Min-max and min-min stackelberg strategies with closed-loop information structure

  • M. JungersEmail author
  • E. Trelat
  • H. Abou-kandil


This paper deals with the min-max and min-min Stackelberg strategies in the case of a closed-loop information structure. Two-player differential one-single stage games are considered with one leader and one follower. We first derive necessary conditions for the existence of the follower to characterize the best response set of the follower and to recast it, under weak assumptions, to an equivalent and more convenient form for expressing the constraints of the leader’s optimization problem. Under a standard strict Legendre condition, we then derive optimality necessary conditions for the leader of both min-max and min-min Stackelberg strategies in the general case of nonlinear criteria for finite time horizon games. This leads to an expression of the optimal controls along the associated trajectory. Then, using focal point theory, the necessary conditions are also shown to be sufficient and lead to cheap control. The set of initial states allowing the existence of an optimal trajectory is emphasized. The linear-quadratic case is detailed to illustrate these results.

Key words and phrases

Stackelberg strategy game theory multi-criteria optimization closed-loop information structure bilevel optimization problem 

2000 Mathematics Subject Classification

91A65 49N70 49N90 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.CRAN UMR CNRS 7039Vandoeuvre cedexFrance
  2. 2.Universite d’Orleans, UFR Sciences Federation Denis Poisson Mathematiques, Laboratoire MAPMOOrleans Cedex 2France
  3. 3.SATIE ENS CACHANCachan CedexFrance

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