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Entropic Regularization in Hierarchical Games

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Abstract

We study hierarchical games where the second stage consists of a finite noncooperative game. To ensure that the lower level problem admits solutions, its mixed extension is considered. By using the Shannon entropy, a regularization scheme for the two-stage game is introduced and some properties are presented, as the asymptotic subgame perfectness.

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References

  1. Başar T, Olsder GJ (1999) Dynamic noncooperative game theory. Reprint of the second 1995 edition. Classics in Applied Mathematics, 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA

  2. Chinchuluun A, Pardalos PM, Migdalas A, Pitsoulis L (2008) Pareto optimality, game theory and equilibria. Springer Optimization and Its Applications series, Vol. 17, Springer New York

  3. von Stackelberg H (1952) Marktform und Gleichgewicht. Julius Springer, Vienna (1934). English Edition: The theory of the market economy. Peacock, A. (ed.), London, William Hodge

  4. Selten R (1965) Stackelberg-Nash-Cournot Equilibria: characterizations and computations. J Inst Theor Econ 121:301–324

    Google Scholar 

  5. Mallozzi L, Morgan J (2005) Oligopolistic markets with leadership and demand functions possibly discontinuous. J Optim Theory Appl 125:393–407

    Article  Google Scholar 

  6. Sherali HD, Soyster AL, Murphy FH (1983) Stackelberg-Nash-Cournot equilibria: characterizations and computations. Oper Res 31:253–276

    Article  Google Scholar 

  7. Dempe S (2020) Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography, In S. Dempe, A. Zemkoho (eds): Bilevel Optimization, Advances and Next Challenges. Springer Optimization and Its Applications series, Vol. 161, Springer Nature Switzerland AG

  8. Grammatico S (2017) Dynamic control of agents playing aggregative games with coupling constraints. IEEE Trans Automat Control 62:4537–4548

    Article  Google Scholar 

  9. Hu M, Fukushima M (2015) Multi-leader-follower games: models, methods and applications. J Oper Res Soc Japan 58:1–23

    Google Scholar 

  10. Leitmann G (1978) On generalized Stackelberg strategies. J Optim Theory Appl 26:637–643

    Article  Google Scholar 

  11. Colson B, Marcotte P, Savard G (2007) An overview of bilevel optimization. Ann Oper Res 153:235–256

    Article  Google Scholar 

  12. Nguyen KC, Alpcan T, Başar T (2009) Security games with incomplete information, Proceedings - 2009 IEEE International Conference on Communications, ICC Dresden 14–18 July, 1–6

  13. Birbil I, Fang SC, Han J (2004) Entropic regularization approach for mathematical programs with equilibrium constraints. Comput Oper Res 31:2249–2262

    Article  Google Scholar 

  14. Korzhyk D, Yin Z, Kiekintveld C, Conitzer V, Tambe MM (2011) Stackelberg vs. Nash in security games: an extended investigation of interchangeability, equivalence, and uniqueness. J Artif Intell Res 41:297–327

  15. Shamma JS, Arslan G (2004) Unified convergence proofs of continuous time fictitious play. IEEE Trans. Automat. Control 49:1137–1141

    Article  Google Scholar 

  16. Sinha A, Fang F, An B, Kiekintveld C, Tambe M (2018) Stackelberg security games: looking beyond a decade of success. Proc. of the 27th International Joint Conference on Artificial Intelligence (IJCAI-18),Stockholm, Sweden, July 13–19, 5494–5501. Research Collection School of Information Systems

  17. Chinchuluun A, Pardalos PM, Huang HX (2009) Multilevel (Hierarchical) Optimization: Complexity Issues, Optimality Conditions, Algorithms. In D. Gao, H. Sherali (eds) Advances in Applied Mathematics and Global Optimization, Springer, 197-221

  18. Migdalas A, Pardalos PM, Värbrand P (1998) Multilevel Optimization: Algorithms and Applications. Nonconvex Optimization and Its Applications series, Vol. 20, Springer US

  19. Pardalos PM, Deng X (1997) Complexity Issues in Hierarchical Optimization. In B. Mirkin, F.R. McMorris, F.S. Roberts, A. Rzhetsky (eds) Mathematical Hierarchies and Biology, DIMACS Series, Vol. 37, AMS 219–224

  20. Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge

    Book  Google Scholar 

  21. Mallozzi L, Sacco A (2021) Stackelberg-Nash equilibrium and quasi harmonic games. Ann Oper Res. https://doi.org/10.1007/s10479-020-03916-x

    Article  Google Scholar 

  22. Caruso F, Lignola MB, Morgan J (2020) Regularization and approximation methods. In S. Dempe, A. Zemkoho (eds): Bilevel Optimization, Advances and Next Challenges. Springer Optimization and Its Applications series, Vol. 161, Springer Nature Switzerland AG

  23. Peyré G, Cuturi M (2019) Computational optimal transport with applications to data science. Found Trends Mach Learn 11(5–6):355–607

    Article  Google Scholar 

  24. Rockafellar RT, Wets RJB (1998) Variational analysis, Springer

  25. Cominetti R, San Martín J (1994) Asymptotic analysis of the exponential penalty trajectory in Linear Programming Math Progr 67:169–187

  26. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–428

    Article  Google Scholar 

  27. Rosen J (1965) Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33:520–534

    Article  Google Scholar 

  28. Mangasarian OL, Stone H (1964) Two-Person Nonzero-Sum Games and Quadratic Programming. J Math Anal Appl 9:348–355

    Article  Google Scholar 

  29. Kuhn HW (1961) Algorithm for Equilibrium Points in Bimatrix Games. Proc Natl Acad Sci USA 47:1657–1662

    Article  Google Scholar 

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Funding

Financial support by GNAMPA-INDAM (Project 2020/2021: Problemi di ottimizzazione con vincoli via trasporto ottimo e incertezza) is gratefully acknowledged.

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Authors and Affiliations

  1. Dipartimento di Matematica e Applicazioni, Università di Napoli Federico II, Via Claudio 21, Naples, 80125, Italy

    Lina Mallozzi

  2. Department of Industrial and Systems Engineering, Center for Applied Optimization, University of Florida, Gainesville, FL, 32611-6595, USA

    Panos M. Pardalos

Authors
  1. Lina Mallozzi
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  2. Panos M. Pardalos
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Correspondence to Lina Mallozzi.

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Mallozzi, L., Pardalos, P.M. Entropic Regularization in Hierarchical Games. Oper. Res. Forum 3, 12 (2022). https://doi.org/10.1007/s43069-022-00130-2

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