AI helps you reading Science

AI generates interpretation videos

AI extracts and analyses the key points of the paper to generate videos automatically


pub
Go Generating

AI Traceability

AI parses the academic lineage of this thesis


Master Reading Tree
Generate MRT

AI Insight

AI extracts a summary of this paper


Weibo:
We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991

Settling the Complexity of 2-Player Nash-Equilibrium

Electronic Colloquium on Computational Complexity (ECCC), (2005)

Cited by: 488|Views111
EI
Full Text
Bibtex
Weibo

Abstract

We prove that nding a solution of two player Nash Equilibrium is PPAD-complete.

Code:

Data:

0
Introduction
  • In 1944, Morgenstern and von Neumann [43] initiated the study of game theory and its applications to economic behavior.
  • In [46], Papadimitriou proved that Bimatrix, the problem of finding a Nash equilibrium in a two-player game with rational payoffs is member of PPAD.
  • In a complexity-theoretic breakthrough, Daskalakis, Goldberg and Papadimitriou [18] proved that the problem of computing a Nash equilibrium in a game among four or more players is complete for PPAD.
Highlights
  • In 1944, Morgenstern and von Neumann [43] initiated the study of game theory and its applications to economic behavior
  • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time
  • The P-Matrix Linear Complementary Problem is computationally harder than convex programming unless every problem in PPAD is solvable in polynomial time
  • Because the two-player Nash equilibrium enjoys several structural properties that Nash equilibria with three or more players do not have, our result enables us to answer some other long-standing open questions in mathematical economics and operations research
  • If Bimatrix is in smoothed polynomial time under uniform or Gaussian perturbations, for all ǫ > 0, there exists a randomized algorithm to compute an ǫ-approximate Nash equilibrium in a two-player game with expected time O poly(m, n, 1/ǫ) or O poly(m, n, log max(m, n)/ǫ), respectively
  • There remains a complexity gap in the approximation of two-player Nash equilibria: Lipton, Markakis and Mehta [40] show that an ǫ-approximate Nash equilibrium can be computed in nO-time, while this paper shows that, for ǫ of order 1/poly(n), no algorithm can find an ǫ-approximate Nash equilibrium in poly(n, 1/ǫ)-time, unless PPAD is contained in P
Results
  • The authors show that from each discrete Brouwer function f , the authors can build a two-player game G and a polynomial-time map Π from the Nash equilibria of G to the fixed points of f .
  • For any c > 0, the problem of computing an n−c-approximate Nash equilibrium of a two-player game is PPAD-complete.
  • For any c > 0, the problem of finding the first (1 + c) log n bits of an exact Nash equilibrium in a two-player game, even when the payoffs are integers of polynomial magnitude, is polynomial-time equivalent to Bimatrix.
  • Let Expc-Bimatrix denote the following search problem: Given a rational and positively normalized bimatrix game (A, B), compute a 2−cn-approximate Nash equilibrium of (A, B), if A and B are n × n matrices;
  • Let Polyc-Bimatrix denote the following search problem: Given a rational and positively normalized bimatrix game (A, B), compute an n−c-approximate Nash equilibrium of (A, B), if A and B are n × n matrices.
  • For a positive integer P , the authors will use P -Bit-Bimatrix to denote the search problem of computing the first P bits of the entries of a Nash equilibrium in a rational bimatrix game.
  • The authors define the perturbation models in the smoothed analysis of Bimatrix and show that if the smoothed complexity of Bimatrix is polynomial, the authors can compute an ǫ-approximate Nash equilibrium of a bimatrix game in randomized poly(n, 1/ǫ) time.
Conclusion
  • The following lemma shows that if the smoothed complexity of Bimatrix is low, under uniform or Gaussian perturbations, one can quickly find an approximate Nash equilibrium.
  • If Bimatrix is in smoothed polynomial time under uniform or Gaussian perturbations, for all ǫ > 0, there exists a randomized algorithm to compute an ǫ-approximate Nash equilibrium in a two-player game with expected time O poly(m, n, 1/ǫ) or O poly(m, n, log max(m, n)/ǫ) , respectively.
  • As every two-player game has a Nash equilibrium, this reduction implies that every generalized circuit with K nodes has a 1/K3-approximate solution.
Funding
  • We would like to thank everyone who asked about the smoothed complexity of the Lemke-Howson algorithm, especially John Reif for being the first player to ask us this question. Xi Chen’s work was supported by the Chinese National Key Foundation Plan (2003CB317807, 2004CB318108), the National Natural Science Foundation of China Grant 60553001 and the National Basic Research Program of China Grant (2007CB807900, 2007CB807901)
  • Xiaotie Deng’s work was supported by City University of Hong Kong for his research
  • Shang-Hua Teng’s work was supported by the NSF grants CCR-0311430 and ITR CCR-0325630
Reference
  • John Reif, Nicole Immorlica, Steve Vavasis, Christos Papadimitriou, Mohammad Mahdian, Ding-Zhu Du, Santosh Vempala, Aram Harrow, Adam Kalai, Imre Barany, Adrian Vetta, Jonathan Kelner and a number of other people asked whether the smoothed complexity of the Lemke-Howson algorithm or Nash Equilibria is polynomial, 2001–2005.
    Google ScholarFindings
  • T. Abbott, D. Kane, and P. Valiant. On the complexity of two-player win-lose games. In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 113–122, 2005.
    Google ScholarLocate open access versionFindings
  • K.J. Arrow and G. Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22:265–290, 1954.
    Google ScholarLocate open access versionFindings
  • I. Barany, S. Vempala, and A. Vetta. Nash equilibria in random games. In FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, pages 123–131, 2005.
    Google ScholarLocate open access versionFindings
  • L. Blum, M. Shub, and S. Smale. On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines. Bulletin of the AMS, 21(1):1–46, July 1989.
    Google ScholarLocate open access versionFindings
  • K.-H. Borgwardt. The average number of steps required by the simplex method is polynomial. Zeitschrift fur Operations Research, 26:157–177, 1982.
    Google ScholarLocate open access versionFindings
  • L.E.J. Brouwer. Uber Abbildung von Mannigfaltigkeiten. Mathematische Annalen, 71:97– 115, 1910.
    Google ScholarLocate open access versionFindings
  • X. Chen and X. Deng. On algorithms for discrete and approximate Brouwer fixed points. In STOC ’05: Proceedings of the 37th Annual ACM Symposium on Theory of computing, pages 323–330, 2005.
    Google ScholarLocate open access versionFindings
  • X. Chen and X. Deng. On the complexity of 2D discrete fixed point problem. In ICALP ’06: Proceedings of the 33rd International Colloquium on Automata, Languages and Programming, pages 489–500, 2006.
    Google ScholarLocate open access versionFindings
  • X. Chen and X. Deng. 3-Nash is PPAD-complete. In Electronic Colloquium in Computational Complexity, TR05-134, 2005.
    Google ScholarLocate open access versionFindings
  • X. Chen, X. Deng, and S.-H. Teng. Sparse games are hard. In Proceedings of the 2nd Workshop on Internet and Network Economics, pages 262–273, 2006.
    Google ScholarLocate open access versionFindings
  • X. Chen, L.-S. Huang, and S.-H. Teng. Market equilibria with hybrid linear-Leontief utilities. In Proceedings of the 2nd Workshop on Internet and Network Economics, pages 274–285, 2006.
    Google ScholarLocate open access versionFindings
  • X. Chen and S.-H. Teng. Paths beyond local search: A nearly tight bound for randomized fixed-point computation. arXiv, 2007. http://arxiv.org/abs/cs.GT/0702088.
    Locate open access versionFindings
  • X. Chen, S.-H. Teng, and P.A. Valiant. The approximation complexity of win-lose games. In SODA ’07: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 2007.
    Google ScholarLocate open access versionFindings
  • B. Codenotti, A. Saberi, K. Varadarajan, and Y. Ye. Leontief economies encode nonzero sum two-player games. In SODA ’06: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 659–667, 2006.
    Google ScholarLocate open access versionFindings
  • A. Condon, H. Edelsbrunner, E. Emerson, L. Fortnow, S. Haber, R. Karp, D. Leivant, R. Lipton, N. Lynch, I. Parberry, C. Papadimitriou, M. Rabin, A. Rosenberg, J. Royer, J. Savage, A. Selman, C. Smith, E. Tardos, and J. Vitter. Challenges for theory of computing: Report of an NSF-sponsored workshop on research in theoretical computer science. SIGACT News, 30(2):62–76, 1999.
    Google ScholarLocate open access versionFindings
  • V. Conitzer and T. Sandholm. Complexity results about nash equilibria. In In Proceedings of the International Joint Conference on Artificial Intelligence (IJCAI), 2003.
    Google ScholarLocate open access versionFindings
  • C. Daskalakis, P.W. Goldberg, and C.H. Papadimitriou. The complexity of computing a Nash equilibrium. In STOC ’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 71–78, 2006.
    Google ScholarLocate open access versionFindings
  • C. Daskalakis, A. Mehta, and C.H. Papadimitriou. A note on approximate Nash equilibria. In Proceedings of the 2nd Workshop on Internet and Network Economics, pages 297–306, 2006.
    Google ScholarLocate open access versionFindings
  • C. Daskalakis and C.H. Papadimitriou. Three-player games are hard. In Electronic Colloquium in Computational Complexity, TR05-139, 2005.
    Google ScholarLocate open access versionFindings
  • X. Deng, C. Papadimitriou, and S. Safra. On the complexity of price equilibria. Journal of Computer and System Sciences, 67(2):311–324, 2003.
    Google ScholarLocate open access versionFindings
  • T. Feder, H. Nazerzadeh, and A. Saberi. Approximating nash equilibria using small-support strategies. Stanford, 2006.
    Google ScholarFindings
  • K. Friedl, G. Ivanyos, M. Santha, and F. Verhoeven. On the black-box complexity of Sperner’s lemma. In Proceedings of the 15th International Symposium on Fundamentals of Computation Theory, pages 245–257, 2005.
    Google ScholarLocate open access versionFindings
  • I. Gilboa and E. Zemel. Nash and correlated equilibria: Some complexity considerations. Games and Economic Behavior, 1(1).
    Google ScholarLocate open access versionFindings
  • P.W. Goldberg and C.H. Papadimitriou. Reducibility among equilibrium problems. In STOC ’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pages 61–70, 2006.
    Google ScholarLocate open access versionFindings
  • M.D. Hirsch, C.H. Papadimitriou, and S. Vavasis. Exponential lower bounds for finding Brouwer fixed points. Journal of Complexity, 5:379–416, 1989.
    Google ScholarLocate open access versionFindings
  • C. A. Holt and A. E. Roth. The Nash equilibrium: A perspective. PNAS, 101(12):3999–4002, March 2004.
    Google ScholarLocate open access versionFindings
  • L.-S. Huang and S.-H. Teng. On the approximation and smoothed complexity of Leontief market equilibria. In Electronic Colloquium in Computational Complexity, TR06-031, 2006.
    Google ScholarLocate open access versionFindings
  • D. Johnson. The NP-completeness column: Finding needles in haystacks. ACM Transactions on Algorithms, (to appear), April 2007.
    Google ScholarLocate open access versionFindings
  • S. Kakutani. A generalization of Brouwer’s fixed point theorem. Duke Mathematical Journal, 8:457–459, 1941.
    Google ScholarLocate open access versionFindings
  • R. Kannan and T. Theobald. Games of fixed rank: A hierarchy of bimatrix games. In SODA ’07: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 2007.
    Google ScholarLocate open access versionFindings
  • N. Karmarkar. A new polynomial time algorithm for linear programming. Combinatorica, 4:373–395, 1984.
    Google ScholarLocate open access versionFindings
  • M. Kearns, M. Littman, and S. Singh. Graphical models for game theory. In Proceedings of the Conference on Uncertainty in Artificial Intelligence, pages 253–260, 2001.
    Google ScholarLocate open access versionFindings
  • L.G. Khachian. A polynomial algorithm in linear programming. Doklady Akademia Nauk, SSSR 244:1093–1096, English translation in Soviet Math. Dokl. 20, 191–194, 1979.
    Google ScholarLocate open access versionFindings
  • V. Klee and G.J. Minty. How good is the simplex algorithm? In O. Shisha, editor, Inequalities – III, pages 159–175. Academic Press, 1972.
    Google ScholarFindings
  • S. Kontogiannis, P. Panagopoulou, and P. Spirakis. Polynomial algorithms for approximating Nash equilibria of bimatrix games. In Proceedings of the 2nd Workshop on Internet and Network Economics, pages 286–296, 2006.
    Google ScholarLocate open access versionFindings
  • C.E. Lemke. Bimatrix equilibrium points and mathematical programming. Management Science, 11:681–689, 1965.
    Google ScholarLocate open access versionFindings
  • C.E. Lemke and J.T. Howson, Jr. Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics, 12:413–423, 1964.
    Google ScholarLocate open access versionFindings
  • R.J. Leonard. Reading Cournot, reading Nash: The creation and stabilisation of the Nash equilibrium. Economic Journal, 104(424):492–511, 1994.
    Google ScholarLocate open access versionFindings
  • R.J. Lipton, E. Markakis, and A. Mehta. Playing large games using simple strategies. In Proceedings of the 4th ACM conference on Electronic commerce, pages 36–41, 2004.
    Google ScholarLocate open access versionFindings
  • N. Megiddo. A note on the complexity of P-matrix LCP and computing an equilibrium. Research Report RJ6439, IBM Almaden Research Center, San Jose, 1988.
    Google ScholarLocate open access versionFindings
  • N. Megiddo and C.H. Papadimitriou. On total functions, existence theorems and computational complexity. Theoretical Computer Science, 81:317–324, 1991.
    Google ScholarLocate open access versionFindings
  • O. Morgenstern and J. von Neumann. Theory of Games and Economic Behavior. Princeton University Press, 1947.
    Google ScholarFindings
  • J. Nash. Equilibrium point in n-person games. Porceedings of the National Academy of the USA, 36(1):48–49, 1950.
    Google ScholarLocate open access versionFindings
  • J. Nash. Noncooperative games. Annals of Mathematics, 54:289–295, 1951.
    Google ScholarLocate open access versionFindings
  • C.H. Papadimitriou. On inefficient proofs of existence and complexity classes. In Proceedings of the 4th Czechoslovakian Symposium on Combinatorics, 1991.
    Google ScholarLocate open access versionFindings
  • C.H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences, pages 498–532, 1994.
    Google ScholarLocate open access versionFindings
  • C.H. Papadimitriou. Algorithms, games, and the internet. In STOC ’01: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 749–753, 2001.
    Google ScholarLocate open access versionFindings
  • T. Sandholm. Issues in computational vickrey auctions. International Journal of Electronic Commerce, 4(3):107 – 129, March 2000.
    Google ScholarLocate open access versionFindings
  • R. Savani and B. von Stengel. Exponentially many steps for finding a Nash equilibrium in a bimatrix game. In FOCS ’04: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 258–267, 2004.
    Google ScholarLocate open access versionFindings
  • H. Scarf. The approximation of fixed points of a continuous mapping. SIAM Journal on Applied Mathematics, 15:997–1007, 1967.
    Google ScholarLocate open access versionFindings
  • H. Scarf. On the computation of equilibrium prices. In W. Fellner, editor, Ten Economic Studies in the Tradition of Irving Fisher. New York: John Wiley & Sons, 1967.
    Google ScholarLocate open access versionFindings
  • E. Sperner. Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes. Abhandlungen aus dem Mathematischen Seminar Universitat Hamburg, 6:265–272, 1928.
    Google ScholarLocate open access versionFindings
  • D.A. Spielman and S.-H. Teng. Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time. Journal of the ACM, 51(3):385–463, 2004, also in STOC ’01: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing.
    Google ScholarLocate open access versionFindings
  • D.A. Spielman and S.-H. Teng. Smoothed analysis of algorithms and heuristics: Progress and open questions. In L. Pardo, A. Pinkus, E. Suli and M.J. Todd, editor, Foundations of Computational Mathematics, pages 274–342. Cambridge University Press, 2006.
    Google ScholarLocate open access versionFindings
  • J. von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen, 100:295–320, 1928.
    Google ScholarLocate open access versionFindings
  • R. Wilson. Computing equilibria of n-person games. SIAM Journal on Applied Mathematics, 21:80–87, 1971.
    Google ScholarLocate open access versionFindings
  • Y. Ye. Exchange market equilibria with Leontief’s utility: Freedom of pricing leads to rationality. In Proceedings of the 1st Workshop on Internet and Network Economics, pages 14–23, 2005.
    Google ScholarLocate open access versionFindings
Author
Your rating :
0

 

Tags
Comments
数据免责声明
页面数据均来自互联网公开来源、合作出版商和通过AI技术自动分析结果,我们不对页面数据的有效性、准确性、正确性、可靠性、完整性和及时性做出任何承诺和保证。若有疑问,可以通过电子邮件方式联系我们:report@aminer.cn
小科