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# Settling the Complexity of Two-Player Nash Equilibrium

FOCS, pp.261-272, (2006)

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Abstract

Even though many people thought the problem of finding Nash equilibria is hard in general, and it has been proven so for games among three or more players recently, it's not clear whether the two-player case can be shown in the same class of PPAD-complete problems. We prove that the problem of finding a Nash equilibrium in a two-player ga...More

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Introduction

- The classical lemma of Sperner [11], which is the combinatorial characterization behind Brouwer’s fixed point theorem, states that any admissible 3-coloring of any triangulation of a triangle has a trichromatic triangle
- It defines a search problem 2D-SPERNER of finding such a triangle in an admissible 3-coloring for an exponential size triangulation, typical of problems in PPAD, a complexity class introduced by Papadimitriou to characterize mathematical structures with the path-following proof technique [10].
- Its 3-dimensional analogue 3D-SPERNER is the first natural problem proved

Highlights

- The classical lemma of Sperner [11], which is the combinatorial characterization behind Brouwer’s fixed point theorem, states that any admissible 3-coloring of any triangulation of a triangle has a trichromatic triangle
- In examination into the PPAD-completeness proof of problem 3D-SPERNER, we found that the main idea is to embed complete graphs in 3-dimensional search spaces [10]
- Edges in the complete graph correspond to independent paths which are composed of neighboring triangles or tetrahedrons in the standard subdivision
- Such an embedding is obviously impossible in the plane, as complete graphs with order no less than 5 are not planar
- Our new proof techniques may provide helpful insight into the study of other related problems : Can we show more problems complete for PPA and PPAD? For example, is 2D-TUCKER [10] PPAD-complete? Can we find a natural complete problem for either PPA or PPAD that doesn’t have an explicit Turing machine in the input? For example, is SMITH [10] PPA-complete? and most importantly, what is the relationship between complexity classes PPA, PPAD and PPADS?

Conclusion

- All the PPAD-completeness proofs of Sperner’s problems before rely heavily on embeddings of complete graphs in the standard subdivisions.
- Edges in the complete graph correspond to independent paths which are composed of neighboring triangles or tetrahedrons in the standard subdivision.
- Such an embedding is obviously impossible in the plane, as complete graphs with order no less than 5 are not planar.
- The authors' new proof techniques may provide helpful insight into the study of other related problems : Can the authors show more problems complete for PPA and PPAD? For example, is 2D-TUCKER [10] PPAD-complete? Can the authors find a natural complete problem for either PPA or PPAD that doesn’t have an explicit Turing machine in the input? For example, is SMITH [10] PPA-complete? and most importantly, what is the relationship between complexity classes PPA, PPAD and PPADS?

Reference

- X. Chen and X. Deng. On Algorithms for Discrete and Approximate Brouwer Fixed Points. In STOC 2005, pages 323–330.
- X. Chen and X. Deng. On the Complexity of 2D Discrete Fixed Point Problem. ECCC, TR06-037, 2006.
- X. Chen, X. Deng, and S.-H. Teng. Computing Nash equilibria: approximation and smoothed complexity.. ECCC, TR06-023, 2006.
- C. Daskalakis, P.W. Goldberg, and C.H. Papadimitriou. The Complexity of Computing a Nash Equilibrium. STOC, 2006.
- K. Friedl, G. Ivanyos, M. Santha, and Y. Verhoeven. Locally 2-dimensional Sperner problems complete for the Polynomial Parity Argument classes. submitted.
- K. Friedl, G. Ivanyos, M. Santha, and Y. Verhoeven. On the complexity of Sperner’s Lemma. Research report NI05002-QIS at the Isaac Newton Institute for Mathematical studies.
- M. Grigni. A Sperner lemma complete for PPA. Inform. Process. Lett., 77(56):255–259, 2001.
- M.D. Hirsch, C. Papadimitriou and S. Vavasis. Exponential lower bounds for finding Brouwer fixed points. J.Complexity, 5:379–416, 1989.
- C.H. Papadimitriou. On graph-theoretic lemmata and complexity classes. In In Proceedings 31st Annual Symposium on Foundations of Computer Science, pages 794–801, 1990.
- C.H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. JCSS, pages 498–532, 1994.
- E. Sperner. Neuer Beweis fur die Invarianz der Dimensionszahl und des Gebietes. Abhandlungen aus dem Mathematischen Seminar Universitat Hamburg, 6:265–272, 1928.

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