May 18, 2020

The Game Theory of Chess

From curious seven-year-old playing chess with his elder sister to representing India at the World Youth Chess Championships, chess has been an integral and transformative part of my life.

What initially began as a hobby has evolved to become a way of life for me.

I’ve been playing competitive chess for the past 7 years, and I’ve always been fascinated by how the game is so seemingly convoluted yet at the same time, straightforward.

The manner in which the mind of a chess player functions whilst playing the game very strongly mimics the certain concepts of an on the up and up branch of Economics referred to as Game Theory.

In a nutshell, Game Theory is applied mathematics, mostly to social circumstances and more importantly, situations where payoffs are interdependent towards each other.

For example, in a game like chess, where one’s position is affected by a move that their opponent makes. Chess, in theoretical terms, falls into the category of combinatorial games.

Combinatorial Game Theory, popularly abbreviated as CGT, studies strategies and mathematics of two-player games of perfect knowledge such as chess.

An important distinction between this subject and classical game theory is that game players are assumed to move in sequence rather than simultaneously, so there are no information-hiding strategies. Your opponent doesn’t have any hidden pieces up to his sleeve now, does he?

Thus, one can say that there is no hidden information in a game like chess. What I find most engaging about this branch of economics is how the games analyzed tend to represent real-life decision-making situations.

For example, chess imitates life and vice versa. Both in life and in chess, you have a multitude of scenarios laid out in front of you, but you can only choose one. Once your choice is made in life, it cannot be undone. Similarly, once you make your move, you can’t take it back.

Games where everyone has perfect information (both players have knowledge of the rules of the game) and that involve no chance (the outcome of the game relies on the skill set of the player, and not on chance) are called combinatorial games.

However, (On Numbers and Endgames, Noam D. Elkis) Combinatorial Game Theory does not apply directly to chess, because the winner of a chess game is in general not determined by who makes the last move, and indeed a game may neither be won nor lost at all but drawn by infinite play.

Infinite play (a concept wherein the game being played does not have a foreseeable end, and the game is rendered endless with no side having won) does not occur in actual games. Instead, the game is drawn when it is apparent that neither side will be able to checkmate against reasonable play.

The major drawback, when applying CGT to chess is that the 8 × 8 chessboard is too small to decompose into many independent subgames (a subgame refers to a part of the game that ensues after a specific sequence of starting moves have been played), or rather that some of the chess pieces are so powerful and influence a chief portion the board’s area that even a decomposition into two weakly interacting subgames (e.g. a kingside attack and a center break) generally break down in a few moves.

Nonetheless, a rather captivating situation arises when one uses certain principles of game theory in order to determine the outcome of a game by mapping out a specific set of moves, in accordance with a game tree that specifies each movers’ advantages and disadvantages.

One can parallel this to how a chess player calculates. What does a chess player do while “calculating moves”?

If one takes a minute to think about it, the seemingly complicated mathematical process of calculating moves essentially encompasses this exercise: once your opponent makes his move, you analyze the position on the board in front of you by mapping out all the scenarios which you deem relevant and you create a mental image of plans of attack and defense.

At the same time, you anticipate your opponent’s response to every move that you will make.

In order to better illustrate the concept explained above, I’ve mentioned some examples below.

1. Game Trees: What do you do when you’re playing chess? You plan, you strategize.

Your brain whirrs into motion and you think “if my opponent plays his Bishop here I’ll respond with this move, if he places his king on that square I will give him a check.”

Consciously or subconsciously, you prepare a plan of strategy, and your ultimate outcome is to win or draw the game, which conditional to the circumstances of the game during that particular moment.

Thus, when a player prepares a “plan of strategy”, he forms the mental image of a game tree, which incorporates his responses to the opponent’s moves.

2. Backward Induction: The method of looking ahead and reasoning back to determine behavior in sequential-move games (e.g. chess, poker) is known as rollback.

Because this reasoning requires working backward, one step at a time, it is also referred to as backward induction. When formulating a plan, a chess player considers relevant responses from his opponent to the move he is planning to make, thus using the technique of backward induction.

3. Rollback Equilibrium: When all players choose their optimal strategies determined by using rollback analysis (with the help of backward induction), the set of strategies derived is referred to as the rollback equilibrium of the game; and the outcome that arises from playing these strategies is the rollback equilibrium outcome. One can associate this concept with the execution of your optimal plan of strategy against your opponent’s.

Therefore, while playing chess, a player, be it consciously or subconsciously, makes use of certain fundamental ideas of Game Theory.

I like to believe that a “complex” game like chess cannot be "solved” using game theory, though the possibility exists, considering the furtherance of even stronger chess engines (like Houdini and Fritz, among others). 

Having said that, the thought-process which involves the decision-making of a chess player unequivocally echoes numerous aspects of game theory.



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