Banerjee, Chassang and Snowberg have an under-appreciated paper, “Decision-Theoretic Approaches to Experiment Design and External Validity”, that anyone who designs experiments should think about.
1. Bayesians do not (if making policy decisions themselves) randomize
Suppose you were a Bayesian and trying to maximize expected utility. There exists some set of ways to assign individuals to the treatment group that would maximize your expected utility. Randomizing could sometimes deviate from that set of ways (e.g. if you are unlucky enough to have imbalance between the treatment and control group along some observable characteristics). Therefore, randomizing would not be optimal. This is in the same spirit as Kasy (2013).
2. Priors matter
Apart from randomizing sometimes leading to failure to obtain balance, randomizing could also not be optimal for some priors. They provide the example of a superintendent who believes that whether a student is from a poor or privileged background is the main determinant of educational outcomes and who believes that those who go to private schools do better because they tend to be from privileged backgrounds, but who is open to testing whether private schools are helpful in and of themselves. The superintendent has the chance to enroll a single student in a private school. Clearly, they would not learn much by enrolling a privileged student in a private school — to learn the most, they should enroll a poor student.
3. The optimal experimental design depends on how the decisions are made
The experimenter may not be making the policy decision themselves. Rather, they may be trying to convince others (or trying to convince some small part of themselves that is uncertain, in the Knightian sense). Thus, when designing the experiment they need to place some weight on how much they want to convince themselves, given their own priors, vs. how much they want to convince someone else (or themselves, under ambiguity), given the other person’s priors. This depends on how the decisions are being made, e.g., whether it is a group of people with quite varied priors making a decision.
4. Randomizing is optimal when faced with an adversarial audience (given a sufficiently large sample size and assuming a maximin objective)
Suppose you care about the worst case scenario: a decision-maker whose priors are such that given the experimenter’s chosen design they have a greater chance of picking the wrong policy than anyone else with different priors.
In this situation, a randomized experiment is best (so long as the sample size is sufficiently large). It is not targeted towards people with any particular priors, and given that it is not targeted towards people with any particular priors, it also leaves less room for error. Optimizing for some priors generally means making decisions worse for people with other priors. The qualifying statement that the sample size must be sufficiently large is included because with small samples comes greater loss of power from randomization compared with the optimal deterministic experiment (again, think of covariate balance).
This paper is nice because, among other things, it helps explain why academics who face skeptical audiences randomize while firms that do not face an adversarial audience but merely wish to learn for the sake of their own decision-making will experiment in a smaller and more targeted way, especially when the costs per participant are high. A key assumption in the current framework is the maximin objective, which may not always be what we care about.