The Math Behind Chess: Game Theory

The title of the most popular and universal board game invariably goes to chess. With a relatively simple setup, it is highly accessible to beginners, yet is capable of ensnaring the most brilliant minds with its intricate complexities. We watch in wonder as grandmasters duel each other across the board, astonishing us with their ingenuity and creativity. Yet this begs the question: does there exist a single optimal strategy for both players; and thus, is the outcome of every “perfect” game definite? The short answer is yes.

The theoretical approach and Zermelo’s Theorem

The mathematical approach to analyzing multi-player situations such as chess is known as game theory. Essentially, game theory is the study of games, situations in which players, parties involved in the situation, must make utterly logical decisions in order to maximize their own benefit. Game theory has a variety of practical applications, such as in diplomacy, psychology, and economics. 

Chess, among other board games such as Go and Checkers, is an example of a constant-sum perfect information game. This is a special type of game in which both parties necessarily have conflicting interests (players must beat their opponent in order to win), and have complete knowledge of what their opponent is doing. In such situations, a tree diagram can be drawn to illustrate all possible permutations of moves. 

For instance, in chess, white has twenty possible opening moves, and for each one of these moves, there are a number of corresponding options for black. By continuing to mark out all viable moves of a player for each previous move of the opponent, we can theoretically create an all-encompassing tree diagram depicting all possible permutations of moves in chess. Thus, using the result of each permutation (win/loss/draw), we will be able to deduce the optimal strategy for both players. And by examining these optimal strategies, we can determine the inevitable outcome of a perfectly-played chess game.

Yet this raises the question: why will both “perfect” players abide by this single set of moves? The answer lies in that while examining every single possible move at any given point in a game, players must play the best move in order to maximize winning chances; if not, then the players are invariably disadvantaged. This results in both players conforming to a single optimal strategy. 

In fact, German mathematician Ernst Zermelo proposed in 1913 that any finite perfect information game has a single optimal strategy for both players, and thus the result of the game (win/loss/draw) is definite. This later became known as the Zermelo’s Theorem.

The practical limitations

In practicality, however, the story is utterly different. A model would have to process every single possible permutation of moves in a game of chess, and identify the optimal strategy for both players. The complexity of this calculation is unfathomable. It is estimated that the total number of possible permutations of moves for chess is around 10123! For comparison, there are only around 1080 atoms in the universe! Yet chess is not the most complex board game that has yet been invented; Xiangqi (chinese chess) and Go have as many as 10150 and 10360 sets of moves respectively. Indeed, it is little wonder why none of the state-of-the-art computers today have been able to complete these calculations in a reasonable amount of time.

It is important to note the differences between this approach to chess and existing chess algorithms. While modern-day chess engines also look at the “tree diagram” of different possible moves, they often only look around ten moves into the future. Hence, while they are sufficient to beat the highest-rated human players, they are still very far from being able to “solve” chess. 

Yet while, using our current technology, we are unable to derive the ultimate solution to chess, perhaps it is enough to remember that this singular solution exists. Some day in the distant future, we may be able to calculate the single optimal strategy of chess, and to discover the epitome of a game that has outwitted us for millenia. 

Works Cited:

Amir, R., & Evstigneev, I. V. (2016). On Zermelo’s theorem. ArXiv [Math.CO]. https://doi.org/10.48550/ARXIV.1610.07160 

Brams, S. J., & Davis, M. D. (2023). game theory. In Encyclopedia Britannica.

Klosky, D. (n.d.). Chess - chessprogramming wiki. Chessprogramming.org. Retrieved March 17, 2023, from https://www.chessprogramming.org/Chess