Combinatorial game theory

Combinatorial game theory is the study of two-player games of perfect information — games like Go, Chess, and Nim — in which no element of chance is involved and both players know the complete state of the game at all times. The field was transformed by John Conway's discovery that every such game could be assigned a numerical value drawn from the surreal numbers, creating a bridge between strategic play and abstract algebra.

Unlike classical game theory, which focuses on equilibria and mixed strategies, combinatorial game theory asks what a position is worth and how that value decomposes under the operations of game combination. This shift from equilibrium to valuation makes it a natural partner for algorithmic game theory and the analysis of computational complexity in strategic settings.