Developing reward-risk aversion distributionally robust contract design models under ambiguous output probabilities

In this study, we address robust contract design problem, where the principal is ambiguous about effort-contingent multi-output probability distribution of the agent. To model this type of distributionally robust contract design problem, the Wasserstein ambiguity set is employed to characterize the ambiguous multi-output probability distribution. Two decision criteria are adopted by the principal to evaluate the designed robust contract under partial distribution information, the first is the worst-case expected criterion, and the second is the worst-case risk criterion. Furthermore, the concept of distributionally robust incentive compatibility is defined with respect to a pair of ambiguity sets. In virtue of two decision criteria, this paper develops a new reward-risk distributionally robust contract design model as well as its extension models based on different risk measures and globalized distributionally robust incentive compatibility condition. The distributionally robust counterpart and globalized distributionally robust counterpart problems of the developed distributionally robust contract design models are linear programming or mixed-integer linear programming models. According to the structural characteristic of the resulting mixed-integer linear programming model, a new tailored Benders decomposition algorithm is designed. At the end of this paper, an inventory decision with backorder problem is addressed and some numerical experiments are performed to demonstrate the influences of three risk measures on robust optimal contracts. The computational results demonstrate that the developed distributionally robust contract design models can facilitate the principal to make the informed contract decisions.