A continuation method for Nash equilibria in structured games

  • Authors:

  • Ben Blum;Christian R. Shelton;Daphne Koller

  • Affiliations:

  • Department of Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA;Department of Computer Science and Engineering, University of California, Riverside, Riverside, CA;Department of Computer Science, Stanford University, Stanford, CA

  • Venue:
  • Journal of Artificial Intelligence Research
  • Year:
  • 2006

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Abstract

Structured game representations have recently attracted interest as models for multi-agent artificial intelligence scenarios, with rational behavior most commonly characterized by Nash equilibria. This paper presents efficient, exact algorithms for computing Nash equilibria in structured game representations, including both graphical games and multi-agent influence diagrams (MAIDs). The algorithms are derived from a continuation method for normal-form and extensive-form games due to Govindan and Wilson; they follow a trajectory through a space of perturbed games and their equilibria, exploiting game structure through fast computation of the Jacobian of the payoff function. They are theoretically guaranteed to find at least one equilibrium of the game, and may find more. Our approach provides the first efficient algorithm for computing exact equilibria in graphical games with arbitrary topology, and the first algorithm to exploit fine-grained structural properties of MAIDs. Experimental results are presented demonstrating the effectiveness of the algorithms and comparing them to predecessors. The running time of the graphical game algorithm is similar to, and often better than, the running time of previous approximate algorithms. The algorithm for MAIDs can effectively solve games that are much larger than those solvable by previous methods.