Combinatorial game theory

Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a position evolves through alternating moves, each governed by well-defined rules, with the aim of achieving a specific winning condition. Unlike economic game theory, combinatorial game theory generally avoids the study of games of chance or games involving imperfect information, preferring instead games in which the current state and the full set of available moves are always known to both players.^[1] However, as mathematical techniques develop, the scope of analyzable games expands, and the boundaries of the field continue to evolve.^[2] Authors typically define the term "game" at the outset of academic papers, with definitions tailored to the specific game under analysis rather than reflecting the field’s full scope.

Combinatorial games include well-known examples such as chess, checkers, and Go, which are considered complex and non-trivial, as well as simpler, "solved" games like tic-tac-toe. Some combinatorial games, such as infinite chess, may feature an unbounded playing area. In the context of combinatorial game theory, the structure of such games is typically modeled using a game tree. The field also encompasses single-player puzzles like Sudoku, and zero-player automata such as Conway's Game of Life—although these are sometimes more accurately categorized as mathematical puzzles or automata, given that the strictest definitions of "game" imply the involvement of multiple participants.^[3]

A key concept in combinatorial game theory is that of the solved game. For instance, tic-tac-toe is solved in that optimal play by both participants always results in a draw. Determining such outcomes for more complex games is significantly more difficult. Notably, in 2007, checkers was announced to be weakly solved, with perfect play by both sides leading to a draw; however, this result required a computer-assisted proof.^[4] Many real-world games remain too complex for complete analysis, though combinatorial methods have shown some success in the study of Go endgames. Analyzing a position using combinatorial game theory involves identifying the optimal sequence of moves for both players until the game's conclusion, but this process becomes prohibitively difficult for anything beyond simple games.

It is useful to distinguish between combinatorial "mathgames"—games of primary interest to mathematicians and scientists for theoretical exploration—and "playgames," which are more widely played for entertainment and competition.^[5] Some games, such as Nim, straddle both categories. Nim played a foundational role in the development of combinatorial game theory and was among the earliest games to be programmed on a computer.^[6] Tic-tac-toe continues to be used in teaching fundamental concepts of game AI design to computer science students.^[7]