One of the big ideas of modern science is that small things can sometimes have huge, irreversible consequences. These so-called tipping points crop up in all walks of life, from economics and finance to human health and the environment. Indeed, they are a fundamental feature of what scientists call complex systems.
In an effort to better understand tipping points, scientists have invested much time and effort into simulating complex systems like the climate and the economy, and then gathering data to test their models.
This work has allowed them to predict many of the tipping points that could have profound impacts on humanity, such as the collapse of ecosystems, climate patterns and trade networks. These models can even help identify steps to avoid these disasters (although whether humans will take these steps is another question).
But amid all this progress, one relatively simple complex system has been somewhat neglected. This is the powerful network of forces that come into play during a game of chess. Instead, scientists have taken a different approach to mastering chess, first with the computational brute force of computers like IBM's Deep Blue and more recently with AI systems like Deepmind's AlphaZero. In the process, the “science of chess” has been largely forgotten.
Now that is changing thanks to the work of Marc Barthelemy at the Universite Paris-Saclay in France who has begun to study chess once again as a complex system. He says an overlooked phenomenon of the game is the tipping points that lead to major changes in the fortunes of one player or the other. By studying the effect tipping points have on the game, Barthelemy hopes to revive interest in a “science of chess”.
Chess has long been a favorite model system for computer scientists. Its simple board, well-defined moves, and clear rules make it relatively easy to simulate. And yet this seeming simplicity masks a game of huge complexity. “The study of chess offers a rich intersection between computational science and complex systems analysis,” says Barthelemy.
Mathematically, he continues, chess can be represented as a decision tree where each branch leads to a win, loss, or draw. The challenge, then, lies in selecting the best move amid this vast combinatorial complexity.
“It is thus surprising that complexity science — and in particular, statistical physics — has had relatively little to say about this system,” says Barthelemy. The time is ripe to change this, he says, because of the vast databases of chess games that are now available to study.
Barthelemy’s new approach is to study the tipping points within a game, where a single move has a decisive influence on the outcome of the game. “These critical positions are often unstable, and a small mistake can lead to dramatic shifts in the game’s trajectory,” he says.
In this way, he introduces a concept he calls the “fragility score” of any position on the board. The positions of greatest fragility often lead to pivotal moves that largely determine the outcome of the game.
A key idea here is the network of forces at play at any point in the game. Each piece on the board projects its power in a well defined pattern — bishops along diagonals, rooks along ranks and files, knights in their famous L-shape and so on. At each point in the game, these patterns criss-cross the board, often combining in attack and defence on specific squares and pieces. Any piece that interacts with many other pieces clearly plays an important role in that game.
Barthelemy's insight is that this pattern of forces forms a kind of network in which pieces are nodes and the interactions with other pieces are edges. This characterization then allows Barthelemy to bring to bear on chess all the mathematical tools of network science.
One key idea in network science is betweenness centrality — defined as how often a node lies on the shortest path between all pairs of nodes in a network. In a chess game, this is a measure of how easily a piece can influence other pieces on the board.
So for any board position, the betweenness centrality identifies the most influential pieces.
An important question, then, is how heavily important pieces are under attack. Once again, this can be easily calculated for all the pieces in any board position.
This is what Barthelemy calls the fragility of the position. It measures how easily important pieces can be removed from the board. “By tracking the total fragility score throughout the game, we can identify critical moments when positions become particularly fragile,” he says.
Barthelemy has done this for more than 20,000 games played by the world’s top players over the last century.
It turns out that the most fragile positions occur around move 16 and that the key pieces in these positions are most often pawns, followed by knights.
This analysis reveals an unexpected pattern. “We observe a surprising universality: the average fragility score is the same for all players and for all openings,” says Barthelemy. “Moreover, in famous games, the maximum fragility often coincides with pivotal moments, characterized by brilliant moves that decisively shift the balance of the game.”
The actual patterns of play follow a universal pattern too. Barthelemy says that fragility builds in the 8 moves before the most fragile position and then remains elevated for about 15 moves after that.
This gradual decay of fragility is consistent with the shift from a middlegame to endgame, where small inaccuracies can dramatically shift the balance of power.
That’s interesting work that provides a new perspective on the dynamics associated with board positions in chess. Most players already use a sophisticated of tally of forces to determine the relative strength of their position.
An interesting question is whether fragility scores provide any useful shortcuts for players making these assessments or whether they lead to more accurate board assessments. “These insights offer a valuable tool for both players and engines to assess critical moments in chess,” he says.
But how this can be done and exploited, Barthelemy does not say. It will clearly take time to determine the strategic significance of this kind of thinking. And if important, worth keeping under wraps while it is exploited for all its worth!
Ref: Fragility of Chess positions: measure, universality and tipping points : arxiv.org/abs/2410.02333
