On Nash-Equilibria of Approximation-Stable Games

  • Pranjal Awasthi
  • Maria-Florina Balcan
  • Avrim Blum
  • Or Sheffet
  • Santosh Vempala
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6386)


One reason for wanting to compute an (approximate) Nash equilibrium of a game is to predict how players will play. However, if the game has multiple equilibria that are far apart, or ε-equilibria that are far in variation distance from the true Nash equilibrium strategies, then this prediction may not be possible even in principle. Motivated by this consideration, in this paper we define the notion of games that are approximation stable, meaning that all ε-approximate equilibria are contained inside a small ball of radius Δ around a true equilibrium, and investigate a number of their properties. Many natural small games such as matching pennies and rock-paper-scissors are indeed approximation stable. We show furthermore there exist 2-player n-by-n approximation-stable games in which the Nash equilibrium and all approximate equilibria have support Ω(log n). On the other hand, we show all (ε,Δ) approximation-stable games must have an ε-equilibrium of support \(O(\frac{\Delta^{2-o(1)}}{\epsilon^{2}}{\rm log n})\), yielding an immediate \(n^{O(\frac{\Delta^{2-o(1)}}{\epsilon^2}log n)}\)-time algorithm, improving over the bound of [11] for games satisfying this condition. We in addition give a polynomial-time algorithm for the case that Δ and ε are sufficiently close together. We also consider an inverse property, namely that all non-approximate equilibria are far from some true equilibrium, and give an efficient algorithm for games satisfying that condition.


Nash Equilibrium Approximation Stability General Game True Equilibrium Disjoint Support 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Pranjal Awasthi
    • 1
  • Maria-Florina Balcan
    • 2
  • Avrim Blum
    • 1
  • Or Sheffet
    • 1
  • Santosh Vempala
    • 2
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.Georgia Institute of TechnologyAtlantaUSA

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