Computing pure Bayesian-Nash equilibria in games with finite actions and continuous types

  • Authors:

  • Zinovi Rabinovich;Victor Naroditskiy;Enrico H. Gerding;Nicholas R. Jennings

  • Affiliations:

  • Department of Computer Science, Bar-Ilan University, Ramat Gan, 52900, Israel and Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, United Kingdom;Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, United Kingdom;Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, United Kingdom;Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, United Kingdom and Department of Computing and Information Technology, King Abdulaziz University, Saudi Arabia

  • Venue:
  • Artificial Intelligence
  • Year:
  • 2013

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Abstract

We extend the well-known fictitious play (FP) algorithm to compute pure-strategy Bayesian-Nash equilibria in private-value games of incomplete information with finite actions and continuous types (G-FACTs). We prove that, if the frequency distribution of actions (fictitious play beliefs) converges, then there exists a pure-strategy equilibrium strategy that is consistent with it. We furthermore develop an algorithm to convert the converged distribution of actions into an equilibrium strategy for a wide class of games where utility functions are linear in type. This algorithm can also be used to compute pure @e-Nash equilibria when distributions are not fully converged. We then apply our algorithm to find equilibria in an important and previously unsolved game: simultaneous sealed-bid, second-price auctions where various types of items (e.g., substitutes or complements) are sold. Finally, we provide an analytical characterisation of equilibria in games with linear utilities. Specifically, we show how equilibria can be found by solving a system of polynomial equations. For a special case of simultaneous auctions, we also solve the equations confirming the results obtained numerically.