A Shadow Variable Approach to Causal Decision Making with One-sided Feedback

We study a class of decision-making problems with one-sided feedback, where outcomes are only observable for specific actions. A typical example is bank loans, where the repayment status is known only if a loan is approved and remains undefined if rejected. In such scenarios, conventional approaches to causal decision evaluation and learning from observational data are not directly applicable. In this paper, we introduce a novel value function to evaluate decision rules that addresses the issue of undefined counterfactual outcomes. Without assuming no unmeasured confounders, we establish the identification of the value function using shadow variables. Furthermore, leveraging semiparametric theory, we derive the efficiency bound for the proposed value function and develop efficient methods for decision evaluation and learning. Numerical experiments and a real-world data application demonstrate the empirical performance of our proposed methods.