\epsilon}_{\mathbb{w}_{n},m_{n}}\\}$
are mutually independent and jointly independent of $\mathbb{W}_{n}$. Then,
for any $n^{*}$ there exist $\mu_{n^{*}}\in\mathbb{R}$ and $\sigma_{n^{*}}>0$
such that

$\displaystyle\sqrt{n^{\prime}}\bigg{(}\sum\limits_{c=1}^{n^{\prime}}f_{n^{*}}^{\prime}(\mathbb{W}_{n^{*}}(c),\epsilon_{\mathbb{W}_{n^{*}},c})-\mu_{n^{*}}\bigg{)}\underset{n^{\prime}\to\infty}{\to}\mathcal{N}(0,\sigma_{n^{*}}^{2}).$
(S48)

###### Proof.

We want to apply a central limit theorem for dependent variables [Theorem
27.4]billingsley_probability_1995. This requires convergence of a different
alpha mixing coefficient, namely the strong alpha mixing coefficient. Let
$X_{1},X_{2},\ldots$ be some stationary stochastic process. Let
$\alpha_{\mathbb{P}_{0}}$ be the limit of $\alpha_{\mathbb{P}_{n}}$. We define
the strong alpha $k$-mixing coefficient of the $X_{i}$ as

$\displaystyle\alpha_{k}:=\alpha_{\mathbb{P}_{0}}((X_{1},\ldots,X_{n^{\prime}}),(X_{n^{\prime}+k},\ldots))=\sup_{\begin{subarray}{c}A\in\sigma(X_{1},\ldots,X_{n^{\prime}}),\\\
B\in\sigma(X_{n^{\prime}+k},\ldots)\end{subarray}}\mid\mathbb{P}_{0}^{F}(A\cap
B)-\mathbb{P}_{0}^{F}(A)\mathbb{P}_{0}^{F}(B)\mid.$

The strong alpha $k$-mixing coefficient $\alpha_{k}$ is the alpha mixing
coefficient of random variables $(X_{1},\ldots,X_{n^{\prime}})$ and
$(X_{n^{\prime}+k},\ldots)$. This condition differs from Condition $C2.3$,
which is not required for this result.

To apply the billingsley_probability_1995 central limit theorem, we need the
stochastic process $\\{X_{i}\\}_{i=1}^{\infty}$ to be stationary and
$\alpha_{k}=O({k}^{-5})$. In particular, this holds if the stochastic process
is $m$-dependent for any $m\in\mathbb{N}$, that is,
$(X_{1},\ldots,X_{n^{\prime}})$ is independent of
$(X_{n^{\prime}+k},\ldots,X_{n^{\prime}+k+l})$ for any $l\in\mathbb{N}$
whenever $k\geq m$. We want this condition to hold for the stochastic process
given by
$\\{f_{n}^{\prime}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c})\\}_{c=1}^{\infty}$.
This sequence is $n$-dependent by construction, and thus verifies
$\alpha_{k}=O({k}^{-5})$. Furthermore, the sequence is stationary by
definition of the sub-clusters beyond $n$; a negligible proportion can be
allowed to not be identically distributed, which we guarantee by sub-cluster
positivity conditions $C2.1$ for the degenerate $\frac{n-m_{n}}{n}$ proportion
of sub-clusters and $C2.2$ for the $m_{n}$ first sub-clusters.

∎

######  Lemma S7.

Consider an asymptotic law $\mathbb{P}_{0}$ following conditions $D$ and sub-
cluster positivity condition $C2$. For each $n$, there exists a transformation
$\boldsymbol{\epsilon}_{\mathbb{w}_{n}}\equiv\\{{\epsilon}_{\mathbb{w}_{n},1},\dots,{\epsilon}_{\mathbb{w}_{n},m_{n}}\\}$
of $\mathbf{O}_{n}$ and a fixed function $f_{n}$ such that the following
properties hold:

  * (1)

For each $c\in\\{1,\dots,m_{n}\\}$,
${\epsilon}_{\mathbb{w}_{n},c}\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\mathbb{W}_{n}(c)$;

  * (2)

The elements of $\boldsymbol{\epsilon}_{\mathbb{w}_{n}}$ are mutually
independent; and

  * (3)

$\tilde{q}_{n}(a\mid
l)=\frac{1}{m_{n}}\sum\limits_{c=1}^{m_{n}}\frac{m_{n}}{\mathbb{L}_{n}(l)}f_{n}(\mathbb{W}_{n}(c),{\epsilon}_{\mathbb{w}_{n},c})$.

###### Proof.

Let $\mathbb{B}_{n,c}(a,l)$ denote the random variable for a cluster of size
$n$ indicating the number of individuals in sub-cluster $c$ with treatment
level $a$ and covariate level $l$, and let $\mathbf{L}_{n,c}$ denote the
matrix of covariates for individuals in sub-cluster $c$.

By definition then we have that:

$\tilde{q}_{n}(a\mid
l)=\frac{1}{m_{n}}\sum\limits_{c=1}^{m_{n}}\frac{m_{n}}{\mathbb{L}_{n}(l)}\mathbb{B}_{n,c}(a,l).$

We have by sub-cluster positivity condition $C2.4$ that
$\mathbb{B}_{n}\equiv\mathbb{B}^{G_{n}+}_{n}$ for a sub-cluster regime
$G_{n}\in\Pi^{F}_{n,m_{n}}(\mathbf{G}_{m})$. We thus also have that
$\mathbb{B}_{n,c}(a,l)=\mathbb{B}^{G_{n}+}_{n,c}(a,l)$. Note that by condition
$C2.1$, sub-clusters have size bounded by $m$. A defining feature of sub-
cluster regimes is that treatment in each cluster is assigned according to a
regime in the $m\times m$ matrix of regimes $\mathbf{G}_{m}$ \- the regime
implemented in sub-cluster $c$ corresponds to the element in $\mathbf{G}_{m}$
in the row indicated by cluster size, $\mathbb{W}_{n}(c)$, and the column
indicated by the number of treatment units available to that sub-cluster,
$\kappa_{n,c}$. Therefore,
$\mathbb{B}^{G_{n}+}_{n,c}(a,l)\equiv\mathbb{B}^{G_{\mathbb{W}_{n}(c)}^{*}+}_{\mathbb{W}_{n}(c)}(a,l)$,
with
$G_{\mathbb{W}_{n}(c)}^{*}\equiv[\mathbf{G}_{m}]_{\mathbb{W}_{n}(c),\kappa_{n,c}}$.
By definition, then we have that:

$\displaystyle\mathbb{B}_{n,c}(a,l)=\sum\limits_{i=1}^{\mathbb{W}_{n}(c)}I\Bigg{(}[\mathbf{G}_{m}]_{\mathbb{W}_{n}(c),\kappa_{n,c},i}(\mathbf{L}_{n,c})=a,L_{i}=l\Bigg{)}.$
(S49)

First we prove property $(1)$ of Lemma S7. Note that $\mathbf{L}_{n,c}$ is a
function of $\epsilon_{\mathbf{L}_{n}}\in\boldsymbol{\epsilon}_{n}$, and that
$\mathbb{W}_{n}(c)$ is a function of $\epsilon_{\mathbf{W}_{n}}$. Take
$\epsilon_{\mathbb{w}_{n},c}\subset\\{\boldsymbol{\epsilon}_{n},\kappa_{n,c}\\}$
to be the subset of error terms that generate $\kappa_{n,c}$,
$\mathbf{L}_{n,c}$, and $L_{i}$, which with $\mathbb{W}_{n}(c)$ comprise the
random arguments to the function generating $\mathbb{B}_{n,c}(a,l)$, the right
hand side of (S49). Due to sub-cluster positivity condition $C2.5$ and $C2.6$
then $\mathbb{W}_{n}(c)\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}\epsilon_{\mathbb{w}_{n},c}$. Thus, we have
property $(1)$ of Lemma S7.

Second, we prove property $(2)$ of Lemma S7. Note that each $L_{i}$ in the
right hand side of (S49) is an element of
$\mathbf{L}_{n,c}\equiv\\{f_{L}(\epsilon_{L_{i}}):W_{i}=c\\}$ and that
$\\{\mathbf{L}_{n,1},\dots,\mathbf{L}_{n,m_{n}}\\}$ form a partition of
$\mathbf{L}_{n}$. Let
$\epsilon_{\mathbf{L}_{n,c}}\equiv\\{\epsilon_{L_{i}}:W_{i}=c\\}$ and so
$\epsilon_{\mathbb{w}_{n},c}\equiv\Big{\\{}\epsilon_{\mathbf{L}_{n,c}},\kappa_{n,c}\Big{\\}}.$
Due to the mutual independence of the elements of $\epsilon_{\mathbf{L}_{n}}$,
via condition $A1$ (conditional noninterference) for all $n$ (by condition
$D$), and sub-cluster positivity condition $C2.5$, then the elements of
$\\{\epsilon_{\mathbb{w}_{n},1},\dots,\epsilon_{\mathbb{w}_{n},m_{n}}\\}$ are
mutually independent. Thus we have property $(2)$ of Lemma S7.

Finally, we prove property $(3)$ of Lemma S7, where it suffices to show that
$\mathbb{B}_{n,c}(a,l)\equiv
f_{n}(\mathbb{W}_{n}(c),{\epsilon}_{\mathbb{w}_{n},c})$. Note that by
condition $A2$ (structural invariance) for all $n$ (by condition $D$), the
elements of $\mathbf{L}_{n,c}$ are generated from
$\epsilon_{\mathbf{L}_{n,c}}$ via the same structural equations $f_{L}$ that
are invariant in $c$. Note also that $\mathbf{A}_{n,c}$ is equivalent to
$\mathbf{A}^{G_{\mathbb{W}_{n}(c)}^{*}+}_{\mathbb{W}_{n}(c)}$ which is
generated by the function $[\mathbf{G}_{m}]_{\cdot,\cdot}(\cdot)$ taking
arguments, $\mathbb{W}_{n}(c)$, $\kappa_{n,c}$ and
$\mathbf{L}_{n,c}\equiv\\{f_{L}(\epsilon_{L_{i}}):W_{i}=c\\}$. Composing
$[\mathbf{G}_{m}]_{\cdot,\cdot}(\cdot)$ and the functions $f_{L}$ generating
$\mathbf{L}_{n,c}$, we have a function of $\epsilon_{\mathbb{w}_{n},c}$ and
$\mathbb{W}_{n}(c)$ that is invariant in $c$. Since $\mathbb{B}_{n,c}(a,l)$ is
a function of $\\{\mathbf{A}_{n,c},\mathbf{L}_{n,c}\\}$, and since
$\\{\mathbf{A}_{n,c},\mathbf{L}_{n,c}\\}$ are each at most $c$-invariant
functions of $\epsilon_{\mathbb{w}_{n},c}$ and $\mathbb{W}_{n}(c)$, then we
have that $\mathbb{B}_{n,c}(a,l)$ is a function of
$\epsilon_{\mathbb{w}_{n},c}$ and $\mathbb{W}_{n}(c)$ that is invariant in
$c$, which we may denote by $f_{n}$. Thus we have property $(3)$ of Lemma S7.
Thus concludes the proof of Lemma S7.

∎

Lemma S7 shows that

$\displaystyle\tilde{q}_{n}(a\mid
l)=\frac{1}{m_{n}}\sum\limits_{c=1}^{m_{n}}\frac{m_{n}}{\mathbb{L}_{n}(l)}f_{n}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c}).$

Moreover, by dividing and multiplying by $n$,

$\displaystyle\tilde{q}_{n}(a\mid l)$
$\displaystyle=\frac{1}{n}\sum\limits_{c=1}^{m_{n}}\frac{n}{\mathbb{L}_{n}(l)}f_{n}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c})$
$\displaystyle=\sum\limits_{c=1}^{m_{n}}f_{n}^{\prime}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c})$
$\displaystyle=\sum\limits_{c=1}^{n}f_{n}^{\prime}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c}),$

where
$f^{\prime}_{n}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c}):=\frac{1}{\mathbb{L}_{n}(l)}f_{n}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c})=\frac{1}{\mathbb{L}_{n}(l)}\mathbb{B}_{n,c}(a,l)\leq
1$ for $c=1,\ldots,m_{n}$ and
$f^{\prime}_{n}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c}):=0$ for
$c=m_{n}+1,\ldots,n$.

Furthermore, by Lemma S6, for any $n^{*}$ there exist
$\mu_{n^{*}}\in\mathbb{R}$ and $\sigma_{n^{*}}>0$ such that

$\displaystyle\sqrt{n^{\prime}}\bigg{(}\sum\limits_{c=1}^{n^{\prime}}f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(c),\epsilon_{\mathbb{W}_{n^{*}},c})-\mu_{n^{*}}\bigg{)}\underset{n^{\prime}\to\infty}{\to}\mathcal{N}(0,\sigma_{n^{*}}^{2}).$
(S50)

In fact, assuming that the first sub-cluster is non-degenerate in the sense of
Condition $C2.2$, we have that:

$\displaystyle\sigma_{n^{*}}^{2}$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]$
$\displaystyle+2\sum\limits_{k=1}^{\infty}\text{Cov}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}),f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k})]$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]$
$\displaystyle+2\sum\limits_{k=1}^{n^{*}-1}\text{Cov}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}),f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k})]$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]$
$\displaystyle+2\sum\limits_{k=1}^{n^{*}-1}\mathbb{E}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k})]$
$\displaystyle-\mathbb{E}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]\mathbb{E}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k})]$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{1})]+2\sum\limits_{k=1}^{n^{*}-1}\sum\limits_{\begin{subarray}{c}\mathbb{w}_{n^{*}}(j),\varepsilon_{j},\\\
j\in\\{1,1+k\\}\end{subarray}}f^{\prime}_{n^{*}}(\mathbb{w}_{n^{*}}(1),\varepsilon_{1})f^{\prime}_{n^{*}}(\mathbb{w}_{n^{*}}(1+k),\varepsilon_{1+k})$
$\displaystyle\cdot(\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1)=\mathbb{w}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}=\varepsilon_{1},\mathbb{W}_{n^{*}}(1+k)=\mathbb{w}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k}=\varepsilon_{1+k})$
$\displaystyle-\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1)=\mathbb{w}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}=\varepsilon_{1})\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1+k)=\mathbb{w}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k}=\varepsilon_{1+k}))$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]+O(n^{*})$
$\displaystyle\cdot\sup_{\mathbb{w}_{n^{*}}(1),\mathbb{w}_{n^{*}}(1+k),\varepsilon_{1},\varepsilon_{1+k}}(\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1)=\mathbb{w}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}=\varepsilon_{1},\mathbb{W}_{n^{*}}(1+k)=\mathbb{w}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k}=\varepsilon_{1+k})$
$\displaystyle-\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1)=\mathbb{w}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1}=\varepsilon_{1})\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1+k)=\mathbb{w}_{n^{*}}(1+k),\epsilon_{\mathbb{W}_{n^{*}},1+k}=\varepsilon_{1+k}))$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]+O(n^{*})\cdot\alpha_{\mathbb{P}_{0}}(\mathbb{W}_{n^{*}}(1),\mathbb{W}_{n^{*}}(1+k))$
$\displaystyle-\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1)=\mathbb{w}_{n^{*}}(1))\mathbb{P}_{0}(\mathbb{W}_{n^{*}}(1+k)=\mathbb{w}_{n^{*}}(1+k))$
$\displaystyle=\text{Var}_{\mathbb{P}_{0}}[f^{\prime}_{n^{*}}(\mathbb{W}_{n^{*}}(1),\epsilon_{\mathbb{W}_{n^{*}},1})]+o(n^{*})$
$\displaystyle=o(n^{*}).$ (S51)

The first equality follows from the statement of Theorem 27.4 in
billingsley_probability_1995; the second equality follows from the
$n$-dependence property; the third equality follows from the definition of
covariance; the fourth equality follows from the definition of expectations,
where we write $\varepsilon_{j}$ for an instantiation of
$\epsilon_{\mathbb{W}_{n^{*}},j}$; the fifth equality follows from the
definition of $\sup$, condition $C2.1$, and as $f_{n}^{\prime}\leq 1$; the
sixth equality follows from
$(\epsilon_{\mathbb{W}_{n},1},\ldots,\epsilon_{\mathbb{W}_{n},m_{n}})\mathchoice{\mathrel{\hbox
to0.0pt{$\displaystyle\perp$\hss}\mkern
2.0mu{\displaystyle\perp}}}{\mathrel{\hbox
to0.0pt{$\textstyle\perp$\hss}\mkern 2.0mu{\textstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptstyle\perp$\hss}\mkern
2.0mu{\scriptstyle\perp}}}{\mathrel{\hbox
to0.0pt{$\scriptscriptstyle\perp$\hss}\mkern
2.0mu{\scriptscriptstyle\perp}}}(\mathbb{W}_{n}(1),\ldots,\mathbb{W}_{n}(m_{n}))$,
and
$\mathbb{P}_{0}(\epsilon_{\mathbb{W}_{n},1}=\epsilon_{1}^{\prime})\mathbb{P}_{0}(\epsilon_{\mathbb{W}_{n},1+k}=\epsilon_{1+k}^{\prime})\leq
1$; the seventh equality follows from condition $C2.3$; the last equality
follows as
$\underset{n\to\infty}{\lim}f_{n}^{\prime}(\mathbb{W}_{n}(c),\epsilon_{\mathbb{W}_{n},c})=\underset{n\to\infty}{\lim}\frac{\mathbb{B}_{n,c}(a,l)}{\mathbb{L}_{n}(l)}=0$
by condition $C2.1$.

Hence,

$\displaystyle\mathbb{E}_{\mathbb{P}_{0}^{F}}[(\tilde{q}_{n}(a\mid
l)-\bar{q}_{0}(a\mid l))^{2}]$
$\displaystyle=\mathbb{E}_{\mathbb{P}_{0}^{F}}[(\tilde{q}_{n}(a\mid
l)-\mu_{n}+\mu_{n}-\bar{q}_{0}(a\mid l))^{2}]$
$\displaystyle=\mathbb{E}_{\mathbb{P}_{0}^{F}}[(\tilde{q}_{n}(a\mid
l)-\mu_{n})^{2}+(\mu_{n}-\bar{q}_{0}(a\mid l))^{2}$
$\displaystyle+2(\tilde{q}_{n}(a\mid l)-\mu_{n})(\mu_{n}-\bar{q}_{0}(a\mid
l))]$
$\displaystyle=\frac{1}{n}(\mathbb{E}_{\mathbb{P}_{0}^{F}}[n(\tilde{q}_{n}(a\mid
l)-\mu_{n})^{2}]-\sigma_{n}^{2})+\frac{\sigma_{n}^{2}}{n}+(\mu_{n}-\bar{q}_{0}(a\mid
l))^{2}$ $\displaystyle+2(\mu_{n}-\bar{q}_{0}(a\mid
l))\mathbb{E}_{P_{0}^{F}}[\tilde{q}_{n}(a\mid l)-\mu_{n}]$
$\displaystyle=\frac{1}{n}o(1)+o(1)+o(1)^{2}+2o(1)o(1)$ $\displaystyle=o(1),$

where the first equality comes from adding and subtracting $\mu_{n}$; the
second equality comes from expanding the square; the third equality comes from
linearity of expectation, multiplying and dividing $(\tilde{q}_{n}(a\mid
l)-\mu_{n})^{2}$ by $n$, adding and subtracting $\sigma_{n}^{2}$, and taking
constants out of expectations; the fourth equality comes from Equations (S48)
and (S51) and the continuous mapping theorem, and the fact that
$\mu_{n}\to\bar{q}_{0}(a\mid l)$ as $\bar{q}_{0}(a\mid
l):=\underset{n\to\infty}{\lim}\tilde{q}_{n}(a\mid l)$ and
$\mu_{n}=\mathbb{E}_{\mathbb{P}_{0}^{F}}[\tilde{q}_{n}(a\mid l)]+o(1)$ by
Equation (S48); the last equality follows from the definition of
$o(1)\underset{n\to\infty}{\to}0$. ∎

##  Appendix D Additional remarks and supporting materials

In this appendix, we present additional remarks and additional supporting
materials for the main text. Section D.1 provides tables of numerical results
for the analysis of ICU data presented in Section 6. Section D.2 provides
directed acyclic graphs illustrating the conditions defining model
$\mathcal{M}^{AB}_{n}$. Section D.3 provides a remark illustrating a
robustness property of optimal rank-preserving CRs that is not enjoyed by
conventional optimal IRs for limited resource settings considered previously
in the literature. Section D.4 provides simple elaborations to definitions of
optimal CRs (from Proposition 3 in Section 4) to accommodate the more
realistic settings in which treatment may be harmful for some non-empty
$v$-subset of the population with positive probability.

###  D.1 Data Analysis

In this appendix, Tables S1 \- S3 provide numerical results supporting Figures
2 and 3 for the data analysis in Section 6. Figures S3 and S4 illustrate the
function that permits translation of the asymptotic optimal $V$-rank-
preserving regime $\mathbf{G}_{0}^{\mathbf{opt}_{1}}$ into a individualized
dynamic treatment regime, conditional on the cumulative density function of
$\Delta_{0}(V_{i})$, an individual’s average treatment effect conditional on
$V_{i}$. Figure S5 provides the analogous illustration for the candidate
asymptotic $V$-rank-preserving regime $\mathbf{G}_{0}^{{\Lambda}_{1}}$, with
the key distinction being that, under $\mathbf{G}_{0}^{{\Lambda}_{1}}$,
individuals will continue to be assigned treatment so long as a treatment unit
is available, whereas under $\mathbf{G}_{0}^{\mathbf{opt}_{1}}$, individuals
with negative CATEs will no longer be assigned treatment.

$\kappa^{*}$ | $\mathbf{G}_{0}^{\mathbf{opt}_{1}}$ | $\mathbf{G}_{0}^{\Lambda_{1}}$ | $\mathbf{G}_{0}^{\mathbf{opt}_{2}}$ | $\mathbf{G}_{0}^{\Lambda_{2}}$
---|---|---|---|---
0.00 | 0.695 (0.685, 0.705) | 0.695 (0.685, 0.705) | 0.695 (0.685, 0.705) | 0.695 (0.685, 0.705)
0.05 | 0.700 (0.690, 0.710) | 0.684 (0.672, 0.696) | 0.696 (0.686, 0.706) | 0.695 (0.686, 0.704)
0.10 | 0.711 (0.701, 0.721) | 0.683 (0.670, 0.696) | 0.697 (0.688, 0.707) | 0.695 (0.686, 0.705)
0.15 | 0.703 (0.693, 0.714) | 0.679 (0.665, 0.692) | 0.699 (0.689, 0.709) | 0.696 (0.686, 0.706)
0.20 | 0.705 (0.695, 0.716) | 0.678 (0.664, 0.692) | 0.702 (0.692, 0.712) | 0.699 (0.689, 0.708)
0.25 | 0.711 (0.700, 0.722) | 0.673 (0.659, 0.687) | 0.705 (0.695, 0.715) | 0.701 (0.692, 0.710)
0.30 | 0.714 (0.703, 0.725) | 0.676 (0.662, 0.690) | 0.707 (0.697, 0.718) | 0.703 (0.694, 0.712)
0.35 | 0.715 (0.704, 0.727) | 0.675 (0.660, 0.690) | 0.707 (0.697, 0.718) | 0.705 (0.696, 0.715)
0.40 | 0.716 (0.705, 0.728) | 0.675 (0.660, 0.690) | 0.708 (0.696, 0.719) | 0.708 (0.698, 0.718)
0.45 | 0.722 (0.710, 0.734) | 0.674 (0.659, 0.690) | 0.709 (0.697, 0.720) | 0.710 (0.699, 0.721)
0.50 | 0.718 (0.705, 0.730) | 0.671 (0.655, 0.687) | 0.711 (0.699, 0.723) | 0.712 (0.701, 0.724)
0.55 | 0.718 (0.705, 0.730) | 0.675 (0.659, 0.692) | 0.712 (0.700, 0.725) | 0.713 (0.701, 0.726)
0.60 | 0.718 (0.705, 0.730) | 0.673 (0.657, 0.690) | 0.712 (0.700, 0.725) | 0.711 (0.699, 0.723)
0.65 | 0.718 (0.705, 0.730) | 0.676 (0.660, 0.693) | 0.713 (0.700, 0.725) | 0.708 (0.696, 0.720)
0.70 | 0.718 (0.705, 0.730) | 0.674 (0.657, 0.691) | 0.713 (0.700, 0.725) | 0.705 (0.692, 0.717)
0.75 | 0.718 (0.705, 0.730) | 0.673 (0.656, 0.691) | 0.713 (0.700, 0.725) | 0.701 (0.689, 0.714)
0.80 | 0.718 (0.705, 0.730) | 0.677 (0.660, 0.695) | 0.713 (0.700, 0.725) | 0.698 (0.685, 0.712)
0.85 | 0.718 (0.705, 0.730) | 0.680 (0.662, 0.697) | 0.713 (0.700, 0.725) | 0.695 (0.681, 0.709)
0.90 | 0.718 (0.705, 0.730) | 0.685 (0.667, 0.702) | 0.713 (0.700, 0.725) | 0.692 (0.677, 0.707)
0.95 | 0.718 (0.705, 0.730) | 0.685 (0.668, 0.703) | 0.713 (0.700, 0.725) | 0.689 (0.673, 0.705)
1.00 | 0.718 (0.705, 0.730) | 0.685 (0.668, 0.702) | 0.713 (0.700, 0.725) | 0.686 (0.669, 0.703)
$\widetilde{\mathbb{E}}[\overline{A}_{0}]\approx 0.27$ | 0.711 (0.700, 0.722) | 0.675 (0.661, 0.689) | 0.706 (0.696, 0.717) | 0.702 (0.693, 0.711)
Table S1: Estimated values of the expected cluster-average potential outcome under large-cluster regimes, $\mathbb{E}[\overline{Y}^{\mathbf{G}_{0}}_{0}]$, with 95% confidence intervals, under varying resource constraints ($\kappa^{*}$). Estimates for parameters defined by $\mathbf{G}_{0}^{\mathbf{opt}_{1}}$ and $\mathbf{G}_{0}^{\Lambda_{1}}$ use semiparametric efficient methods adapted from Luedtke and van der Laan (2016). Estimates for parameters defined by $\mathbf{G}_{0}^{\mathbf{opt}_{2}}$ and $\mathbf{G}_{0}^{\Lambda_{2}}$ use online-learning-based methods adapted from Luedtke and Van Der Laan (2016b). See Appendix B.4 for details on inferential procedures. $\kappa_{n^{*}=20}$ | IPW† | g-formula‡ | Doubly-robust∘
---|---|---|---
0 | 0.700 (0.690, 0.709) | 0.705 (0.695, 0.714) | 0.695 (0.685, 0.705)
1 | 0.701 (0.692, 0.709) | 0.704 (0.695, 0.713) | 0.695 (0.686, 0.704)
2 | 0.702 (0.693, 0.710) | 0.703 (0.695, 0.712) | 0.695 (0.686, 0.705)
3 | 0.703 (0.694, 0.711) | 0.703 (0.694, 0.712) | 0.696 (0.686, 0.70 6)
4 | 0.703 (0.694, 0.711) | 0.703 (0.695, 0.712) | 0.699 (0.689, 0.708)
5 | 0.704 (0.695, 0.712) | 0.704 (0.695, 0.712) | 0.701 (0.692, 0.710)
6 | 0.704 (0.695, 0.713) | 0.704 (0.695, 0.713) | 0.703 (0.694, 0.712)
7 | 0.705 (0.695, 0.713) | 0.705 (0.695, 0.714) | 0.705 (0.696, 0.715)
8 | 0.705 (0.695, 0.714) | 0.705 (0.696, 0.714) | 0.708 (0.698, 0.718)
9 | 0.705 (0.695, 0.714) | 0.705 (0.695, 0.715) | 0.710 (0.699, 0.721)
10 | 0.704 (0.694, 0.714) | 0.704 (0.694, 0.715) | 0.712 (0.701, 0.724)
11 | 0.702 (0.692, 0.713) | 0.703 (0.692, 0.714) | 0.713 (0.701, 0.726)
12 | 0.700 (0.689, 0.710) | 0.701 (0.690, 0.712) | 0.711 (0.699, 0.723)
13 | 0.696 (0.685, 0.707) | 0.698 (0.688, 0.710) | 0.708 (0.696, 0.720)
14 | 0.692 (0.681, 0.703) | 0.695 (0.684, 0.707) | 0.705 (0.692, 0.717)
15 | 0.688 (0.676, 0.699) | 0.691 (0.679, 0.703) | 0.701 (0.689, 0.714)
16 | 0.683 (0.671, 0.695) | 0.687 (0.674, 0.699) | 0.698 (0.685, 0.712)
17 | 0.678 (0.666, 0.691) | 0.683 (0.670, 0.696) | 0.695 (0.681, 0.709)
18 | 0.674 (0.659, 0.688) | 0.679 (0.665, 0.693) | 0.692 (0.677, 0.707)
19 | 0.669 (0.653, 0.685) | 0.675 (0.660, 0.691) | 0.689 (0.673, 0.705)
20 | 0.665 (0.648, 0.681) | 0.671 (0.655, 0.688) | 0.686 (0.669, 0.703)
Table S2: Estimated values of the expected cluster-average potential under regimes defined by the rank function, $\Lambda_{2}$, with 95% confidence intervals, under varying resource constraints ($\kappa_{n^{*}}$). $\dagger$ Estimates of finite cluster parameters $\mathbb{E}[\overline{Y}^{G^{\Lambda_{2}}_{n^{*}=20}}_{n^{*}=20}]$ using a semi-parameteric IPW estimator of the individual-level g-formula, based on parametric models for $\overline{q}_{0}$ with 95% confidence intervals computed using 500 nonparametric bootstrap samples; $\ddagger$ estimates of $\mathbb{E}[\overline{Y}^{G^{\Lambda_{2}}_{n^{*}=20}}_{n^{*}=20}]$ using a parametric plug-in estimator of the individual-level g-formula based on parametric models for $Q_{Y}$ with 95% confidence intervals computed using 500 nonparametric bootstrap samples; $\circ$ Estimates of the analogous large-cluster parameter $\mathbb{E}[\overline{Y}^{\mathbf{G}^{\Lambda_{2}}_{0}}_{0}]$ using online-learning-based methods adapted from Luedtke and Van Der Laan (2016b). $\kappa_{n^{*}}$ |  $x=$6 |  $x=$7 |  $x=$8 |  $x=$9 |  $x=$10
---|---|---|---|---|---
0 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.012 (0.010, 0.015) | 0.031 (0.026, 0.037)
1 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.012 (0.010, 0.015) | 0.030 (0.026, 0.036)
2 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.012 (0.009, 0.014) | 0.030 (0.025, 0.035)
3 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.011 (0.009, 0.014) | 0.029 (0.025, 0.035)
4 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.011 (0.009, 0.014) | 0.029 (0.025, 0.034)
5 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.009, 0.014) | 0.029 (0.024, 0.034)
6 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.009, 0.013) | 0.028 (0.024, 0.034)
7 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.009, 0.013) | 0.028 (0.024, 0.034)
8 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.009, 0.013) | 0.028 (0.024, 0.034)
9 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.008, 0.013) | 0.028 (0.023, 0.034)
10 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.004) | 0.011 (0.008, 0.014) | 0.028 (0.023, 0.034)
11 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.003 (0.003, 0.005) | 0.011 (0.009, 0.014) | 0.029 (0.024, 0.035)
12 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.001) | 0.004 (0.003, 0.005) | 0.012 (0.009, 0.015) | 0.030 (0.025, 0.037)
13 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.002) | 0.004 (0.003, 0.006) | 0.013 (0.010, 0.016) | 0.033 (0.026, 0.040)
14 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.002) | 0.005 (0.003, 0.006) | 0.014 (0.011, 0.018) | 0.035 (0.028, 0.043)
15 | 0.000 (0.000, 0.000) | 0.001 (0.001, 0.002) | 0.005 (0.004, 0.007) | 0.015 (0.012, 0.020) | 0.038 (0.031, 0.047)
16 | 0.000 (0.000, 0.001) | 0.002 (0.001, 0.002) | 0.006 (0.004, 0.008) | 0.017 (0.013, 0.022) | 0.041 (0.033, 0.051)
17 | 0.000 (0.000, 0.001) | 0.002 (0.001, 0.003) | 0.007 (0.005, 0.009) | 0.019 (0.014, 0.025) | 0.045 (0.036, 0.055)
18 | 0.001 (0.000, 0.001) | 0.002 (0.001, 0.003) | 0.008 (0.005, 0.011) | 0.021 (0.016, 0.028) | 0.048 (0.038, 0.060)
19 | 0.001 (0.000, 0.001) | 0.003 (0.002, 0.004) | 0.009 (0.006, 0.012) | 0.023 (0.017, 0.031) | 0.052 (0.041, 0.065)
20 | 0.001 (0.000, 0.001) | 0.003 (0.002, 0.005) | 0.010 (0.006, 0.014) | 0.026 (0.018, 0.035) | 0.056 (0.043, 0.070)
$\kappa_{n^{*}}$ |  $x=$11 |  $x=$12 |  $x=$13 |  $x=$14 |  $x=$15
0 | 0.066 (0.057, 0.075) | 0.115 (0.105, 0.125) | 0.164 (0.157, 0.171) | 0.192 (0.191, 0.192) | 0.179 (0.170, 0.186)
1 | 0.065 (0.057, 0.073) | 0.114 (0.104, 0.123) | 0.164 (0.156, 0.170) | 0.191 (0.191, 0.192) | 0.179 (0.172, 0.186)
2 | 0.064 (0.056, 0.072) | 0.112 (0.103, 0.122) | 0.163 (0.156, 0.170) | 0.191 (0.190, 0.192) | 0.180 (0.172, 0.187)
3 | 0.063 (0.056, 0.072) | 0.112 (0.103, 0.121) | 0.162 (0.155, 0.169) | 0.191 (0.190, 0.192) | 0.181 (0.173, 0.187)
4 | 0.062 (0.055, 0.071) | 0.111 (0.102, 0.120) | 0.161 (0.154, 0.168) | 0.191 (0.190, 0.192) | 0.181 (0.174, 0.188)
5 | 0.062 (0.055, 0.070) | 0.110 (0.101, 0.120) | 0.161 (0.154, 0.168) | 0.191 (0.190, 0.192) | 0.182 (0.174, 0.188)
6 | 0.061 (0.054, 0.070) | 0.109 (0.100, 0.120) | 0.160 (0.153, 0.168) | 0.191 (0.190, 0.192) | 0.182 (0.175, 0.188)
7 | 0.061 (0.053, 0.070) | 0.109 (0.100, 0.119) | 0.160 (0.153, 0.168) | 0.191 (0.189, 0.192) | 0.183 (0.175, 0.189)
8 | 0.061 (0.053, 0.070) | 0.109 (0.099, 0.120) | 0.160 (0.152, 0.168) | 0.191 (0.189, 0.192) | 0.183 (0.175, 0.189)
9 | 0.061 (0.053, 0.070) | 0.109 (0.099, 0.120) | 0.160 (0.152, 0.168) | 0.192 (0.190, 0.193) | 0.183 (0.174, 0.190)
10 | 0.061 (0.053, 0.071) | 0.110 (0.099, 0.121) | 0.161 (0.153, 0.170) | 0.192 (0.190, 0.193) | 0.183 (0.174, 0.190)
11 | 0.063 (0.054, 0.073) | 0.112 (0.101, 0.124) | 0.163 (0.154, 0.171) | 0.193 (0.191, 0.193) | 0.182 (0.172, 0.189)
12 | 0.065 (0.056, 0.076) | 0.115 (0.103, 0.127) | 0.166 (0.156, 0.174) | 0.193 (0.192, 0.194) | 0.179 (0.169, 0.188)
13 | 0.069 (0.058, 0.080) | 0.119 (0.107, 0.131) | 0.168 (0.159, 0.176) | 0.193 (0.191, 0.194) | 0.176 (0.166, 0.186)
14 | 0.073 (0.062, 0.084) | 0.123 (0.111, 0.136) | 0.171 (0.163, 0.178) | 0.193 (0.190, 0.193) | 0.173 (0.161, 0.183)
15 | 0.077 (0.066, 0.089) | 0.128 (0.116, 0.140) | 0.174 (0.166, 0.180) | 0.192 (0.188, 0.193) | 0.168 (0.156, 0.179)
16 | 0.082 (0.070, 0.094) | 0.133 (0.120, 0.145) | 0.177 (0.169, 0.182) | 0.190 (0.185, 0.192) | 0.163 (0.150, 0.175)
17 | 0.086 (0.073, 0.100) | 0.137 (0.124, 0.150) | 0.179 (0.171, 0.184) | 0.188 (0.182, 0.192) | 0.159 (0.145, 0.171)
18 | 0.091 (0.077, 0.106) | 0.142 (0.127, 0.155) | 0.180 (0.173, 0.185) | 0.186 (0.178, 0.191) | 0.154 (0.138, 0.168)
19 | 0.096 (0.080, 0.112) | 0.146 (0.131, 0.159) | 0.182 (0.175, 0.185) | 0.184 (0.174, 0.190) | 0.148 (0.131, 0.165)
20 | 0.101 (0.084, 0.118) | 0.150 (0.134, 0.163) | 0.183 (0.176, 0.184) | 0.181 (0.170, 0.189) | 0.143 (0.125, 0.161)
$\kappa_{n^{*}}$ |  $x=$16 |  $x=$17 |  $x=$18 |  $x=$19 |  $x=$20
0 | 0.130 (0.119, 0.141) | 0.071 (0.062, 0.081) | 0.028 (0.023, 0.033) | 0.007 (0.005, 0.008) | 0.001 (0.001, 0.001)