Learning to Personalize Treatments When Agents Are Strategic

There is increasing interest in allocating treatments based on observed individual data: examples include targeted marketing, individualized credit offers, and heterogenous pricing. Treatment personalization introduces incentives for individuals to modify their behavior to obtain a better treatment. This shifts the distribution of covariates, requiring a new definition for the Conditional Average Treatment Effect (CATE) that makes explicit its dependence on how treatments are allocated. We provide necessary conditions that treatment rules under strategic behavior must meet. The optimal rule without strategic behavior allocates treatments only to those with a positive CATE. With strategic behavior, we show that the optimal rule can involve randomization, allocating treatments with less than 100\% probability even to those with a positive CATE induced by that rule. We propose a dynamic experiment based on Bayesian Optimization that converges to the optimal treatment allocation function without parametric assumptions on individual strategic behavior.