Mini-max incentive strategy for leader–follower games under uncertain dynamics

Abstract

This paper studies the problem of designing an incentive strategy for a leader–follower dynamic game affected by some sort of uncertainties. As is traditionally understood in the standard theory of incentives, the leader has complete knowledge of the game parameters, including the follower's performance index. So then he can compute the strategy that will lead the game to the global optimum that is favourable for him. Most of the current work is devoted to this situation. Nevertheless, such an assumption is unrealistic. This paper proposes an incentive scheme in which the game's dynamic depends on an unknown value that belongs to a finite set. The solution of the incentive strategy is computed in terms of the worst-case scenario, of the team's optimal solution. Based on the Robust Maximum Principle, the new incentive is presented in the form of a mini-max feedback control. Two numerical examples illustrate the effectiveness of the approach.

Keywords:

• Dynamic games
• incentive theory
• robust optimisation

AMS CLASSIFICATIONs:

• 91A65
• 49L12
• 49N10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Notes on contributors

Celeste Rodríguez-Carreón

Celeste Rodríguez-Carreón received her BS degree in mathematics in 2008, and a PhD degree in industrial physics engineering in 2013 at the Faculty of Physics and Mathematics Science from the Autonomous University of Nuevo Leon, Mexico. She is currently at the same faculty. Her research interests include differential games, stochastic systems, and robust control problems.

Manuel Jiménez-Lizárraga

Manuel Jiménez-Lizárraga received his BSc in electrical engineering from Culiacan Institute of Technology, and the MSc and PhD degrees in automatic control from CINVESTAVIPN, in Mexico, in 1996, 2000, and 2006, respectively. He was a postdoctoral fellow at the ECE Department of the Ohio State University in 2009 and a visiting scholar at Heuristique el Diagnostic des Systemes Complexes at Universite de Technologie de Compiegne in 2015. He is a member of the Mexican Academy of Sciences and is currently with the Faculty of Physical and Mathematical Sciences of the UANL. His research interests include differential games, robust, optimal and sliding mode control and applications such as UAVs guidance and control.

César Emilio Villarreal

César Emilio Villarreal received his doctoral degree in mathematics from the Centro de Investigacion y de Estudios Avanzados del Instituto Politecnico Nacional (CINVESTAV IPN), Mexico, in 1998. He is a professor at the Universidad Autonoma de Nuevo Leon (UANL), Mexico. His research interests include stochastic processes, probability and ergodic theory.

Ignacio Quiroz-Vázquez

Ignacio Quiroz-Vázquez received his BSc as a administrator mechanical engineer and master’s degree in engineering with orientation in mechanics in 1999 and 2006 respectively. He is currently a PhD student at the Faculty of Physics and Mathematics Science from the Autonomous University of Nuevo Leon, Mexico.