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Nonlinear dynamics in combinatorial games: renormalizing Chomp.

Eric J Friedman1, Adam Scott Landsberg

  • 1School of ORIE, Cornell University, Ithaca, New York 14853, USA.

Chaos (Woodbury, N.Y.)
|July 7, 2007
PubMed
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We introduce a novel method linking combinatorial games to nonlinear dynamics. This approach analyzes the geometric growth in games like Chomp, offering new insights into game theory and dynamical systems.

Area of Science:

  • Mathematics
  • Physics
  • Game Theory

Background:

  • Combinatorial games like Chomp, Chess, Checkers, and Go remain largely unsolved.
  • Understanding complex dynamics and chaos is central to nonlinear dynamics.

Purpose of the Study:

  • To develop a new approach connecting combinatorial games with nonlinear dynamics.
  • To analyze the underlying geometric structure and growth in the game of Chomp.

Main Methods:

  • Adapted a renormalization procedure from physics to analyze game growth.
  • Transformed a combinatorial game into a dynamical system.

Main Results:

  • Revealed connections between combinatorial games and nonlinear dynamics concepts (scaling, chaos, universality).

Related Experiment Videos

  • Provided a complete probabilistic description of optimal play in Chomp.
  • Answered a longstanding question about Chomp's winning opening move.

    • Conclusions:

    • The developed methodology offers a mathematical framework for studying combinatorial games.
    • This approach bridges the gap between combinatorial game theory and modern dynamical systems theory.