Journal of Optimization Theory and Applications

, Volume 168, Issue 1, pp 348–374 | Cite as

Optimal Affine Leader Functions in Reverse Stackelberg Games

Existence Conditions and Characterization
  • Noortje Groot
  • Bart De Schutter
  • Hans Hellendoorn


A generalizing analysis is made in order to ease the solvability of the generally complex single-leader–single-follower reverse Stackelberg game. This game is of a hierarchical nature and can therefore be implemented as a structure for multi-level decision-making problems, like in road pricing. In particular, a leader function of the affine type is analyzed in order to procure a systematic approach to solving the game to optimality. To this end, necessary and sufficient existence conditions for this optimal affine leader function are developed. Compared to earlier results reported in the literature, differentiability of the follower objective functional is relaxed, and locally strict convexity of the sublevel set at the desired reverse Stackelberg equilibrium is replaced with the more general property of an exposed point. Moreover, a full characterization of the set of optimal affine leader functions that is derived, which use in the case of secondary optimization objectives as well as for a constrained decision space, is illustrated.


Stackelberg games Hierarchical decision making Existence conditions 

Mathematics Subject Classification

91A35 91A65 



Research supported by the European Union Seventh Framework Programme [FP7/2007-2013] under Grant agreement no. 257462 HYCON2 Network of Excellence, and by the European COST Action TU1102.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Noortje Groot
    • 1
  • Bart De Schutter
    • 1
  • Hans Hellendoorn
    • 1
  1. 1.Delft Center for Systems and ControlDelft University of TechnologyDelftThe Netherlands

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