In this study, we investigate the algebraic structures and complexities of several two-player perfect-information games using Krohn-Rhodes algebraic automata theory and computer algebra. We examine Nim, Subtraction Game, Tic-Tac-Toe, and the \(3\times 3\) Hex game, taking advantage of the small state space. Our approach involves a precise formulation of state-space representation and a detailed algebraic analysis of best-play automata generated using minimax tree searches and backtracking algorithms. Notably, our findings do not contradict established theoretical frameworks but offer new insights, particularly in the case of a specific Nim game setup that refutes an existing conjecture. Although our study found a reliable upper bound for the complexity of Tic-Tac-Toe and 3 \(\times \) 3 Hex, the existence of some lower-complexity best-play automata remains an open question. Overall, this research attempts to bridge the gap between algebraic automata theory and classical game theory.