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Structure theorems for game trees.

Srihari Govindan1, Robert Wilson

  • 1Economics Department, University of Western Ontario, London, ON, Canada N6A 5C2.

Proceedings of the National Academy of Sciences of the United States of America
|June 13, 2002
PubMed
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This study explores equilibrium outcome correspondences in extensive-form games. Researchers prove an analog to previous findings by perturbing strategy spaces, enabling computation of approximate equilibria.

Area of Science:

  • Game Theory
  • Mathematical Economics

Background:

  • The Nash equilibrium correspondence in normal-form games has a graph homeomorphic to its payoff space domain.
  • The analogous property does not hold for equilibrium outcome correspondences in extensive-form games.

Purpose of the Study:

  • To investigate the topological properties of equilibrium outcome correspondences in extensive-form games.
  • To develop an algorithm for computing equilibria in perturbed extensive-form games.

Main Methods:

  • Perturbing the space of behavior strategies in extensive-form games to ensure nonzero path probabilities.
  • Analyzing the structure of the equilibrium outcome correspondence graph under these perturbations.
  • Applying the findings to construct an equilibrium computation algorithm.

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Main Results:

  • A counterexample shows the non-analogous behavior for extensive-form games without perturbations.
  • An analog is proven for perturbed behavior strategy spaces, where every path has nonzero probability.
  • The graph of the unperturbed correspondence is the closure of a finite union of subsets, each diffeomorphic to an open subset of the payoff space.

Conclusions:

  • The study provides a theoretical framework and computational method for approximating equilibria in extensive-form games.
  • Perturbation techniques are crucial for understanding the topological structure of equilibrium correspondences.
  • The developed algorithm offers a practical application for computing game equilibria.