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Computing Nash Equilibria: Approximation and Smoothed Complexity

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Xi Chen, Xiaotie Deng, Shang-Hua Teng

Computational Complexity · Submitted 11 Feb 2006

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P.

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arXiv IDcs/0602043 (version 2)
Pith IDDZNWDR6T, Pith's permanent record of this paper and its review
CitationShang-Hua Teng, Xiaotie Deng, Xi Chen. "Computing Nash Equilibria: Approximation and Smoothed Complexity." arXiv:cs/0602043 (2006). https://pith.science/paper/DZNWDR6T
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What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.62 · human participants 0.02 · controlled comparison 0.01 · randomized assignment 0.02 · states quantitative result 0.80 · compares with prior results 0.20 · mentions code or data availability 0.02.

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arXiv listing abstract v2. Results and the main claim are underlined; .

We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. In other words, no algorithm with time polynomial in n and 1/\epsilon can compute an \epsilon-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. Furthermore, we prove, unless PPAD is in RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/\sigma under perturbations with magnitude \sigma. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. result 0.97 0.93 0.77 asserted 1.00
s1 In other words, no algorithm with time polynomial in n and 1/\epsilon can compute an \epsilon-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. interpretation 0.56 0.70 0.61 asserted 0.99
s2 Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. method 0.58 0.95 0.06 asserted 1.00
s3 Furthermore, we prove, unless PPAD is in RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/\sigma under perturbations with magnitude \sigma. result 0.98 0.92 0.18 asserted 1.00
s4 Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria. interpretation 0.98 0.65 0.13 asserted 0.99

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We show that the BIMATRIX game does not have a fully polynomial-time approximation scheme, unless PPAD is in P. In other words, no algorithm with time polynomial in n and 1/\epsilon can compute an \epsilon-approximate Nash equilibrium of an n by nbimatrix game, unless PPAD is in P. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional cube with a constant side-length, such as on an n-dimensional cube with side-length 7, and show that they are PPAD-complete. Furthermore, we prove, unless PPAD is in RP, that the smoothed complexity of the Lemke-Howson algorithm or any algorithm for computing a Nash equilibrium of a bimatrix game is polynomial in n and 1/\sigma under perturbations with magnitude \sigma. Our result answers a major open question in the smoothed analysis of algorithms and the approximation of Nash equilibria.

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