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Best-response dynamics, playing sequences, and convergence to equilibrium in random games.

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The best-response dynamic converges to Nash equilibrium in most games when players choose actions randomly. However, convergence is rare with a fixed, cyclic order of play.

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Area of Science:

  • Game Theory
  • Computational Economics
  • Mathematical Economics

Background:

  • The best-response dynamic is a fundamental concept in game theory for analyzing strategic interactions.
  • Understanding convergence properties is crucial for predicting outcomes in normal-form games.
  • The impact of player action-updating sequences on convergence remains an active area of research.

Purpose of the Study:

  • To investigate the performance of the best-response dynamic across all normal-form games.
  • To determine the influence of playing sequence on convergence to Nash equilibrium.
  • To compare convergence rates under fixed cyclic versus random playing sequences.

Main Methods:

  • Analysis using a random games approach.
  • Asymptotic analysis of game dynamics.
  • Evaluation across all possible normal-form games.

Main Results:

  • Convergence to a pure Nash equilibrium is extremely sensitive to the playing sequence.
  • With a fixed cyclic playing order, the best-response dynamic converges in a vanishingly small fraction of large games.
  • With a random playing sequence, the dynamic converges to a pure Nash equilibrium in almost all large games where one exists.

Conclusions:

  • Randomized playing sequences significantly enhance the convergence of the best-response dynamic to Nash equilibria.
  • The choice of playing sequence is a critical factor in the stability and predictability of game dynamics.
  • These findings have implications for understanding emergent behavior in complex strategic environments.