On the Equilibrium of a Class of Leader-Follower Games With Decision-Dependent Chance Constraints

In this paper, we study the existence of equilibrium in a single-leader-multiple-follower game with decision-dependent chance constraints (DDCCs), where decision-dependent uncertainties (DDUs) exist in the constraints of followers. DDUs refer to the uncertainties impacted by the leader's strategy, while players cannot capture the exact probability distributions. To address such problems, we first use decision-dependent ambiguity sets under moment information and Cantelli's inequality to transform DDCCs into second-order cone constraints. This simplifies the game model by estimating the probabilistic bounds without requiring the exact distribution information. For equilibrium existence, most results either assume fixed strategy spaces or study coupled strategy spaces among followers (e.g., Stackelberg generalized Nash games). In contrast, DDUs extend the follower strategy spaces to a hierarchical form of decision dependence, i.e., the leader's strategy endogenously reshapes the followers' strategy spaces. This transforms the best responses of followers into the marginal map, whose regularity properties (upper semicontinuity, convexity, and compactness) are no longer guaranteed. To fill this gap, we prove that the strategy space of the follower is a continuous set-valued map with convex and compact values under mild assumptions. This ensures that the marginal map satisfies the regularity properties required by the fixed-point theorem, allowing us to establish the existence of equilibrium. Finally, a numerical example is provided to show the impact of DDUs on the equilibrium of such a game.