Notions of Anonymity, Fairness and Symmetry for Finite Strategic-Form Games

TL;DR

This paper surveys various notions of anonymity, fairness, and symmetry in finite strategic-form games, analyzing their relationships and providing a framework for constructing and ordering symmetric games.

Contribution

It introduces a comprehensive survey of symmetry and anonymity notions, formalizes game isomorphisms as groupoids, and offers methods to construct and classify symmetric games.

Findings

Game bijections and isomorphisms form groupoids.

Matchings characterize strategy triviality.

Framework for constructing and ordering symmetric games.

Abstract

In this paper we survey various notions of anonymity and symmetry for finite strategic-form games present in relevant literature, and discuss notions of fairness; 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 examples that range all combinations of surveyed symmetry notions, which when combined with other results in this paper gives the precise relationship between the various symmetry notions.