  Designing individualized allocation of treatments so as to maximize the
equilibrium welfare of interacting agents has many policy-relevant
applications. Focusing on sequential decision games of interacting agents, this
paper develops a method to obtain optimal treatment assignment rules that
maximize a social welfare criterion by evaluating stationary distributions of
outcomes. Stationary distributions in sequential decision games are given by
Gibbs distributions, which are difficult to optimize with respect to a
treatment allocation due to analytical and computational complexity. We apply a
variational approximation to the stationary distribution and optimize the
approximated equilibrium welfare with respect to treatment allocation using a
greedy optimization algorithm. We characterize the performance of the
variational approximation, deriving a performance guarantee for the greedy
optimization algorithm via a welfare regret bound. We implement our proposed
method in simulation exercises and an empirical application using the Indian
microfinance data (Banerjee et al., 2013), and show it delivers significant
welfare gains.
