Gradient-Based Algorithms for Finding Nash Equilibria in Extensive Form Games

  • Andrew Gilpin
  • Samid Hoda
  • Javier Peña
  • Tuomas Sandholm
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4858)


We present a computational approach to the saddle-point formulation for the Nash equilibria of two-person, zero-sum sequential games of imperfect information. The algorithm is a first-order gradient method based on modern smoothing techniques for non-smooth convex optimization. The algorithm requires O(1/ε) iterations to compute an ε-equilibrium, and the work per iteration is extremely low. These features enable us to find approximate Nash equilibria for sequential games with a tree representation of about 1010 nodes. This is three orders of magnitude larger than what previous algorithms can handle. We present two heuristic improvements to the basic algorithm and demonstrate their efficacy on a range of real-world games. Furthermore, we demonstrate how the algorithm can be customized to a specific class of problems with enormous memory savings.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Andrew Gilpin
    • 1
  • Samid Hoda
    • 2
  • Javier Peña
    • 2
  • Tuomas Sandholm
    • 1
  1. 1.Computer Science Department, Carnegie Mellon University 
  2. 2.Tepper School of Business, Carnegie Mellon University 

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