Single-Stage Causal Incentive Design via Optimal Interventions

We introduce Causal Incentive Design (CID), a framework that applies causal inference to canonical single-stage principal-agent problems (PAPs) characterized by bilateral private information. Within CID, the operating rules of PAPs are formalized using an additive-noise causal graphical model (CGM). Incentives are modeled as interventions on a function space variable, Γ, which correspond to policy interventions in the principal-follower causal relation. The causal inference target estimand V(Γ) is defined as the expected value of the principal's utility variable under a specified policy intervention in the post-intervention distribution. In the context of additive-Gaussian independent noise, the estimand V(Γ) decomposes into a two-layer expectation: (i) an inner Gaussian smoothing of the principal's utility regression; and (ii) an outer averaging over the conditional probability of the follower's action given the incentive policy. A Gauss-Hermite quadrature method is employed to efficiently estimate the first layer, while a policy-local kernel reweighting approach is used for the second. For offline selection of a single incentive policy, a Functional Causal Bayesian Optimization (FCBO) algorithm is introduced. This algorithm models the objective functional γ↦V(γ) using a functional Gaussian process surrogate defined on a Reproducing Kernel Hilbert Space (RKHS) domain and utilizes an Upper Confidence Bound (UCB) acquisition functional. Consequently, the policy value V(γ) becomes an interventional query that can be answered using offline observational data under standard identifiability assumptions. High-probability cumulative-regret bounds are established in terms of differential information gain for the proposed FBO algorithm. Collectively, these elements constitute the central contributions of the CID framework, which integrates causal inference through identification and estimation with policy search in principal-agent problems under private information. This approach establishes a causal decision-making pipeline that enables commitment to a high-performing incentive in a single-shot game, supported by regret guarantees. Provided that the data used for estimation is sufficient, the resulting offline pipeline is appropriate for scenarios where adaptive deployment is impractical or costly. Beyond the methodological contribution, this work introduces a novel application of causal graphical models and causal reasoning to incentive design and principal-agent problems, which are central to economics and multi-agent systems.