Abstract
In this paper we characterize the feedback equilibrium of a general infinite-horizon Stackelberg-Nash differential game where the roles of the players are mixed. By mixed we mean that one player is a leader on some decisions and a follower on other decisions. We prove a verification theorem that reduces the task of finding equilibrium strategies in functional spaces to two simple steps: First solving two static Nash games at the Hamiltonian level in a nested version and then solving the associated system of Hamilton-Jacobi-Bellman equations. As an application, we study a novel manufacturer-retailer cooperative advertising game where, in addition to the traditional setup into which the manufacturer subsidizes the retailer’s advertising effort, we also allow the reverse support from the retailer to the manufacturer. In this representative case, we find an equilibrium that can be expressed by a solution of a set of algebraic equations. We then conduct an extensive numerical study to assess the impact of model parameters on the equilibrium.
Highlights
Heinrich Freiherr von Stackelberg [33] introduced a concept of equilibrium in games, which is known as the Stackelberg equilibrium or the Stackelberg solution
We investigate a cooperative advertising game in a manufacturerretailer supply chain where there are no clear leaders and followers in the following sense: Based on the observed market share, at each instant of time both players first simultaneously decide their support rates of the other party’s advertising effort, and based on the announced support rates as well as the market share determined according to an extension of the model in [27], simultaneously decide their own advertising effort
We provide a theoretical framework to study a class of stochastic StackelbergNash differential games
Summary
Heinrich Freiherr von Stackelberg [33] introduced a concept of equilibrium in games, which is known as the Stackelberg equilibrium or the Stackelberg solution. This representative example offers a new insight into the mixed leadership perspectives on the cooperative advertising decisions between a manufacturer and a retailer In this respect, we investigate a cooperative advertising game in a manufacturerretailer supply chain where there are no clear leaders and followers in the following sense: Based on the observed market share, at each instant of time both players first simultaneously decide their support rates (lead decisions) of the other party’s advertising effort, and based on the announced support rates as well as the market share determined according to an extension of the model in [27], simultaneously decide their own advertising effort (follow-up decisions). Based on this result, we study a cooperative advertising game in a manufacturerretailer supply chain, and obtain a system of algebraic equations for a specific equilibrium where the players’ profits are affine functions of the market share.
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