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An impossibility theorem in game dynamics.

Jason Milionis1, Christos Papadimitriou1, Georgios Piliouras2

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Proceedings of the National Academy of Sciences of the United States of America
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Summary
This summary is machine-generated.

Game theory's Nash equilibrium (NE) is not always reachable. This study proves that deterministic player behaviors cannot guarantee convergence to NE from all starting points in repeated games, showing limitations in predicting long-term behavior.

Keywords:
Nash equilibriumdynamical systemsgame theorysolution concept

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Area of Science:

  • Game Theory
  • Dynamical Systems Theory
  • Computational Complexity

Background:

  • The Nash equilibrium (NE) is a core concept in game theory, representing a stable outcome where no player benefits from unilaterally changing their strategy.
  • Despite the universal existence of NE, it remains unknown if deterministic player strategies can guarantee convergence to NE from any initial state in repeated games.

Purpose of the Study:

  • To investigate whether deterministic game dynamics can always converge to a Nash equilibrium from all starting points.
  • To explore the predictability of Nash equilibria and their approximations in repeated games using dynamical systems theory.

Main Methods:

  • Application of dynamical systems theory, specifically Conley index theory, to analyze the convergence properties of game dynamics.
  • Development of impossibility results for convergence to both exact and approximate Nash equilibria.

Main Results:

  • Proved a general impossibility result: certain games exist where all deterministic dynamics fail to converge to a Nash equilibrium from all initial conditions.
  • Demonstrated a stronger impossibility result for approximate Nash equilibria, showing that for a significant set of games, convergence to approximations is not guaranteed for substantial bounds.
  • Identified degenerate games as key to the impossibility findings, with conjectures extending the results to non-degenerate games under computational complexity assumptions.

Conclusions:

  • The concepts of Nash equilibrium and its approximations, while broadly applicable, are fundamentally incomplete predictors of long-term player behavior in repeated games.
  • Deterministic dynamics do not universally guarantee convergence to Nash equilibria, highlighting limitations in theoretical game dynamics and predictive modeling.