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Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization




In a two-stage Stackelberg game, depending on the leader's information about the choice of the follower among his optimal responses, one can associate different types of mathematical problems. We present formulations and solution concepts for such problems, together with their possible connections in bilevel optimization, and we illustrate the crucial issues concerning these solution concepts. Then, we discuss which of these issues can be positively or negatively answered and how managing the latter ones by means of two widely used approaches: regularizing the set of optimal responses of the follower, via different types of approximate solutions, or regularizing the follower's payoff function, via the Tikhonov or the proximal regularizations. The first approach allows to obviate the lack of existence and/or stability through approximating problems, whose solutions exist under not restrictive conditions and enable to construct a surrogate solution to the original problem. The second approach permits to overcome the non-uniqueness of the follower's optimal response, by constructing sequences of Stackelberg games with a unique second-stage solution which approximate in some sense the original game, and to select among the solutions by using a constructive method with behavioural motivations.

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  • Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2019. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," CSEF Working Papers 541, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
  • Handle: RePEc:sef:csefwp:541

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    References listed on IDEAS

    1. John C. Harsanyi & Reinhard Selten, 1988. "A General Theory of Equilibrium Selection in Games," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262582384.
    2. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2019. "Subgame Perfect Nash Equilibrium: A Learning Approach via Costs to Move," Dynamic Games and Applications, Springer, vol. 9(2), pages 416-432, June.
    3. Alain Haurie & Jacek B Krawczyk & Georges Zaccour, 2012. "Games and Dynamic Games," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 8442, November.
    4. H. Bonnel & J. Morgan, 2006. "Semivectorial Bilevel Optimization Problem: Penalty Approach," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 365-382, December.
    5. M. Lignola & Jacqueline Morgan, 2012. "Approximate values for mathematical programs with variational inequality constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 485-503, October.
    6. Maria Carmela Ceparano & Jacqueline Morgan, 2017. "Equilibrium selection in multi-leader-follower games with vertical information," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 526-543, October.
    7. Laurent Drouet & Alain Haurie & Francesco Moresino & Jean-Philippe Vial & Marc Vielle & Laurent Viguier, 2008. "An oracle based method to compute a coupled equilibrium in a model of international climate policy," Computational Management Science, Springer, vol. 5(1), pages 119-140, February.
    8. Dockner,Engelbert J. & Jorgensen,Steffen & Long,Ngo Van & Sorger,Gerhard, 2000. "Differential Games in Economics and Management Science," Cambridge Books, Cambridge University Press, number 9780521637329.
    9. Jacqueline Morgan, 2005. "Approximations and Well-Posedness in Multicriteria Games," Annals of Operations Research, Springer, vol. 137(1), pages 257-268, July.
    10. Jonathan F. Bard, 1983. "An Algorithm for Solving the General Bilevel Programming Problem," Mathematics of Operations Research, INFORMS, vol. 8(2), pages 260-272, May.
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