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# The Complexity of Computing a Nash Equilibrium

Electronic Colloquium on Computational Complexity, no. 1 (2009): 195-259

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摘要

In 1951, John F. Nash proved that every game has a Nash equilibrium [Ann. of Math. (2), 54 (1951), pp. 286-295]. His proof is nonconstructive, relying on Brouwer's fixed point theorem, thus leaving open the questions, Is there a polynomial-time algorithm for computing Nash equilibria? And is this reliance on Brouwer inherent? Many algorit...更多

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简介

- In a recent CACM article, Shoham [23] reminds them of the long relationship between Game Theory and Computer Science, going back to John von Neumann at Princeton in the 1940s, and how this connection became stronger and more crucial in the past decade due to the advent of the Internet: Strategic behavior became relevant to the design of computer systems, while much economic activity takes place on computational platforms.

Game Theory is about the strategic behavior of rational agents. - Three other well-known games, called chicken, prisoner’s dilemma, and penalty shot game, respectively, are shown in Figure 2; the penalty shot game is zero-sum, but the other two are not.
- All these games have two players; Game Theory studies games with many players, but these are harder to display.1 rock paper scissors rock (0, 0) (1, −1) (−1, 1)

重点内容

- In a recent CACM article, Shoham [23] reminds us of the long relationship between Game Theory and Computer Science, going back to John von Neumann at Princeton in the 1940s, and how this connection became stronger and more crucial in the past decade due to the advent of the Internet: Strategic behavior became relevant to the design of computer systems, while much economic activity takes place on computational platforms.

Game Theory is about the strategic behavior of rational agents - If computing a particular kind of equilibrium is an intractable problem, of the kind that take lifetimes of the universe to solve on the world’s fastest computers, it is ludicrous to expect that it can be arrived at in real life. This consideration suggests the following important question: Is there an efficient algorithm for computing a mixed Nash equilibrium? In this article, we report on results that indicate that the answer is negative—our own work [7, 8, 9, 14] obtained this for games with 3 or more players, and shortly afterwards, the papers [3, 4] extended this—unexpectedly—to the important case of 2-player games
- To show that Nash is complete for PPAD, we show how to convert an end of the line graph into a corresponding game, so that from an approximate Nash equilibrium of the game we can efficiently construct a corresponding end of the line
- That is, is there an algorithm for -Nash equilibria which runs in time polynomial in the game size, if we allow arbitrary dependence of its running time on 1/ ? Such an algorithm would go a long way towards alleviating the negative implications of our complexity result
- Polynomial-time approximation scheme has been given in [6], but the complexity of the problem is unknown for more general low-degree graphs

结论

- The authors' hardness result for computing a Nash equilibrium raises concerns about the credibility of the mixed Nash equilibrium as a general-purpose framework for behavior prediction
- In view of these concerns, the main question that emerges is whether there exists a polynomial-time approximation scheme (PTAS) for computing approximate Nash equilibria.
- PTAS has been given in [6], but the complexity of the problem is unknown for more general low-degree graphs
- Another positive recent development [6] has been a PTAS for a broad and important class of games, called anonymous

引用论文

- V. Bubelis. “On Equilibria in Finite Games,” International Journal on Game Theory, 8(2):65–79 (1979).
- X. Chen and X. Deng. “3-NASH is PPAD-Complete,” Electronic Colloquium in Computational Complexity, TR05-134, 2005.
- X. Chen and X. Deng. “Settling the Complexity of 2-Player Nash-Equilibrium,” Proceedings of FOCS, 2006.
- X. Chen, X. Deng and S. Teng. “Computing Nash Equilibria: Approximation and Smoothed Complexity,” Proceedings of FOCS, 2006.
- V. Conitzer and T. Sandholm. “Complexity Results about Nash Equilibria,” Proceedings of IJCAI, 2003.
- C. Daskalakis and C. H. Papadimitriou. “Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games,” Proceedings of FOCS, 2008.
- C. Daskalakis, P. W. Goldberg and C. H. Papadimitriou. “The Complexity of Computing a Nash Equilibrium,” Proceedings of STOC, 2006.
- C. Daskalakis, P. W. Goldberg and C. H.
- C. Daskalakis and C. H. Papadimitriou. “Three-Player Games Are Hard,” Electronic Colloquium in Computational Complexity, TR05-139, 2005.
- K. Etessami and M. Yannakakis. “On the Complexity of Nash Equilibria and Other Fixed Points (Extended Abstract),” Proceedings of FOCS, 113-123, 2007.
- E. Friedman and S. Shenker. “Learning and Implementation on the Internet,” Rutgers University, Department of Economics, 1997.
- M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the theory of NP-Completeness. Freeman, 1979.
- I. Gilboa and E. Zemel. “Nash and Correlated Equilibria: Some Complexity Considerations,” Games and Economic Behavior, 1989.
- P. W. Goldberg and C. H. Papadimitriou. “Reducibility Among Equilibrium Problems,” Proceedings of STOC, 2006.
- S. Hart and Y. Mansour. “How Long to Equilibrium? The Communication Complexity of Uncoupled Equilibrium Procedures” Proceedings of STOC, 2007.
- M. Kearns, M. Littman and S. Singh. “Graphical Models for Game Theory,” Proceedings of UAI, 2001.
- B. Knaster, C. Kuratowski and S. Mazurkiewicz, “Ein Beweis des Fixpunktsatzes fur n-dimensionale Simplexe,” Fundamenta Mathematicae, 14: 132–137, 1929.
- C. E. Lemke and J. T. Howson, Jr. “Equilibrium Points of Bimatrix Games,” SIAM Journal of Applied Mathematics, 12: 413–423, 1964.
- R. Lipton, E. Markakis and A. Mehta. “Playing Large Games Using Simple Strategies,” Proceedings of the ACM Conference on Electronic Commerce, 2003.
- J. Nash. “Noncooperative Games,” Annals of Mathematics, 54: 289-295, 1951.
- C. H. Papadimitriou. “On the Complexity of the Parity Argument and Other Inefficient Proofs of Existence,” Journal of Computer and System Sciences, 48(3): 498–532, 1994.
- H. E. Scarf. “The Approximation of Fixed Points of a Continuous Mapping,” SIAM Journal of Applied Mathematics, 15: 1328–1343, 1967.

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