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Some Aspects of the Stackelberg Leader/Follower Model

  • L. Mallozzi
  • R. Messalli
  • S. Patrì
  • A. Sacco
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 141)

Abstract

The paper presents different aspects of the Stackelberg Leader/Follower model. Generalizations of the model introduced by von Stackelberg (Marktform und Gleichgewicht, Julius Springer, Vienna, 1934) are discussed, and some related open questions are enlightened.

Keywords

Game theory Stackelberg model Bi-level optimization 

Notes

Acknowledgements

This work has been supported by STAR 2014 (linea 1) “Variational Analysis and Equilibrium Models in Physical and Social Economic Phenomena,” University of Naples Federico II, Italy and by GNAMPA 2016 “Analisi Variazionale per Modelli Competitivi con Incertezza e Applicazioni.”

References

  1. 1.
    Amir, R., Grilo, I.: Stackelberg versus Cournot equilibrium. Games Econ. Behav. 26, 1–21 (1999)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bagwell, K.: Commitment and observability in games. Games Econ. Behav. 8, 271–280 (1995)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic, Dordrecht (1998)CrossRefGoogle Scholar
  4. 4.
    Başar, T., Olsder, G.J.: Dynamic noncooperative game theory. Reprint of the second 1995 edition. Classics in Applied Mathematics, vol. 23. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1999)Google Scholar
  5. 5.
    Başar, T., Srikant, R.: Stackelberg network game with a large number of followers. J. Optim. Theory Appl. 115, 479–490 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ben Abdelaziz, F., Ben Brahim, M., Zaccour, G.: R&D equilibrium strategies with surfers. J. Optim. Theory Appl. 136, 1–13 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Breton, M., Alj, A., Haurie, A.: Sequential Stackelberg equilibria in two-person games. J. Optim. Theory Appl. 59, 71–97 (1988)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cao, D., Leung, L.C.: A partial cooperation model for non-unique linear two-level decision problems. Eur. J. Oper. Res. 140(1), 134–141 (2002)CrossRefGoogle Scholar
  9. 9.
    Ceparano, M.C., Morgan, J.: Equilibria for multi–leader multi–follower games with vertical information: existence results. In: CSEF Working Paper, vol. 417 (2015)Google Scholar
  10. 10.
    Chakrabarti, S., Gilles, R.P., Lazarova, E.A.: Strategic behavior under partial cooperation. Theor. Decis. 71, 175–193 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Chinchuluun, A., Pardalos, P.M., Huang, H.X.: Multilevel (hierarchical) optimization: complexity issues, optimality conditions, algorithms. In: Gao D., Sherali, H. (eds.) Advances in Applied Mathematics and Global Optimization, pp. 197–221. Springer, Berlin (2009)CrossRefGoogle Scholar
  12. 12.
    Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 235–256 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D’Amato, E., Daniele, E., Mallozzi, L., Petrone, G.: Equilibrium strategies via GA to Stackelberg games under multiple follower’s best reply. Int. J. Intell. Syst. 27, 74–85 (2012)CrossRefGoogle Scholar
  14. 14.
    Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52, 333–359 (2003)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Denisova, L., Garnaev, A.: Fish wars: cooperative and non-cooperative approaches. The Czech Econ. Rev. 2, 28–41 (2008)zbMATHGoogle Scholar
  16. 16.
    Floudas, C.A., Pardalos, P.M.: Encyclopedia of Optimization. Springer, New York (2008)zbMATHGoogle Scholar
  17. 17.
    Gal-Or, E.: First mover and second mover advantages. Int. Econ. Rev. 26(3), 649–653 (1985)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hamilton, J., Slutsky, S.: Endogenous timing in duopoly games: Stackelberg or Cournot equilibria. Games Econ. Behav. 2, 29–46 (1990)CrossRefGoogle Scholar
  19. 19.
    Hörtnagl, T., Kerschbamer, R.: How the value of information shapes the value of commitment or: why the value of commitment does not vanish. EconPaper Repec (2014)Google Scholar
  20. 20.
    Kulkarni, A.A., Shanbhag, U.V.: An existence result for hierarchical Stackelberg v/s Stackelberg games. IEEE Trans. Autom. Control. 60(12), 3379–3384 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Leitmann, G.: On generalized Stackelberg strategies. J. Optim. Theory Appl. 26, 637–643 (1978)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lieberman, M.B., Montgomery, D.B.: First-mover advantages. Strateg. Manage. J. 9(S1), 41–58 (1988)CrossRefGoogle Scholar
  23. 23.
    Lignola, M.B., Morgan J.: Topological existence and stability for Stackelberg problems. J. Optim. Theory Appl. 84, 145–169 (1995)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lignola, M.B., Morgan J.: Stability of regularized bilevel programming problems. J. Optim. Theory Appl. 93(3), 575–596 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Loridan, P., Morgan J.: New results on approximate solution in two-level optimization. Optimization 20(6), 819–836 (1989)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Loridan, P., Morgan J.: Weak via strong Stackelberg problem: new results. J. Global Optim. 8, 263–287 (1996)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lu, J., Shi, C., Zhang, G.: On bilevel multi-follower decision making: general framework and solutions. Inf. Sci. 176, 1607–1627 (2006)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Luh, P.B., Chang, T.S., Ning, T.: Three-level Stackelberg decision problems. IEEE Trans. Autom. Control. AC-29, 280–282 (1984)CrossRefGoogle Scholar
  29. 29.
    Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  30. 30.
    Maggi, G.: The value of commitment with imperfect observability and private information. RAND J. Econ. 30(4), 555–574 (1999)CrossRefGoogle Scholar
  31. 31.
    Mallozzi, L., Morgan J.: ε-mixed strategies for static continuous Stackelberg problem. J. Optim. Theory Appl. 78(2), 303–316 (1993)Google Scholar
  32. 32.
    Mallozzi, L., Morgan, J.: Weak Stackelberg problem and mixed solutions under data perturbations. Optimization 32, 269–290 (1995)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Mallozzi, L., Morgan, J.: Hierarchical systems with weighted reaction set. In: Di Pillo, G., Giannessi, F. (eds.), Nonlinear Optimization and Applications, pp. 271–282. Plenum Publ. Corp., New York. ISBN: 0-306-45316-9 (1996)Google Scholar
  34. 34.
    Mallozzi, L., Morgan, J.: Mixed strategies for hierarchical zero-sum games. In: Altman E., Pourtallier O. (eds.) Advances in Dynamic Games and Applications. Annals of the International Society of Dynamic Games, vol. 6, pp. 65–77. Birkhauser, Boston (2001)CrossRefGoogle Scholar
  35. 35.
    Mallozzi, L., Morgan, J.: Oligopolistic markets with leadership and demand functions possibly discontinuous. J. Optim. Theory Appl. 125(2), 393–407 (2005)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Mallozzi, L., Morgan, J.: On approximate mixed Nash equilibria and average marginal function for two-stage three players games. In: Dempe, S., Kalshnikov V. (eds.) Optimization with Multivalued Mapping. Springer Optimization and Its Applications, vol. 2, pp. 97–107. Springer, New York (2006)CrossRefGoogle Scholar
  37. 37.
    Mallozzi, L., Tijs, S.: Conflict and cooperation in symmetric potential games. Int. Game Theory Rev. 10(3), 1–12 (2008)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Mallozzi, L., Tijs, S., Voorneveld, M.: Infinite hierarchical potential games. J. Optim. Theory Appl. 78(2), 303–316 (2000)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Marcotte, P., Blain, M.A.: Stackelberg-Nash model for the design of deregulated transit system. In: Hamalainen, R.H., Ethamo, H.K. (eds.) Dynamic Games in Economic Analysis. Lecture Notes in Control and Information Sciences, vol. 157, pp. 21–28. Springer, Berlin (1991)Google Scholar
  40. 40.
    Migdalas, A.: When is a Stackelberg equilibrium Pareto optimum? In: Pardalos, P. et al. (eds.) Advances in Multicriteria Analysis, pp. 175–181. Kluwer Academics, Dordrecht (1995)CrossRefGoogle Scholar
  41. 41.
    Migdalas, A., Pardalos, P.M.: Editorial: hierarchical and bilevel programming. J. Global Optim. 8(3), 209–215 (1996)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Migdalas, A., Pardalos, P.M., Varbrand, P. (eds.): Multilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrecht (1998)zbMATHGoogle Scholar
  43. 43.
    Miller, T.C., Friesz, T.L., Tobin, R.L.: Equilibrium Facility Location on Networks. Springer, Berlin (1996)CrossRefGoogle Scholar
  44. 44.
    Morgan, J., Raucci, R.: Lower semicontinuity for approximate social Nash equilibria. Int. J. Game Theory. 31, 499–509 (2002)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Morgan, J., Várdy, F.: An experimental study of commitment and observability in Stackelberg games with observation costs. Game Econ. Behav. 49, 401–423 (2004)CrossRefGoogle Scholar
  46. 46.
    Morgan, J., Várdy, F.: The value of commitment in contests and tournaments when observation is costly. Game Econ. Behav. 60, 326–338 (2007)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Morgan, J., Várdy, F.: The fragility of commitment. Manag. Sci. 59(6), 1344–1353 (2013)CrossRefGoogle Scholar
  48. 48.
    Nakamura, T.: One-leader and multiple-follower Stackelberg games with private information. Econ. Lett. 127, 27–30 (2015)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Nan, G., Mao, Z., Yu, M., Li, M., Wang, H., Zhang, Y.: Stackelberg game for bandwidth allocation in cloud-based wireless live-streaming social networks. IEEE Syst. J. 8(1), 256–267 (2014)CrossRefGoogle Scholar
  50. 50.
    Ochea, M.I., de Zeeuw, A.: Evolution of reciprocity in asymmetric international environmental negotiations. Environ. Resour. Econ. 62(4), 837–854 (2015)CrossRefGoogle Scholar
  51. 51.
    Oechssler, J., Schlag, K.H.: Loss of Commitment? An Evolutionary Analysis of Bagwell’s Example. Working Paper (2013)Google Scholar
  52. 52.
    Okuguchi, K., Szidarovszky, F.: The Theory of Oligopoly with Multi-Product Firms. Springer, Berlin (1990)CrossRefGoogle Scholar
  53. 53.
    Olsder, G.J.: Phenomena in inverse Stackelberg games, part 1: static problems. J. Optim. Theory Appl. 143(3), 589–600 (2009)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Pensavalle, C., Pieri, G.: Stackelberg problems with followers in the grand coalition of a TU-game. Decisions Econ. Finan. 36(1), 89–98 (2013)MathSciNetCrossRefGoogle Scholar
  55. 55.
    Sheraly, H.D., Soyster, A.L., Murphy, F.H.: Stackelberg-Nash-Cournot equilibria: characterizations and computations. Oper. Res. 31, 253–276 (1983)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Várdy, F.: The value of commitment in Stackelberg games with observation costs. Games Econ. Behav. 49, 374–400 (2004)MathSciNetCrossRefGoogle Scholar
  57. 57.
    Vincente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Global Opt. 5, 291–306 (1994)MathSciNetCrossRefGoogle Scholar
  58. 58.
    Vives, X.: Information and competitive advantage. Int. J. Ind. Organ. 8, 17–35 (1990)CrossRefGoogle Scholar
  59. 59.
    Vives, X.: Strategic supply function competition with private information. Econometrica. 79(6), 1919–1966 (2011)MathSciNetCrossRefGoogle Scholar
  60. 60.
    von Stackelberg, H.: Marktform und Gleichgewicht. Julius Springer, Vienna (1934). In: Peacock, A. (ed.) The Theory of the Market Economy, English Edition. William Hodge, London (1952)Google Scholar
  61. 61.
    Voorneveld, M., Mallozzi, L., Tijs, S.: Sequential production situations and potentials. In: Patrone, F., Garcia-Jurado, I., Tijs, S. (eds.) Game Practice: Contributions from Applied Game Theory. Theory and Decision Library C, vol. 23, pp. 241–258. Kluwer Academic Publishers, Boston (2000)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics and ApplicationsUniversity Federico IINaplesItaly
  2. 2.Department of Economics and StatisticsUniversity Federico IINaplesItaly
  3. 3.Department of Methods and Models for Economics, Territory and FinanceSapienza UniversityRomeItaly

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