Hard-to-solve bimatrix games

Econometrica

Volumn 74, Issue 2, 2006, Pages 397-429

  Savani, Rahul ^a   Von Stengel, Bernhard ^a  

^a LONDON SCHOOL OF ECONOMICS   (United Kingdom)

Author keywords

Bimatrix game; Computational complexity; Lemke Howson algorithm; Nash equilibrium; Support enumeration

Indexed keywords

EID: 33645018035     PISSN: 00129682     EISSN: 14680262     Source Type: Journal    
DOI: 10.1111/j.1468-0262.2006.00667.x     Document Type: Article

References (43)

1. ADLER, I., AND N. MEGIDDO (1985): "A Simplex Algorithm whose Average Number of Steps Is Bounded Between Two Quadratic Functions of the Smaller Dimension," Journal of the Association for Computing Machinery, 32, 871-895.
2. BÁRÁNY, I., S. VEMPALA, AND A. VETTA (2005): "Nash Equilibria in Random Games," in Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS). New York: IEEE Press, 123-131.
3. CODENOTTI, B., AND D. ŠTEFANKOVIČ (2005): "On the Computational Complexity of Nash Equilibria for (0, 1) Bimatrix Games," Information Processing Letters, 94, 145-150.
4. CONITZER, V., AND T. SANDHOLM (2003): "Complexity Results about Nash Equilibria," in Proceedings of the 18th International Joint Conference on Artificial Intelligence (IJCAI-03) 765-771.
5. COTTLE, R. W., J.-S. PANG, AND R. E. STONE (1992): The Linear Complementarity Problem. San Diego, CA: Academic Press.
6. DANTZIG, G. B. (1963): Linear Programming and Extensions. Princeton, NJ: Princeton University Press.
7. DICKHAUT, J., AND T. KAPLAN (1991): "A Program for Finding Nash Equilibria," The Mathematica Journal, 1 (4), 87-93.
8. FATHI, Y. (1979): "Computational Complexity of LCPs Associated with Positive Definite Symmetric Matrices," Mathematical Programming, 17, 335-344.
9. GALE, D. (1963): "Neighborly and Cyclic Polytopes," in Convexity, Proceedings of Symposia in Pure Mathematics, Vol. 7, ed. by V. Klee. Providence, RI: American Mathematical Society, 225-232.
10. GAREY, M. R., AND D. S. JOHNSON (1979): Computers and Intractability: A Guide to the Theory of NP-Completeness. San Francisco, CA: Freeman.
11. GILBOA, I., AND E. ZEMEL (1989): "Nash and Correlated Equilibria: Some Complexity Considerations," Games and Economic Behavior, 1, 80-93.
12. GOLDFARB, D. (1983): "Worst-Case Complexity of the Shadow Simplex Algorithm," Report, Department of Industrial Engineering and Operations Research, Columbia University.
13. _ (1994): "On the Complexity of the Simplex Method," in Advances in Optimization and Numerical Analysis, Mathematics and Its Applications, Vol. 275. Dordrecht: Kluwer, 25-38.
14. GRAHAM, R. L., D. E. KNUTH, AND O. PATASHNIK (1994): Concrete Mathematics. Reading, MA: Addison-Wesley.
15. GRÜNBAUM, B. (2003): Convex Polytopes (Second Ed.). New York: Springer-Verlag.
16. KLEE, V., AND P. KLEINSCHMIDT (1987): "The d-Step Conjecture and Its Relatives," Mathematics of Operations Research, 12, 718-755.
17. KLEE, V., AND G. J. MINTY (1972): "How Good Is the Simplex Algorithm?" in Inequalities, III, ed. by O. Shisha. New York: Academic Press, 159-175.
18. LEMKE, C. E. (1965): "Bimatrix Equilibrium Points and Mathematical Programming," Management Science, 11, 681-689.
19. LEMKE, C. E., AND J. T. HOWSON, JR. (1964): "Equilibrium Points of Bimatrix Games," Journal of the Society for Industrial and Applied Mathematics, 12, 413-423.
20. MCKELVEY, R. D., A. MCLENNAN, AND T. L. TUROCY (2005): "Gambit: Software Tools for Game Theory," Version 0.2005.06.13. Available at http://econweb.tamu.edu/gambit.
21. MCLENNAN, A., AND R. TOURKY (2005): "From Imitation Games to Kakutani," Unpublished Manuscript, University of Minnesota.
22. MEGIDDO, N. (1986): "On the Expected Number of Linear Complementarity Cones Intersected by Random and Semi-Random Rays," Mathematical Programming, 35, 225-235.
23. MORRIS, W. D., JR. (1994): "Lemke Paths on Simple Polytopes," Mathematics of Operations Research, 19, 780-789.
24. MOTWANI, R., AND P. RAGHAVAN (1995): Randomized Algorithms. Cambridge, U.K.: Cambridge University Press.
25. MURTY, K. G. (1978): "Computational Complexity of Complementary Pivot Methods," Mathematical Programming Study, 7: Complementary and Fixed Point Problems. Amsterdam: North-Holland, 61-73.
26. _ (1980): "Computational Complexity of Parametric Linear Programming," Mathematical Programming, 19, 213-219.
27. NASH, J. F. (1951): "Non-Cooperative Games," The Annals of Mathematics, 54, 286-295.
28. PAPADIMITRIOU, C. H. (1994): Computational Complexity. Reading, MA: Addison-Wesley.
29. _ (2001): "Algorithms, Games, and the Internet," in Proceedings of the 33rd Symposium on the Theory of Computing. New York: ACM Press, 749-753.
30. PORTER, R. W., E. NUDELMAN, AND Y. SHOHAM (2004): "Simple Search Methods for Finding a Nash Equilibrium," in Proceedings of the 19th National Conference on Artificial Intelligence. Cambridge, MA: AAAI/MIT Press, 664-669.
31. QUINT, T., AND M. SHUBIK (1997): "A Theorem on the Number of Nash Equilibria in a Bimatrix Game," International Journal of Game Theory, 26, 353-359.
32. SAVANI, R. (2004): "Challenge Instances for NASH," Research Report LSE-CDAM-2004-14, CDAM, London School of Economics.
33. _ (2005): "Finding Nash Equilibria of Bimatrix Games," Ph.D. Thesis, London School of Economics.
34. SAVANI, R., AND B. VON STENGEL (2004): "Exponentially Many Steps for Finding a Nash Equilibrium in a Bimatrix Game," Research Report LSE-CDAM-2004-03, CDAM, London School of Economics.
35. [Extended abstract in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS). New York: IEEE Press, 258-267.]
36. SHAPLEY, L. S. (1974): "A Note on the Lemke-Howson Algorithm," Mathematical Programming Study, 1: Pivoting and Extensions, ed. by M. Balinski. Amsterdam: North-Holland, 175-189.
37. SMALE, S. (1983): "On the Average Number of Steps of the Simplex Method of Linear Programming," Mathematical Programming, 27, 241-262.
38. TODD, M. J. (2001): "The Many Facets of Linear Programming," Mathematical Programming Series B, 91, 417-436.
39. VON STENGEL, B. (1999): "New Maximal Numbers of Equilibria in Bimatrix Games," Discrete and Computational Geometry, 21, 557-568.
40. _ (2002): "Computing Equilibria for Two-Person Games," in Handbook of Game Theory, Vol. 3, ed. by R. J. Aumann and S. Hart. Amsterdam: North-Holland, 1723-1759.
41. VON STENGEL, B., A. H. VAN DEN ELZEN, AND A. J. J. TALMAN (2002): "Computing Normal Form Perfect Equilibria for Extensive Two-Person Games," Econometrica, 70, 693-715.
42. VOROB'EV, N. N. (1958): "Equilibrium Points in Bimatrix Games," Theory of Probability and Its Applications, 3, 297-309.
43. ZIEGLER, G. M. (1995): Lectures on Polytopes. New York: Springer-Verlag.