Incentive-Aware Synthetic Control: Accurate Counterfactual Estimation via Incentivized Exploration
Authors: Dung Daniel Ngo
Keegan Harris
Anish Agarwal
Vasilis Syrgkanis
Steven Wu
Venue: Accepted by TMLR ()

Abstract: Synthetic control methods (SCMs) are a canonical approach used to estimate treatment effects from panel data in the internet economy. We shed light on a frequently overlooked but ubiquitous assumption made in SCMs of ``overlap'': a treated unit can be written as some combination---typically, convex or linear---of the units that remain under control. We show that if units select their own interventions, and there is sufficiently large heterogeneity between units that prefer different interventions, overlap will not hold. We address this issue by proposing a recommender system which incentivizes units with different preferences to take interventions they would not normally consider. Specifically, leveraging tools from information design and online learning, we propose an SCM that incentivizes exploration in panel data settings by providing incentive-compatible intervention recommendations to units. We establish this estimator obtains valid counterfactual estimates without the need for an a priori overlap assumption. We extend our results to the setting of synthetic interventions, where the goal is to produce counterfactual outcomes under all interventions, not just control. Finally, we provide two hypothesis tests for determining whether unit overlap holds for a given panel dataset.

Keegan Harris* Published in Transactions on Machine Learning Research (02/2026)
Incentive-Aware Synthetic Control: Accurate
Counterfactual Estimation via Incentivized Exploration
Daniel Ngo
Goondung.ngo@jpmmchee
J.P. Morgan Chase AT Research
keegen.harris@äeereeeeey edu
University of California, Berkeley
Anish Agarwal Columbia University
aa5194 1.519//cclumbia.. edu
Vasilis Syrgkanis Stanford University
usyrgk:Sstanford.com edu
Zhiwei Steven Wu Carnegie Mellon University
#stevenwu@cmmu. edu
Reviewed on OpenReview: https: // openreview. net/ forum? id= koc.kolnnjjj P5c
Abstract
Synthetic control methods (SCMs) canonical approach used to estimate treatment effects from panel data in the internet economy. We shed light but ubiquitous assumption made in SCMs of "overlap": a treated unit can be written as some combination typically, convex or linear-of the units that remain under control. We show that if units select their own interventions, and there is sufficiently large heterogeneity between units that prefer different interventions, overlap will not hold. We address this issue by proposing recommender system which incentivizes units with different preferences to take interventions they would not normally consider. Specifically, leveraging tools from information design and online learning, propose SCM that incentivizes exploration in panel data settings by providing incentive-compatible intervention recommendations to units. We establish this estimator obtains valid counterfactual estimates without the need for an priori overlap assumption. We extend our results to the setting of syuthetic interventions, where the goal is to produce counterfactual outcomes under all interventions, not just control. Finally, we provide two hypothesis tests for determining whether unit overlap holds for given panel dataset.
frequently overlooked
1 Introduction
A ubiquitous task the internet cconomy is estimate counterfactual outcomes for a group units (c.g. people, products, geographic regions) under different interventions (e.g. marketing campaigus, discount levels, legal regulations) over time. Such multi-dimmensional data often referred to panel dato, where the different units may be thought of as rows of matrix and the time-steps as columns. A prominent framework for counterfactual inference using panel data is that of synthetic control methods (SCMs) Abadie & Gardeazabal (2003); Abadie et al. (2010), which aim to estimate the counterfactual outcome under control, i.e. no treatment, for units that were treated.
"Equal contribution.
Work done during visit CMU.

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In the SCM framework there is pre-interuention time period, during which all units are under control (i.e. under no intervention). After the pre-intervention period there is post-inlervention time period, where each unit is given exactly one intervention from a set of possible interventions (which includes control).
The goal of SCMs is to estimate the counterfactual outcomes under control for a treated unit during the post-intervention period. To do this, SCMs broadly follow two steps:
1. during the pre-intern interveeenno period, linearly regress the outcomes of treated unit against the out- comes of the units that remain under control, i.e. the donor units;
2. use the learned linear model and the outcomes of the donor units during the post-intervention period to produce counterfactual trajectory of the treated unit under control.
While SCMs were originally applied to public policy domains (e.g. Abadic et al. 2010 Freire (0018)
they are now often used to estimate counterfactuals in various internet domains, ranging from surge pricing for ride-sharing applications Uber (2019) to the effectiveness of advertisements across different geographic regions EBBy 2022)
From the description of SCMs laid out above, it is clear that for the counterfactual estimate to be valid, the treated unit's potential outcomes under control should be linearly expressible by the observed outcomes of the donor units 2
To provide statistical guarantees on the performance of SCMs, such intuition has traditionally been made formal by making an overlap assumption on the relationship between the outcomes of the donor units and the test unit; for instance of the following form:
Unit Overlap Assumption. Denote the potential outcome for unit i under intercention d at time t by (d) Vi,t E R. For a given unit i and intervention d, there enists a set of weights wlf.d) E RR such that
E Bl
Σ jEINal
Bly,
(d)) for all te [T].
where T is the namber of time-steps, Na is the namber of donor units who have received intervention d. The expectation is laken with respect to the randomness in the unil oulcomes.
More broadly, previous works that provide theoretical guarantees for SCMs assume there exists some underlying mapping (t.d) (e.g. linear or convex) through which the outcomes of a treated unit (unit i) may be expressed by the outcomes the Na donor units. Since such a condition appears to be necessary in order to make valid counterfactual inference, assumptions of this nature are ubiquitous when proving statistical guarantees about SCMs (e.g, Abadie et al. (2010): Amjad et al. (2018); Agarwal et al. (2020b 2023)
However, despite their ubiquity, such overlap assumptions may not hold in all domains in which one would like to apply SCMs. For example, consider a streaming service with two service plans (i.e. interventions): yearly subscription (the treatment) and pay-as-you-go model (the control). Suppose that all streamers (the units in this example) are initially signed up for the pay-as-you-go model, and may choose to switch to the subscription model after a trial period (the pre-intervention period). Furthermore, suppose that the streaming service wants to determine the effectiveness of its subscription program on user engagement (the unit outcome of interest) over the next year (the post-intervention period).
Under this setting, the subpopulation of streamers who self-select the subscription plan is most likely those who believe they will consume large amounts of content the platform. In contrast, those who pay-as-they-go most likely believe they will consume less content. The two subpopulations may have very different experiences under the two business plans (due to their differing viewing habits), leading to two different sets of potential outcomes that may not overlap, i.e., are not linearly expressible in terms each other. This could make difficult to draw conclusions about the counterfactual user engagement levels of one subpopulation using the realized engagement levels of the other.
Soc Section 2.1 for more in depth description of the SCM framework consider.
2See Section for mathematical justification of this fact.
We discuss The role of the Unit Overlap Assumption and related conditions in the literature in Section 1.1
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In this work, our goal is to leverage tools from information design to incentivize the exploration of different treatments by non-overlapping unit subpopulations in order to obtain valid counterfactual estimates using SCMs. Specifically, we adopt tools and techniques from the literature on incentivizing exploration in multi-armed bandits (o.g. Mansour et al. p0200g Slivkins (0024) to show how the principal-the platform running the SCM can leverage knowledge gained from previous interactions with similar units to persuade the current unit to take an intervention they would not normally take. The principal achieves this by sending a signal or recomeenndatin to the unit with information about which intervention is best for them. The principal's recommendation policy is designed to be compatible i.. it is in the unit's best interests to follow the intervention recommended to them by the policy. In our streaming service example, incentive-coppatibee signalling may correspond to recommending a service plan for each user based on their usage of the platform in a way that guarantees that. units are better off in expectation when purchasing the recommended service plan. Our procedure ensures that the Unit Overlap Assumption becomes satisfied over time, which enables the principal to do valid counterfactual inference using off-tho-shelf SCMs after they have interacted with sufficiently many units.
Overview of paper. We introduce online learning model where the principal observes panel data from a population of units with differing preferences. We show that the Unit Overlap Assumption is indeed a necessary condition to obtain valid counterfactual estimates in our setting. To achieve overlap, we introduce a recommender system for incentivizing exploration when there are two interventions: treatment and control. At high level, we adapt the "hidden exploration" paradigm from incentivized exploration in bandits to the panel data setting: First, randomly divide units into "exploit" units and "explore" units. For all exploit units, recommend the intervention that maximizes their estimated expected utility, given the data seen far. For every explore unit, recommend the intervention for which the principal would like the Unit Overlap Assumption to be satisfied. Units are not aware of their exploit/explore designation by the algorithm, and the explore probability is chosen to be low enough such that the units have an incentive to follow the principal's recommendations. After recommendations have been made to sufficiently-mant that with high probability, the [nit Overlap Assumption is satisfied for all units and either intervention of our choosing. This enables the use of existing SCMs to obtain finite sample guarantees for counterfactual estimation for all units under control after running Algorithm 1
units, we show
We extend our methods to the setting of synthetic interventions Agarwal et al. 2020b). in which the principal may wish to estimate counterfactual outcomes under different treatments in addition to control (Appendix D). We also provide two hypothesis lests for checking whether the [nii Overlap Assumption holds for a given treated unit and set of donor units. Finally, we empirically evaluate the performance of our recommender system. We find that it enables consistent counterfactual outcome estimation when the Unit Overlap Assumption does not hold a priori while existing SCMs alone do not.
1.1 Related work
Causal inference and synthetic control methods. Popular methods for counterfactual inference using
panel data include SCMs Abadie & Gardeazabal 2003); Abadie et al. 2010), differenee-innddiffeeeess Angrist
& Pischke [2009]; Ashenfelter & Card (1984); Bertrand et al.
(2004).
and clustering-basee methods Zhang
let al. (2019) Dwivedi et al. (2022) Within the literature on synthetic control, our work builds off of
the line of work on robust synthetic control Amjad et al. (2018 2019); Agarwal et al. (2200a b 2023
which assumes outcomes are generated via a latent factor model (e.g. Chamberlain (1984); Liang & Zeger
[1866] Arellano & Honore| (2000)) and leverages principal component regression (PCR) Jollifie (9822);
Massy 1965]; Agarwal et al.
[2019 2020a) to estimate unit counterfactual outcomes. Our work falls in
the small- line of work at the intersection of SCMs and online learning Chen 20233 Farias
ct al. (2022): : Agarwal ct al.
(2023), although we are the first to consider unit incentives in this sctting.
Particularly relevant to our work is the model used by Aggawal et al. (2023). ; which extends the finite sample guarantees from PCR in the panel data setting to online settings. While Harris et al. (2024) also consider incentives in SCMs (albeit in an offline setting), they only allow for a principal who can assign interventions to units (e.g. can force compliance). As a result, the strategizing they consider is that of units who modify their pre-intervention outcomes in order to be assigned more desirable intervention. Tn
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Protocol: Incentivizing Exploration for Synthetic Control For each unit i e [n]:
1. Principal observes pre-intervention outcomes Viære
2. Principal recommends an intervention de {0,1]
3. Unit i chooses intervention d; E {0,1}
4. Principal observes post-intervention outcomes Vispost
Figure 1: Summary of our setting.
contrast, we consider principal who cannot assign interventions to units but instead must persuade units to take different interventions by providing them with incentive-compatible recommendationss.
Unit overlap and similar conditions. Our notion of unit overlap comes directly from the literature on robust synthetic control (see the previous paragraph for references). However, it is important note that analogous assumptions appear more broadly in the literature on synthetic control. For example, Abadic et al. (010 requires that the treated unit's s pre-treatment outcomes and covariates lie in the convex hull of the donor units pool. Beyond synthetic control, similar assumptions to the Unit Overlap Assumption are also prevalent in the literature on other matching-based estimators such as difference- iinrenceein-diffrennes differences (Donald & Lang 2007) or clustering-basee based methods (Zhang et al., 2019). Finally, it is worth noting that in causal inference, the term "overlap" also commonly used to refer to propensity score overlap, which assumes that the probability unit receives treatment is bounded in (0, 1). This is in contrast to the unit autcome/latent factor overlap that we consider.
Incentivized exploration. Our work draws on techniques from the growing literature on incentivizing Mansour el al. 2020 2022: Sellke & Slivkins 2023 Immorlica et al. Sellke 2023: Hu et al. 2022 Ngo et al. 2021 In IE, a principal interacts with a sequence of myopic agents over time. Each agent takes an action, and the principal can observe the outcome of each action. While each individual agent would prefer to take the action which appears be sillit-maximizingg (i.e. exploit), the principal would like to incentivize agents to explore different actions in order to benefit the population in aggregate. Motivation for IE includes online rating platforms and recommendation systems which rely on information collected by users to provide informed recommendations about various products and services.
exploration (TE) (Keeer et al., 2013
2020
2 Preliminaries
Notation. Subscripts are used to index the unit and time-step, while superscripts are reserved for interventions. We use to index units, to index time-steps, and d index interventions. For N, use the shorthand [x] (1.2...., and [æ]o == {0,1. 1}. y(t] denotes the i-th component of vector y, where indexing starts at 1. We sometimes use the shorthand T; := T - To for T,To E Noo such that T > To. ( denotes the set of possible probability distributions over X. O₂ is shorthand for the vector [0, 0, Rd. Finally, we use the notation a A b= minfa, b) and a v b. max (a,b. b).
2.1 Our panel data setting
We consider an online setting in which the principal interacts with sequence units one-by-one for T time steps each. As is standard in the literature on synthetic control, posit that there pre-intervention period of To time-steps, for which each unit i [n] is under the same intervention, i.e., under control. After the pre-intervention period, the principal either recommends the treatment the unit suggests that they remain under control (possibly using knowledge of the outcomes for units j < f). We denote the principal's recommended intervention to unit by d. {0, 1}, where denotes the treatment and the
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While we assumee the existence of such structure in the data, we do not assume that the principal knows or gets observe u(d),, V;: or e(dd) In the literature on synthetic control, the goal of the principal is estimate unit-specific counterfactual outcomes under different interventions. Specifically, our target causal Published in Transactions on Machine Learning Research (02/2026)
control. After receiving the recommendation, unit i chooses intervention d, {0, 1} and remains under intervention d, for the remaining T - To time-steps We use Vipre == [....... (0) 1 E R to denote unit i's pre-treatment outcomes under control, and Yi == W. -VrT] E T- To to refer to unit i's post-inlervention potential outcomes under intervention d. We denote the set of possible pre-trealment outcomes by Vrre. See Figure | for a summary of our setting.
Ti
(d)
In order to impose structure how unit outcomes related for different units, time-steps, and interven- tions, we assume that outcomes are generaled via the following latent factor model, a popular assumplion in the literature on synthetic control (see references in Section 1.1p.
Assumption 2.1 (Latent. Factor Model). Suppose the outcome for unit at time t under trealment E (0, 1 takes the factorized form E(gi,.) {u, ( .v),, Vi, Ely depends only on the lime-slep L and intervention d, E lR" is lalent veclor which only depends on wnil i, (d) and eit is zero-mean sub- Gabssins random noise with variance at most o², For simplicity, we assume that
(d)
(d)] + Ei,t where u,!) E R is latent vector which
(d)
BBy. S 1, ViE [n].t E e
parameter is the (counterfactual) average expected post-intervention outcome
Definition 2.2. (Average expected post-intervention outcome) The average expected post- post-intervention out- come of unit under intervention d is El = 7, Ei Ely,,,, where the expectation is taken with (d)
respect to (ei, TOSEST.
2.2 Recommendations and beliefs
The data generated by the interaction between the principal and unit may be characterized by the tuple (Vi.pre: de, d4, y (indss) where yi.pre are unit i's pre-treatment outcomes, de is the intervention recommended to unit i, d: is the intervention taken by unit i, and Vipost are unit i's post-intervention outcomes under intervention d,
Definition 2.3 (Interaction History). The interaction history at unit the sequence of outcomes, recom- mendations, and interventions for all units [i [i- 1]. Formally, H, == ((ys.rredd dg. dj. y ( We denote the set of all possible histories at unit i as Hi:
(dz)
We refer to the way the principal assigns interventions to units as a recommendation policy. Definition 2.4 (Recommendation Policy). vecommendation policy n; H. x Vpre - () (possibly stochastic) mapping from histories and pre- ee-treetment outcomes to interventions.
We assume that before the first unit arrives, the principal commits to method for computing the sequence of recommendation policies { which is fully known to all units. Whenever is clear from the context, we use the shorthand d, n: Vi.re) to denote the recommendation of policy n, to unit i.
In addition to having corresponding latent factor, each unit has associated belief over the effectiveness of each intervention. This is made formal through the following definitions.
Definition 2.5 (Unit Belief). Unit i has prior belief Pv,, which is a joint distribution over potential post- intervention outcomes (E(y (posll expected overage post-intervention oulcome, with respect to their prior Pv.-
id) E(y(! [(y!!) (poel) We the shorthand pa; = Ep., Iol to refer to unit 's (
Definition 2.6 (Unit Type). Unit is of type E (0,1) if = arg de pe. We denote the set of all wnils of lype as (T) and the dimension of the latent subspuce spanned by type wnils as Tr = span(v; j E T()).
We cnvision sceuario where both interventions available units and they can choose between them. 6Qur results hold in the more general setting where the target causal parameter is any Knear function of the vector of expected intteintervnntion outcomes. We focus the average as it a common quantity of interest..
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In other words, unit's type is the intervention they would take according to their prior without any additional information about the effectiveness of each intervention. We consider the setting in which the possible latent factors associated with each type lie in mutually orthogonal subspaces (i.e. the Unit Overlap Assumption is not satisfied).Se Section 3 for a simple theoretical example of such a data-generating process and Section for an empirical example. If principal could get the Unit. Overlap Assumplion to be satisfied, they could use existing synthetic control methods off-the-shelf to obtain valid finite sample guarantces.
As is standard in the literature on IE, we assume that units Bayesian-rational and each unit knows its place in the sequence of n units (i.e,, their index i E [huusThus given recommendation d, unit i selects their intervention dj such that dj E arg maxde(0.1) Ep,, W [di), i.e. they select the intervention de which maximizes their utility in expectation over their prior Pv,, conditioned on receiving recommendation d; from the principal.
Definition 2.7 (Bayesian Incentive-Compent Cncentive-ommpatibllltyy.. We say that. a recomsmendation d is Bayesian incentive-compatibl (BIC) for unit i if, conditional on receiving intervention recommendation & = =d, unit i's average expected post-intervention outcome under intervention d is at least as large as their average expected post-interventton outcome under any other intervention. Formally, Ep., W wi.poeel |d: = d 2 0 for every d' E (0,1). A recommenddation policy is BIC if the above. condition holds for every intervention which is recommended with non-zero probability.
It is worth clarifying the behavioral interpretation of Definition 2.7 Formally, Definition 2.7 assunnes that units are Bayesian-rational, undersland the recommendation algorithm, and correctly condition on the infor- mation conveyed by receiving a particular recommendation. This assumption provides clean and standard benchmark that ensures rocommendations are followed and allows us to focus on the statistical question of inducing overlap. At the same time, we do not view this assumption as requiring literal Bayesian reasoning by agents in practice. In many real-world systems, users may instead follow recommendations with high but imperfect probability, rely on coarse beliefs about the system's reliability, or exhibit heterogeneous levels of sophistication. To this end, explore setting in which units follow recommendations imperfectly in Ap- pendix G and we find that our recommendation schemes are empirically robust to unit non-compliance.
3 The necessity of overlap
Suppose that the principal would like to estimate counterfactuals for some unit under intervention d. We begin by showing that the Unit Overlap Assumption must be satisfied for this unit under intervention in order for any algorithm to obtain consistent counterfactual estimates under the setting described in Section 2
Theorem 3.1. Fix time horizon T, pre-intervention time period To, and number of donor units (0) For any algorithm used to eslimate the average post-inlervention oulcome under control for test unil, there ezists a problem instance such that the produced estimate has constant error whenever the Unit Overlap Assumption is not satisfied for the test unit under control, even as T, To, (0) +0..
The proof of Theorem 3.1 proceeds by constructing family of problem instances under which the Unit Overlap Assumption not satisficd, before showing that no algorithm can consistently estimate B0 11,ost on all instances in the family.
Proof. Consider the setting in which all type 0 units have latent factor vo = |0 1J_ and all type 1 units have latent factor Vi [1 T. Suppose that u [1 TJT if t mod 2 = 0 and u, (0) 10 1] if L mod 2 2 1 if S To- For all t > To, let u, = [H 1j_ for some H in range [-c, c], where e > O. Furthermore, suppose that there no noise in the unit outcomes. Under this setting, the (expected) outcomes for type 0 units under control are
(0 if t mod 2 = 0. To
E (0) =
1 if t mod 2 = 1,t < To
if t> To
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and the outcomes for type units under control are
1 if t mod 2 = 0, < To
E Elyi.1 [0)
if mod 2 = 1, To
H if To.
Suppose that H Uni and consider principal who wants to estimate Elyi.ped using just the set of outcomes prel, Elyo,postl and E[yi,pre]. Since these outcomes do not contain any information about H, any estimator Elyi.poel cannot be function of H and thus will be at least constant distance away from the true average post-intervention outcome Plyi.poel in expectation over H. That is,
Eh
E EL
postl
Ely
(0) post
EE H E Elyi (.stttss)
1 (c + EE (poap e
2c
Note that choice of T and To was arbitrary, and that cstimation does not improve as the number of donor units increases since (1) there is no noise in the outcomes and (2) all type 0 units have the same latent factor. Therefore any estimator for postl is inconsistent in expectation over H as n T, To + co. This implies our desired result because estimation error constant in expectation over problem instances, there must exist at least one problem instance with constant. estimation error.
3.1 On estimating unit-specific counterfactual outcomes
To infer the unit outcomes in the post-intervention period from outcomes in the pre-intervention period, we require the following linear spon inclusion condition to hold on the latent factors associated with the pre- and post-intervention time-steps:
Assumption 3.2 (Lincar Span Inclusion). For each intervention d E (0,1) and time > To, we assume that u. E span(u (0) uu
(0)
Much like the Unit Overlap Assumption Assumption 3.2 is ubiquitous in the literature robust synthetic control (see, e.g. Agarwal et al. (2020b 2023 Harris et al. (2024)). In essence, Assumption 3.2 requires that the pre-intervention period be "sufficiently informative" about the post-intervention period. Indeed, As- sumption 3.2 may be thought of as the "transpose" of the Unit Overlap Assumption] where the "overlap" condition is required to hold on the time-steps instead of on the units [ To gain intuition for why such assumption is necessary, consider the limiting case in which ut = 0, for all t < To- Under such a setting, all expected unit outcomes in the pre-intervention time period will be 0, regardless of the underlying unit latent factors, which makes it impossible for the principal to infer anything non-trivial about a unit's post-intervention outcomes from looking at their pre-intervention outcomes alone.
Vertical vs. horizontal regression. The following proposition is an important structural result for our main algorithm, as it will allow us to transform the problem of estimating unit counterfactual outcomes from a regression over donor units (verlical regression) to a regression over pre-intervention outcomes (horizontal regression). While Proposition 3.3 known in the literature on learning from panel data (e.g., (Agarwal
ct al. 2023 Harris al. 2024 we state it here for completeness.
Proposition 3.3. Under Assumption 2.1 and 3.2, there exists slope vector ((d) E To such that the average expected post-intervention outcome of wriit i under intervention d given by Elyi = ((((d) Elyi,,re)) We assume that the principal has knowledge of a volid upper- bound I on the tn-m of g(d), i.e. | ||2 - for d [0,
Note that Assumption 3.2 substitute for the Unit. Overlap Assumption by Theorem 3.1

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ALGORITHM 1: Incentivizing Exploration for Synthetic Control: Type 1 units
First stage length No, batch size L, number of batches B, failure probability 6, gap C' E (0,1) Provide no recommendation to first No units.
Input:
for batch 1,2,..., .......,. B do
Select an explore index is E [L] uniformly at random.
for j = 1,2..... I do
if j = in then
Recommend intervention dNo+(0-1). = 0 to unit No + (b 1) T. +
else
if p - ploy. 2 C then
Recommend intervention dNo+ (b-1)) = 0 to unit No + (b - 1) - +
else
Recommend intervention dN, (b-1))-)) to unit No + (b (b- 1) L J.
Given historical data of the form (y j.post }=1 we estimate ((d) 0, using some estimation (d)
(dy)
procedure, which in turn allows us to estimate Blyl,post as Elpipoeel := T; (0 Hi.pre) for d (0,1). While this is not a synthetic control method in the traditional sense, as it does not construct a "synthetic rol trajectory from the trajectories of the donor units, it turns out that the point estimate produces (and assumptions it requires) are cquivalent that of a vertical regression- basod SCM many scttings (including ours). See Shen al. (2022) for more background on the similarities and differences between using horizontal vegression-based approaches (like ours) and vertical regression-basee approaches (like traditional SCMs) in panel data settings. We choose to use horizontal regression- approach because it is easier to work with in our setting, where units arrive sequentially (recall Figure Dp. Importantly, we would still need the Unit Overlap Assumption to be satisfied for intervention d if we used a vertical regression- based method.
For the reader familiar with bandit algorithus,, may be useful to draw the following connection between our setting and that of linear contextual bandits. In particular, it is sometimes helpful think of E[gi,prel as the "context" of unit i, and Bly as the principal's expected reward of assigning intervention d given Elys,pre]- Since we observe Vt.pre instead of pryt we cannot apply linear contextual bandit algorithms to our panel data setting out-of-the-box. However as we will show in Section 4 one can combine ideas from incentivizing exploration in contextual bandits with principled ways of handling the noise in the pre-intervention outcomes to ivien exploration in panel data settings.
4 Incentivized exploration for synthetic control
In this section, turn our focus to incentivizing type 1 units (i.e. units who priori prefer the treatment) to remain under control in the post-intervention period. The methods we present may also be applied to incentivize type units (i.e. units who a priori prefer the control) to take the treatment. We focus on incentivizing control amongst type units in order to be in line with the literature on synthetic control, which aims to estimate counterfactual unit outcomes under control.
The goal of the principal is design sequence of recommendation policies that convinces enough units of type 1 to select the control in the post-treatment period, such that the Unit Overlap Assumption is satisfied for both interventions for all units of type 1, At high level, the principal is able to incentivize units to explore because they have more information than any one particular unit (as they view the realized history H; before making a recommendation), and so they can selectively reveal information in order to persuade units to take interventions which they would not normally take. Our algorithm (Algorithm I inspired by the "detail free" algorithm for incentivizing exploration in multi-armed bandits Mansour et al. (2020). Furthermore, through our discussion in Section 3 one may view Algorithm | as an algorithm for incentivizing exploration in linear contextual bandit settings with measurement. error in the context.
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The recommendation policy in Algorithm i split into two stages. In the first stage, the principal provides no recommendations to the first No units. Left to their own devices, these units take their preferred intervention according to their prior belief: type 0 units take control, and type units take the treatment. We choose Ny such that it is large enough for the Unit Overlap Assumption to be satisfied for all units of type 1 under the treatment with high probability!
In the second stage, we use the initial data collected during the first stage to construct an estimator of the average expected post-intervention outcome for type 1 units under treatment using PCR, as described in Section 3.1 By dividing the total number of units in the second stage into phases of L rounds each, the principal can randomly "hide" one explore recommendation amongst L- - exploit recommendations. When the principal sends an exploit recommendation to unit i, they recommend the intervention which would result in the highest estimated average iostiintervnntion outcome for unit i, where unit i's post-intervention outcomes under treatment are estimated using the data collected during the first stage. On the other hand, the principal recommends that unit take the control whenever they send an explore recommendation.
Thus under Algorithm I if type unit receives recommendation to take the treatment, they will always follow the recommendation since they can infer they must have received an exploit recommendation. However if type unit receives a recommendation to take the control, they will be unsure if they have received an explore or an exploit recommendation, and can therefore be incentivized to follow the recommendation as long as L is set to be large enough (i.e. the probability of an explore recommendation sufficiently low). This works because the sender's estimate of Ely!posd is more informed than unit i's prior belief, due to the history of outcomes H, that the principal has observed.
Our alm requires that bounds on various aggregate statistics about the underlying unit population are common knowledge amongst the principal and all units. Such assumptions are standard in the literature on incentivized exploration.
We are now ready to present our main result: way to initialize Algorithm 1 ensure that the Unit. Overlap Assumption is satisfied with high probability.
Theorem 4.1 (Informal; detailed version in Theorem C.5 Under Assumption C.3. if the number of initial units No and the batch size L are chosen to be sufficiently large, then Algorithm satisfies the BIC property (Defiinition 2.7 J.
Moreover, if the wumber of batches B is chosen to be sufficiently large, then the conditions of the Unil Overlap Assumption will be satisfied simultaneously for all type 1 units under control w.h.p.
C.3 posits that the principal and units know valid upper- and lower-bounds on various population-level statistios about the underlying unit population (e.g., the fraction of units of each type, the prior mean for each type and intervention). As these are population- statistics, in practice, the principal may be able to learn these bounds from samples of the data and broadcast them to the unit population. See Appendix C.2 for more details.
Summary of Assumption C.3 Assumption
Proof Sketch. See Appendix C for complete proof details. At a high level, the analysis follows by expressing the compliance condition for type 1 units as different cases depending on the principal's recommendation. In particular, a type 1 unit could receive recommendation d, = for two reasons: (1) Under event Éc,, when control is indeed the better intervention according to the prior and the observed outcomes of previous units, or (2) when the unit is randomly selected as an explore unit. Using bounds on the probabilitics of these two events occurring, we can derive a condition on the minimum phase length L such that a unit's expected gain from exploiting (when the event fc,, happens) exceeds the expected (unit) loss from exploring. We then further simplify the condition on the phase length L so that it is computable by the principal by leveraging existing finite sample guarantees for PCR using the samples collected in the first stage when no recommendations are given.
Thee Unit Overlap Assumption will always be satisfied w.l.p. for type units under the treatment after sceing sufficiently- many type units. The hard part is convincing enough type units to remain under control so that the Unit Overlap Assumption is also satisfied for control.
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Theorem 4.1 says that after running Algorithm [ with optimally-chosen parameters, the Unit Overlap
Assumption will be satisfied for all type 1 units under both interventions with high probability. Therefore [ on sufficiently-many units, the principal can use off- off-the-shelf SCMs (e.g. Agarwal 2020b: 2023 to obtain counterfactual estimates with valid confidence intervals for all type 1 units under rontrol (w.h.p.). Note that all parameters required to run Algorithm [ are computable by the principal polynomial time. See Example C.6 for a concrete instantiation of Algorithm
after running Algorithm
ct al.
5
Testing whether the Unit Overlap Assumption holds
So far, our focus has been on designing algorithms to incentivize subpopulations of units to take interventions for which the Unit Overlap Assumption does not initially hold. In this section study the complementary problem of designing procedure for determining whether the Unit. Overlap Assumption holds for given test unit and set of donor units. In particular, we provide two simple hypothesis tests for deciding whether the Unit Overlap Assumption holds, by leveraging cxisting finitc sample guarantces for PCR. Our first hypothesis test is non-asymplotic, and thus is valid under any amount of data. While our second hypothesis test is asymptotic and therefore valid only in the limit, we anticipate that it may be of more practical use.
5.1 A non- con-asymptotic hypothesis test
Our first hypothesis test. proceeds as follows: Using the PCR techniques described in Appendix [ linear relationship between the first To/2 time-steps in the pre-intervention time period and the average outcome in the second half of the pre-intervention time period using data from the donor units. Using this learned relationship, second half of the pre-intervention time period. If the difference between the two is larger than the confidence interval given by PCR (plus an additional term to account for the noise), we can conclude that the Unit Overlap Assumption is not satisfied with high probability.
we learn
then compare our estimate for the test unit with their true average outcome in the
In order for our hypothosis test to be valid we require the following assumption, which is analogous to As- sumption 3.2
Assumption 5.1 (Linear Span Inclusion, revisited). For every time-step t such that To/2 < t S To, 2/2)
assume that u, E span(u, (0)
While Assumption 3.2 requires that the post-intervention latent factors for any intervention fall within the linear span of the pre-intervention latent factors, Assumption 5.1 requires that the second half of the pre-intervention latent factors fall within the linear span of the first half. This assumption allows us to obtain valid confidence bounds when regressing the second half of the pre-intervention outcomes on the first half. Let Vi.pre' := [y........ /2,TTa/23 Pi.pre" == (0)
V. "... and Vi.pre" := 2 Lbb E/=6//211 (0) To
Under Assumption 5.1 we can estimate pree) as [[rr,pr"]] == (0 Vi.pre)), where O is computed using the PCR techniques in Appendix
We are now ready to introduce our hypothesis test. In what follows, we will leverage the high-probability confidence interval of Agarwal et al. (2023) (Theorem C.I). Under Assumption 5.1 we can apply Theo- rem C.1 out-of-the-box to estimate E[yt.rre] using the first To/2 time-steps as the pre-intervention period and the next. T2/2 2 time-steps as the post-intervention period.
Hypothesis test. Consider the hypothesis that Unit Overlap Assumption is not satisfied for test unit n.
1. Using donor units I and test unit n, compute E [o via the PCR method of Appendix B
2. Let a(8) be the confidence bound given in Theorem C.1 s.t. [B[yt,pee' Elm,,pre']] a(6)with
probability at least 1-6 8. Specifically, compute a()) + 20 ing Theorem C.1 with the first Ta/2 time-steps as the pre-intervention period and the next To/2 time-steps as the iost-intervention period.
log(1/8) for confidence level 8 by apply-
To
3. If [8 [Tn.prrr] Dn.pre"! > a16)) + V
log(1/5) then accept the hypothesis. Otherwise, reject.
To
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Theorem 5.2. Under Assumption 5.1 if the Unit Overlap Assumption is satisfied for unit n, then
E B'yn.pre' Dnn.ren S 2 dence interval which is defined in Theorem C.1 when using the first To/2 time-steps as the pre-intervention period and the next T2/2 lime-steps us the post-intervention period.
[og(1/6) with probability 1 - O(o), where a(ō) is the high-probability confi-
To
Therefore we can conclude that if [Blphgreer] Di.preet > a(6) + 20 Assumption will not be satisfied for unit w.h.p.
log(1/0) To
then the Unit Overlap
5.2 An asymptotic hypothesis test
Next we present hypothesis test which, while only valid in the limit, may be of more practical when compared to the non-asymptotic hypothesis test of Section 5.1 At high level, asymptotic hypothesis test leverages Assumption 5.1 and the guarantees for synthetic interventions (SI) in Agarwal et al. (2020b) to determine whether the Unit Overlap Assumption holds for a given test unit n. In particular, their results imply that under Assumption 5.1 and the [nit Overlap Assumption a rescaled version of E [[u.pree'] - Elyn.pre"] converges in distribution to the standard normal disttribution as To, of Ti - oo. While the rescaling amount depends on quantities which are computable, relatively simple application of the continuous mapping theorem and Slutsky's theorem imply that we can compute a valid test statistic which converges in distribution to the standard normal.
Notation. In what follows, let (n,0) E IR [4] be the linear relationship between the test unit and donor units I which is known to exist under the Unit Overlap Assumption i.c. [[va = Eiez w(n.0) Bly...) Let ((.00) be the projection of w(n.0) onto the subspace spanned by the latent factors of the test units. Let (),,0) be the estimate of ()(.00) and ā the estimate of the standard deviation of the noise terms {ea : E I,t E [TT} given by SC. Finally, let 2a be the z score at significance level a (0,1), i.c. = P(a 2a); where (),
Hypothesis test. Consider the hypothesis that the Unit Overlap Assumption is not satisfied for test unit n.
1. Using donor units I and test unit n, compute E [r.."r and (3,00) using ST.
2. Accept the hypothesis only if 2m lla
[r.ere In.pre" 20.95-
Theorem 5.3 (Informal; detailed version in Theorem F.1. Under Assumplion 5.1 if (1.((n 1| is sufficientll
large and @(n,0) + ((n00) at a sufficiently fast rate, then [Z], To, Ti + oo the Asymptotic Hypothesis Test falsely accepts the hypothesis with probability at most 5%.
6 Numerical simulations
In this section, we complement our theoretical results with a numerical comparison to standard SCMs which do not take incentives into consideration.
Experiment Setup. We consider the setting of Section 4 There two interventions and unit types. Type 1 units prefer the treatment and type 0 units prefer the control. Our goal is to incentivize type units to take the control so we can obtain accurate counterfactual estimates of all units under control. See Appendix G for details about our data-generating process.
We apply the asymptotic hypothesis test of Section 5.2 using all type units as the donor units and a single type unit as the test unit. We accept the null hypothesis, as the test statistic /TT a1 1|2 [
~ 2.24, which much larger than the value of 20.95 Rs 1.96 needed for acceptance. That is, the asymptotic test (correctly) returns that the Unit Overlap Assumption is not satisfied for a test unit of type under control.
Using Theorem C.5 (the formal version of Theorem 4.1 we can calculate a lower bound on the phase length I. of Algorithm such that the BIC condition (Definition 2.7 is satisfied for units of type 1 who
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Prediction error for type 1 units under cantrol. r- belief gap-0.20
Prediction error for type units uncer centrol, r-6, belief gap-0.40
incentio aware synthhttic control
incertive awere synthetic
synthelic control
syntheik control
12
10-
Predicion eror
0.8
Prediction
0.4
0.0
300
400
100
200
(a)
Prior mean gap 0.2
(b) Prior mean gap 0.4
Prediction error for type 1 units under control. B, belief gap-0.60
Prediction error for type units uncer control, r-6, belief gap-0.80
incentivo aware synthetic control synthetic control
incertive aware synthetic control syntheik control
1.2
1.0
Predecion weoor
Preduction
11
0.2
0.9
300
Number units
Number of units
(c) Prior mean gap 0.6
(d) Prior mean gap 0.8
Figure 2: Counterfactual estimation error for units of type under control using Algorithm 1 (bluc) and synthetic control without incentives (orange) with different gaps between the prior mean reward of control and treatment. Results are averaged over 50 runs, with the shaded regions representing one standard deviation.
are recommended the control. We run three different sets of simulations and compare the estimation error Eynngsst - Eys.posel for a new type unit n under control when incentivizing exploration using Algorithm I (blue) the estimation error without incentivizing exploration (orange). For both our method and the incentive-unwware ablation, we use the adaptive PCR method of Agarwal al. (2023) to estimate counterfactuals. All experiments repeated 50 times and report both the average prediction error and the standard deviation for estimating the post-treatment outcome of type units under control.
Varying Strength of Belief. In Figure D we explore the effects of changing the units' "strength of beliefs". Specifically, we vary the gap between the prior mean outcome under control and treatment for type 1 units by setting /4 14 (0) {0.2, 0.4, 0.8). As the gap decreases it becomes easier to persuade units to explore, and therefore our counterfactual estimates improve.
Varying Latent Factor Size. In Figure we vary r (2,4,6,8) while keeping all other parameters of the problem instance constant. Increasing r increases the batch size L, so we see a modest decrease performance increases.
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where the FPR is Published in Transactions on Machine Learning Research (02/2026)
6.1 Hypothesis test
In order to test the validity of the asymptotic hypothesis test in Section 5.2 we use synthetic data for which we know the underlying dala-generaling process.
Specifically, we create two types of synthetic datasets: one where the Unit Overlap Assumption holds ("UOA data"), and one where docs not ("non-UOA data"). For simplicity, we consider a sctting where there are two interventions and two unit. types.
For non-UOA data, we use the same data-generating process as described above, where if unit is of type (resp. type 0), generate latent vector Vi = [0 0 v,[1] e/fr/]] (resp. Vi [n:[1] ... ni[r/2] 0 ... OJ), where Vjj E [r/2] wg[i] ~ Unif 1). This dataset then contains 500 units of each type. On the other hand, UOA data contains 1000 Type 0 units only, where the latent vector is generated the same way as Type O units in non-UOA data. The pro-treatment and post-treatment latent factors are generated in the same manner as above.
Using this data-generating process, we generate 500 datasets of each type (for a total of 1000 datasots), and run the asymptotic hypothesis test in Section 5.2 for each dataset. If accept the null hypothesis, i.e, the Unit Overlap Assumption is predicted not to be satisfied according to our test, then say the 'predicted" label of the dataset is 1; otherwise, the 'predicted' label is O. We say that dataset has (true) label if the Unit Overlap Assumption is not satisfied, and label 0 if it is.
We report the true positive rate (TPR) and false positive rate (FPR) for this experiment. The resulting confusion matrix is:
475
25
22
478
with the TPR = 0.956 and FPR = 0.05. This experimental result supports Theorem 5.3 at most 0.05.
7 Conclusion
We study the problem of non-compliance when estimating counterfactual outcomes using panel data. Our focus is on synthetic control methods, which canonically require an overlap assumption on the unit outcomes in order to provide valid finite sample guarantees. We shed light on this often overlooked assumption, and provide an algorithm for incentivizing units to explore different interventions using tools from information design and online learning. After running our algorithm, the [nii Overlap Assumption will be satisfied with high probability, which allows for the principal obtain valid finite sample guarantees for all units when using off- off-th SCMs to estimate counterfactual outcomes under control. We also extend our algorithm to satisfy the Unit Overlap Assumption for all units and all interventions when there are more than two interventions, and we provide two hypothesis tests for determining if the Unit Overlap Assumption hold for a given test unit and set of donor units. Finally, we complement our theoretical findings with numerical simulations, and observe that our procedure for estimating counterfactual outcomes produces significantly more accurate estimates when compared to existing methods which do not take unit incentives into considcration.
There are several exciting directions for future research. In some multi-armed bandit settings, lower bounds are known the sample complexity of incentivizing exploration (Sellke & Slivkins (2023) ) Sellke (2033
It would be interesting to prove analogous bounds on the number of units required to incentivize sufficient exploration for the Unit Overlap Assumption to be satisfied for all units and all interventions in our setting, or clustering- based algorithms for
Another avenue for future research is to design infferenceeii-diffeeenn incentivizing exploration in other causal inference settings. Finally would be interesting to relax other assumptions typically needed for synthetic control, such as the linear span inclusion assumption on the time-step latent. factors.
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Acknowledgments
We would like to thank the anonymous reviewers for their helpful feedback. For part of this work, DN was Ph.D. student at the University of Minnesota and KH was a Ph.D. student at Carnegie Mellon University. VS was supported by NSF Award ITS-2337916 and 7SW was supported by an NSF CAREER Award.
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A Additional related work
There has been recent interest in causal inference on characterizing optimal treatment policies which must satisfy some additional constraint(s). For example, Luedtke & van der Laan (016) study a setting in which the treatment supply is limited. Much like us, Qiu et al. (2021) consider a setting in which intervening on the treatment not possible, but encouraging treatment feasible. However their focus is on the setting in which the treatment is limited resource, so their goal is to encourage treatment to those who would benefit from it the most. Since they do not consider panel data and place different behavioral assumptions the individuals under intervention, the tools and techniques use to persuade units in our set differ significantly from theirs. Finally, Qiu et al. (2022); Sun et al. (2021) consider settings where there is some uncertain cost associated with treatment.
Within the literature on IE, the work most related to ours on a conceptual level that Li Slivkins (2022,, who consider the problem of incentivizing exploration in clinical trial settings. Like us, Li & Slivkins 2022) study setting in which the principal would like to incentivize agents to explore different treatments. However, their goal is to estimate population-level statistics about each intervention, while we are interested in estimating unil-specific counterfactuals under different inlerventions using panel data. As a result, while our high-level motivations are somewhat similar, the tools and techniques we use to obtain our results differ significantly. On a technical level, our mechanisms are somewhat similar to the initial exploration phase in Mansour et al. (2020); although we consider a more general setting where unit outcomes may vary over time, in contrast to the simpler multi-armed bandit setting they consider.
Finally, our work is conceptually related to the literature causal bandits (Lattimore et al. 2016). which is a sequential decision-making selling where act ions correspond to interventions on causal graph. As we mention in Section B our setting may be viewed as linear contextual bandit setting, where interventions are actions and the principal observes a noisy version of the context. Within the literature on causal bandits, papers on linear causal bandits (where the underlying causal system is assumed to be linear) are perhaps the most similar to our work (e-g- Yan & Tajer (2024); Van et al. (2024): Varici al. (2023)). However unlike this line of work, we do not study regret minimization under causal graph, but instead focus on the problem of identificatinn in the panel data setting. Moreover, our focus is the regime in which compliance cannot be enforced and units are free to self-select their interventions.
B Background principal component regression
The specific algorithm use to estimate ((d) is called principal component regression (PCR). At high level, PCR "de-noises" the matrix of covariates (in our case, the set of all pre-treatment outcomes for all donor units seen so far who underwent intervention d) using hard singular value thresholding, before regressing the (now-denoised) covariates on the outcomes of interest (in our case, the average post-intervention outcome under intervention d).
Let Z(dd) be the set of units who have received intervention d before unit i arrives, and let n; (d) be the number of such units. We use
pre.i S.re J I TR" (xx xTo
denote the matrix of pro-treatment outcomes corresponding to the subset of units who have undergone intervention d before unit arrives, and
T
(dd post,i
Σ [=
Uu j I((0)
E E
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be the vector of the sum of post- inottiintervnntion outcomes for the subset of units who have undergone intervention d before unit arrives. We denote the singular value decomposition of pre, as
140 ATD
(d) (d) (d)
pre,i
82 ui
(d) NTo where {&) Jē=1 are the singular values of yrd) pre,at and u and (lo) are orthonormal column vectors. We
assume that the singular values are ordered such that $1 (Y pre,)) 2 2 8 (n A2o (Ypres)) 2 0. For some
threshold value r, we use
(d)) pre,i
(d) n
(2) (P(L))
t=1
to refer to the truncation of ylu, to its top r singular values. We define the projection matrix onto the subspace spanned by the top r right singular vectors ((d) Rrxr given by P(d Ei= (((i) (v!
Equipped with this notation, we are now ready to define the procedure for estimating 0 (d) using (regularized) principal component regression.
Definition B.1 (Regularized Principal Component Regression). Given regularization parameter p 0 and plod +PP(d) i,r Then, regularized PCR
trancation level E N, for d E (0,1} and i 2 L, let vid0 (Drcc, estimates ((d) as
0.00 = -vx(0))
Pid) pre,a isposts
where g(d) is defined as in Proposition 9.3.
The average post-interuention outcome for unit i under inter- (d)
vention d may then be estimated as Ely. D.ppppst
( :Vipre)
Ti
Observe that if p 0, then Di (pld)) pre,s
(()) and recover the non-regularized version of PCR. pre.i Under the Unit Overlap Assumption and Assumption uses (regularized) PCR to obtain consistent estimates for counterfactual post-intervention outcomes under different interventions. Intuitively, Unit Overlap Assumption is required for consistent estimation because there needs to sufficient information about the test unit contained in the data from donor units in order accurately make predictions about the test unit using a model learned from the donor units' data.
3.2 work robust synthetic control (see Section I.1
C Appendix for Section 4 Incentivized exploration for synthetic control
C.1 Causal parameter recovery
Thcorem C.1 (Thcorem G.3 Agarwal al. (2023) ). Let 8 (0,1) be an arbitrary confidence parameter and > 0 be chosen be sufficiently small. Further, assume that Assumption 2.1 and Unit Overlop Assumption are satisfied, there is some io 2 such that (an)(X;;)(d)) r, and (n);dd) for all 2 in
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Then, with probability at least O(k6), simultaneously for all interventions d [klo:
(d) E [iiipe
Elsisposst
(d)
- a()),
T
L(v74 12/ (Z,(d))
v 2errr(c)
where a(8)
(di;(d)
(T' -o) sii; (d)
VT To or (z;
247
12T Zu (d) /3T0
err,
+ (T To) (d);
(T - To) (
VT To o, (Z)
Lo / /2lo((k/0)
Lo 1 (k
24 Z, (z)(
T- To
sur, (d) To)
snr, (T) TT)
20 /err,(d) log(k/5)
or (.(
where poppp
(d), vir) is the estimated average post-interuention outcome for unit under
T-1n
intervention d and 0
< L.
Corollary C.2. Given gap and the same assumplions as in Theorem S E is at least - 8, where
C.1
the probabilily that
Bipippoad - Elyspoosl
log(n V
exp ((V
(Z((d)) (1+ F) (V vTo)
D
10
2D
with
|
(V74+12 (d))
A 3oT
To
-I-I5
((()
V21Ta
12 p(()
(d)
F
To
T-To
VT'T6
D
(1)
0V148
24
2x
T To
T T-10
VT
((4)
E
To
= A+ D(Vn + (To) + or, (Z)
Proof. We bogin by setting the right-hand side of Theorem C.1 to be The goal is to write the failure probability d as function of e. Then, using the notations above, can write
log((K/0) (ni; + (olo())/))
(d)r + (dr;(d) +
First, we take a look at the signal- ratio sur; (d). By definition, we have:
(dr = or (Z,(d))
U
(Zi(d)
Vnlai + To (lg
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Hence,
1
vn(d) + +To + (og( 0, (d,
(d)
Hence, we can write an upper bound on e as: Observe that since (d); 2 2, we have (d)r, (nr;
A+F (Vlo)(k/0) (d) +
(d) + EVloo(k/8)
(A F) (Vnld) VT))
+ (A+ F) (log(log(((1))0
Tra (Z,(d)
(d)
D ( vd+ log(log(n(1))/)) (kog(k/5) E log(k/6)
o, (Z,(
Since log(x) is strictly increasing function for >> 0, we simplify the above expression as:
6 5 (A+ (++F)(nnee (v To) Gra (Z;(d))
(A+ F)
log( (00k/)))))//
+
rra (Z:())
D(vn vdly
log (11)))))())))///)
D logg( log(log(n/*)////5 + E FV log log(lor(n(0))///b ora (Z;(d)
Ore (d)
Subtracting the first term from both sides and multiplying by or (d)(),), we have:
(A + /Tb)
(A+ F) + D(vn + val logglo + D log (ogg(n (ykk/s))
+ Fo,, (Z)( (Z,,,Z log((or(n
= -A+ DDv V + Eo,, (Z,(d)) log( (on(nn)((()))v4/)) + D log()on(n ( v2k88)
Let a A+ F+ + V + Eo (d))), we can rewrite the inequality above as:
(z(((d))c - - F)(n()))()) + +To) s ay (og + D log(loz(n (988/))
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Then, we can complete the square and obtain:
a² Tra (Zi(d)) (A (V + VTo)
a²
log log(loz(n()))k//) + DV log(los(n (1000)()000/vk/)))
+
4D2
4D
(v log( (ogglooiin (00(11)((((0))088/)) +
2
rra (d) (A F)(Vn()) VTo)
a
2D
4D
log( log(log(n(0))w/ppp
a
Tra (Z,()) (A ()(V)()))))))) + (TT))
a²
2D
4D
logg( 10)))))(((00))1//)))
rr (Z,(d)) (A + F)( nld) + (To)
a
a
D
4C2
2D
Tra (Z,()e (A + F) (v(())
n(d) +Vo))
a
log(ing(( (Ing(n /1//6//))
D
4D2
2D
exp (110 Ora (Zi(d))
ld) 1iVVk
F) (Z (A+
To
a
D
4D
2D
log(n(a))ve V
8 85
(Vor (2,)) (V
exp
1
10
0²
2D
C.2 Proof of Theorem 4.1
Assumption C.3. (Unit Knowledge for Synthetic Controlj We assume that the following are common knowl- edge among all wrils and the principal:
1. Valid upper- and lower-bounds on the fraction of type units in the population, f.c. if the true proportion of type 1 unils is Pı E (0, the principal and wrils know pt. PH E (0,1) such that pr. S P S pn.
2. Valid upper- and lower-bounds on the prior mean for each receiver type and intervention, i.c. volues (dd.F.), ((d. e IR such that y(d.T.1) S pez S p(d.r.h) for all units E [[], receiver types T E {0, and interventions d 10.
(d)
3. For some C 1), lower bound on the smallest probability of the event Eo,, over the priors and latent
factors of type 1 units, denoted by mingeaa) Prp, 2 Sc O, where εc,, := {M1 2 Di.post + C} and
(0) (()
D.p is the average post-inlervention oulcome for wnil under intervention 1.
4. A sufficient namber of obscrvations No needed such that the Unit Overlap Assumption is satisfied for type units under the treatment with probability at least 1 - 6, for some ó E 10,
At high level, part (3) requires that (i) there is non-zero probability under each unit i's prior that (0) Rippost 2 Pisp and (ii) the principal will be able to infer a lower bound on this probability given the set of observed outcomes for units in the first phase 3
8Such "fighting chance" assumptions very common (and are necessary) in the literature incentivized exploration in bandits.
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First, note that in the first stage of Algorithm the principal does not provide any recommendation to the units and instead lets them pick their preferred intervention. The goal of this first stage is to ensure the linear span inclusion assumption (Unit Overlap Assumption) is satisfied for type 1 units and intervention 1. This condition equivalent to having enough samples type 1 units such that the set of latent vectors (zz)) spans the latent vector space St. We invoke the following theorem from Vershynin (2018) that shows span( ((1) = S1 with high probability:
Validity of Assumption C.3
Theorem C.4 (Theorem 4.6.1 of Vershynin (2018) ). Let A be dent, mean zero, sub-gaussion isolropic random vectors in IR. Then for any 1 2 we have with probabilily at least 1 - exp(
matriz whose rows A, are indepen-
Vm
()()))))))))))))))))))))))))))() $1()) vm k k2( (V++ +t)
(1)
where K max; || and ever is absolute constant.
Hence, after observing NO samples of type units taking intervention 1, the linear span inclusion assumption is satisfied with probability at least - exp
(- [NO N
(-(x (-VN Ove
2
Vri
Theorem C.5. Suppose the there are two interventions, and assume that Assumplion C.3 holds for some constant gap C E (0, If No is chosen to be large enough such that Unit Overlap Assumption is satisfied for all wnits of hype 1 under trealment with probability at least. 1 80, then Algorithm with parameters 6, B,C C is BIC for all units of type 1 if
(1) pv.
(0)
L 1+ max ()
(C- ov
[log(1/6r,) Ti
Pr
Pr.
Posp 2 C - a(ôPCR) ov
21og(1/6.) T;
25
where a(8PCR) is the high-probabilte confidence bound defined in Theorem C.1 6 6e, + do + ôPCR and de, (0, 1) is the failure probability of the Chernoff- HoefJding bound the overage of sub- Gaussaassnn random moise (G, (1)
11-1641 and
log( (1( V
Pren S
exp ( / (1C2)) +vTb)
a² 4D2 2D
and K. (y pre,No) =
(yee.o)) is the condition number of the matrir of observed pre-interoention The remaining voriables in 12 pre.N, W3To
outcomes for the subset of the first No units who have taken the treatment. ören are defined as A 3
T(VT4+ 12V6x( No.
J2 To
2T/211, T-16
T-1n
T-1o
VT-T6'
+ VT-T, Moreover if the number of batches B is chosen to be large enough such that with probobility at least 8, d, 0 E (1)) No+ rank j (jjeTa)) then the Unit Overlap Assumption will
= Tav118 + 2AN (O. LVG T-Tp
pre.Ng
20 E
V2 T- and = A+F D( NOO + (TD) + or (Y
pre, No ).
T T-10
be satisfied for all type units under control with probability at least 1 28.
Proof. At particular time step t, unit i of type 1 can be convinced to pick control if Ev, Foppost Di d 0) pr = 0j 2 0. There are two possible disjoint events under which unit i is recommended intervention O: either intervention 0 is better empirically, i.e. ( 2 T C, intervention is better and unit i is
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chosen for exploration. Hence, have
(0) Ev, past Vispost 1d 0r
Elwiipoat Vi.post 1 2 Eluippoae! + C] 2 Blu + C7
L ( + - Ev, Wi - -iipoarl
L
= -(ee Ev, lv 1 E Vi + c1) Prim Bloipood +C (1-1) 1 --) + ( - m!
Rearranging the terms and taking the maximum over all units of type gives the lower bound on phase length L:
(0) [ M p (c Pr.. (o,,))
Me.
(Me.
E.,
To complete the analysis, we want to find lower bound for the terms in the denominator. That is, we want to lower bound
Pee Evgg Wopp 114 2 Blyhpood c1) Pr(u (0) 2 Blyhppood + C] Us
Let Eo denote the event that Unit Overlap Assumption is satisfied for type 1 units under treatment. From Assumption C.3 we know that event occurs with probability at least 1 ó for some dg (0, 1). Let &POR denote the event that the high-probability concentration bound holds for units of type and intervention 1. Finally, let Ee, denote the event where the Chernoff- Heeeroof-Hoeffding bound holds on the average of the sub-Gaussian noise çie that is with probability at least 8c,, we have:
(1)
1
Σ t=
(1) G
2 log (1)6)))))))
S
T
Define another clean event & where the events Eo. &e, and EPCR happen simultancously with probability least 1 - 8, where do ÖPCR + deg- Then, we have:
(p!0) Ev,
W. (c,,)) Pr(.))
(m(0) Ev, lu! www.eeoeleeeeoooo Ec {l) PriEc, {] + (M - Ev, (1))
Deeeetee (e),,,,,) Pr(Ec,,, -E]
2 (m!0) Br, Rc,, { Pr Pr(Ec, i E] 26
(since Pr[-E] < ô and u - Er, M.ppart 2 22)
We proceed to lower bound the expression above as follows:
Ev, ( Ec, E]
Eyg
Di V.pperl 1n
Elyipnst + C7, Byy.ppoc
Ely/gnasel
S a((PCR);
T
Σ (=16+1
T
fit
(1)
S
()
T
2 log( 1
Egg
(1) Vi.post 11
Elylppocl C a(8pcn),
S Sa
Ti
Ti
T=++1
2log (
pe. E(vi]
(1) Vi 14
Vi. C a(8poR)
Vo -
Ti
2 211(1/)).)
> C - (1-p(ppR)---- -0
Ti
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Furthermore, write the probability of the joint event εc, as:
Pr(Ec,t:1 5
Pr
Pe: E8 Wi + C7, E Mi poxel Woor
S a(ŌPCR),
1
(1)
I
2 log (1/ô..)
T
t=
2 2log(1/8g)
Pr
Pe:
(1) Dippast
2 C a(rron) - -o
Ti
Hence, we can derive the following lower bound on the denominator of L:
(1) Me - Ev, Sespostl&c..
2 2log(1 2log(1/6a) (1/6c) C - ((0PpRR)) -
2 2log(1/6g 28
Vo
Pr
Mr. (0)
Pi.post 2 C' a(ÖPCR) ov T
Ti
Applying this lower bound to the expression of L and taking the maximum over type 1 units, have:
(0) pv:
(1)
pv.
ph..
L 1 14 max (EZ()
C a(dren) - ov 2lor(2/6..) 71
Pr
(0)
Vipost 2 - a(6ren) - ov log(1/)6.))
(1)
71 26
Example C.6. Consider the following data-generating process: Suppose there are two receiver types, each with equal probability; type units prefer the treatment and type units prefer control. Let Unif[0.25, 0.75). If unit is a type 1 unit, they have latent factor V; = [U, 0]. Otherwise they have latent factor V, = [0 v,]. Suppose that the pre-inlervention period is of length To = and the post-intervention period is of length Ti 1. We leave ui (0) (0) (0) (1) unspecified.
U2 U3 U33
Since 1, we only need to incentivize a single type 1 unit to take the control in order for the Unat Overlop Assumption to be satisfied for type units under control. Therefore, for a given confidence level 6, it suffices to set No B log(1/6), After the first No time-steps, the principal will know all of the parometers necessary to compute the batch size L.
Since Ti 1, unit i's prior is over [E EE?.. For simplicity, suppose that (1) all type 1 units believe (0) Unif[0, 0.5] and Elg ~ Unif [o, 1] and (2) there no noise in the post-intervention outcome. that El
Under this setting, the event Ec,s simplifies to Ec. = {0.25 < El + cc for any unit and constant
C 1). Therefore
Sc = Pr Pl S 0.25 C7 = max (0,0.25 C}.
D Extension to synthetic interventions
Our focus far has been to incentivize units who priorí prefer the (single) treatment to take the control in order to obtain accurate counterfactual estimates for all units under control. In this section, extend Algo- rithm to the setting where there are multiple treatments. Now our goal is to incentivize enough exploration across all interventions such that the Unit Overlap Assumption] is satisfied for every intervention and thus our counterfactual estimates are valid simultaneously for all units in the population under every intervention. In order do so, we use tools from the literature on synthetic interventions, which is generalization of the synthetic control framework to allows for counterfactual estimation under different treatments, in addition
to control Agarwal et al. 20020b 2023
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ALGORITHM 2: Incentivizing Exploration for Synthetic Interventions: sub-type r units
Input: Group size L, number of batches B, failure probability 6, fixed gap C' (0,1)
Provide no recommendation to first No units.
// Loop through explore interventions
for f {= T 1]. - do
for batch b = 1,2, B do
Select an explore index is E [L] uniformly at random.
for j = 1.2....
do
if = is then
Recommend intervention dNo + (b- 1).
else
if Elyipoal +C Blyipoool < (l(.T..) - C for every intervention d E (8-1,... [[k] then
Recommend intervention dNo +N (b-1)-L++
elsc
Recommend intervention dNo+(O-1I.
=[1]
Our setup is the same as in Section 4 with the only difference being that each unit may choose one of k > interventions after the pre-treatment period. We assume that Assumption 2.1 Definition 2.4 Definition 2.5 and Assumption 3.2 are all extended to hold under & interventions. Under this setting, a unit's beliefs induce a preference ordering over different interventions. We capture this through the notion of a unit sub-type, which a generalization of our definition of unit lype to the setting with interventions.
Definition D.1 (Unit Sub-type). A unit sub-type T E [K] preference ordering over interventions. Unit is of sub-type if for every K S k,
[|K| = arg
max AET
(d)
Unit i's type is [1]. We denote the set of all units of sub-type, T as T(r) and the dimension of the latent subspace sponned by sub-type unils
== span(v, :j T(T)).
Intuitively, a unit sub-type is just the preference ordering over interventions such that intervention T[a] is preferred to intervention T(6] for < 6 Since different unit sub-types have different preference orderings over interventions, we design our algorithm (Algorithm 2 to be BIC for all units of a given sub-type (in contrast to Algorithm which BIC for all units of given type).
While there can be up kt possible sub-types, recall that the unit latent space has dimension r. Therefore, there can be at most linearly independent subspaces that matter for overlap. As result, if we want to satisfy overlap for all k! subtypes, we only need to run Algorithm 2 for enough unit subtypes to span this r dimensional space. 10
For a given sub-type т, the goal of Algorithm 2 is to incentivize sufficient exploration for the Unit Overlap Assumption to be satisfied for all interventions with high probability. As was the case in Section 4 the algorithm will still be split into two phases, and units will not be given any rocommendation in the first phase. The main difference compared to Algorithm I is how the principal chooses the exploit intervention in the second phase.
When there are only two interventions, the principal can compare their estimate of the average counterfactual outcome under treatment to the lower bound on the prior-mean average counterfact outcome for control for type 1 units (and vice versa for type 0 units). However, a more complicated procedure is required to maintain Bayesian incentive-compattblit when there are more than two interventions.
Note that it is without loss of generality to any tie-breaking is done
that units have preference ordering the interventions, long known and deterministic way.
Ths is because if the Unit Overlap Assumption is satisfied for specific unit sub-type whose latent factors lic within particular subspace, it also satisfied for all other units whose latent factors lie within the same subspace, and there are of course y different subspaces by Assumption 2.1
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Instead, in Algorithm the principal will set the exploit intervention as follows: For receiver sub-type T; A Algorithm 2 sets intervention & to be the exploit intervention for unit i only if the sample average of all interventions d E ( [[]] are both (1) larger than the sample average of intervention T[1] by some constant gap C' and (2) less than the lower bound on the prior-mean average counterfactual outcome n4 by C. If no such intervention satisfies both conditions, intervention T[1] is chosen to be the exploit intervention.
We require the following "common knowledge" assumption for Algorithm 2 which is analogous to Assump- tion C.3
A Assumption D.2. We assume that the following are common knowledge among all units and the principal:
Volid upper- and lower-bounds on the fraction of sub-type T units in the population, i.e. if the true proportion of sub-type units is pr (), the principal knows Lr Pr,H H E (0,1) such that
Pr.t. S Pr Pr.
Valid upper- and lower-bounds on the prior mean for each receiver sub-type and intervention, t.e. values k)) g(d.T.)) e IR such that
((dTT S S.:
(1) S ((
for all veceiver sub-lypes T, interventions d E [k]o and all wnils E T(T).
For some C' (0, 1), a lower bound on the smallest probability of the event Ec,s over the prior and lalent faclors of sub-lype unils, denoled by min Prp.prrr..... (Ec,)) So 0, where
$D/ (66 Ec. = (Vj E [Koo Vippost +C S Vippost S M4 (T(K)) -C}
(TDI
and posp is the average post-intervention outcome for unit under intervention E E [k]o-
((E)
4- A sufficient number of observations No needed such that the Unit Overlap Assumption is satisfied for hupe I units under intervention T[1] with probability at least 1 1 6, for some E (0, 1).
Theorem D.3 (Informal; detailed version in Theorem E. Under Assumption D.2 if the ramber of initial units No and batch size L are chosen to be sufficiently large, then Algorithm satisfies the BIC property (Desinilion 7.
Moreover, if the number of batches B is chosen to be sufficiently large, then the Unit Overlap Assumption will be satisfied for all units of sub-type T under all interventions with high probability.
See Appendix for the proof. The main difference compared to the analysis of Algorithm I is in the definition of the "exploit" intervention: First, when there are interventions, an intervention is chosen as the exploit intervention only when the sample average of cach intervention d E 1...,...,,,pp is (i) larger than the sample average of intervention r[1], and (ii) smaller than the prior-mean average outcome of intervention (with some margin C".. Second, instead of choosing between a "previously best" intervention (i.c., an intervention Tii] < l with the highest sample average) and intervention é to be the 'exploit' inter- vention, Algorithm always choose between T[1] and f. This is because in the former case, conditional on any given intervention being chosen as the exploit intervention, it is an open technical challenge to show that the exploit intervention will have a higher average expected outcome compared to all other interventions
E Appendix for Section D Extension to synthetic interventions
Theorem E. Suppose that Assumption D. holds for some constant gap C E (0,1). If the number of
initial wnils No is chosen to be large enough such that Unii Overlop Assumplion is salisfied for all unils of
This observation is in line with work on frequentist-based BIC algorithms incentivizing exploration in bandits. See Section 6.2 of Mansour al. (2020) more details.
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subtype T under intervention r[1] with probability - 6, then Algorithm for all unils of sublype if:
with parameters ö, 8, B, C is BIC
(((11) p4g (7(1)) [R))
L 1214 + max (ET
(c- 2a(6rcn) - 20 21og(1/6..)
(1-5)
5- Pr [Ec,s] 26
Moreover, if the number of batches B is chosen to be large enough such that with probability al least we have rank([Ely,,pp d, = o r[]], 1 Assumption will be satisfied for all units of subtype T under all interventions with probability at least 1 O
(prel/((y,,.)"lprree then the Unit Overlap
Proof. According to our recommendation policy, unit i is either recommended intervention T[1] or interven- tion We will prove that in either case, unit will comply with the principal's recommendation.
((0) Let c, denote the event that the exploit intervention for unit i is intervention é. Formally, we have
Elyiposd S 44 cf
2 = B Blp.podd post
Ely(post and iscC
I min
max
When unit is recommended intervention & When unit i E 7(r) is recommended intervention ł, argue that this unit will not switch to any other intervention j * L. Because of our ordering of the prior mcan reward, any intervention j > has had no sample collected by type t unit and fi, S pé. Hence, we only need to focus the cases where j < l. For the recommendation policy to be BIC, we need to show that
(TDI)
(7(4)
Biu P.peatt ā Prlā=
There are two possible disjoint events under which unit recommended intervention &: either de- termined to be the 'cxploit' intervention or unit i is chosen as an 'explore' unit. Since being chosen as an explore unit does not imply any information about the rewards, we can derive that conditional on being in an 'explore' unit, the expected gain for unit i to switch to intervention j is simply pa: - - (+bl) On the other hand, if intervention l is the 'exploit' intervention, then event C, lend- side of the BIC condition above as:
(2)
(()) has happened. Hence, we can rewrite the
E El ooblll ā E) Pr(d = f Eiu sl c.
Pr( (1- (u -m?)
Rearranging the terms, we have the following lower bound the phase length L for the algorithm to be BIC for type units:
(TD
L 2 1+
Elpes
D.
post cc.
T 20
Pr
To complete the analysis, we need to lower bound the denominator of the expression above. Consider the following event C: C=jvj (Vj E [k]
[bblTlal a(dren))
Let Eg denote the event that Unit Overlap Assumption is satisfied for units of type T under intervention
[[1]. From Assumption D.2 we know that event En occurs with probability at Icast 1 - Sy for some do E
(0. 1). Furthermore,
Ege denote the event where the Chernoff- Cheernof-Hooffding bound on the noise sequences
(Tbl) pT
To+1 holds, that is with probability at least 1 Ee,, we have:
2 logg11/8_) (1/
(TDD) Gie
< σ
T
+1
27

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Then, can define clean event & where the events C, & and & happens simultaneously with probability at least 1 - 6, where = 5 + deg + ōg-
Note that since
(((())) Di.p
I 1, we can rewrite the denominator as:
El (6 y post ca.l Pr(ee
Elu (0) y (
20
El Pr C, 81
Eiu
20 vlrli C -8 Pr(E -
C
(0) ((Tl)) 20
8l
Pr P
El
28
C,
Define an event & on the true average expected post-intervention outcome of each intervention as follows:
8/9
Dispost z min Tbal) - 22(0PCR) 2
(ril))
21og(1/5a
iejet
Ti
and
Visp I !!! - C 222(@PCR)) - 20 isjet
max (TiI))
2 log(1 (1/5c,) Ti
We can observe that under event &, event (0 is implied by event E/0 Hence, we have:
Pr[e, e & Pr(e, E
We can rewrite the left-hand side as:
Pr[C, - Pr(e! (1 61-6) Pr(e!!
Substituting these expressions using elt) for the ones using Ec,s in the denominator gives:
2 log 11g(1/)).) C 2a(6PCR) 1
E 1
Pr E10
(1 (1-68 (1-5) Pr(E 25
Ti
Applying this lower bound to the expression of L and taking the maximum over type 1 units, we have:
(T]1))
(T(A)
L 1 max (EZO)
C 20(ŌPCR) - 20
2 2log(1/6,,)
(1 (1-8)P Pr
((()) C,
26
Ti
When unit is recommended intervention T[1]: When unit i gets recommended intervention r[1], they know that they are not in the explore group. Hence, the event Ec,, did not happen, and the BIC condition, in this case, can be written as: for any intervention T # [[],
Siypp post Dipport
20 - Pr P 2 0
Similar the previous analysis on the recommendation of intervention e, suffices to only consider inter- ventions T[J] < l. We have:
(TD)) Dispost
Pr(
[(0)
Ri (((b/1)) E (T(I) (Tbi))
Dist [ Pr
Pr(
C.i.
(7(1)) - (T(I)) Ely (Tbi) (7|1)) 210)
C.l [
By definition, we have pr:
(T(1))
2 pm:
(7b1)
Hence, suffices to show that for any intervention <<][j] < E, we
have:
(TBI)) 2(e)
C
Pr(ced
26
28

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Published in Transactions on Machine Learning Research (02/2026)
Observe that with the event defined above, can write:
E L poott
(+
(T(I)) @00
Pr(ec.l
post C,
E b post
(rb7))
post C, E) Pr (
(Tl)) ((0)
C. 8]
E Dispost ((((1)) 6(0) C,3 -C Pr) -
(T())
(+i)))
((TlI)) 200)
1 Mi post
Dr.poet C,
8l Pr C.. E]
26
When event 200)
c,2 happens, we know that E (TLT
z E [oppost post + C. Furthermore, when event & happens, we
know that Po,p
E ( a(ôren) - o (1/(6_)
T1 and Ei post
((T1!) a(òron) oV 11
2log(1/6.,]
a()) Hence, when these two events C, and & happen simultaneously, we have (TLL)) 2 Ni.post +C - 2a(6POR)
(TUII)
21 Therefore, the lower bound can be writlen as:
11
2 2oog(1 (1)6a)
E
Ely Poptt
(TII)) Pr C.N 2 Mist C.
C- (0(OFO)R)) - V
Pr(e( C.a E] 20
Ti
Similar to the previous analysis when unit i gets recommended intervention E, we have:
Ely ElyiTonn pippnse Ec,,l E,,, Pr(Ec,]) (c- (C22((óren) - 20
2 2ogg(1 (1/g.)))))
(1 (1- Pr(e 28
Ti
Choosing large enough C' such that the right-hand side is non-negative, we conclude the proof.
F Appendix for Section 5 Testing whether the Unit Overlap Assumption holds
Theorem 5.2. Under Assumption 5.1
if the Unit Overlap Assumption is satisfied for unit n, then with probability 1 1 O where a(5) is the high-probability confi-
[lnppree - Dn.prel S a()) + 20 dence interval which is defined in Theorem C.1 when using the first To/2 time-steps as the pre-intervention period and the next 2 time-steps the jost-intervention period.
log(1/6) To
Proof.
[[iiprro Vi.pre" 5 Di Dipp"e
E Ely
[E [Ti
-Vippee!
$ (0)
+
2
0
To
T2/222+1
with probability at least 1- 06),, where the second inequality follows from Theorem C.1 from a Hoeffding bound.
The result follows
Theorem F.1. Under Assumption 5.1 if
3/2 log (To
min mon [Z] T/ [Z/2/2 o(1)
(2)
[2000)))22
and
T Σ T- TET
Elv!001 (2(m.0) (mm000)))) (1)
(3)
(5)
1l2
then as To, + oc the Asymptotic Hypothesis Test falsely accepts the hypothesis with probability at most 5%, where 7² is defined as in Equation (4) and (6100) is defined as in Equation (5) in Agarwal et al. (2020b).
Condition requires that the E norm of ((,0)) sufficiently large, and condition $3 requires that the estimation error of (v.00) decreases sufficiently fast. See Section 5.3 of Agarwal et al. 2020bb for more details.
29

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Published in Transactions on Machine Learning Research (02/2026)
Proof. We use æ 4 (resp. I 9 y) x converges in probability (resp, distribution) to y. Let o' denote the true standard deviation 6i By Theorem of Agarwal et al. (2020b) we know that as [I] To, - oc,
VT 1. of 100 1|2
(E(ūn.pre") - [[ S 100.1)
2. σ 2 of and
3. @(..0) E w(..))
By the contimuuus mapping theorem, we know that 2 and [
1
4
We can rewrite
|
E [[um.prer] E[un,pre"] as Elūn.pre"] In.pre" + En.pre"; and we know that Enprre" 2 0. Applying Slutksy's theorem several times obtains the desired result.
G Appendix for Section 6 Numerical simulations
Experimental Setup. If unit i is of type (resp. type 0), generate latent factor V;
10 0 v.[1] ... v:[r/2]] (resp. V; [v:[1] 1,[r/2] 0 0)), where Vj E [r/2] vili] Unif(0, 1). We consider an online setting where 500 units of alternating types arrive sequentially 12 We consider a
pre-intervention time period of length To = 100 with latent factors as follows:
If mod? 2 generate latent factor u, [0- u, ( [1] ....... u, [r/]], where E [r/2] ui (0) [$] ~ (.25
If mod 2 = 1, generate latent factor
(0) [1] wt " |r/ 0..00........ 0J, where vl E [r/2] [ Unif[0.25, 0.75). [uz
We consider a gost-intervention period of length T₃ = 100 and generate the following post-intervention latent factors:
If d = 0, set u, (d) [ut [1(=)(1 [1]
(4) i,
[r]] :t [To + 1,7], where V [r] u4 ((] Unif 1]. (d)
If d = 1, we set u
(d) = [ut [1] ... (d) [r] / E [To + 1,7], where V E [r] 4 [6] Unif[-1,0]
(d) Finally, outcomes are generated by adding independent Gaussian noise f NO 0.01) to each inner product of latent factors.
Non-compliance setting Following the standard in the literature IE, we assume that all units are Bayesian-rational (Definition 2.7 That is, given a recommendation d, unit i will select intervention d; that maximizes their utility in expectation over their prior conditional on the recommendation d, However, in practice, units may exhibit sub-rational behaviors, where they neglect the principal's recommendation and act according to their initial prior. In Figure 4 we examine the performance of Algorithm 1 in the presence of such non-compliance behavior.
We use the same data-generatingg process as described above. When running Algorithm D we let a random p- p-proportion of the units ignore the principal's recommendation and choose their initially preferred inter- vention, Le., type 1 units will select the treatment and type O units will select the control. In Figure 1 we vary the non-compliance parameter p (0, 0.2, 0.4, 0.6, 0.8) and calculate the counterfactual estimation error for units of type 1 under control. As the non-compliance level increases, it becomes more challenging to persuade units to explore, and the counterfactual estimation error increases.
12e alternating arrival of different unit types is done only convenience. The order of unit. arrival is unknown to the algorithms.
30

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Frediction error for tyae 1 units under control, r-4
Prediction error for type units under control, r-6
Incertive awere synthetic
0.8
syntheii cortrol
Predicion eroor
0.4
0.3
Prediction
0.4
0.1
0.2
-8 Incentive aware synthetic control
syrthwric control
300
400
100
200
(a) 4-dimensional latent factor
(b) 6-dimensional latent factor
Prediction error for type 1 units under control, r-8
Prediction error for type 1 units under cantrol, r-10
Incentive aware synthetic control synthwtic control
3.0
Incentive aware synthetic control syrthelic control
1.6
2.5
2.4
2.0
Predecton
3.0
0.8
0.0
300 Number units
Number of units
(c) 8-dimensional latent factor
(d) 10-dimensional latent factor
Figure 3: Counterfactual estimation error for units of type under control using Algorithm 1 (bluc) and synthetic control without incentives (orange) with increasing number of units and different lengths of the latent factors. Results are averaged over 50 runs, with the shaded regions representing one standard deviation.
Violation of Linear Span Inclusion assumption 3.2 Similar to prior work in the literature on synthetic control, our thcoretical framework relies on the lincar span inclusion assumption (Assumption
3.2 which is necessary to ensure the counterfact.aal estimation procedure in Appendix [ accurate. In Figure we investigate the performance of our algorithm when this assumption does not hold.
Specifically, we make the following modifications to the data-generee generatinnnggggg process. If unit i is of type 1 (resp.
type 0), we generate a latent factor V, = [0 ... 0 wz[1] ... ve[(r - 1)/2) 1] (resp. v; = [m][1] vi[(r 1)/2] 0 ... 1]), where Vj E [(r - 1)/2] : w ~ Uni 1). In the pre-intervention time period of length To = 100, the time latent factors are generated with a padded entry
the end as follows:
tf mod 2 = 0, generate latent factor u
0...0
(0) [1] ... oz o 1)/2] 0), where V E [(r 1)/2)
(r- 11/2
4 1 Unif 0.75).
Iff mod 2 = 1, generate latent factor u,
(0) [1] ... 2 (0) [[r - 1)/2] 0...0 0J, where AA E [(r 1)/2)
11/2
(r-11/2
(0) " [ Unif(0.25, 0.75).
31

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Published in Transactions on Machine Learning Research (02/2026)
Prediction error for type units under control
p-0.0
IASC pm0.2
1.4
IA-SC p=0.4
[A-SC p=0.6
12
[A-SC p-0.8
Prediction error
10
100
200
500
Number of units
Figure 4: Counterfactual estimation error for units of type under control using Algorithm 1 with increasing number of units and different non-compliance level {0.0, 0.2, 0.4, 0.6, 0.8). Results are averaged over 50 runs, with the shaded regions representing one standard error.
In the post-intervention period of length Ti = 100, the time-and-inttrrvention latent factors are generated as follows.
I = 0, with probability p, set u, [ub (d) []] (1) [r] E To 1,7], where A E [r] u u [4 Unif(o, 1]; and with probability p, we sot u, (d) [u, [1]
(d)
u [r 0 [To + 1,7), where
vl E [r - 1] [09 Unif(0. 1].
If = 1, with probability p, set u, jui 1] u, [r]
(d) (d)
(4)
t E To + 1,77, where V E [r] u !
(4)
Unif[-1, 1, 0]: with probability p, we set u,
(d) [u (1) [1] u [r 0) To + 1,7], where
(4)
vê E [r 1] (1) [1] Uni
( does not lie in the span of the pre- That is, with probability the post-intervention latent factor u, intervention latent factors {uj (0) ................................ ........ 1 )- In Figure 5 as the probability p of post-intervention latent factors not lying in the pre-intervention span increases, ie., more severe violation of Assumption 3.2 the counterfactual estimation error docs not decrease to zero and instead may increase as more units arrive.
32

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Published in Transactions on Machine Learning Research (02/2026)
Prediction error for type units under control, LSI violation
IA-CC p-0.0
IA-SC p-0.2
0.35
IA-S p-0.4
I-SC p-0.6
IA-SC p-0.8
Prediction enor
0.25
18
0.15
0.05
100
200
300
400
500
Number units
Figure 5: Counterfactual estimation error for units of type under control using Algorithm | with increasing number of units and probability lincar span inclusion violation p E (0,0.2,0.4, 0.6, 0.8). Results are averaged over runs, with the shaded regions representing one standard error.
33


Reviews:
[{"review_id": "eZlC776F3n", "reviewer": "Reviewer_zhae", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies counterfactual inference with panel data when treatment assignment is endogenous: units have heterogeneous preferences over interventions and self-select, so the usual synthetic control identification condition (“unit overlap”) can fail exactly in the cases of practical interest.  The first main contribution is a sharp necessity result: if the Unit Overlap Assumption fails for the target unit/intervention, then no algorithm can consistently recover the counterfactual (Theorem 3.1 provides an explicit impossibility construction).  The second contribution is methodological: the authors design an “incentive-aware synthetic control” recommendation policy that creates overlap by persuading some units to take otherwise suboptimal interventions, adapting the “hidden exploration” paradigm from incentivized exploration so that recommendations remain Bayesian incentive-compatible.  They show that after interacting with sufficiently many units, overlap holds with high probability, enabling valid counterfactual estimation via off-the-shelf robust SCM/PCR guarantees.  Finally, the paper adds practical diagnostics (two hypothesis tests for overlap) and extends the approach to multiple interventions via synthetic interventions, supported by simulation evidence.", "claims_and_evidence_yn": "Yes", "claims_explanation": "In fact, due to my lack of expertise in synthetic control method, I'm unable to fully understand and verify the correctness of the results. The writing does look sound at a glance.", "audience_yn": "Yes", "audience_explanation": "A meaningful subset of TMLR’s audience in online learning and sequential experimentation should find this paper relevant. The main conceptual move is to recast a canonical panel-data counterfactual task (synthetic control) as a problem of data acquisition under endogenous participation: when users self-select among interventions, the platform may never observe the variation needed for reliable counterfactual estimation. The paper’s response—an incentive-aware recommendation policy that implements “hidden exploration” while maintaining Bayesian incentive compatibility—connects synthetic control to core themes in bandits and incentivized exploration: how to design exploration policies when feedback is filtered through strategic or preference-driven behavior.\n\nThe paper also speaks to recommender-systems and platform readers by framing counterfactual identification as a platform design issue rather than a purely statistical one. In settings where interventions correspond to product choices, plans, or policies offered to heterogeneous users, the paper argues that purely observational data can be non-overlapping precisely because the recommendation environment is already optimized for user utility. The proposed mechanism provides a principled way to inject controlled exploration via recommendations so that the platform can learn counterfactual outcomes, making the contribution legible to “internet economy” contexts where learning and incentive alignment are inseparable.", "requested_changes": "The authors should try to rewrite the paper to be more accessible for readers without background knowledge about synthetic control.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Df0fBWiMmK", "reviewer": "Reviewer_gKLn", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper offers novel contributions to counterfactual causal estimation by bringing ideas from online active interventions. Most importantly, in my opinion, is that it provides an impossibility result showing that without unit overlap, no estimator can consistently recover counterfactual outcomes in latent-factor panel models with self-selected interventions. Motivated by these observations, the paper introduces an incentive-based exploration control that encourages units to try interventions they would not usually choose. This promotes creating intervention overlaps. This is a departure from prior SCM studies, which assume the experimenter controls interventions and do not model strategic agents.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Overall, the contribution is well-motivated and technically sound, and I would like to recommend it for publication. Before that, however, there are a couple of issues that I invite the authors to consider for revising the manuscript.\n\n1-- The key ideas about how online explorations are induced and the two-step intervention strategy are primarily presented in the introduction. However, most of the statements are not written clearly enough. This took me a few times reading the paper to fully understand some of the statements fully. For instance, the second paragraph of the introduction has a key role, and the choice of words and references in the description of the two stages is not clear enough.\n\n2-- The scope of the paper has some indirect, but important, connections to contextual bandits, linear bandits, and causal bandits. I recommend that the authors include a discussion of connections to the relevant literature, especially recent advances in causal bandits.\n\n3-- The streaming service example could be an interesting case for identifying gaps in the literature. However, as written, the message is unclear. I encourage the authors to rewrite it more in line with the key elements of the model (i.e., how pre-intervention, intervention, and post-intervention phases manifest in this example).\n\n4-- The paper argues that the Unit Overlap Assumption is necessary for consistent counterfactual estimation. While the constructed example seems correct, the authors should more explicitly position this result within identifiability discussions in generalized SCM, existing overlap/common-support assumptions used in SCM inference papers, impossibility results in causal inference, and panel identification. Currently, the claims are as if SCM literature has entirely ignored the necessity of overlap, whereas several papers mention related conditions, though not with this formal impossibility structure. Clarifying what is completely novel and what the result adds to prior insights would improve the contribution's credibility.\n\n5-- The presentation of the extension to more than two interventions is a bit too busy and complex. I suggest that the authors provide a discussion on why the ordering structure (sub-types) is natural, when such a structure arises in real applications, and how exploration interacts across multiple treatments in a more intuitive way.", "audience_yn": "Yes", "audience_explanation": "It will be of interest to those working on causality, especially in the potential outcome framework.", "requested_changes": "Please refer to comments 1--5 earlier. They all provide recommendations for changes and modifications.", "broader_impact_concerns": "There is no broader statement section. Nevertheless, I do not notice any ethical implications.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "r5qVikogcc", "reviewer": "Reviewer_xHZa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper studies synthetic control methods in settings where units self-select into interventions, so the usual “unit overlap” assumption may systematically fail. The authors formally show that without overlap, SCMs cannot recover certain counterfactuals even with infinite data, propose an incentive-aware exploration policy that actively creates overlap while remaining Bayesian incentive-compatible, and provide diagnostics to test for overlap plus simulations that illustrate both the failure of standard SCM and the benefits of their approach.\n\nThe main contributions are:\n- Demonstrating the necessity of unit overlap for SCM.\n- Introducing an incentive-aware exploration mechanism to restore overlap.\n- Providing practical diagnostics for overlap and simulation-based evidence.\n\n# Strengths\n- Clarifies a fundamental limitation of SCM in self-selection settings.\nBy proving that unit overlap is not just convenient but necessary for consistent SCM estimation, the paper sharpens our understanding of when SCM can and cannot be trusted. This is especially valuable because SCM is often used in observational contexts where self-selection is strong, yet the overlap assumption is rarely interrogated. The explicit counterexample with non-overlapping latent types is simple but powerful: it provides a concrete mental model practitioners can use to reason about whether their own setting might violate overlap.\n- Innovative integration of causal inference and incentive design.\nThe work brings together synthetic control, latent factor models, and incentive-compatible exploration. Rather than trying to “fix” SCM post hoc, it changes the data-collection process via recommendations so that SCM’s assumptions become true over time. This is conceptually elegant: the platform uses early data to learn which units are likely to belong to different latent types, then deliberately sends some of them to “unlikely” interventions in a way that still respects their incentives. The mechanism design angle is non-standard in SCM and opens a promising direction: designing recommendation policies that are both user-aligned and statistically informative.\n\n- Solid theoretical development with clear logic and connections to prior work.\nThe paper’s main theorems are built on a clear latent factor framework and make sensible use of known results from robust SCM and PCR. The move from vertical SCM to a horizontal regression view is particularly helpful: it simplifies the online setting and links the problem to contextual bandit intuition. Overall, the theoretical narrative is coherent: it starts from a stark impossibility, then shows how changing incentives and recommendations can get around it.\n\n- Useful diagnostics via overlap testing.\nThe two proposed tests for unit overlap give practitioners a practical handle on a condition that is otherwise quite abstract. In SCM applications today, overlap is often assumed without being checked. The paper’s tests provide a way to monitor whether individual units are in- or out-of-support for a given donor pool. Even aside from the specific incentive mechanism, these diagnostics could be adopted in existing SCM workflows as a pre-check or sensitivity tool to flag units whose counterfactuals are inherently hard to identify.\n\n- Synthetic experiments that closely match the theoretical story.\nThe simulations are carefully designed to instantiate the latent-type, non-overlapping-subspace structure that motivates the theory and to mimic the online arrival of units under self-selection. As a result, the empirical plots directly reflect the claims: baseline SCM stalls at a nonzero error, whereas the incentive-aware algorithm drives error down; the overlap tests behave as predicted by the theorems. This tight theory–experiment alignment makes it easy for readers to see the mechanisms at work and trust that the formal results are not just artifacts of overfitting a proof.\n\n# Weaknesses\n- Strong and somewhat opaque assumptions for incentive compatibility and parameter tuning.\nThe guarantees for the main algorithm hinge on technical assumptions and knowledge of various bounds (on latent factors, noise, etc.) that both the platform and the agents are assumed to share. In the current presentation, these requirements are mostly relegated to the appendix and not fully unpacked in the main text. For a reader thinking about deployment, it is not obvious how realistic it is to know or estimate these quantities, nor how sensitive BIC and overlap guarantees are to misspecification. This makes the theoretical results feel somewhat “brittle”: mathematically clean, but with unclear robustness when the assumed bounds are only roughly known.\n\n- Idealized behavioral model for users.\nThe BIC notion presumes users are fully Bayesian, know the recommendation policy, and condition correctly on receiving a recommendation when deciding whether to follow it. Real users on platforms may be myopic, heterogeneous in sophistication, or simply not pay close attention to recommendation logic. The paper does not explore how much deviation from the idealized BIC setting the mechanism can tolerate while still creating enough exploration to restore overlap. As a result, there is a gap between the theoretical guarantees (which are very clean) and the behavioral complexity of actual recommendation or pricing environments.\n\n- Narrow structural model and limited exploration of model misspecification.\nThe core intuition is built around units belonging to distinct latent “types” with non-overlapping subspaces under different interventions. While this is a compelling stylized model, the paper does relatively little to examine more nuanced situations: partly overlapping subspaces, many types, or smoother heterogeneity structures. Similarly, the simulations closely mirror the assumed latent factor structure, so we don’t see how the method behaves when the model is mis-specified (e.g., true dynamics are nonlinear, or the latent rank is under- or over-estimated). This limits our ability to judge how robust the approach is outside of the carefully crafted theoretical regime.\n\n- Empirical evaluation confined to synthetic data.\nAll empirical evidence is based on synthetic panels designed to match the theory. While this is appropriate for validating the math, it means the paper does not demonstrate its methods on any real or semi-synthetic SCM benchmarks (e.g., policy interventions, marketing campaigns, pricing experiments). Without at least a small real-world or semi-synthetic example, it’s hard to assess issues like data sparsity, unmodeled covariates, or noisy behavioral responses that often arise in practice. This weakens the case for immediate practical impact, even though the conceptual contribution is strong.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The theoretical claims are well supported within their modeling framework, and the simulations convincingly illustrate the main phenomena under those assumptions, but the empirical backing for real-world robustness is limited.", "audience_yn": "Yes", "audience_explanation": "The work is highly relevant to the SCM and causal inference community and online learning community.", "requested_changes": "1. Currently Theorem 4.1 refers to Assumption C.3 with little intuition in the main text. Could you summarize in the main body:\n- What exactly does Assumption C.3 require (in qualitative terms)?\n- Which moments/parameters of the population must the principal and units know (or upper/lower bound)?\n- Why is it reasonable in intended applications?\n\nSimilarly, could you give a high-level description of how $N_0$, $L$, $B$ in Alg.1 are chosen from observable quantities, with a concrete example or at least an order-of-magnitude sketch.\n\n2. Add more discussion of behavioral assumptions and BIC interpretation. The BIC argument assumes Bayesian-rational agents who know the algorithm and can condition correctly on receiving a given recommendation. Could you add a dedicated subsection discussing:\n\n- To what extent this aligns with realistic user behavior in systems like recommender or pricing systems.\n- Whether “approximate” BIC (users loosely trusting recommendations) suffices for the algorithm to still produce overlap.\n- Any robustness to misspecified beliefs or heterogeneous sophistication.\n\n3. It would be good to have an experiment that probes robustness beyond the idealized model. Without necessarily requiring a full real-world dataset, it would be valuable to have a simulation that breaks some assumptions in a controlled way, e.g.:\n- Latent subspaces for types only partially orthogonal.\n- Some units misfollowing recommendations (e.g., follow with probability <1).\n- Mild violations of linear span inclusion.\n\nand report whether the algorithm still substantially improves estimation relative to baseline SCM.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)