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Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising

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


Alain Bensoussan

Shaokuan Chen

Suresh P. Sethi

Chi Chung Siu

Sheung Chi Phillip Yam


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. We find an equilibrium that can be expressed by a solution of a set of quartic algebraic equations. We then conduct an extensive numerical study to assess the impact of model parameters on the equilibrium.


Bensoussan, A., Chen, S., Chutani, A., Sethi, S. P., Siu, C. C., & Yam, S. C. P. (2019). Feedback Stackelberg-Nash equilibria in mixed leadership games with an application to cooperative advertising. SIAM Journal on Control and Optimization, 57(5), 3413-3444.

Journal Article Type Article
Acceptance Date Jul 19, 2019
Online Publication Date Oct 17, 2019
Publication Date Oct 17, 2019
Deposit Date Dec 14, 2018
Publicly Available Date Jul 23, 2019
Journal SIAM Journal on Control and Optimization
Print ISSN 0363-0129
Electronic ISSN 1095-7138
Publisher Society for Industrial and Applied Mathematics
Peer Reviewed Peer Reviewed
Volume 57
Issue 5
Pages 3413-3444
Keywords Stackelberg differential game, Nash differential game, feedback solution, coop- erative advertising AMS Subject Classification: 49N90 Applications of optimal control and differential games; 49N70 Differential games; 90B60 Marketing, advertising *
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