Recently popularised by the show ‘The Queen's Gambit’, chess is a game of strategy, calculation and precision where two players compete for the final win. The history of chess has been sighted as far back as a millennium, but the game only gained momentum in 1886 when the official World Chess Championship was hosted. In the 20th century, with significant improvements to technology, the ‘Clay Blitz’ chess computer program defeated 2268-rated player Joe Sentef for the first time in tournament play - a milestone in computational engineering. But how far can economic theory go in ‘solving’ the 8x8 board game?

Nobel Prize Winner John Forbes Nash Jr. was an American mathematician, renowned for his contributions to applied mathematics and most importantly, game theory. His hypothesis proved that players can always arrive at an optimal outcome (‘Nash Equilibrium’) in a finite game. In the real world, game theory can be applied to myriad fields from business strategies to classical examples like poker. This idea is commonly applied to chess as we shall see below. In a nutshell, chess is generally classified as a non-cooperative two player zero-sum game. 

Take for instance the simple zero-sum game of matching pennies, where if two players match player A will win while player B loses. When they don’t match, the opposite is true. Hence, it follows that one player wins ie. +1 only if another player loses i.e -1. The total utility of the game will always be 0 no matter what the players play. 

Let’s take a minute to apply this to chess. Every game of chess has three possible outcomes: win-loss(1,0), draw-draw(0.5,0.5), loss-win(0,1). The total utility of the game will always be 1 ie 1-sum game, but this distinction isn’t extremely relevant to games nature. We see that the payoff matrix can easily be altered to win-loss(1,-1), draw (0,0) and (-1,1). In more technical terms, chess is a game that occurs in perfect knowledge as both players look at the same chessboard and players move after one another (sequentially). 

Exploring Nash’s theory of equilibrium, we see that chess should be solvable. Counter arguments such as chess being an infinite game can be refuted by the game’s rules that determine a draw. For instance, the ‘rule of 3’ forces a draw as the same position cannot be repeated thrice in a single game. 

In practice, we see that the minds of chess players intuitively showcase backward induction, , and rollback equilibrium to reason and strategize iteratively, essentially reasoning back through moves already played to find the optimum move. Thus, each player creates a mental plan to counter their opponent’s moves which is essentially a game tree with nodes representing the various paths the game could take. 

Before delving into the complexities of chess, let’s look at a more simple and familiar game - Tic Tac Toe (Noughts and Crosses). A 3 by 3 matrix played by 2 players with a relatively simpler goal. Today’s improved data science and modern technology have used brute force to prove that the game of tic-tac-toe will always end in a draw if dominant strategies are used by both players (no silly blunders). Quote literally, game theory has ‘solved’ tic-tac-toe. However, despite its simplicity, this game has 255,168 game tree nodes, or possible arrangements on the board.

On the other hand, chess is defined by its 8 by 8 board that has 32 pieces which can all move in their own unique way. There are millions of possible Nash Equilibriums, calculations and tree nodes for this game. We see over 288 billion move combinations after just four moves a piece. Modern technology is far from solving chess’s 10120 game variations or 1043 board combinations.  In fact, there are still extraordinary moves, invisible to engines, that have been played by grandmasters. Information theorist Claude Shannon argued that it would take 1090 years to solve chess in 1951. 

However, during his period, technology was just beginning to develop. Nowadays, elegant mathematical theories fused with empirical data has led to a massive growth in the knowledge acquired by economists and game theoreticians. For example, we know that central pawns and knights have the lowest survival rates in games or where notable chess players have moved their pieces the most. The smartest engines are able to capture previous game knowledge forming patterns and algorithms that eliminate worse moves quicker. 

Another interesting question is whether the game of chess is a forced win or draw. A forced win would typically mean that white (player who plays the first move) has a strictly dominant strategy that could force black to resign. The idea that playing first has an intrinsic advantage holds merit. A robust body of empirical evidence proves that white consistently wins more often than black at a rate of 52% to 56%. Adams and Rauzer suggested that white may be winning after playing e4 while Berliner claimed that d4 may win, in spite of a perfect play from black. Yet, George Walker wrote that the very first move of the game is of little worth and insignificant to the realm of possibilities the game could take on. 

The more common consensus is a forced draw where both players achieve an optimal strategy and deviating would lead to a loss. As shown by the graph above, the higher rated (skilled) players draw much more frequently than a lower rated player. This would suggest that, at a certain level of proficiency yet not achieved, most games would end in a draw as both players incorporate the best strategies. The common saying “greater the risk, greater the reward” may have an impact in chess too, as the most outrageous moves are those that are not studied. These moves increase the stakes in the game and thereby increase the possibility of a miscalculation or loss. Logically, grandmasters may be especially risk averse as their reputation is at stake which could suggest they are playing to draw. Most grandmasters play tried and tested classical openings like the ‘Sicilian Defence’ or the ‘Scotch Game’ where the optimal strategy has been long analysed.  This concern of a “draw death” was expressed by Bobby Fischer, who believed that chess is ultimately a draw. 

In the 21st century, AI catalysed by innovation and science has been getting smarter at an exponential rate. Humans have also been getting more intelligent, as depicted by the world’s increasing average IQ. As human capital further mixes with technological progress, perhaps chess will be ‘solved’ sooner than anticipated and become the “tic-tac-toe” for the next generation. Until then, Magnus Carlson - our current world chess champion - deserves the glory of being crowned as a mastermind, and a mastermind beyond the realm of today’s AI.

Aishi Basu.