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, Volume 5, Issue 3, pp 173–210 | Cite as

Generalized Nash equilibrium problems

  • Francisco Facchinei
  • Christian Kanzow
Invited survey

Abstract

The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field.

Keywords

Generalized Nash equilibrium problem Equilibrium Jointly convex constraints Nikaido–Isoda-function Variational inequality Quasi-variational inequality 

MSC classification (2000)

90C30 91A10 91B50 

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References

  1. 1.
    Adida E, Perakis G (2006a) Dynamic pricing and inventory control: uncertainty and competition. Part A: existence of Nash equilibrium. Technical Report, Operations Research Center, Sloan School of Management, MITGoogle Scholar
  2. 2.
    Adida E, Perakis G (2006b) Dynamic pricing and inventory control: uncertainty and competition. Part B: an algorithm for the normalized Nash equilibrium. Technical Report, Operations Research Center, Sloan School of Management, MITGoogle Scholar
  3. 3.
    Altman E and Wynter L (2004). Equilibrium games and pricing in transportation and telecommunication networks. Netw Spat Econ 4: 7–21 CrossRefGoogle Scholar
  4. 4.
    Altman E, Pourtallier O, Haurie A and Moresino F (2000). Approximating Nash equilibria in nonzero-sum games. Int Game Theory Rev 2: 155–172 Google Scholar
  5. 5.
    Antipin AS (2000a). Solution methods for variational inequalities with coupled contraints. Computat Math Math Phys 40: 1239–1254 Google Scholar
  6. 6.
    Antipin AS (2000b). Solving variational inequalities with coupling constraints with the use of differential equations. Differential Equations 36: 1587–1596 Google Scholar
  7. 7.
    Antipin AS (2001). Differential equations for equilibrium problems with coupled constraints. Nonlinear Anal 47: 1833–1844 CrossRefGoogle Scholar
  8. 8.
    Arrow KJ and Debreu G (1954). Existence of an equilibrium for a competitive economy. Econometrica 22: 265–290 CrossRefGoogle Scholar
  9. 9.
    Aubin JP (1993). Optima and equilibria. Springer, Berlin Google Scholar
  10. 10.
    Aubin JP and Frankowska H (1990). Set-valued analysis. Birkhäuser, Boston Google Scholar
  11. 11.
    Başar T, Olsder GJ (1989) Dynamic noncooperative game theory, 2nd edn. Academic Press, London (reprinted in SIAM Series “Classics in Applied Mathematics”, 1999)Google Scholar
  12. 12.
    Bassanini A, La Bella A and Nastasi A (2002). Allocation of railroad capacity under competition: a game theoretic approach to track time pricing. In: Gendreau, M and Marcotte, P (eds) Transportation and networks analysis: current trends, pp 1–17. Kluwer, Dordrecht Google Scholar
  13. 13.
    Baye MR, Tian G and Zhou J (1993). Characterization of existence of equilibria in games with discontinuous and non-quasiconcave payoffs. Rev Econ Stud 60: 935–948 CrossRefGoogle Scholar
  14. 14.
    Bensoussan A (1974). Points de Nash dans le cas de fonctionnelles quadratiques et jeux differentiels lineaires a N personnes. SIAM J Control 12: 460–499 CrossRefGoogle Scholar
  15. 15.
    Berridge S, Krawczyk JB (1997) Relaxation algorithms in finding Nash equilibria. Economic working papers archives, http://econwpa.wustl.edu/eprints/comp/papers/9707/9707002.absGoogle Scholar
  16. 16.
    Bertrand J (1883) Review of “Théorie mathématique de la richesse sociale” by Léon Walras and “Recherches sur les principes mathématiques de la théorie des richesses” by Augustin Cournot. J des Savants 499–508Google Scholar
  17. 17.
    Breton M, Zaccour G and Zahaf M (2005). A game-theoretic formulation of joint implementation of environmental projects. Eur J Oper Res 168: 221–239 CrossRefGoogle Scholar
  18. 18.
    Cavazzuti E and Pacchiarotti N (1986). Convergence of Nash equilibria. Boll UMI 5B: 247–266 Google Scholar
  19. 19.
    Cavazzuti E, Pappalardo M and Passacantando M (2002). Nash equilibria, variational inequalities, and dynamical systems. J Optim Theory Appl 114: 491–506 CrossRefGoogle Scholar
  20. 20.
    Chan D and Pang JS (1982). The generalized quasi-variational inequality problem. Math Oper Res 7: 211–222 Google Scholar
  21. 21.
    Contreras J, Klusch MK and Krawczyk JB (2004). Numerical solution to Nash-Cournot equilibria in coupled constraints electricity markets. IEEE Trans Power Syst 19: 195–206 CrossRefGoogle Scholar
  22. 22.
    Cournot AA (1838). Recherches sur les principes mathématiques de la théorie des richesses. Hachette, Paris Google Scholar
  23. 23.
    Dafermos S (1990). Exchange price equilibria and variational inequalities. Math Programming 46: 391–402 CrossRefGoogle Scholar
  24. 24.
    van Damme E (1996). Stability and perfection of Nash equilibria, 2nd edn. Springer, Berlin Google Scholar
  25. 25.
    Dasgupta P and Maskin E (1986a). The existence of equilibrium in discontinuous economic games, I: theory. Rev Econ Stud 53: 1–26 CrossRefGoogle Scholar
  26. 26.
    Dasgupta P and Maskin E (1986b). The existence of equilibrium in discontinuous economic games, II: applications. Rev Econ Stud 53: 27–41 CrossRefGoogle Scholar
  27. 27.
    Debreu G (1952). A social equilibrium existence theorem. Proc Natl Acad Sci 38: 886–893 CrossRefGoogle Scholar
  28. 28.
    Debreu G (1959). Theory of value. Yale University Press, New Haven Google Scholar
  29. 29.
    Debreu G (1970). Economies with a finite set of equilibria. Econometrica 38: 387–392 CrossRefGoogle Scholar
  30. 30.
    Dirkse SP and Ferris MC (1995). The PATH solver: a non-monotone stabilization scheme for mixed complementarity problems. Optim Methods Softw 5: 123–156 CrossRefGoogle Scholar
  31. 31.
    Ehrenmann A (2004) Equilibrium problems with equilibrium constraints and their application to electricity markets. Ph.D. Dissertation, Judge Institute of Management, The University of Cambridge, CambridgeGoogle Scholar
  32. 32.
    Facchinei F, Kanzow C (2007) Globally convergent methods for the generalized Nash equilibrium problem based on exact penalization. Technical Report, University of Würzburg, Würzburg, (in press)Google Scholar
  33. 33.
    Facchinei F and Pang JS (2003). Finite-dimensional variational inequalities and complementarity problems. Springer, New York Google Scholar
  34. 34.
    Facchinei F and Pang JS (2006). Exact penalty functions for generalized Nash problems. In: Di Pillo, G and Roma, M (eds) Large-scale nonlinear optimization., pp 115–126. Springer, Heidelberg CrossRefGoogle Scholar
  35. 35.
    Facchinei F, Fischer A and Piccialli V (2007a). On generalized Nash games and variational inequalities. Oper Res Lett 35: 159–164 CrossRefGoogle Scholar
  36. 36.
    Facchinei F, Fischer A, Piccialli V (2007b) Generalized Nash equilibrium problems and Newton methods. Math Programming (in press); doi: 10.1007/s10107-007-0160-2Google Scholar
  37. 37.
    Facchinei F, Piccialli V, Sciandrone M (2007c) On a class of generalized Nash equilibrium problems. DIS Technical Report, “Sapienza” Università di Roma, Rome, (in press)Google Scholar
  38. 38.
    Flåm SD (1993). Paths to constrained Nash equilibria. Appl Math Optim 27: 275–289 CrossRefGoogle Scholar
  39. 39.
    Flåm SD (1994). On variational stability in competitive economies. Set Valued Anal 2: 159–173 CrossRefGoogle Scholar
  40. 40.
    Flåm SD, Ruszczyński A (1994) Noncooperative convex games: computing equilibrium by partial regularization. IIASA Working Paper 94-42, LaxenburgGoogle Scholar
  41. 41.
    Fudenberg D and Tirole J (1991). Game theory. MIT Press, Cambridge Google Scholar
  42. 42.
    Fukushima M (2007). A class of gap functions for quasi-variational inequality problems. J Ind Manage Optim 3: 165–171 Google Scholar
  43. 43.
    Fukushima M and Pang JS (2005). Quasi-variational inequalities, generalized Nash equilibria and multi-leader-follower games. Comput Manage Sci 2: 21–56 CrossRefGoogle Scholar
  44. 45.
    Gabriel S, Smeers Y (2006) Complementarity problems in restructured natural gas markets, in Recent Advances in Optimization. Lectures Notes in Economics and Mathematical Systems 563, Springer, Heidelberg, pp 343–373Google Scholar
  45. 44.
    Gabriel SA, Kiet S and Zhuang J (2005). A mixed complementarity-based equilibrium model of natural gas markets. Oper Res 53: 799–818 CrossRefGoogle Scholar
  46. 46.
    Garcia CB and Zangwill WI (1981). Pathways to solutions, fixed points and equilibria. Prentice-Hall, New Jersey Google Scholar
  47. 47.
    Gürkan G, Pang JS (2006) Approximations of Nash equilibria. Technical Report, Department of Mathematical Sciences, Rensselaer Polytechnic Institute, TroyGoogle Scholar
  48. 48.
    Harker PT (1991). Generalized Nash games and quasi-variational inequalities. Eur J Oper Res 54: 81–94 CrossRefGoogle Scholar
  49. 49.
    Harker PT and Hong S (1994). Pricing of track time in railroad operations: an internal market approach. Transport Res B 28: 197–212 CrossRefGoogle Scholar
  50. 50.
    Haurie A and Krawczyk JB (1997). Optimal charges on river effluent from lumped and distributed sources. Environ Model Assess 2: 93–106 Google Scholar
  51. 51.
    von Heusinger A, Kanzow C (2006) Optimization reformulations of the generalized Nash equilibrium problem using Nikaido–Isoda-type functions. Technical Report, Institute of Mathematics, University of Würzburg, WürzburgGoogle Scholar
  52. 52.
    von Heusinger A, Kanzow C (2007) SC1 optimization reformulations of the generalized Nash equilibrium problem. Technical Report, Institute of Mathematics, University of Würzburg, WürzburgGoogle Scholar
  53. 53.
    Hobbs B and Pang JS (2007). Nash-Cournot equilibria in electric power markets with piecewise linear demand functions and joint constraints. Oper Res 55: 113–127 CrossRefGoogle Scholar
  54. 54.
    Hobbs B, Helman U, Pang JS (2001) Equilibrium market power modeling for large scale power systems. IEEE Power Engineering Society Summer Meeting. pp 558–563Google Scholar
  55. 55.
    Hotelling H (1929). Game theory for economic analysis. Econ J 39: 41–47 CrossRefGoogle Scholar
  56. 56.
    Hu X, Ralph D (2006) Using EPECs to model bilevel games in restructured electricity markets with locational prices. Technical Report CWPE 0619Google Scholar
  57. 57.
    Ichiishi T (1983). Game theory for economic analysis. Academic, New York Google Scholar
  58. 58.
    Jiang H (2007). Network capacity management competition, Technical Report. Judge Business School at University of Cambridge, UK Google Scholar
  59. 59.
    Jofré A and Wets RJB (2002). Continuity properties of Walras equilibrium points. Ann Oper Res 114: 229–243 CrossRefGoogle Scholar
  60. 60.
    Kesselman A, Leonardi S, Bonifaci V (2005) Game-theoretic analysis of internet switching with selfish users. In: Proceedings of the first international workshop on internet and network economics. WINE 2005, Lectures Notes in Computer Science 3828:236–245Google Scholar
  61. 61.
    Kočvara M and Outrata JV (1995). On a class of quasi-variational inequalities. Optim Methods Softw 5: 275–295 CrossRefGoogle Scholar
  62. 62.
    Krawczyk JB (2000) An open-loop Nash equilibrium in an environmental game with coupled constraints. In: Proceedings of the 2000 symposium of the international society of dynamic games. Adelaide, pp 325–339Google Scholar
  63. 63.
    Krawczyk JB (2005). Coupled constraint Nash equilibria in environmental games. Resour Energy Econ 27: 157–181 CrossRefGoogle Scholar
  64. 64.
    Krawczyk JB (2007). Numerical solutions to coupled-constraint (or generalised Nash) equilibrium problems. Comput Manage Sci 4: 183–204 CrossRefGoogle Scholar
  65. 65.
    Krawczyk JB and Uryasev S (2000). Relaxation algorithms to find Nash equilibria with economic applications. Environ Model Assess 5: 63–73 CrossRefGoogle Scholar
  66. 66.
    Laffont J and Laroque G (1976). Existence d’un équilibre général de concurrence imparfait: Une introduction. Econometrica 44: 283–294 CrossRefGoogle Scholar
  67. 67.
    Leyffer S, Munson T (2005) Solving multi-leader-follower games. Argonne National Laboratory Preprint ANL/MCS-P1243-0405, IllinoisGoogle Scholar
  68. 68.
    Margiocco M, Patrone F and Pusilli Chicco L (1997). A new approach to Tikhonov well-posedness for Nash equilibria. Optimization 40: 385–400 CrossRefGoogle Scholar
  69. 69.
    Margiocco M, Patrone F and Pusilli Chicco L (1999). Metric characterization of Tikhonov well-posedness in value. J Optim Theory Appl 100: 377–387 CrossRefGoogle Scholar
  70. 70.
    Margiocco M, Patrone F and Pusilli Chicco L (2002). On the Tikhonov well-posedness of concave games and Cournot oligopoly games. J Optim Theory Appl 112: 361–379 CrossRefGoogle Scholar
  71. 71.
    Morgan J, Scalzo V (2004) Existence of equilibria in discontinuous abstract economies. Preprint 53-2004, Dipartimento di Matematica e Applicazioni R. CaccioppoliGoogle Scholar
  72. 72.
    Morgan J, Scalzo V (2007) Variational stability of social Nash equilibria. Int Game Theory Rev 9 (in press)Google Scholar
  73. 73.
    Munson TS, Facchinei F, Ferris MC, Fischer A and Kanzow C (2001). The semismooth algorithm for large scale complementarity problems. INFORMS J Comput 13: 294–311 CrossRefGoogle Scholar
  74. 74.
    Myerson RB (1991). Game theory. Analysis of conflict. Harvard University Press, Cambridge Google Scholar
  75. 75.
    Nash JF (1950). Equilibrium points in n-person games. Proc Natl Acad Sci 36: 48–49 CrossRefGoogle Scholar
  76. 76.
    Nash JF (1951). Non-cooperative games. Ann Math 54: 286–295 CrossRefGoogle Scholar
  77. 77.
    Neumann J (1928). Zur Theorie der Gesellschaftsspiele. Math Ann 100: 295–320 CrossRefGoogle Scholar
  78. 78.
    Morgenstern O and Neumann J (1944). Theory of games and economic behavior. Princeton University Press, Princeton Google Scholar
  79. 79.
    Nikaido H (1975). Monopolistic competition and effective demand. Princeton University Press, Princeton Google Scholar
  80. 80.
    Nikaido H and Isoda K (1955). Note on noncooperative convex games. Pac J Math 5: 807–815 Google Scholar
  81. 81.
    Nishimura K and Friedman J (1981). Existence of Nash equilibrium in n person games without quasi-concavity. Int Econ Rev 22: 637–648 CrossRefGoogle Scholar
  82. 82.
    Outrata JV, Kočvara M and Zowe J (1998). Nonsmooth approach to optimization problems with equilibrium constraints. Kluwer, Dordrecht Google Scholar
  83. 83.
    Pang JS (2002) Computing generalized Nash equilibria. ManuscriptGoogle Scholar
  84. 84.
    Pang JS and Qi L (1993). Nonsmooth equations: motivation and algorithms. SIAM J Optim 3: 443–465 CrossRefGoogle Scholar
  85. 85.
    Pang JS and Yao JC (1995). On a generalization of a normal map and equation. SIAM J Control Optim 33: 168–184 CrossRefGoogle Scholar
  86. 86.
    Pang JS, Scutari G, Facchinei F, Wang C (2007) Distributed power allocation with rate contraints in Gaussian frequency-selective interference channels. DIS Technical Report 05-07, “Sapienza” Università di Roma, RomeGoogle Scholar
  87. 87.
    Puerto J, Schöbel A, Schwarze S (2005) The path player game: introduction and equilibria. Preprint 2005-18, Georg-August University of Göttingen, GöttingenGoogle Scholar
  88. 88.
    Qi L (1993). Convergence analysis of some algorithms for solving nonsmooth equations. Math Oper Res 18: 227–244 Google Scholar
  89. 89.
    Qi L and Sun J (1993). A nonsmooth version of Newton’s method. Math Programming 58: 353–367 CrossRefGoogle Scholar
  90. 90.
    Rao SS, Venkayya VB and Khot NS (1988). Game theory approach for the integrated design of structures and controls. AIAA J 26: 463–469 Google Scholar
  91. 91.
    Reny PJ (1999). On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67: 1026–1056 CrossRefGoogle Scholar
  92. 92.
    Robinson SM (1993a). Shadow prices for measures of effectiveness. I. Linear model. Oper Res 41: 518–535 Google Scholar
  93. 93.
    Robinson SM (1993b). Shadow prices for measures of effectiveness. II. General model. Oper Res 41: 536–548 CrossRefGoogle Scholar
  94. 94.
    Rockafellar RT and Wets RJB (1998). Variational analysis. Springer, Berlin Google Scholar
  95. 95.
    Rosen JB (1965). Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 33: 520–534 CrossRefGoogle Scholar
  96. 96.
    Schmit LA (1981). Structural synthesis—its genesis and development. AIAA J 19: 1249–1263 CrossRefGoogle Scholar
  97. 97.
    Scotti SJ (1995) Structural design using equilibrium programming formulations. NASA Technical Memorandum 110175Google Scholar
  98. 98.
    Stoer J and Bulirsch R (2002). Introduction to numerical analysis, 3rd edn. Springer, New York Google Scholar
  99. 99.
    Sun LJ and Gao ZY (2007). An equilibrium model for urban transit assignment based on game theory. Eur J Oper Res 181: 305–314 CrossRefGoogle Scholar
  100. 100.
    Tian G and Zhou J (1995). Transfer continuities, generalizations of the Weierstrass and maximum theorems: a full characterization. J Math Econ 24: 281–303 CrossRefGoogle Scholar
  101. 101.
    Tidball M and Zaccour G (2005). An environmental game with coupling constraints. Environ Model Assess 10: 153–158 CrossRefGoogle Scholar
  102. 102.
    Uryasev S and Rubinstein RY (1994). On relaxation algorithms in computation of noncooperative equilibria. IEEE Trans Automatic Control 39: 1263–1267 CrossRefGoogle Scholar
  103. 103.
    Vincent TL (1983). Game theory as a design tool. ASME J Mech Trans Autom Design 105: 165–170 CrossRefGoogle Scholar
  104. 104.
    Vives X (1994). Nash equilibrium with strategic complementarities. J Math Econ 19: 305–321 CrossRefGoogle Scholar
  105. 105.
    Walras L (1900) Éléments d’économie politique pure. LausanneGoogle Scholar
  106. 106.
    Wei JY and Smeers Y (1999). Spatial oligopolistic electricity models with Cournot generators and regulated transmission prices. Oper Res 47: 102–112 Google Scholar
  107. 107.
    Zhou J, Lam WHK and Heydecker BG (2005). The generalized Nash equilibrium model for oligopolistic transit market with elastic demand. Transport Res B 39: 519–544 CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Computer and System Sciences “A. Ruberti”“Sapienza” Università di RomaRomeItaly
  2. 2.Institute of MathematicsUniversity of WürzburgWürzburgGermany

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