Constant rank bimatrix games are PPAD-hard

Research output: Chapter in Book/Report/Conference proceedingConference contribution


The rank of a bimatrix game (A,B) is defined as rank(A + B). Computing a Nash equilibrium (NE) of a rank-0, i.e., zero-sum game is equivalent to linear programming (von Neumann'28, Dantzig'51). In 2005, Kannan and Theobald gave an FPTAS for constant rank games, and asked if there exists a polynomial time algorithm to compute an exact NE. Adsul et. al. (2011) answered this question affirmatively for rank-1 games, leaving rank-2 and beyond unresolved. In this paper we show that NE computation in games with rank ≥ 3, is PPAD-hard, settling a decade long open problem. Interestingly, this is the first instance that a problem with an FPTAS turns out to be PPAD-hard. Our reduction bypasses graphical games and game gadgets, and provides a simpler proof of PPAD-hardness for NE computation in bimatrix games. In addition, we get: • An equivalence between 2D-Linear-FIXP and PPAD, improving a result by Etessami and Yannakakis (2007) on equivalence between Linear-FIXP and PPAD. • NE computation in a bimatrix game with convex set of Nash equilibria is as hard as solving a simple stochastic game [12]. • Computing a symmetric NE of a symmetric bimatrix game with rank ≥ 6 is PPAD-hard. • Computing a 1/poly(n) -approximate fixed-point of a (Linear- FIXP) piecewise-linear function is PPAD-hard. The status of rank-2 games remains unresolved.

Original languageEnglish (US)
Title of host publicationSTOC 2014 - Proceedings of the 2014 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)9781450327107
StatePublished - 2014
Externally publishedYes
Event4th Annual ACM Symposium on Theory of Computing, STOC 2014 - New York, NY, United States
Duration: May 31 2014Jun 3 2014

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Other4th Annual ACM Symposium on Theory of Computing, STOC 2014
Country/TerritoryUnited States
CityNew York, NY


  • 2D-Brouwer
  • Constant rank
  • LCP
  • Linear-FIXP
  • PPAD-hard

ASJC Scopus subject areas

  • Software


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