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Combinatorial games with a pass: a dynamical systems approach.

Rebecca E Morrison1, Eric J Friedman, Adam S Landsberg

  • 1Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, USA. rebeccam@ices.utexas.edu

Chaos (Woodbury, N.Y.)
|January 10, 2012
PubMed
Summary
This summary is machine-generated.

Introducing a pass move into combinatorial games like Nim and Chomp significantly alters game complexity. This study analyzes these changes using dynamical systems, revealing how passes impact game structure and behavior.

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Area of Science:

  • Combinatorial Game Theory
  • Dynamical Systems Theory
  • Computational Complexity

Background:

  • The impact of a "pass" move on combinatorial games has been a long-standing open question.
  • Understanding game behavior changes is crucial for analyzing game complexity and strategy.

Purpose of the Study:

  • To investigate how the introduction of a pass move affects the behavior and structure of combinatorial games.
  • To provide a theoretical framework for predicting the impact of pass moves.

Main Methods:

  • Modeling combinatorial games as dynamical systems using recursion relations.
  • Analyzing two specific games: 3-pile Nim and 3-row Chomp.
  • Employing numerical stability analysis to understand game behavior.

Main Results:

  • The pass move dramatically increases the complexity of 3-pile Nim.
  • The pass move has a minimal impact on the behavior of 3-row Chomp.
  • Identified structural connections between games with passes and generic (perturbed) games.

Conclusions:

  • Recasting games as dynamical systems provides insights into the effect of pass moves.
  • The framework allows for prediction of how pass moves alter game complexity.
  • Game-specific analysis is essential, as the impact of a pass move varies significantly.