An Algebraic Perspective on Game Theory

Abstract

Even though the fields of game theory and algebra are well-established, limited literature analyzes games using modern algebraic theories. In 2009, Rhodes published a game complexity study titled \textit{Complexity of Games} via \textit{Applications of Automata Theory and Algebra}, which uses Krohn-Rhodes theory to analyze the best-play automata of the two-player complete-information sequential games. It defines the key terminologies, derives the automata representations from complete game trees, and proposes several theorems and conjectures. Rhodes' work heavily inspires this thesis as we expand on the established theoretical frameworks and challenge some of Rhodes' claims.

We begin by presenting the essential terminologies of the semigroup theory and Krohn-Rhodes theory. We then examine the existing methods for complexity analysis and propose novel algorithms to calculate the complexity lower bound for transformation semigroups. These algorithms underwent multiple iterations of improvements and contributed considerably to the analysis in the subsequent chapters.

A major component of the thesis is a direct expansion of Rhodes' work with respect to the two-player complete-information sequential games. We represent the gameplay machines identical to Rhodes'; however, we also consider symmetries in the structures of games to reduce the state space. We then developed practical methodologies to evaluate the algebraic structure of several simple games, such as Tic-Tac-Toe and used them to verify some of Rhodes' conjectures.

Another major component of the thesis is the algebraic analysis of extensive-form games, which encompass a much broader category of games (including the complete-information sequential games). This exploration encountered many challenges, in particular, the analysis of nondeterminism associated with mixed strategy profiles. We proposed the usage of classic powerset construction of the nondeterministic finite automata, but it turned out to be highly problematic due to state explosion and the possibility of collapsing into a single `superset' state. To overcome these issues, we proposed an alternative construction that captures the nondeterminism without the drawbacks of the powerset construction. This construction enables us to analyze the algebraic structure and complexity of games effectively.

Overall, this thesis serves to bridge the gap between the classical game theory and the algebraic automata theory.