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Nonatomic games on Loeb spaces.

M A Khan1, Y Sun

  • 1Department of Economics, Johns Hopkins University, Baltimore, MD 21218.

Proceedings of the National Academy of Sciences of the United States of America
|December 24, 1996
PubMed
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This study introduces hyperfinite Loeb spaces as a novel framework for nonatomic games, ensuring the existence of pure-strategy Nash equilibria even with uncountable action sets. This advances noncooperative game theory by resolving prior existence issues in nonatomic settings.

Area of Science:

  • Game Theory
  • Mathematical Economics
  • Measure Theory

Background:

  • Noncooperative game theory traditionally models agent negligibility using nonatomic measure spaces like the unit interval with Lebesgue measure.
  • Existing models face challenges, as pure-strategy Nash equilibria do not exist in nonatomic games with uncountable action sets.

Purpose of the Study:

  • To establish a satisfactory existence theory for pure-strategy Nash equilibria in nonatomic games.
  • To introduce hyperfinite Loeb spaces as a suitable mathematical framework for modeling these games.
  • To present a unified treatment for large games and games with incomplete information.

Main Methods:

  • Formulation of nonatomic games on hyperfinite Loeb spaces.
  • Demonstration of the existence of pure-strategy Nash equilibria within this new framework.

Related Experiment Videos

  • Exploration of desirable properties of games analyzed using hyperfinite Loeb spaces.

    • Main Results:

    • Successfully established a robust existence theory for pure-strategy Nash equilibria in nonatomic games.
    • Demonstrated that hyperfinite Loeb spaces resolve the non-existence problem associated with uncountable action sets.
    • Provided a synthetic treatment applicable to both large games and games with incomplete information.

    Conclusions:

    • Hyperfinite Loeb spaces offer a theoretically sound and practically applicable foundation for nonatomic game theory.
    • This approach overcomes limitations of previous models, enabling a broader understanding of strategic interactions.
    • The framework facilitates unified analysis of diverse game structures, including large and incomplete information settings.