Studies in Alternating and Simultaneous Combinatorial Game Theory

Abstract

Combinatorial game theory has a beautiful algebraic structure. Games form an abelian group under the disjunctive sum and the normal play winning convention. However, not all games can be easily analyzed under this framework. In many cases, we must restrict properties to subclasses of games in order to have any useful analysis.

In this thesis, exact values are either hard to obtain or they are so complicated that they obscure the underlying structure. To aid with the analysis, techniques that approximate the value are used. Tools used for approximations include the reduced canonical form and outcome classes, particularly when values were challenging to calculate. We also present a method to construct game boards for games where initial positions are not naturally defined. Lastly, we develop a framework for simultaneous play combinatorial games, which requires approximation tools from economic game theory. We prove that the profile determines equality under extended normal play and continued conjunctive sum, while the economic game value determines equality for scoring play under the continued conjunctive sum.