Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models

Abstract

In this paper we prove a sufficient maximum principle for general stochastic differential Stackelberg games, and apply the theory to continuous time newsvendor problems. In the newsvendor problem a manufacturer sells goods to a retailer, and the objective of both parties is to maximize expected profits under a random demand rate. Our demand rate is an Itô-Lévy process, and to increase realism information is delayed, e.g., due to production time. We provide complete existence and uniqueness proofs for a series of special cases, including geometric Brownian motion and the Ornstein-Uhlenbeck process, both with time variable coefficients. Moreover, these results are operational because we are able to offer explicit solution formulas. An interesting finding is that more precise information may be a considerable disadvantage for the retailer.

Stochastic Leader-Follower Differential Game with Asymmetric Information

“…Thus the usual approach to deal with game problems, such as [6][7][8]10], where the two players act as equivalent roles, does not apply. Second, the game problem has the asymmetric information between the two players, which was not considered in [3,13,14]. In detail, the information available to the follower is based on some sub-σ-algebra of that available to the leader.…”

“…Any process triple q * Á ðÞ; R * Á ðÞ; w * Á ðÞ ðÞ satisfying the above is called a Stackelberg equilibrium. In Øksendal et al [3], a time-dependent newsvendor problem with time-delayed information is solved, based on stochastic differential game (with jump-diffusion) approach. But it cannot cover our model.…”

Design of optimal quantity discounts for multi-period bilevel production planning under uncertain demands

“…They especially studied theoretical analysis with a general model in a single-period model. Øksendal et al 31 analyzed the time-dependent newsvendor models in a Stackelberg game. They offered an analytical method for the multi-period model in the Stackelberg game.…”

Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information

“…The coefficients μ, σ, and γ are assumed to satisfy standard conditions ensuring that (2.1) has a unique solution (see Øksendal and Sulem (2007)). At time t − δ a retailer and a manufacturer negotiate a contract for items to be delivered at time t, where δ > 0 is the delay time.…”