#  Picasso: A Sparse Learning Library for High Dimensional Data Analysis in R
and Python††thanks: Jason Ge and Xingguo Li contributed equally. Tuo Zhao is
the corresponding author.

Jason Ge†<EMAIL_ADDRESS>Xingguo Li§<EMAIL_ADDRESS>Haoming Jiang⋄
<EMAIL_ADDRESS>Han Liu‡<EMAIL_ADDRESS>Tong Zhang‡
<EMAIL_ADDRESS>Mengdi Wang†<EMAIL_ADDRESS>Tuo Zhao⋄
<EMAIL_ADDRESS>
$\dagger$Department of Operations Research and Financial Engineering,
Princeton University
§Computer Science Department, Princeton University
⋄School of Industrial and Systems Engineering, Georgia Tech
‡Tencent Artificial Intelligence Lab

###### Abstract

We describe a new library named picasso 111More details can be found in our
Github page: https://github.com/jasonge27/picasso, which implements a unified
framework of pathwise coordinate optimization for a variety of sparse learning
problems (e.g., sparse linear regression, sparse logistic regression, sparse
Poisson regression and scaled sparse linear regression) combined with
efficient active set selection strategies. Besides, the library allows users
to choose different sparsity-inducing regularizers, including the convex
$\ell_{1}$, nonvoncex MCP and SCAD regularizers. The library is coded in C++
and has user-friendly R and Python wrappers. Numerical experiments demonstrate
that picasso can scale up to large problems efficiently.

##  1 Overview

Sparse Learning arises due to the demand of analyzing high-dimensional data
such as high-throughput genomic data (Neale et al., 2012) and functional
Magnetic Resonance Imaging (Liu et al., 2015). The pathwise coordinate
optimization is undoubtedly one the of the most popular solvers for a large
variety of sparse learning problems. By leveraging the solution sparsity
through a simple but elegant algorithmic structure, it significantly boosts
the computational performance in practice (Friedman et al., 2007). Some recent
progresses in (Zhao et al., 2017; Li et al., 2017) establish theoretical
guarantees to further justify its computational and statistical superiority
for both convex and nonvoncex sparse learning, which makes it even more
attractive to practitioners.

We recently developed a new library named picasso, which implements a unified
toolkit of pathwise coordinate optimization for solving a large class of
convex and nonconvex regularized sparse learning problems. Efficient active
set selection strategies are provided to guarantee superior statistical and
computational preference. Specifically, we implement sparse linear regression,
sparse logistic regression, sparse Poisson regression and scaled sparse linear
regression (Tibshirani, 1996; Belloni et al., 2011; Sun and Zhang, 2012). The
options of regularizers include the $\ell_{1}$, MCP, and SCAD regularizers
(Fan and Li, 2001; Zhang, 2010). Unlike existing libraries implementing
heuristic optimization algorithms such as ncvreg or glmnet (Breheny, 2013;
Friedman et al., 2010), our implemented algorithm picasso have strong
theoretical guarantees that it attains a global linear convergence to a unique
sparse local optimum with optimal statistical properties (e.g. minimax
optimality and oracle properties). See more details in Zhao et al. (2017); Li
et al. (2017).

##  2 Algorithm Design and Implementation

The algorithm implemented in picasso is mostly based on the generic pathwise
coordinate optimization framework proposed by Zhao et al. (2017); Li et al.
(2017), which integrates the warm start initialization, active set selection
strategy, and strong rule for coordinate preselection into the classical
coordinate optimization. The algorithm contains three structurally nested
loops as shown in Figure 1:

(1) Outer loop: The warm start initialization, also referred to as the
pathwise optimization scheme, is applied to minimize the objective function in
a multistage manner using a sequence of decreasing regularization parameters,
which yields a sequence of solutions from sparse to dense. At each stage, the
algorithm uses the solution from the previous stage as initialization.

(2) Middle loop: The algorithm first divides all coordinates into active ones
(active set) and inactive ones (inactive set) by a so-called strong rule based
on coordinate gradient thresholding (Tibshirani et al., 2012). Then the
algorithm calls an inner loop to optimize the objective, and update the active
set based on efficient active set selection strategies. Such a routine is
repeated until the active set no longer changes

(3) Inner loop: The algorithm conducts coordinate optimization (for sparse
linear regression) or proximal Newton optimization combined with coordinate
optimization (for sparse logistic regression, Possion regression, scaled
sparse linear regression, sparse undirected graph estimation) only over active
coordinates until convergence, with all inactive coordinates staying zero
values. The active coordinates are updated efficiently using an efficient
“naive update” rule that only operates on the non-zero coefficients. Better
efficiency is achieved by the “covariance update” rule. See more details in
(Friedman et al., 2010). The inner loop terminates when the successive descent
is within a predefined numerical precision.

The warm start initialization, active set selection strategies, and strong
rule for coordinate preselection significantly boost the computational
performance, making pathwise coordinate optimization one of the most important
computational frameworks for sparse learning. The numerical evaluations show
that picasso is highly scalable and efficient.

The library is implemented in C++ with the memory optimized using sparse
matrix output, and called from R and Python by user-friendly interfaces.
Linear algebra is supported by the Eigen3 library (Guennebaud et al., 2010)
for portable high performance computation. The implementation is modularized
so that the algorithm in src/solver/actnewton.cpp works with popular sparsity-
inducing regularizer functions and any convex objective function that exhibits
restricted strong convexity property (Zhao et al., 2017). Users can easily
extend the package by writing customized objective function subclass and
regularizer function subclass following the virtual function interfaces of
class ObjFunction and class RegFunction in include/picasso/objective.hpp.

Figure 1: The pathwise coordinate optimization framework with 3 nested loops:
(1) Warm start initialization; (2) Active set selection, and strong rule for
coordinate preselection; (3) Active coordinate minimization.

##  3 Example of R User Interface

We illustrate the user interface by analyzing the eye disease data set in
picasso.

    > library(picasso); data(eyedata) # Load the data set
    > out1 = picasso(x,y,method="l1",type.gaussian="naive",nlambda=20,
    +                lambda.min.ratio=0.2) # Lasso
    > out2 = picasso(x,y,method="mcp", gamma = 1.25, prec=1e-4)  # MCP regularizer
    > plot(out1); plot(out2) # Plot solution paths

The program automatically generates a sequence of regularization parameters
and estimate the corresponding solution paths based on the $\ell_{1}$ and MCP
regularizers respectively. For the $\ell_{1}$ regularizer, we set the number
of regularization parameters as 20, and the minimum regularization parameter
as 0.2*lambda.max. For the MCP regularizer, we set the concavity parameter as
$\gamma=1.25$, and the pre-defined accuracy as $10^{-4}$. Here nlambda and
lambda.min.ratio are omitted, and therefore set by the default values
(nlambda=100 and lambda.min.ratio=0.05). We further plot two solution paths in
Figure 2.

##  4 Numerical Simulation

To demonstrate the superior efficiency of our library, we compare picasso with
a popular R library ncvreg (version 3.9.1) for nonconvex regularized sparse
regression, the most popular R library glmnet (version 2.0-13) for convex
regularized sparse regression, and two R libraries scalreg-v1.0 and
flare-v1.5.0 for scaled sparse linear regression. All experiments are
evaluated on an Intel Core CPU i7-7700k 4.20GHz and under R version 3.4.3.
Timings of the CPU execution are recored in seconds and averaged over 10
replications on a sequence of 100 regularization parameters. All algorithms
are compared on the same regularization path and the convergence threshold are
adjusted so that similar objective gaps are achieved.

We compare the timing performance and the optimization performance in Table 4.
We choose the problem size to be $(n=3000,d=30000)$, where $n$ is the number
of observation and $d$ is the dimension of the parameter vector. We tests the
algorithms for both well-conditioned cases and ill-conditioned cases. The
details of data generation can be found in the R library vignette. Here is our
summary:

(1) For sparse linear regression using any regularizer and sparse logistic
regression using the $\ell_{1}$ regularizer, all libraries achieve almost
identical optimization objective values, and picasso slightly outperforms
glmnet and ncvreg in the timing performance.

(2) For sparse logistic regression using nonconvex regularizers, picasso
achieves comparable objective value with ncvreg, and significantly outperforms
ncvreg in timing performance. We also remark that picasso performs stably for
various settings and tuning parameters. However, ncvreg may converge very slow
or fail to converge for sparse logistic regression using nonconvex
regularizers, especially when the tuning parameters are relatively small
(corresponding to denser estimators), as the ill-conditioned SCAD case shows.

(3) For scaled Lasso, in order to make other competitors (flare and scalreg)
converges in resonable time, we swtich to a smaller problem size
$(n=1000,d=10000)$. We see that picasso much more time saving than flare and
scalreg.

Table 1: Average timing performance (in seconds) with standard errors in the
parentheses and achieved objective values.

Sparse Linear Regression
---
|  well-conditioned |  ill-conditioned
$\ell_{1}$ | picasso | 1.61(0.03)s | 27.691 | 3.62(0.02)s | 32.543
glmnet | 4.15(0.03)s | 27.692 | 9.43(0.01)s | 32.537
| ncvreg | 5.92(0.01)s | 27.690 | 6.66(0.01)s | 32.536
SCAD | picasso | 1.56(0.01)s | 27.668 | 3.74(0.01)s | 33.133
ncvreg | 5.61(0.01)s | 27.673 | 7.05(0.01)s | 33.156
MCP | picasso | 1.47(0.02)s | 27.161 | 1.89(0.02)s | 32.468
ncvreg | 4.07(0.03)s | 27.161 | 2.56(0.01)s | 32.468
Sparse Logistic Regression
$\ell_{1}$ | picasso | 2.03(0.01)s | 0.363 | 2.10(0.03)s | 0.327
glmnet | 16.32(0.12)s | 0.363 | 20.31(0.02)s | 0.327
| ncvreg | 4.04(0.01)s | 0.363 | 62.89(0.04)s | 0.327
SCAD | picasso | 4.25(0.01)s | 0.227 | 4.35(0.02)s | 0.172
ncvreg | 11.47(0.04)s | 0.278 |  error ($>$300s) |
MCP | picasso | 4.32(0.01)s | 0.221 | 4.38(0.02)s | 0.165
ncvreg | 9.37(0.08)s | 0.248 | 6.99(0.01)s | 0.242
Scaled Lasso
$\ell_{1}$ | picasso | 0.36(0.01)s | 4.454 | 0.15(0.01)s | 5.495
flare | 5.23(0.06)s | 5.188 | 297.36(2.77)s | 5.959
| scalreg | 40.20(0.63)s | 4.492 | 49.12(10.98)s | 5.507
|  |  |  |  |

---
Figure 2: The solution paths of $\ell_{1}$ (up) and MCP (down) regularizers.

##  5 Conclusion

The picasso library demonstrates significantly improved computational and
statistical performance over existing libraries for nonconvex regularized
sparse learning such as ncvreg. Besides, picasso also shows improvement over
the popular libraries for convex regularized sparse learning such as glmnet.
Overall, the picasso library has the potential to serve as a powerful toolbox
for high dimensional sparse learning. We will continue to maintain and support
this library.

## References

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  * Breheny (2013) P Breheny.  ncvreg: Regularization paths for scad-and mcp-penalized regression models.  _R package version_ , pages 2–6, 2013.
  * Fan and Li (2001) J. Fan and R. Li.  Variable selection via nonconcave penalized likelihood and its oracle properties.  _Journal of the American Statistical Association_ , 96(456):1348–1360, 2001.
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  * Liu et al. (2015) Han Liu, Lie Wang, and Tuo Zhao.  Calibrated multivariate regression with application to neural semantic basis discovery.  _Journal of Machine Learning Research_ , 16:1579–1606, 2015\.
  * Neale et al. (2012) Benjamin M Neale, Yan Kou, Li Liu, Avi Ma’Ayan, Kaitlin E Samocha, Aniko Sabo, Chiao-Feng Lin, Christine Stevens, Li-San Wang, Vladimir Makarov, et al.  Patterns and rates of exonic de novo mutations in autism spectrum disorders.  _Nature_ , 485(7397):242, 2012.
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# Optimal multi-action treatment allocation:
A two-phase field experiment to boost immigrant naturalization

Achim Ahrens _Correspondence:_ Achim Ahrens, Immigration Policy Lab, ETH
Zurich. Leonhardshalde 21, CH–8092, +41 44 632 82 06.
<EMAIL_ADDRESS>(Ahrens<EMAIL_ADDRESS>(Stampi-Bombelli<EMAIL_ADDRESS>(Hangartner),
<EMAIL_ADDRESS>(Kurer). _Acknowledgements:_ We thank our partners
from the City of Zurich for their collaboration and support. We are grateful
to Zhengyuan Zhou who has provided helpful feedback and to Teresa Freitas
Monteiro for comments. We also thank seminar participants at ETH Zurich,
Princeton University and Stanford University as well as participants at the
PolMeth Europe conference. All remaining errors are our own. _Funding:_ This
research was supported by the Stiftung Mercator Schweiz and by the _nccr – on
the move_ program, which is funded by the Swiss National Science Foundation
(grant no. 51NF40-182897). _Data:_ The authors provide replication code
through the Journal of Applied Econometrics Data Archive. The data is owned by
the City of Zurich and the Canton of Zurich. It is not publicly available for
confidentiality reasons. Data access requests should be directed to the City
of Zurich’s population office and the Canton of Zurich’s municipal office.
Alessandra Stampi-Bombelli  Selina Kurer  Dominik Hangartner

##  1 Introduction

Policymakers frequently need to select among alternative treatment options.
While one of the stated aims of empirical research is to provide new insights
to inform decision-making processes, the primary focus is usually on
estimating averages of treatment effects rather than providing direct guidance
on how to design assignment mechanisms for alternative treatments. In
practice, the empirical researcher specifies a statistical model and estimates
the efficacy of each treatment using an experimental or observational sample,
while the decision maker assigns the treatment, interpreting the point
estimates as if they were true. This approach, termed _as-if_ maximization by
[57], tends to yield one-size-fits-all rules assigning the same treatment to
the wider population. Such one-size-fits-all policies seem inefficient given
that treatment effects frequently exhibit relevant effect heterogeneity across
observations and the increasing availability of administrative data providing
rich individual characteristics.

Policy learning provides a framework for directly estimating statistical
decision rules, so-called policy rules, which prescribe treatments to
individuals based on their observed characteristics (also known as profiling
or targeting). While its origins date back to statistical decision theory [75,
65], the seminal work of [55] sparked a flourishing literature in econometrics
which has developed methods for estimating statistical treatment rules,
initially focusing on data drawn from randomized control trials [55, 66, 67,
39], but subsequently also covering observational data under unconfoundedness
assumptions ([56, 4, 78]; see [40] for a review). While applied research using
policy learning is still relatively scarce, previous work has revealed the
potential for data-driven treatment allocation across a variety of domains,
including active labor market programs [53, 23, e.g.], vaccines accounting for
spill-over effects [48], deforestation-reducing policies [2], anti-malaria
subsidies under budget constraints [9], energy use information campaigns [43,
27] and maximizing fundraising [13].

In this pre-registered study, we co-design and evaluate an individualized
treatment allocation program with the goal of facilitating the naturalization
of eligible immigrants in the City of Zurich, Switzerland. An expanding body
of literature is utilizing close naturalization referendums or temporal
discontinuities created by policy reform to enable credible comparisons
between naturalized and non-naturalized immigrants to demonstrate that
acquiring host-country citizenship offers long-term integration benefits for
immigrants, and, indirectly, to host societies. These benefits span various
integration dimensions, including employment and earnings [26, 36], political
efficacy, knowledge, and participation [34], as well as social incorporation
and cooperation [46, 35, 21]. Yet, despite these benefits, naturalization
rates remain low in many countries, with a median annual naturalization rate
(number of naturalized immigrants divided by number of immigrants) of 1.9% in
Europe and 3.1% in the U.S. [76]. Against this background, policymakers at the
national level in Estonia, Ireland, Latvia, North Macedonia, Spain, and at the
local level in Germany, Italy, Switzerland, and the United States, have
deployed information campaigns to boost citizenship applications and fully
reap the integration benefits of naturalization [42].

Informed by existing research [11, 5, 60, e.g.,] and the insights of
integration and naturalization bureaucrats of the City of Zurich, this study
considers interventions that address three specific hurdles blocking eligible
immigrants’ path to citizenship. These hurdles include: (i) the perceived
complexity of the application process, (ii) knowledge gaps about the
requirements for naturalization, and (iii) the feeling of not being welcome to
naturalize. To address the first two hurdles, we co-designed specific
information letters with the City of Zurich. For the third hurdle, a letter
sent by the Mayor of City of Zurich encouraged immigrants to apply. In line
with recent recommendations by [32], we opted for three separate treatment
letters with accompanying flyers to ensure that each letter is short and easy
to understand. Addressing all hurdles in a combined treatment letter with
several flyers is likely counterproductive due to the limited time and
attention that recipients devote when reading the letters.111A large
literature in behavioral economics stresses that information processing is
costly and provides evidence that individuals often fail to translate all
available information into optimal decisions (for recent reviews, see [37, 24,
54]).

Since it is unknown which treatment letter is optimal for maximizing the
individual application probabilities, and given that the optimal treatment
choice may differ among individuals, we derive a multi-action policy rule.
This policy rule is structured as a decision tree, which is referred to as a
‘policy tree.’ Policy trees are introduced by [4] for binary and by [78] for
multi-valued treatments. In our context, the policy tree selects one treatment
from a set of three treatment options for each eligible immigrant based on
their individual characteristics including residency, nationality and age. The
treatment options are incorporated into three different letters with enclosed
flyers sent out by the City of Zurich. Thus, by applying policy learning, we
allow the optimal content and framing of the information provision to vary
with observed immigrant characteristics. The policy rule is chosen to maximize
the application rate for naturalization, the first step in the process of
acquiring Swiss citizenship.

Policy trees possess several strengths that make them a particularly promising
method for immigrant naturalization and other sensitive policy contexts.
First, policy trees allow policymakers and researchers to select those
variables that can be used to tailor treatment assignment and, more
importantly, exclude those that should not be used (e.g., protected
characteristics such as religion)—and quantify the costs of exclusion in terms
of foregone treatment efficacy. Second, policy trees make transparent which
variables, and which variable values, guide treatment assignment. This is in
contrast to black-box _plug-in_ rules, providing no insights into what drives
treatment allocation. Related to the second strength is the third: policy
trees are easy to visualize and easy to explain to users of the research—e.g.,
policymakers, case officers, and subjects receiving treatment assignment—even
if they lack training in statistics. Together, transparency and
interpretability are important steps towards satisfying requirements for
explainable Artificial Intelligence (AI), e.g., as outlined in recent
proposals for the regulation of AI by the [20] and [70]. Finally, from a
practical perspective, the so-called offline approach of policy trees, which
learns policies from a single data batch, is often easier to implement in a
public policy context than adaptive approaches training policy rules
dynamically over time (e.g., [14]).

After introducing the methodology of policy learning, we illustrate the
practical feasibility of the targeted assignment rule and evaluate its
benefits using a tailored, two-phase randomized controlled trial. In the first
phase of our field experiment, we randomly allocate 60% of our sample of 5,145
citizenship-eligible immigrants to receive one of three letters addressing
specific naturalization hurdles. Based on first-wave application outcomes and
leveraging observed treatment effect heterogeneity, we estimate the optimal
multi-action policy rule using the estimation framework of [78]. In the second
phase, we field the fitted policy rule on one-half of the remaining sample
while sending random treatment letters to the other half. Adopting terminology
from reinforcement learning, we refer to these two phases as the exploration
phase (aimed at gathering knowledge about treatment efficacy) and the
exploitation phase (aimed at implementing the reward-maximizing strategy),
respectively. We evaluate the performance of the derived policy rule against
random treatment allocation, one-size-fits-all policy rules assigning the same
treatment to everyone, and a model-free _plug-in_ rule assigning the treatment
with the largest estimated treatment effect. We find that policy trees can
capture the vast majority of treatment effect heterogeneity of the more
flexible but less transparent and non-interpretable _plug-in_ rule. Despite
only moderate levels of heterogeneity, the policy tree yields a larger, albeit
insignificant, increase in take-up than each individual treatment.

Our study relates to three fields of empirical research. First, sparked by
methodological advances, especially the advent of causal forests [74, due to],
there is a burgeoning literature estimating heterogeneous treatment effects
using machine learning [19, 51, 50, e.g.,].222Other methods for estimating
conditional average treatment effects using machine learning include [17, 52].
For an overview, see [49, 45]. While studies in this literature emphasize the
potential of estimating heterogeneous effects for improved targeting, they
usually do not explicitly derive interpretable targeting rules. Second, we
build on the expanding literature applying statistical decision rules. The
vast majority of applied studies, including those discussed above (i.e., [53,
23, 9, 2, 48]), only provide backtest results about the ex-post performance of
policy targeting rules. [43] propose a policy-learning framework that allows
participants to self-select their treatment and apply their method to a
residential energy rebate program. Closest to our study are [27] and [13].
[27] follow the methodology of [47] to estimate policy rules for a behavioral
intervention targeted at reducing household electricity usage, but do not
implement the derived policy rules. Similar to us, [13] consider policy trees
in an application to maximizing fundraising and gauge the performance of the
estimated policy tree on out-of-sample data. We add to this literature by
fielding the estimated optimal policy rule in the second phase of our
experiment, which allows us to directly evaluate the performance against other
policy rules. Furthermore, both [13] and [27] focus on the choice between two
treatment options, whereas we are concerned with the more challenging problem
of multi-action policy learning. Third, we contribute to the larger literature
on informational interventions aimed at increasing take-up of government
services and subsidies among eligible people [8, 22, 41, 28, e.g.,]. Beyond
contributing to these three strands, this article aims to make policy learning
accessible to a wider audience by offering an introduction relevant both for
randomized field experiments and applications relying on observational data.

This article proceeds as follows. Section 2 provides an introduction to policy
learning. Section 3 turns to our application. We contextualize our
application, describe the data, the treatments and the study design in
Sections 3.1-3.4. We summarize the results of the exploration and exploitation
phase in Sections 3.5 and 3.6. Section 4 concludes.

##  2 Multi-action policy learning

In this section, we provide a brief review of (multi-action) policy learning,
with a special focus on the policy learning framework of [78]. While we rely
on a randomized experimental design to learn the optimal policy rule in our
application, we also discuss the setting where one has to rely on
unconfoundedness assumptions, thereby illustrating the generality of the
methodological framework.

The aim of policy learning is to formulate a policy rule $\pi(X)$ designed to
maximize the expected value of $Y$, the outcome of interest. A policy rule
assigns a treatment $a$ from the choice set of treatment options
$\mathcal{A}=\\{1,2,\dots,D\\}$ to each individual based on their observed
covariates $X$. Note that $\mathcal{A}$ may include the no-treatment option.
Formally, $\pi(X)$ is a function mapping individual characteristics to one of
the treatment options in $\mathcal{A}$. For example, a policy rule might
assign treatment 1 to every person below age 30, treatment 2 to individuals
aged 30-40, and treatment 3 to individuals older than 40.

###  2.1 Estimating optimal policies

Before we turn to the estimation of optimal policies, it is instructive to
consider a candidate policy rule $\pi^{\prime}(X)$ and assess its
effectiveness. We assume that we have access to the sample
$\\{Y_{i},A_{i},X_{i}\\}$ for $i=1,\ldots,n$, which is drawn from the joint
population distribution $P$. The sample data include the treatment received,
$A_{i}$, the realized outcome, $Y_{i}$, as well as observed individual $i$’s
characteristics $X_{i}$. In our application, the data stems from the
exploration phase of the randomized controlled trial, but the general approach
also extends to observational data.

As typical in the causal effects literature, we assume the existence of the
potential outcomes $\\{Y_{i}(1),Y_{i}(2),\ldots,Y_{i}(D)\\}$, which are the
outcomes if individual $i$ had received treatments 1, 2, …, $D$ [64, 44]. This
allows us to define the expected reward of $\pi^{\prime}(X)$, which is the
expected value of the potential outcomes if the policy rule had been followed,
i.e., $Q(\pi^{\prime}(X_{i}))=E[Y_{i}(\pi^{\prime}(X_{i}))]$ where $E[\cdot]$
denotes the expectation with respect to the population $P$. In non-
experimental settings, the fundamental challenge of estimating the reward of a
candidate policy $\pi^{\prime}(X)$ is that we only observe
$Y_{i}=Y_{i}(A_{i})$ and that individuals might self-select into treatment
options that optimize their expected pay-off.

The offline policy learning literature commonly imposes the following set of
assumptions [47, 78]:

###### Assumption 1

  1. (a)

Unconfoundedness: $Y_{i}(1),\ldots,Y_{i}(D)\\}\perp A_{i}|X_{i}.$

  2. (b)

Overlap: There exists some $\eta>0$ such that $e_{a}(X_{i})\geq\eta$ for any
$a\in\mathcal{A}$ and $X$, where $e_{a}(X_{i})\equiv P(A_{i}=a|X_{i})$ are the
propensity scores for treatment $a$.

  3. (c)

Boundedness: The potential outcomes are contained on a finite interval in
$\mathbb{R}^{D}$.

Unconfoundedness in (a) states that we observe all necessary covariates
allowing us to account for selection biases. The condition is naturally
satisfied by randomized treatment assignments. The overlap assumption in (b)
requires that for any observed individual characteristic $X_{i}$, the
probability $e_{a}(X_{i})$ of taking each action $a$ is greater than zero. The
boundedness assumption in (c) serves the purpose of simplifying mathematical
proofs but can be replaced by weaker assumptions.

Under the stated assumptions, we can evaluate the reward of a candidate policy
$\pi^{\prime}$ by averaging over observations that happen to align with the
candidate policy rule, i.e.,

$\widehat{Q}_{IPW}(\pi^{\prime}(X_{i}))=\frac{1}{n}\sum_{i=1}^{n}\frac{{\mathbbm{1}}{\left\\{A_{i}=\pi^{\prime}\left(X_{i}\right)\right\\}}Y_{i}}{e_{A_{i}}(X_{i})}.$
(1)

where we take the weighted average across observations that align with the
candidate policy rule. We inversely weight by the propensity score
$e_{a}(X_{i})$ to account for selection bias [69, 47].

Suppose that the policymaker suggests a number of policy rules, e.g.,
$\Pi^{\prime}=\\{\pi^{\prime},\pi^{\prime\prime},\pi^{\prime\prime\prime}\\}$
where $\Pi^{\prime}$ is the set of candidate policies. The optimal policy is
the policy that maximizes the expected reward; formally,
${\pi}^{\star}(X_{i})=\arg\max_{\pi\in\Pi^{\prime}}Q(\pi(X_{i}))$.
Accordingly, we can leverage our sample to estimate the optimal policy rule as

$\hat{\pi}(X_{i})=\arg\max_{\pi\in\Pi^{\prime}}\hat{Q}_{IPW}(\pi(X_{i})).$ (2)

The performance of a policy learner $\hat{\pi}(X_{i})$, which estimates
$\pi^{\star}(X_{i})$ from the data, is measured by its regret,
$R(\hat{\pi}(X_{i}))=Q(\pi^{\star}(X_{i}))-Q(\hat{\pi}(X_{i}))$. Thus, regret
measures the difference between the reward of the unobserved optimal policy
and the value of the estimated policy.

Note that we leverage the same covariates $X_{i}$ to adjust for selection
effects in (1) and to form policy rules in (2). There may, however, be good
reasons to use distinct covariate sets in each step. For example, legal or
ethical concerns might mandate the exclusion of protected characteristics from
the policy rule (e.g., gender, nationality, religion). Yet, the inclusion of
these characteristics in the propensity score estimation could be necessary if
prior evidence suggests their potential influence on treatment allocation. In
randomized experiments, a consistent estimation of the reward does not require
covariate adjustment but can enhance statistical precision.

###  2.2 Cross-fitting and double-robust estimation

If the propensity scores $e_{a}(X)$ are known, the regret converges to zero at
$\sqrt{n}$-rate [69, 47]. If the exact assignment mechanism is not known,
which is typically the case in non-experimental settings, we have to estimate
$e_{a}(X)$ from the data. One approach is to estimate $e_{a}(X)$ using the
full sample and plug the estimates into (1). However, this approach generally
yields sub-optimal convergence rates unless we impose strong convergence rate
requirements on the first-step estimator [47]. The sub-optimal performance can
be attributed to the own-observation bias, which arises if the first-step
estimation error from the propensity score estimation is correlated with the
error associated with estimating the reward. To allow for a general class of
data-adaptive nonparametric estimators, including popular supervised machine
learners such as random forests, which are more robust towards unknown data
structures, [78] combine two strategies for policy learning: cross-fitting and
double-robust estimation using augmented inverse-propensity weighting (AIPW).
We discuss each strategy in turn.

To illustrate how cross-fitting addresses the own-observation bias, consider
the simple case where we randomly split the data into two sub-samples,
referred to as auxiliary and main samples. In the first step, we leverage the
auxiliary sample for the estimation of conditional expectation functions
(e.g., the propensity scores). The second step uses out-of-sample predicted
values from the first step on the main sample to estimate the reward. This
sample-splitting approach resolves the own-observation bias since the second
step is, after conditioning on the auxiliary sample, independent from the
first-step estimation error.

Cross-fitting extends this sample-splitting approach by flipping the auxiliary
and main samples, thus effectively using the full sample for both the first
and second-step estimation. Cross-fitting also allows the splitting of the
data into more than two partitions.333The causal machine learning literature
frequently relies on sample splitting approaches, such as cross-fitting; see
for example [16] for the estimation of average treatment effects and [74] for
the estimation of CATE using causal forests. Specifically, to implement cross-
fitting, we randomly split the sample into $K$ folds of approximately equal
size. We use $\hat{e}_{a}^{-k(i)}(X_{i})$ to denote the cross-fitted
propensity score of observation $i$ for treatment $a$. The cross-fitted
predicted value is calculated as the out-of-sample predicted value from
fitting an estimator on all folds but fold $k(i)$, which is the fold that
observation $i$ falls into. Similarly, we introduce
$\hat{\mu}_{a}^{-k(i)}(X_{i})$ which is the cross-fitted predicted value of
the outcome under treatment $a$ using predictors $X_{i}$, i.e., it is a cross-
fitted estimate of $\mu_{a}(X_{i})\equiv E[Y_{i}(a)|X_{i}]$.

Double robust estimation of the reward allows for non-random treatment
assignment under uncounfoundedness. The estimator adjusts for biases arising
from selective treatment allocation by combining the reweighting approach of
inverse-propensity weighting as used in (1) with outcome adjustment (as used
in regression-based adjustment). The advantage over the IPW estimator is that
the resulting double-robustness property guarantees consistency if either the
propensity scores $e_{a}(X_{i})$ or the conditional expectation of outcome
given covariates, i.e., $\mu_{a}(X_{i})$, are correctly specified. Using the
cross-fitted estimates $\hat{e}_{a}^{-k(i)}(X_{i})$ and
$\hat{\mu}_{a}^{-k(i)}(X_{i})$, we can define the cross-fitted AIPW (CAIPW)
estimator of the reward as444The function $\mathbbm{1}\\{\cdot\\}$ denotes the
indicator function.

$\widehat{Q}_{CAIPW}(\pi(X_{i}))=\frac{1}{n}\sum_{i=1}^{n}\left(\frac{Y_{i}-\hat{\mu}_{A_{i}}^{-k(i)}\left(X_{i}\right)}{\hat{e}_{A_{i}}^{-k(i)}\left(X_{i}\right)}\mathbbm{1}\\{A_{i}=\pi(X_{i})\\}+\hat{\mu}_{\pi(X_{i})}^{-k(i)}\left(X_{i}\right)\right).$
(3)

The first term in (3) adjusts the observed outcome by subtracting the
conditional expectation of the outcome under the observed treatment and by
inversely weighting with the propensity scores if the observed treatment,
$A_{i}$, is equal to the treatment recommended by the policy, $\pi(X_{i})$.
The second term adds the conditional expectation of the outcome under the
treatment assigned by the policy. Using the double-robust estimator of the
reward, we can estimate the optimal policy by evaluating
$\widehat{Q}_{CAIPW}(\pi(X_{i}))$ for all candidate policies in
$\Pi^{\prime}$, i.e., we calculate
${\hat{\pi}(X_{i})}_{CAIPW}=\arg\max_{\pi\in\Pi^{\prime}}\widehat{Q}_{CAIPW}(\pi(X_{i}))$.

###  2.3 Policy class

So far, we have assumed a predefined set of candidate policies. In many
applications, however, we wish to learn policies flexibly from the data
instead of relying on a pre-defined set of policy rules. A fully flexible
approach could assign each individual to the treatment for which the estimated
treatment effect is the largest. This plug-in policy rule requires no
functional form restrictions but may be inappropriate when stakeholders wish
to learn about the drivers of treatment efficacy and have hesitations to rely
on a black-box treatment assignment mechanism.555For formal results on plug-in
rules, see [39, 9].

Policy learning allows estimating interpretable treatment rules from the data.
To this end, we must choose a suitable policy class from which we estimate the
optimal policy. In an application to active labor market programs with a
binary treatment and two covariates, [47] discuss three policy classes defined
by the three functional form restrictions:

Quadrant policy rule: $\displaystyle\qquad\pi(X_{i})$
$\displaystyle{=}\mathbbm{1}\\{s_{1}(X_{1i}-\beta_{1})\geq
0\\}\mathbbm{1}\\{s_{2}(X_{2i}-\beta_{2})\geq 0\\}$ Linear policy rule:
$\displaystyle\pi(X_{i})$
$\displaystyle{=}\mathbbm{1}\\{\beta_{0}+\beta_{1}X_{1i}+\beta_{2}X_{2i}\geq
0\\}$ Cubic policy rule: $\displaystyle\pi(X_{i})$
$\displaystyle{=}\mathbbm{1}\\{\beta_{0}+\beta_{1}X_{1i}+\beta_{2}X_{2i}+\beta_{3}X_{2i}^{2}+\beta_{4}X_{2i}^{3}\geq
0\\}$

where $s_{1},s_{2}\in\\{-1,+1\\}$ and $\beta_{j}\in\mathbbm{R}$. The first
rule defines a quadrant in the two-dimensional space spanned by the covariates
$X_{1}$ and $X_{2}$, and assigns the treatment to individuals for which
$X_{1i}$ and $X_{2i}$ lie in that quadrant. The second rule defines a linear
decision boundary. Figure 1(a) and (b) illustrate examples of the quadrant and
linear policy rule. The third rule allows for non-linear decision boundaries
by including quadratic and cubic terms. Compared to the plug-in rule, these
rules exhibit a higher degree of interpretability and, as shown in Figure 1,
they can be easily visualized in a two-dimensional plot.

(a) Quadrant policy rule

(b) Linear policy rule

(c) Policy tree

(d) Policy tree in two-dimensional space

_Note:_ The figure shows three examples of policy rules that assign treatments
based on two covariates $X_{1}$ and $X_{2}$. Figures (a) and (b) illustrate
examples of the quadrant and linear policy rule considered in [47] for binary
treatment choices. Figure (c) and (d) provide two alternative illustrations of
the same policy tree with three treatment options.

Figure 1: Illustrative examples of policy rules with different functional
forms

In a multi-action setting, [78] focus on the class of policy rules that take
the form of (shallow) decision trees. Trees are widely employed as predictive
tools that construct predictions by splitting the feature space into non-
overlapping regions. In the prediction context, classification and regression
trees yield the same prediction for observations falling into the same region.
In the policy context, observations falling into the same region are assigned
the same treatment action. Figure 1(c) shows an example of a policy tree which
assigns treatment options 1, 2 or 3 based on the values of $X_{1}$ and
$X_{2}$. Figure 1(d) visualizes the same tree in a two-dimensional space,
highlighting that policy trees are conceptually similar to the quadrant rule
but can be generalized to multiple treatments. [78] describe how the
optimization of policy trees can be regarded as a mixed integer program and
[68] implement a less costly but approximate optimization algorithm, referred
to as hybrid tree search.

To conclude, the policy learning framework of [78] has two attractive
properties, which are particularly relevant in the context of the the
naturalization campaign discussed in the next section: First, since trees can
be easily visualized, they are transparent and simple to interpret, even
without statistical training, making them attractive in a public policy
context where users of the research vary in statistical literacy and often
view black-box methods with scepticism. Second, the estimation of policy trees
is computationally feasible with readily available software [71, 68].

##  3 Personalizing naturalization campaigns

In this section, we apply the multi-action policy tree to an information
campaign encouraging eligible immigrants residing in the City of Zurich to
apply for Swiss citizenship. We introduce the policy context in Section 3.1,
and discuss data and treatments in Section 3.2 and 3.3. Section 3.4 summarizes
the study design and estimation methodology. Results are presented in Section
3.5 and 3.6.

###  3.1 Background: Immigrant Integration and Citizenship

The integration of immigrants into the host-country fabric and economy is a
central policy issue in many countries across the globe. One promising policy
to foster integration is naturalization, i.e., the process of awarding host-
country citizenship to immigrants [30, 18]. Observational studies relying on
difference-in-difference models and regression discontinuity designs comparing
similar naturalized and non-naturalized immigrants show that acquiring host-
country citizenship can positively impact the integration of immigrants by
increasing their earnings and labor market attachment [61, 58, 26, 36, 73, 31,
25], fostering political efficacy and knowledge [34], spurring cultural
assimilation and cooperation [46, 21], and reducing feelings of isolation and
discrimination [35].666With the exception of [58], who studies immigrants from
Latin American countries in the U.S, the studies referenced above focus on
France, Germany and Switzerland. This might limit external validity since we
expect the benefits of naturalization to be context-dependent and generally
decline with lower naturalization hurdles. While testing this hypothesis
requires more comparative research, [72] provide initial evidence from a
correspondence test varying citizenship in fictitious applications in Sweden
that is consistent with this conjecture. This process can also benefit the
host society by increasing immigrants’ contributions to economic growth,
lowering their dependency on welfare, and, by extension, reducing societal
tensions and strengthening social cohesion [60, 63, for reviews, see].

Despite these potential benefits, naturalization rates remain low in many
countries [10]. What explains this mismatch between the benefits of host-
country citizenship and the low demand for naturalization? Previous evidence
from surveys and qualitative studies suggest that uncertainty about the
eligibility criteria such as residency and language requirements can prevent
immigrants from applying [5, 29]. Other studies highlight that—particularly in
hostile immigration environments—a lack of encouragement by politicians,
public administration, or the general public might deter immigrants [5, 11,
12]. Furthermore, in earlier research using a tailored survey, we find
evidence for informational deficits and the feeling that an application is not
welcome by the host society [38]. Lastly, in countries that unlike Switzerland
do not allow for dual citizenship, immigrants might not be willing to give up
the passport from their origin country to obtain host-country
citizenship.777Whether a person is allowed to retain the previous citizenship
when naturalizing in another country generally depends on the regulations of
both the origin and host country. Switzerland has guaranteed the right to hold
dual (or more) citizenship without restrictions since 1992. Hence, eligible
immigrants seeking Swiss citizenship are subject only to restrictions of their
origin countries. Among the ten largest countries by nationality in our sample
(which jointly amount to 72% of origin countries), only Austria and Spain
generally do not allow for dual citizenship.

To boost naturalization rates, countries, states, and municipalities across
Europe and the U.S. have begun to turn to information campaigns to overcome
hurdles to citizenship acquisition for eligible immigrants. While the content
and scope of these naturalization campaigns vary, they often combine
information provision about the naturalization process and requirements with
an encouragement to apply for citizenship. Yet, despite the growing popularity
of these campaigns across Europe and the U.S., there exists little
experimental research to evaluate its effectiveness. An important exception is
[41], who show that a low-cost nudge informing low-income immigrants about
their eligibility for a fee waiver increased the rate of citizenship
applications by 8.6 percentage points (from 24.5% in the control group to
33.1%). Most similar to our study is [38], who evaluated previous versions of
the naturalization campaign of the City of Zurich and showed that a similarly
low-cost letter (about CHF 1.20 per person, see below) combining information
and encouragement increased naturalization rates by about 2.5 percentage
points (from 6.0% in the control group to 8.5%).

Past naturalization campaigns, including the one by the City of Zurich
mentioned above, have typically relied on a one-size-fits-all approach—despite
the substantial diversity of the immigrant population in terms of, e.g.,
country of origin, language skills, and age. There are good reasons to suspect
treatment effect heterogeneity along various dimensions: Immigrants’
willingness to naturalize and their susceptibility to certain information
letters might depend on their current nationality due to the specific dual
citizenship regulations, the relative benefits in terms of visa requirements
_vis-á-vis_ third countries, the attachment to the home country, and the
attitudes of native citizens towards specific immigrants groups. For example,
immigrants who feel discriminated against might be more likely to be persuaded
by a letter welcoming them to set roots and apply for citizenship in their
host country. Furthermore, language requirements might be less of a concern
for immigrants who speak the same language (such as Austrians and Germans in
Switzerland). Thus, tailoring such campaigns to the specific needs of diverse
immigrants promises to deliver both a deeper understanding of the different
hurdles that immigrants face and to increase the effectiveness of the
campaign.

###  3.2 Data

We draw our data from administrative sources of the Canton of Zurich. The data
includes records of whether and when eligible immigrants submit an application
for Swiss citizenship to the City of Zurich during the study period, which
allows us to define the outcome variable of our analysis. The data also
includes additional covariates which we use to identify and leverage treatment
effect heterogeneity. These covariates are age, gender, nationality, years of
residency in Switzerland, and years of residency in Zurich. The data also
includes an address identifier which allows us to assign the treatment on a
building level to minimize contamination by spill-over effects.

The study sample includes all immigrants in the City of Zurich who satisfy the
following criteria:

  1. 1.

They were born on or before June 30, 2003 (i.e., they must have been at least
18 years of age at the start of the study),

  2. 2.

they arrived in Switzerland on or before June 30, 2011,

  3. 3.

they arrived in Zurich City on or before June 30, 2019,

  4. 4.

they must have possessed a permanent residence permit (C permit) at the time
of randomization (August 2021), and

  5. 5.

they must not have received any information or encouragement letter in the
past.

The first criterion ensures that only adults are in the study. Criteria 2-4
ensure that the entire sample meets the current residency and permit
requirements for citizenship. The sample includes 5,145 individuals.

###  3.3 Treatment letters

Combining insights from the existing literature and our own surveys, we
identify three key barriers to naturalization: (i) perceived complexity of the
naturalization process, (ii) perceived difficulty of and uncertainty about
naturalization requirements and (iii) perception that naturalization is not
welcome. In collaboration with the City of Zurich, we developed three
treatment letters where each letter puts emphasis on one of the hurdles. Each
treatment involves the receipt of a letter sent by representatives of the City
of Zurich. The treatments differ in the sender, content, wording and design of
the letters.The per-unit costs of the three treatments range between 1.20 and
1.50 CHF, and are thus negligible compared to the fiscal benefits of
naturalization.888[36] quantify the long-term effect of naturalization on
immigrants’ earnings at CHF 4,500 per year, which implies an increase in tax
revenues for Swiss municipalities of at least CHF 450 per year. We chose to
develop distinct letters to keep the letters brief and understandable, thus
avoiding the risk of an informational overload [32]. The letters, including
enclosed flyers, were written in German. Appendix A.2 contains copies of the
original letters in German as well as an English translation.

The _Complexity letter_ consists of a short informational cover letter written
by the City Clerk of the City of Zurich (see Appendix A.2.1) and a flyer. The
half-page cover letter informs recipients that they meet the basic
requirements for Swiss citizenship and directs them to sources of further
information about the citizenship application process. The flyer included in
the _Complexity letter_ (shown in Figure A.2.2) attempts to tackle the
perceived complexity of the naturalization process. The left-hand side of the
flyer shows a video screenshot and a QR code that directs readers to the
video, explaining the naturalization process in a simplified way. The right-
hand side encourages readers to scan another QR code redirecting to the
contact and advice webpage999The first QR code redirects to https://www.stadt-
zuerich.ch/portal/de/index/politik_u_recht/einbuergerungen.html (last accessed
on December 7, 2022). The second QR code redirects to https://www.stadt-
zuerich.ch/portal/de/index/politik_u_recht/einbuergerungen/kontakt-und-
beratung.html (last accessed on December 7, 2022). of the City of Zurich’s
citizenship office.

The _Requirements letter_ includes the same short informational cover letter
as the _Complexity letter_ but uses a different flyer addressing the perceived
difficulty of the naturalization process (see Appendix A.2.3). This flyer is
also divided into two sections, each containing a descriptive text and a QR
code. The QR code on the left-hand side redirects to the targeted, free-of-
charge mobile application, which allows immigrants to study for the civics
exam and test their knowledge with practice questions.101010The mobile
application is developed by the City of Zurich and named Einbügerungstest Code
Schweiz, which translates to Naturalization Test Code Switzerland. The section
on the right lists the German language requirements for citizenship and the QR
code redirects to a webpage containing more detailed information on the
language requirements, exam costs, as well as a link to a practice language
exam.111111The website, which the QR code redirected to, moved to
https://www.stadt-
zuerich.ch/portal/de/index/politik_u_recht/einbuergerungen/kenntnisse/sprachlicheanforderungen.html
on October 21, 2022, due to a mistake by the website maintainers. As a
consequence, the QR code broke more than five months after the letter was
dispatched to wave 2 participants. We show in Table A.3, where we only
consider the naturalization applications recorded up to five months after
letter dispatch, that our main results in Table 3.6 are not affected by this
issue. We thus use, in line with the pre-analysis plan, application outcomes
recorded seven months after letter dispatch in the remainder of the study.

The _Welcome letter_ is an information and encouragement letter signed by the
Mayor of the City of Zurich. The _Welcome letter_ attempts to tackle the
hurdle stemming from the perception that naturalization is not welcome [33].
The letter includes only a cover letter (shown in Appendix A.2.4) that is a
little less than one page long and contains three sections. The first section
informs recipients that they meet the basic eligibility requirements for Swiss
citizenship. The second section encourages them to play an active part in
Zurich’s political life by becoming a citizen. The last section briefly
directs to sources for further information about the citizenship application
process and states that the City hopes to see them at the next ceremony for
new citizens. Hence, compared to the other two treatment letters, this letter
puts more emphasis on the emotional and psychological aspects associated with
naturalization and only provides minimal information.

###  3.4 Experimental design and estimation methodology

This section summarizes the pre-registered experimental design, estimation
methodology and evaluation strategy.121212The study was pre-registered at
https://osf.io/9wf4t. In the exploration phase of the project, we randomly
divide the sample of $5{,}145$ eligible immigrants into two groups: Group A
(60% of the sample) receives one of three treatment letters at random from the
City of Zurich in October 2021, while Group B (40%) received no letter. The
randomization design allocates one of the three treatment letters to
individuals in Group A by building address and applied block randomization by
nationality groups. The randomization by building address reduces the risk of
spill-over effects among eligible immigrants living in the same or neighboring
households. The block randomization by nationality group ensures that we have
a roughly equal share of nationalities in Group A (including each subgroup
receiving different letters) and Group B. We block on nationality groups given
the importance of this effect moderator in earlier studies [76]. The letters
for this first wave were delivered on October 8, 2021.

The first-wage application outcomes enable us to estimate the average
treatment effect of treatment letter $d$, i.e., $E[Y_{i}(d)-Y_{i}(0)]$, and
the conditional average treatment effect $E[Y_{i}(d)-Y_{i}(0)|X_{i}]$ where we
use $Y_{i}$ to denote the application outcome recorded at the end of March
2022 and $Y_{i}(d)$ its potential outcome under treatment $d$. The covariates
$X_{i}$ are country group of nationality, age, gender, years lived in Zurich
and years lived in Switzerland, which are constant over the sample period. We
employ causal forests due to [74, 3], a non-parametric method for the
estimation of heterogeneous treatment effects relying on random forests.

The main objective, however, is to leverage the first-wave application
outcomes $Y_{i}$ and individual characteristics $X_{i}$ to fit a multi-action
policy tree based on the estimation methodology of [78] outlined in Section 2.
To select the tree depth, we consider a validation exercise: In each
iteration, we randomly split the wave-1-data (including untreated) into
training and test data with a 60/40 split, and sample from each partition
separately with replacement to construct bootstrapped training and validation
data sets of sizes $n_{1}=4{,}871$ and $n_{2}=1{,}857$. We then fit a policy
tree on the bootstrapped training data and estimate the difference in reward
between alternative policy rules on the bootstrapped validation data.

In the exploitation phase, we field the fitted policy tree on not-yet-treated
individuals in Group B. Specifically, in order to evaluate the performance of
the policy rule, we randomly subdivide Group B into two sub-groups, referred
to as Group B.1 and Group B.2, and send treatment letters to Group B.1 based
on the estimated policy rule while Group B.2 receive a random treatment letter
(with one-third probability for each letter). We randomize by building address
for the random division into Groups B.1 and B.2, as well as for the
randomization of treatments within Group B.2. The City of Zurich delivered the
letters for the exploitation phase on May 6, 2022.131313Note that for
practical reasons, there was a two-month time gap between measuring the
application outcomes in March 2022 and sending out the letter in May.

The evaluation compares the policy tree against no treatment, random treatment
allocation, and conventional one-size-fits-all policy rules that always assign
the same treatment to everyone, ignoring treatment effect heterogeneity. To
this end, we estimate models of the form:

$Y_{it}=W_{it}^{\prime}\beta+f(X_{i},\delta_{t})+\varepsilon_{it}$ (4)

where $Y_{it}$ is the application outcome of eligible immigrant $i$ at the end
of wave $t\in\\{1,2\\}$. We add the wave subscript $t$ to accommodate the two-
wave structure of the data. The outcomes for the evaluation analysis were
recorded approximately 7 months after the date of letter dispatch
$t$.141414The application outcomes for the evaluation analysis were recorded
in May 9 and December 9, 2022, respectively. In Table A.3, we provide
alternative results where we consider all application outcomes until March 21
and October 21, 2022, respectively (see fn. 11). The time-invariant covariates
$X_{i}$ are defined above. $\delta_{t}$ is a dummy for wave $t\in\\{1,2\\}$,
and accounts for seasonal effects and other external shocks that may affect
application rates. The vector $W_{it}$ assigns individuals to treatment
groups, and is defined as
$W_{it}=(\textit{Letter}^{1}_{it},\textit{Letter}^{2}_{it},\textit{Letter}_{it}^{3},\textit{Nothing}_{it},\linebreak[1]\textit{PolicyTree}_{it})$
or
$W_{it}=\left(\textit{Random}_{it},\textit{Nothing}_{it},\textit{PolicyTree}_{it}\right)$,
respectively, where $\textit{Letter}^{j}_{it}$ is set to 1 if the individual
$i$ was randomly assigned to treatment letter $j\in\\{1,2,3\\}$ for wave $t$,
0 otherwise. $\textit{Nothing}_{it}$ is set to 1 if the individual $i$ has
received no treatment in wave $t$, and $\textit{PolicyTree}_{it}$ equals 1 if
individual $i$ has received the treatment letter assigned to them by the
policy tree. Finally, $\textit{Random}_{it}$ is set to 1 if individual $i$ was
randomly assigned to one of the three letters, 0 otherwise.

We estimate (4) by linear regression using only the elementary controls, but
also consider more flexible methods. Namely, we use Post-Double Selection
Lasso [6, PDS-Lasso;] and Double-Debiased Machine Learning [16, DDML;] where
we extend the set of controls by interaction terms and second-order
polynomials.151515For the Post-Double Selection Lasso, we use cluster-robust
penalty loadings of [7]. With regard to DDML, we use 10 cross-fitting folds, 5
cross-fitting repetitions and use stacking with a set of candidate learners
including linear regression, lasso, ridge, random forests and gradient
boosting [1]. We cluster standard errors by building addresses, i.e., the
level at which the treatment was applied.161616We note that the clustered
standard errors do not account for sampling variability arising from the
estimation of the policy rules. The issue is akin to the well-known generated
regressor problem which occurs when a regressor is unobserved and replaced by
a first-step estimate. The generated regressor problem is usually addressed
using standard-error adjustments [62, 59] or, most commonly, using
bootstrapping; see e.g. review in [15] or [77]. Neither of these approaches is
feasible in our setting. Analytical standard errors are, to our knowledge, not
available for this specific problem. Bootstrapping or other resampling
techniques would require us to repeatedly field policy rules fitted on
bootstrapped samples of the data in order to capture the variability in
estimated policy rules, which is practically infeasible. We thus interpret the
standard errors with caution.

###  3.5 Results from the exploration phase: Learning the policy rule

We begin by analyzing the results from the exploration phase of the experiment
using naturalization applications received by the end of March 2022 (i.e.,
wave 1). Descriptive statistics of the wave-1-data are provided in Table 3.5.
We proceed in three steps: estimation of (conditional) averages of treatment
effects, tuning policy trees using a validation exercise and fitting the
policy tree on the full wave-1-data.

|  | Avg. | St.dev. | Min | Max | Obs.
---|---|---|---|---|---|---
Dependent variable: |  |  |  |  |
|  Naturalization application | 0.07 | 0.25 | 0.00 | 1.00 | 4871

Covariates: Age 41.79 10.95 19.00 99.00 4871 Years in Switzerland 14.88 7.44
11.00 67.00 4871 Years in Zurich 9.08 4.06 3.00 20.00 4871 Regions: Americas &
Caribbean 0.05 0.22 0.00 1.00 4871 Asia 0.06 0.25 0.00 1.00 4871 Central-East
Europe 0.04 0.19 0.00 1.00 4871 Germany and Austria 0.37 0.48 0.00 1.00 4871
Italy 0.10 0.30 0.00 1.00 4871 Middle Easy and Northern Africa 0.01 0.12 0.00
1.00 4871 South-East Europe 0.12 0.32 0.00 1.00 4871 Spain and Portugal 0.12
0.33 0.00 1.00 4871 Stateless 0.00 0.04 0.00 1.00 4871 Sub-Saharian Africa
0.02 0.12 0.00 1.00 4871 Western Europe 0.11 0.32 0.00 1.00 4871  Notes: The
table shows summary statistics for covariates and dependent variables measured
until March 2022.

Table 1: Descriptive statistics of wave-1 data

First, we fit a multi-arm causal forest to estimate average treatment effects,
as well as conditional average treatment effects by nationality group and
years lived in Switzerland [74, 3]. Results are displayed in Figure
3.5.171717We removed 274 individuals who moved between October 2021 and March
2022, resulting in an estimation sample of $4{,}871$ individuals. The average
treatment effects for the first-wave sample imply that the _Complexity letter_
increases application rates by $1.08$ p.p. ($s.e.{=}0.91$), the _Requirements
letter_ by $4.33$ p.p. ($s.e.{=}1.04$), and the _Welcome letter_ by $3.51$
p.p. ($s.e.{=}1.03$), relative to the control condition of no letter.181818See
[38] for a discussion the letters’ efficacy in overcoming specific hurdles.

The left panel of Figure 3.5 shows only moderate heterogeneity in treatment
effects by nationality. The _Welcome letter_ appears to have slightly stronger
effects for immigrants from Germany and Austria, consistent with the idea that
Germans and Austrians do not perceive complexity and difficulty as major
hurdles due to their cultural proximity and language. At the same time, the
_Welcome letter_ is also the most effective letter for immigrants from the
Americas, which could indicate that this minority group does not feel very
welcome in Switzerland. The relative effect size of the _Requirements letter_
is particularly large for immigrants from Central-Eastern and South-Eastern
Europe, as well as for stateless immigrants. The right panel of Figure 3.5
indicates that the _Complexity letter_ has the largest effect on application
rates among eligible immigrants who have lived between 13 and 16 years in
Switzerland. In contrast, eligible immigrants who have lived for more than 30
years in Switzerland are especially receptive to the requirements letter,
suggesting that the perceived difficulty of the naturalization process may
discourage some eligible immigrants from applying over long periods. This
effect may also be partially driven by age since we also find the
_Requirements letter_ to have the largest effect among immigrants aged 46 and
above (see Figure 3.5 in the Appendix). Finally, we find that men are slightly
more receptive to the letter treatments overall than women, but the ranking of
treatment letter efficacy is the same (see Figure 3.5).

Figure 2: Average and conditional average treatment effects

Second, we conduct the validation exercise outlined above to assess the out-
of-sample performance of various policy rules and to select the tree depth of
the policy tree. We focus on policy trees with tree depths of 2 and 3, as well
as a hybrid policy tree of depth 4. For comparison, we consider (i) one-size-
fits-all rules that always assign one of the _Complexity_ , _Requirements_ or
_Welcome_ letters, (ii) random allocation of one of the three letters, and
(iii) a model-free plug-in rule that assigns the treatment for which the
estimated reward is the largest. We repeat the exercise 500 times and report
average differences in rewards and bootstrapped standard errors in Table
3.5.191919We opted for this approach rather than $K$-fold cross-validation as
it allows us to match the sizes of the training and validation data to the
actual sample sizes. However, we obtain similar results when applying $K$-fold
cross-validation. The table reports in each column the gain in reward of a
specific policy choice compared to alternative policy rules (shown in rows).
For instance, the coefficient of 1.026 ($s.e.=.99$) in the top-left entry
corresponds to the gain in reward of a one-size-fits-all policy rule assigning
the _Complexity letter_ to everyone relative to a policy rule assigning no
letter. We find that all three policy trees outperform each individual
treatment letter as well as random treatment allocation. Among the three
policy trees, the tree of depth 3 performs marginally better than trees of
depth 2 and 4. As expected, the plug-in rule shows overall the best
performance. However, the plug-in rule provides no insights into the drivers
of treatment effects. The results thus highlight the trade-off between
interpretability and performance but also show that, in this context, the
best-performing policy tree is able to reach more than 85% of the performance
of the plug-in rule.

Table 2: The effect of the policy rule compared randomization, always the same
treatment and no treatment

Notes: The table reports the difference in estimated rewards between policy
rules based on wave-1 data (including untreated immigrants of Group B).
Specifically, each cell corresponds the gain in reward of a specific policy
rule (shown in columns) relative to alternative policy rules (listed in rows).
The results are based on a resampling exercise where we randomly split the
wave-1 data into training and test data using a 60/40 split, and separately
draw $n_{1}=4871$ and $n_{2}=1857$ observations with replacement from the
training and test data. We use 500 repetitions and report the average
difference in rewards and associated bootstrapped standard errors.
Significance levels:
${}^{\star\star\star}\,0.01,{}^{\star\star}\,0.05,{}^{\star}\,0.1$.

Third, in light of the advantages and limited costs of policy trees in this
setting, we opted for implementing the policy tree of depth 3. Following the
approach of [78] as outlined in Section 2, we trained the policy tree on wave
1 data, including Group A (who received a letter in the first wave) and Group
B (who did not receive a letter in the first wave). Since we randomized
treatment assignment in the first wave, we did not need to estimate the
propensity scores but plugged the known treatment shares into (3).202020We
note that in a setting where $e_{a}(X_{i})$ is known, the IPW estimator of the
reward in (1) is also applicable. We find in simulations that the AIPW
estimator using the known propensity scores outperforms the IPW estimator. We
used multi-arm causal forests to estimate the double robust scores, although
other estimators are possible. The fitted policy tree $\hat{\pi}$ of depth
three is displayed in Figure 3.5. The boxes at the bottom of the tree show the
assigned treatment for the wave-1 sample and the wave-2 sample (i.e., Group B)
per terminal node. For instance, the very-left branch assigns individuals who
have spent no more than 12 years in Switzerland, are aged 37 years or younger,
and who are not from Italy to the requirements treatment. 815 individuals in
total and 324 individuals from Group B fall into that category. In total, 139
individuals of Group B are assigned to the _Complexity letter_ , 874
individuals to the _Requirements letter_ and 844 to the _Welcome
letter_.212121We assigned policies for Groups B.1 and B.2 after removing
individuals who either applied without being treated (99 individuals) or moved
out of the municipality of Zurich (101 individuals). The splits in the tree
are based on years in Switzerland, age, and only two nationality indicators,
but no split is based on gender confirming that the relative performance of
each letter is the same for women and men. It is also noteworthy that no
individuals were assigned to receive no letter, which suggests that at least
one of the three letters has a positive effect for every individual.

_Notes:_ The figure shows the policy tree fitted to data from wave 1. The size
of the training sample is $4{,}871$. The numbers at the bottom indicate the
number of individuals assigned to each terminal node in the training sample
and in Group B.

Figure 3: Fitted policy tree

###  3.6 Results from the exploitation phase: Evaluating the policy rule

Table 3.6 shows the results of the evaluation based on estimating versions of
(4) using OLS (columns 1-3), PDS lasso (col. 4-5), and DDML (col. 6-7). The
sample includes only wave 2 in column 1, and both waves in the remaining
columns. The reference group in column 1 is random treatment allocation, while
the base group in columns 2-7 is no treatment. Panel A reports the coefficient
estimates and Panel B compares the policy rule using policy trees against each
individual treatment letter and random treatment allocation.

Table 3: The effect of the policy rule compared randomization, always the same
treatment and no treatment

Panel B. Comparison of Policy tree with: Random 1.794 1.899 1.759 1.495
(1.358) (1.361) (1.359) (1.257) Complexity 2.897 2.745 2.559 (1.539) (1.538)
(1.432) Requirements 1.477 1.294 1.144 (1.503) (1.499) (1.409) Welcome 1.340
1.253 1.083 Sample Wave 2 Wave 1-2 Wave 1-2 Wave 1-2 Wave 1-2 Wave 1-2 Wave
1-2 Estimator OLS OLS OLS PDS lasso PDS lasso DDML DDML Outcome mean 7.69 7.92
7.92 7.92 7.92 7.92 7.92 Observations 1717 6588 6588 6588 6588 6588 6588

Notes: The table reports results from estimating versions of (4) using OLS
(columns 1-3), PDS-Lasso (columns 4-5) and DDML (columns 6-7). Column 1 only
uses data from wave 2; the remaining columns use the full data set. The
reference group in column 1 is random treatment allocation; no treatment in
columns 2-7. Panel A reports the coefficient estimates. Panel B compares the
policy rule using policy trees against always assigning the same treatment to
everyone and random treatment allocation. Covariates include the region of
nationality, age, gender, years lived in Zurich and years lived in
Switzerland.

Standard errors are clustered at building address level. ∗ $p<0.05$, ∗∗
$p<0.01$, ∗∗∗ $p<0.001$

According to the OLS results in columns 1-3, the treatment assignment by
policy tree increased the application rate by $1.79$ ($s.e.{=}1.36$) to $1.90$
p.p. ($1.36$) relative to random treatment, and by around $5.13$ p.p. ($1.61$)
compared to no treatment. Random allocation is associated with an application
rate increase of approximately $3.23$ p.p. ($0.82$). Turning to the individual
treatments, we find that the _Welcome letter_ yields overall the largest
increase in application take-up with an effect size around $3.79$ p.p.
($1.07$), closely followed by the _Requirements letter_ with an effect size
around $3.65$ p.p. ($1.10$). The _Complexity letter_ performs substantially
worse in comparison, with an effect size of $2.23$ ($s.e.{=}1.04$). Panel B
shows that the policy tree performs better than random treatment or each
individual treatment option. The take-up increase compared to the best-
performing individual treatment (the _Welcome letter_) is 1.03 p.p. but
statistically insignificant. The PDS lasso estimates are almost identical and
the DDML estimator yields effect sizes only marginally smaller.222222Appendix
Table A.3 also shows alternative results using logistic regression. The
average marginal effects from logistic regression are almost identical to
those from OLS.

##  4 Conclusion

This paper employs policy trees for assigning eligible immigrants to the
information and encouragement treatment that is most likely to address hurdles
on their path to citizenship and boost their propensity to naturalize. We
evaluate the benefits of this policy rule using a tailored two-phase field
experiment. During the exploration phase, we randomly assign eligible
immigrants to one of three treatment arms or the control group, based on which
we estimate average treatment effects and train the policy tree. We find that
despite its simplicity, the optimal policy tree of depth 3 captures more than
85% of the treatment effect heterogeneity (relative to a model-free plug-in
rule). Next, we move on to the exploitation phase, in which we assign the
subjects that belonged to the control group in the previous phase to either
the policy tree or randomly to one of the three treatments. We find that the
policy tree outperforms the best-performing individual treatment slightly.
While these differences are not statistically significant, it is worth noting
that these benefits persist in a context with at most moderate levels of
treatment effect heterogeneity and come at little additional costs.

Policy trees possess several advantages that make them particularly suited for
policymakers and researchers interested in tailoring treatment assignment to
the specific needs of increasingly diverse populations. Policy trees are
transparent in terms of which variables guide treatment assignment, they are
simple to visualize, and intuitive to communicate even to users of the
research who lack statistical training. While using machine learning to
personalize treatment assignments raises a host of important ethical and
policy questions, we should keep in mind that a one-size-fits-all approach can
often exacerbate existing inequalities. For instance, an earlier information
letter sent out by the City of Zurich had by far the strongest effects among
newly eligible immigrants, which often score higher on multiple integration
dimensions compared to more marginalized immigrants who have been residing in
the host country for decades without naturalizing [76]. For all these reasons,
we believe that policy trees are a well-suited approach to leverage the
potential of tailored treatment assignment in a world where rich background
characteristics are increasingly available.

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##  A Supplementary materials

###  A.1 Additional results on heterogeneous treatment effects

Figure A.4: Policy tree of depth 3

###  A.2 Letters and flyers sent by the City of Zurich

####  A.2.1 Informational cover letter

Figure A.5: Information Cover Letter. Original in German.

Zurich, October 2021
Important information regarding Swiss citizenship
Dear
You have been living in Switzerland for at least ten years, two of which in
Zurich, and are in possession of a permanent residence permit (Permit C). You
thus already fulfill an important requirement to become a Swiss citizen.
Take an active part in democracy
More than 420,000 people live in Zurich, including 130,000 foreigners from all
continents. Foreigners are an important part of our society and contribute and
contribute to a good and attractive community in our city. They all make the
city of Zurich a successful and livable city.
If you are planning your future in Switzerland, it is important that you can
also shape our common future and that you can vote and elect. Because a
democracy is only alive and strong if as many people as possible have a say in
politics. Thanks to the Swiss citizenship, you have this opportunity to have a
voice in politics. That is why the City Council is committed to ensuring that
people who meet the requirements for citizenship also naturalize.
Inform yourself about naturalization
Find out whether you meet the other requirements for naturalization. You can
find out more about naturalization in the information sheet (enclosure) or on
our website (www.stadt-zuerich.ch/einbuergerungen). There you will also find
all the forms for a naturalization application.
Every year, the City Council welcomes the new Swiss citizens of the city of
Zurich at a ceremony. We would be pleased to also welcome you as a Swiss
citizen soon.
Kind regards
Corine Mauch
Mayor
Dr. Claudia Cuche-Curti
City Clerk

Figure A.6: Informational cover letter. English translation.

####  A.2.2 Flyer enclosed in complexity letter

(b) English translation

Figure A.7: Flyer enclosed in complexity letter

####  A.2.3 Flyer enclosed in requirements letter

(b) English translation

Figure A.8: Flyer enclosed in requirements letter

####  A.2.4 Welcome cover letter

Figure A.9: Welcome cover letter. Original in German

Zurich, October 2021
Information on Swiss citizenship
Dear
You have been living in Switzerland for at least ten years, two of which in
Zurich, and are in possession of a permanent residence permit (Permit C). You
thus already fulfill an important requirement to become a Swiss citizen.
Would you like to apply for Swiss citizenship? Then find out whether you also
meet the other requirements for naturalization. You can find out more about
naturalization in the enclosed flyer and on our website (www.stadt-
zuerich.ch/einbuergerungen). On our website, you will also find all the forms
for such an application.
We look forward to reviewing your naturalization application.
Kind regards,
Michael Lamatsch
Deputy City Clerk

Figure A.10: Welcome cover letter. English translation

###  A.3 Alternative results

Table A.4: The effect of the policy rule compared randomization, always the
same treatment and no treatment (alternative results)

Panel B. Comparison of Policy tree with: Random 2.312 2.339 2.205 2.077
(1.229) (1.231) (1.231) (1.132) Complexity 3.540 3.386 3.242 (1.402) (1.405)
(1.296) Requirements 1.713 1.537 1.436 (1.373) (1.371) (1.28) Welcome 1.779
1.702 1.544 Sample Wave 2 Wave 1-2 Wave 1-2 Wave 1-2 Wave 1-2 Wave 1-2 Wave
1-2 Estimator OLS OLS OLS PDS lasso PDS lasso DDML DDML Outcome mean 6.17 6.48
6.48 6.48 6.48 6.48 6.48 Observations 1717 6588 6588 6588 6588 6588 6588

Notes: In this table, we provide alternative results where we consider only
application outcomes recorded until approximately five months after letter
dispatch, i.e., until March 21 (for wave 1) and October 21, 2022 (for wave 2),
respectively. We provide these alternative results to verify the robustness of
the main results in Table 3.6 to the exclusion of application outcomes
recorded after the second QR code of the Requirements Letter broke due to an
error, see fn. 11.

See Table 3.6 for more information.

Standard errors are clustered at building address level. ∗ $p<0.05$, ∗∗
$p<0.01$, ∗∗∗ $p<0.001$

Table A.5: Comparison of marginal effects from OLS and logistic regression

Panel B. Comparison of Policy tree with: Random 0.0179 0.0190 0.0172 0.0190
(.014) (.014) (.014) (.014) Complexity 0.0290 0.0285 (.015) (.016)
Requirements 0.0148 0.0148 (.015) (.015) Welcome 0.0134 0.0143 Sample Wave 2
Wave 1-2 Wave 1-2 Wave 2 Wave 1-2 Wave 1-2 Estimator OLS OLS OLS Logit Logit
Logit Outcome mean .08 .08 .08 .08 .08 .08 Observations 1717 6588 6588 1715
6580 6580

Notes: This table compares OLS and logistic regression estimation results.
Columns (1)-(3) of this table reproduce columns (1)-(3) of the main results in
Table 3 and rely on OLS. Columns (4)-(6) are based on the same specifications
but employ logistic regression and report the average marginal effects. Column
1 and 4 only use data from wave 2; the remaining columns use the full data
set. The reference group in columns 1 and 4 is random treatment allocation; no
treatment in the remaining columns. Panel A reports the coefficient estimates.
Panel B compares the policy rule using policy trees against always assigning
the same treatment to everyone and random treatment allocation. Covariates
include the region of nationality, age, gender, years lived in Zurich and
years lived in Switzerland.

Standard errors are clustered at building address level. ∗ $p<0.05$, ∗∗
$p<0.01$, ∗∗∗ $p<0.001$