Skip to main content
SpringerLink
Log in

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

  • Conference paper
  • First Online:
Experimental Algorithms (SEA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9125))

Included in the following conference series:

Abstract

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size \(2000 \times 2000\). Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria.

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. Anbalagan, Y., Norin, S., Savani, R., Vetta, A.: Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3. In: Chen, Y., Immorlica, N. (eds.) WINE 2013. LNCS, vol. 8289, pp. 15–23. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  2. Bosse, H., Byrka, J., Markakis, E.: New algorithms for approximate Nash equilibria in bimatrix games. Theoretical Computer Science 411(1), 164–173 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  3. Chen, X., Deng, X., Teng, S.-H.: Settling the complexity of computing two-player Nash equilibria. Journal of the ACM 56(3), 14:1–14:57 (2009)

    Google Scholar 

  4. Codenotti, B., Rossi, S.D., Pagan, M.: An experimental analysis of Lemke-Howson algorithm (2008). CoRR, abs/0811.3247

    Google Scholar 

  5. Daskalakis, C., Goldberg, P.W., Papadimitriou, C.H.: The complexity of computing a Nash equilibrium. SIAM Journal on Computing 39(1), 195–259 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Daskalakis, C., Mehta, A., Papadimitriou, C.H.: A note on approximate Nash equilibria. Theoretical Computer Science 410(17), 1581–1588 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fearnley, J., Goldberg, P. W., Savani, R., Sørensen, T. B.: Approximate well-supported nash equilibria below two-thirds (2012). CoRR, abs/1204.0707

    Google Scholar 

  8. Gatti, N., Patrini, G., Rocco, M., Sandholm, T.: Combining local search techniques and path following for bimatrix games (2012). CoRR, abs/1210.4858

    Google Scholar 

  9. Goldberg, L.A., Goldberg, P.W., Krysta, P., Ventre, C.: Ranking games that have competitiveness-based strategies. Theor. Comput. Sci. 476, 24–37 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  10. Goldberg, P.W., Papadimitriou, C.H., Savani, R.: The complexity of the homotopy method, equilibrium selection, and lemke-howson solutions. ACM Trans. Economics and Comput. 1(2), 9 (2013)

    Article  Google Scholar 

  11. Kontogiannis, S.C., Spirakis, P.G.: Well supported approximate equilibria in bimatrix games. Algorithmica 57(4), 653–667 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kontogiannis, S., Spirakis, P.: Approximability of symmetric bimatrix games and related experiments. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 1–20. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  13. Lemke, C.E., Howson Jr, J.: Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12(2), 413–423 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nudelman, E., Wortman, J., Shoham, Y., Leyton-Brown, K.: Run the gamut: a comprehensive approach to evaluating game-theoretic algorithms. In: AAMAS, pp. 880–887 (2004)

    Google Scholar 

  15. Porter, R., Nudelman, E., Shoham, Y.: Simple search methods for finding a nash equilibrium. Games and Economic Behavior, 642–662 (2008)

    Google Scholar 

  16. Sandholm, T., Gilpin, A., Conitzer, V.: Mixed-integer programming methods for finding nash equilibria. In: AAAI, pp. 495–501 (2005)

    Google Scholar 

  17. Savani, R., von Stengel, B.: Hard-to-solve bimatrix games. Econometrica 74(2), 397–429 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Savani, R., von Stengel, B.: Unit vector games (2015). CoRR, abs/1501.02243

    Google Scholar 

  19. Tsaknakis, H., Spirakis, P.G.: An optimization approach for approximate Nash equilibria. Internet Mathematics 5(4), 365–382 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Tsaknakis, H., Spirakis, P.G., Kanoulas, D.: Performance Evaluation of a Descent Algorithm for Bi-matrix Games. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 222–230. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Liverpool, Liverpool, UK

    John Fearnley, Tobenna Peter Igwe & Rahul Savani

Authors
  1. John Fearnley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tobenna Peter Igwe
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Rahul Savani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to John Fearnley .

Editor information

Editors and Affiliations

  1. Université Pierre et Marie Curie LIP6, Paris, France

    Evripidis Bampis

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Fearnley, J., Igwe, T.P., Savani, R. (2015). An Empirical Study of Finding Approximate Equilibria in Bimatrix Games. In: Bampis, E. (eds) Experimental Algorithms. SEA 2015. Lecture Notes in Computer Science(), vol 9125. Springer, Cham. https://doi.org/10.1007/978-3-319-20086-6_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20086-6_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20085-9

  • Online ISBN: 978-3-319-20086-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation