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Settling the complexity of computing two-player Nash equilibria
Open accessAbstract
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. Our result, building upon the work of Daskalakis et al. [2006a] on the complexity of four-player Nash equilibria, settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems: —Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. —The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: —Arrow-Debreu market equilibria are PPAD -hard to compute.
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Cites
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Single and multi-agent deep reinforcement learning for AI-enabled wireless networks: A tutorial
Feriani, A., & Hossain, E. (2021) · IEEE Communications Surveys & Tutorials · doi:10.1109/comst.2021.3063822 · Cited by 410
Conitzer, V., & Oesterheld, C. (2023) · Proceedings of the AAAI Conference on Artificial Intelligence · doi:10.1609/aaai.v37i13.26791 · Cited by 19
Metadata
- Authors
- Chen, X., Deng, X., & Teng, S.-H.
- Venue
- Journal of the ACM
- Published
- 2009-05
- Volume
- 56
- Issue
- 3
- Pages
- 1-57
- ISSN
- 0004-5411 (ISSN-L)
- 1557-735X
- OpenAlex
- W2057913812
- Cited by
- 560
- References
- 38
- FWCI
- 24.14
- Type
- journal article
- Language
- en
- Open access
- green
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