Stackelberg Risk Preference Design

Risk measures are commonly used to capture the risk preferences of decision-makers (DMs). The decisions of DMs can be nudged or manipulated when their risk preferences are influenced by factors like the perception of losses and the availability of information about the uncertainties. In this work, we propose a Stackelberg risk preference design (STRIPE) problem to capture a designer's incentive to influence DMs' risk preferences. STRIPE consists of two levels. In the lower-level, individual DMs in a population, known as the followers, respond to uncertainties according to their risk preference types that specify their risk measures. In the upper-level, the leader influences the distribution of the types to induce targeted decisions and steers the follower's preferences to it. Our analysis centers around the solution concept of approximate Stackelberg equilibrium that yields suboptimal behaviors of the players. We show the existence of the approximate Stackelberg equilibrium. The primitive risk perception gap, defined as the Wasserstein distance between the original and the target type distributions, is important in estimating the optimal design cost. We connect the leader's optimality tolerance on the cost with her ambiguity tolerance on the follower's approximate solutions leveraging Lipschitzian properties of the lower-level solution mapping. To obtain the Stackelberg equilibrium, we reformulate STRIPE into a single-level optimization problem using the spectral representations of law-invariant coherent risk measures. We create a data-driven approach for computation and study its performance guarantees. We apply STRIPE to contract design problems to mitigate the intensity of moral hazard. Moreover, we connect STRIPE with meta-learning problems and derive adaptation performance estimates of the meta-parameter using the sensitivity of the optimal value function in the lower-level.