[Paper Review] Notions of Symmetry for Finite Strategic-Form Games

Nicholas Ham|arXiv (Cornell University)|Nov 18, 2013

Game Theory and ApplicationsDecision Sciences4 citations

TL;DR

This paper introduces and formalizes multiple notions of symmetry in finite strategic-form games, establishing game bijections and isomorphisms as groupoids, using matchings to characterize strategy triviality, and presenting a framework to construct and partially order parameterized symmetric games with diverse examples across symmetry types.

ABSTRACT

In this paper we survey various notions of symmetry for finite strategic-form games; show that game bijections and game isomorphisms form groupoids; introduce matchings as a convenient characterisation of strategy triviality; and outline how to construct and partially order parameterised symmetric games with numerous examples that range all combinations of surveyed symmetry notions.

Motivation & Objective

• To systematize and formalize various notions of symmetry in finite strategic-form games.
• To demonstrate that game bijections and isomorphisms form groupoids under composition.
• To introduce matchings as a structural tool for characterizing strategy triviality.
• To develop a framework for constructing and partially ordering parameterized symmetric games.
• To provide illustrative examples that span all combinations of surveyed symmetry notions.

Proposed method

• Utilizing groupoid theory to model game bijections and isomorphisms, enabling algebraic treatment of symmetry.
• Defining matchings as a combinatorial characterization of strategy triviality in games.
• Introducing parameterized families of symmetric games to explore structural and symmetry properties.
• Applying partial orderings to compare symmetric games based on their symmetry structure.
• Constructing explicit examples that realize all combinations of symmetry notions.
• Using algebraic and combinatorial techniques to analyze game equivalence and symmetry types.

Experimental results

Research questions

• RQ1 How can different notions of symmetry in finite strategic-form games be formally unified and classified?
• RQ2 What algebraic structure underlies the set of game bijections and isomorphisms?
• RQ3 How can matchings be used to characterize strategy triviality in games?
• RQ4 What is the structure of parameterized symmetric games, and how can they be systematically constructed?
• RQ5 How can symmetric games be partially ordered based on their symmetry properties?

Key findings

• Game bijections and isomorphisms form groupoids, providing a coherent algebraic framework for symmetry transformations.
• Matchings offer a concise and effective characterization of strategy triviality in strategic games.
• A systematic method for constructing parameterized symmetric games is established, enabling controlled exploration of symmetry types.
• The partial order on symmetric games allows for comparative analysis of symmetry richness and complexity.
• Numerous examples are provided that realize all combinations of the surveyed symmetry notions, demonstrating structural diversity.
• The framework enables clear distinctions and relationships between various symmetry concepts in finite games.