Feedback Stackelberg--Nash Equilibria in Mixed Leadership Games with an Application to Cooperative Advertising

Alain Bensoussan,Chi Chung Siu,Sheung Chi Phillip Yam,Suresh P. Sethi,Shaokuan Chen,Anshuman Chutani

doi:10.1137/17m1153212

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 i...

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

Introduction

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.

Model and problem formulation

Characterization of a Feedback Stackelberg-Nash Equilibrium

A Model of Cooperative Advertising

Numerical Analysis

Comparison with the Traditional Model

Concluding Remarks