Chess and Game Theory
“Chess grandmasters can see 20 moves ahead.”
“Top players have memorized how to respond to any move.”
These comments and many similar statements are made about top level chess players. It is commonly believed that grandmasters can “see” the game play out, many moves in advance. But in just 20 moves, there are billions of differing move sequences! No player can truly predict or memorize even a fraction of these sequences. While, these statements are not entirely true, the basic premise remains relevant. At a high level, chess players anticipate their opponents moves and plan their moves accordingly. Due to the exponentially increasing number of possible move sequences as players “look ahead,” it is difficult to draw a payoff matrix. However, chess can still be “game-theory-ified”.
Let’s take a look at the opening to a chess match. This is the one combination of pieces that occurs in every chess game. Mastering the opening is necessary to play at the highest level; there are hundreds of different openings, but the most common openings are memorized and the defense to them are studied carefully. While there is no known dominant strategy in chess—and chess is a game of complete information— some openings do occur more frequently. For example “white pawn d4” and “white pawn e4” are two of the most common opening moves—Queen’s Pawn and King’s Pawn openings. These are considered strong moves with high payoffs, as they allow the white to take control of the center of the board with their pawns. A common response to these openings are “black pawn d5” and “black pawn e5”, which contest the center. While these are not the only viable openings or counters, playing “black pawn a6” (moving the corner pawn one space forward) would have a lower payoff, as it would allow white to control the center without effectively developing any of blacks pieces.
King’s and Queen’s Pawn Openings
Although it is difficult to quantify the respective payoffs (lack thereof) of these various moves, it is evident that certain moves have greater or lesser payoff, depending on the context. Players need to make decisions based on these payoffs, and also based on how their opponent will react, similar to how a payoff matrix might be analyzed, just with less tangible payoffs. For example, capturing the opponent’s rook with your queen might seem like it has a high payoff initially, but when you examine the board and realize that your queen can be captured during your opponent’s next turn, your decision making will change.
It is difficult to mathematically model chess moves in a traditional game theory sense. However, specific move sequences, such as the opening of the match, or in trading of pieces, the payoffs, relative to each other, can be used to make a more rational decision, in a manner similar to how a payoff matrix could be analyzed. Grandmasters don’t look exactly 20 moves ahead, but they visualize potential outcomes and compare relative payoffs to make the best possible decision.
https://chesspathways.com/chess-openings/
https://dwheeler.com/chess-openings/