A Note on Approximate Nash Equilibria

  • Constantinos Daskalakis
  • Aranyak Mehta
  • Christos Papadimitriou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4286)


In view of the intractability of finding a Nash equilibrium, it is important to understand the limits of approximation in this context. A subexponential approximation scheme is known [LMM03], and no approximation better than \(1\over 4\) is possible by any algorithm that examines equilibria involving fewer than logn strategies [Alt94]. We give a simple, linear-time algorithm examining just two strategies per player and resulting in a \(1\over 2\)-approximate Nash equilibrium in any 2-player game. For the more demanding notion of well-supported approximate equilibrium due to [DGP06] no nontrivial bound is known; we show that the problem can be reduced to the case of win-lose games (games with all utilities 0–1), and that an approximation of \(5\over 6\) is possible contingent upon a graph-theoretic conjecture.


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  1. [Alt94]
    Althofer, I.: On sparse approximations to randomized strategies and convex combinations. Linear Algebra and its Applications 199 (1994)Google Scholar
  2. [CH78]
    Caccetta, L., Haggkvist, R.: On minimal digraphs with given girth. Congressus Numerantium XXI (1978)Google Scholar
  3. [Cha05]
    Charbit, P.: Circuits in graphs and digraphs via embeddings. Doctoral dissertation, University of Lyon I (2005)Google Scholar
  4. [CD05]
    Chen, X., Deng, X.: Settling the complexity of two-player nash equilibrium. In: FOCS (2006)Google Scholar
  5. [CDT06]
    Chen, X., Deng, X., Teng, S.-H.: Computing nash equilibria:approximation and smoothed complexity. In: ECCC (2006)Google Scholar
  6. [DGP06]
    Daskalakis, C., Goldberg, P., Papadimitriou, C.: The complexity of computing a nash equilibrium. In: STOC (2006)Google Scholar
  7. [GP06]
    Goldberg, P., Papadimitriou, C.: Reducibility among equilibrium problems. In: STOC (2006)Google Scholar
  8. [LMM03]
    Lipton, R., Markakis, E., Mehta, A.: Playing large games using simple strategies. ACM Electronic Commerce (2003)Google Scholar
  9. [Mye03]
    Myers, J.S.: Extremal theory of graph minors and directed graphs. Doctoral dissertation (2003)Google Scholar
  10. [KPS06]
    Kontogiannis, S.C., Panagopoulou, P.N., Spirakis, P.G.: WINE 2006. LNCS, vol. 4286. Springer, Heidelberg (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Constantinos Daskalakis
    • 1
  • Aranyak Mehta
    • 2
  • Christos Papadimitriou
    • 1
  1. 1.University of CaliforniaBerkeleyUSA
  2. 2.IBM Almaden Research CenterSan JoseUSA

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