Skip to main content
SpringerLink
Log in

Differential Evolution as a New Method of Computing Nash Equilibria

  • Chapter
Transactions on Computational Collective Intelligence IX

Part of the book series: Lecture Notes in Computer Science ((TCCI,volume 7770))

Abstract

The Nash equilibrium is one of the central concepts in game theory. Recently it was shown, that problems of finding Nash equilibrium and an approximate Nash equilibrium are PPAD-complete. In this article we present the Differential Evolution algorithm adapted to that problem, and we compare it with two well-known algorithms: the Simplicial Subdivision and the Lemke-Howson. The problem of finding the Nash equilibrium for two players games may be classified as a continuous problem, where two probability distributions over the set of pure strategies of both players should be found. Each deviation from the global optimum is interpreted as the Nash approximation and called the ε-Nash equilibrium. We show that the Differential Evolution approach can be determined as a method, which in successive iterations is capable of obtaining ε value close to the global optimum. We show, that the Differential Evolution may be succesfully used to obtain satisfactory results and it may be easily expanded into n-person games. Moreover, we present results for the problem of computing Nash equilibrium, when some arbitrary set strategies have a non-zero probability of being chosen.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aubin, J.: Mathematical Methods of Game and Economic Theory. North-Holland Publ. Co., New York (1979)

    MATH  Google Scholar 

  2. Aumann, R.: Subjectivity and correlation in randomized strategies. Journal of Mathematical Economics 1, 67–96 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boryczka, U., Juszczuk, P.: Approximate Nash Equilibria in Bimatrix Games. In: Jędrzejowicz, P., Nguyen, N.T., Hoang, K. (eds.) ICCCI 2011, Part II. LNCS, vol. 6923, pp. 485–494. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  4. Boryczka, U., Juszczuk, P., Kosowicz, L.: A Comparative Study of Various Strategies in Differential Evolution. In: Arabas, J. (ed.) KAEiOG 2009 - Evolutionary Computing and Global Optimization, Warszawa, pp. 19–26 (2009)

    Google Scholar 

  5. Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: 47th Symposium Foundations of Computer Science, pp. 261–271 (2006)

    Google Scholar 

  6. Chen, X., Deng, X., Teng, S.-H.: Computing Nash Equilibria: Approximation and Smoothed Complexity. In: 47th Annual IEEE Symposium on Foundations of Computer Science, pp. 603–612 (2006)

    Google Scholar 

  7. Daskalakis, C., Mehta, A., Papadimitriou, C.: A note on approximate Nash equilibria. Journal Theoretical Computer Science 410, 1581–1588 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Etessami, K., Yannakakis, M.: On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract). In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 113–123 (2007)

    Google Scholar 

  9. Gilboa, I.: Nash and correlated equilibria: Some complexity considerations. Games and Economic Behavior 1, 80–93 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  10. Griss, M., Letsinger, R.: Games at Work-Agent-Mediated E-Commerce Simulation. In: Proceedings of the Fourth International Conference on Autonomous Agents (2000)

    Google Scholar 

  11. Hammerstein, P., Selten, R.: Handbook of game theory - Chapter Game theory and evolutionary biology. University of Bonn (1994)

    Google Scholar 

  12. Kaplan, T., Dickhaut, J.: A Program for Finding Nash Equilibria, University of Minnesota, Department of Economics in its series Working papers, number 004

    Google Scholar 

  13. Karaboga, D., Basturk, B.: ABC - A powerful and efficient algorithm for numerical function optimization: artificial bee colony (ABC) algorithm, pp. 459–471. Springer, Netherlands (2007)

    Google Scholar 

  14. Kasemir, K.U.: Detecting ellipses of limited eccentricity in images with high noise levels. Image and Vision Computing 21(7), 221–227 (2003)

    Article  Google Scholar 

  15. Kontogiannis, S.C., Panagopoulou, P.N., Spirakis, P.G.: Polynomial algorithms for approximating Nash equilibria of bimatrix games. Journal Theoretical Computer Science 410, 1599–1606 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Krohn, I., Moltzahn, S., Rosenmuller, J., Sudholter, P., Wallmeier, H.M.: Implementing the modified LH algorithm. Applied Mathematics and Computation 45, 31–72 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. van der Laan, G., Talman, A.J., van Der Heyden, L.: Simplicial variable dimension algorithms for solving the nonlinear complementarity problem on a product of unit simplices using a general labelling. Mathematics of Operations Research, 377–397 (1987)

    Google Scholar 

  18. Lampinen, J., Zelinka, I.: Mixed variable non-linear optimization by differential evolution. In: Proceedings of Nostradamus (1999)

    Google Scholar 

  19. Lampinen, J., Zelinka, I.: On stagnation of the differential evolution algorithm. In: Proceedings of Mendel, 6th International Mendel Conference on Soft Computing (2000)

    Google Scholar 

  20. Lemke, C.E., Howson, J.T.: Equilibrium Points of Bimatrix Games. Society for Industrial and Applied Mathematics 12, 413–423 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lipton, R.J., Markakis, E., Mehta, A.: Playing large games using simple strategies. In: Proceedings of the 4th ACM Conference on Electronic Commerce, pp. 36–41 (2003)

    Google Scholar 

  22. Magoulas, G.D., Vrahatis, M.N., Androulakis, G.S.: Effective backpropagation training with variable stepsize. Neural Netw. 10, 69–82 (1997)

    Article  Google Scholar 

  23. McKelvey, R.D., McLennan, A.M., Turocy, T.L.: Gambit: Software Tools for Game Theory, Version 0.2010.09.01 (2010), http://www.gambit-project.org

  24. McLennan, A.: The Expected Number of Nash Equilibria of a Normal Form Game. Econometrica 73, 141–174 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Milchtaich, I.: Computation of completely mixed equilibrium payoffs in bimatrix games. International Game Theory Review 8, 483–487 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nash, J.F.: Non-cooperative games. Annals of Mathematics 54(2), 286–295 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  27. Neel, J., Mackenzie, A., Menon, R., Dasilva, L., Hicks, J., Reed, J., Gilles, R.: Using game theory to analyze wireless ad hoc networks. IEEE Communications Surveys & Tutorials 7(4), 46–56 (2005)

    Article  Google Scholar 

  28. Nudelman, E., Wortman, J., Shoham, Y., Leyton-Brown, K.: Run the GAMUT: A Comprehensive Approach to Evaluating Game-Theoretic Algorithms. In: Proceedings of the Third International Joint Conference on Autonomous Agents and Multiagent Systems, vol. 2, pp. 880–887 (2004)

    Google Scholar 

  29. Ordeshook, P.: Game theory and political theory. Cambridge University Press (1986)

    Google Scholar 

  30. Pavlidis, N., Parsopoulos, K., Vrahatis, M.: Computing Nash Equilibria Through Computational Intelligence Methods. Journal of Computational and Applied Mathematics, 113–136 (2005)

    Google Scholar 

  31. Porter, R., Nudelman, E., Shoham, Y.: Simple search methods for finding a Nash equilibrium. Games and Economic Behavior, 664–669 (2004)

    Google Scholar 

  32. Price, K., Storn, R., Lampinen, J.: Differential evolution: a practical approach to global optimization. Springer (2005)

    Google Scholar 

  33. Sandholm, T., Gilpin, A., Conitzer, V.: Mixed-integer programming methods for finding Nash equilibria. In: Proceedings of the 20th National Conference on Artificial Intelligence, vol. 2, pp. 495–501 (2005)

    Google Scholar 

  34. Savani, R., von Stengel, B.: Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 258–267 (2004)

    Google Scholar 

  35. Storn, R.: Differential evolution design of an iir-Filter. In: IEEE International Conference on Evolutionary Computation, ICEC 1996, pp. 268–273 (1996)

    Google Scholar 

  36. Storn, R., Price, K.: Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. Journal of Global Optimization 11(4), 341–359 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Tennenholtz, M.: Game Theory and Artificial Intelligence. In: d’Inverno, M., Luck, M., Fisher, M., Preist, C. (eds.) UKMAS 1996-2000. LNCS (LNAI), vol. 2403, pp. 49–58. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  38. Tsaknakis, H., Spirakis, P.G.: An Optimization Approach for Approximate Nash Equilibria. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 42–56. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  39. Widger, J., Grosu, D.: Computing Equilibria in Bimatrix Games by Parallel Support Enumeration. In: International Symposium on Parallel and Distributed Computing, pp. 250–256 (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Computer Science, University of Silesia, ul.Bedzinska 39, Sosnowiec, Poland

    Urszula Boryczka & Przemyslaw Juszczuk

Authors
  1. Urszula Boryczka
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Przemyslaw Juszczuk
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Informatics, Wrocław University of Technology, Str. Wybrzeże Wyspiańskiego 27, 50-370, Wrocław, Poland

    Ngoc Thanh Nguyen

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Boryczka, U., Juszczuk, P. (2013). Differential Evolution as a New Method of Computing Nash Equilibria. In: Nguyen, N.T. (eds) Transactions on Computational Collective Intelligence IX. Lecture Notes in Computer Science, vol 7770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36815-8_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-36815-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36814-1

  • Online ISBN: 978-3-642-36815-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation