simulated samples from the experimental data, and then demonstrate how ‘ex ante’ simulations can be used to decide, for example, on the number of experimental waves. Section 7 concludes with a short discussion.

# **2 Using Adaptive Sampling in Experiments for Policy Choice**

This section gives an overview over the use of adaptive sampling for policy choice and the exploration sampling algorithm proposed by Kasy and Sautmann (2021a). We start with the “basic ingredients” for an adaptive experiment: the objective, an algorithm that builds on the data from each wave to adaptively allocate units to treatment arms, the estimation approach, and constraints that determine whether an adaptive experiment is feasible. The corresponding features of the IVR experiment are described in detail in sections 3 and 4.

This section also discusses the gains from adaptive vs. non-adaptive sampling or adding adaptive waves, and how these gains can be calculated in simulations to choose the number of experimental waves. We return to this in section 6, building on the data collected in Kenya and the estimation results in section 5. To begin with, however, we assume that there are _t_ = 1 _, ..., T_ exogenously given consecutive sample draws (waves) of size _Nt_ available for testing.

**Objective.** In the canonical policy choice problem, there are _K >_ 2 policy options – or treatment arms – labeled _k_ = 1 _,_ 2 _, . . . , K_ . Each arm has unobserved (stationary) average outcome _θ_<sup>_k_</sup> , and the policy maker wants to implement the arm with the highest average outcome. Formally, let _k_<sup>(1)</sup> = argmax _k θ_<sup>_k_</sup> be the true best arm, and _k_<sup>_∗_</sup> the arm that is chosen. We call the loss (per unit) from implementing a suboptimal arm _k_ the policy regret, ∆<sup>_k_</sup> = _θ_<sup>_k_(1)</sup> _− θ_<sup>_k_</sup> . Ex post, the policy maker will select the arm _k_<sup>_∗_</sup> that has the highest average outcome, or lowest policy regret, based on the observed data.

It is assumed that the outcomes of the experimental units are observed at the end of each period _t_ . This means we can learn from wave _t_ and adjust the allocation of units to treatment arms in wave _t_ + 1, i.e. use adaptive sampling. In the policy choice problem, the policymaker’s wants to implement an adaptive sampling strategy that maximizes welfare, that is, minimizes the expected policy regret from the final choice given the true (unobserved) vector of average outcomes: _E_ [∆<sup>_k∗_</sup> _|θ_ ]. Adaptivity increases the efficiency of learning for a given objective by over-sampling some arms based on what was learned, at the expense of other arms (and other objectives).

_Remark: Other Objectives._ Large literatures consider sampling for specific learning goals.

The classical multi-arm bandit problem (MAB) considers the objective to maximize average outcomes during the ongoing experiment, or equivalently to minimize in-sample regret, which introduces the wellknown exploration-exploitation trade-off (e.g Lai and Robbins, 1985; Bubeck and Cesa-Bianchi, 2012). The policy choice problem of choosing the arm with the highest average outcome can be seen as a special case 6