Are some chess positions more critical than others? This guest blog post by Dr. Marc Barthelemy answers this question, using a "fragility score" which quantifies the tension in chess positions. Dr. Barthelemy is a French theoretical physicist based at the Institute of Theoretical Physics, CEA-Saclay, France. Comments about this post may be directed to marc.barthelemy@ipht.fr .

Chess and Complexity Science: Understanding the Fragility of Positions by Marc Barthelemy, PhD

Chess is more than just a game; from being a testbed for artificial intelligence to a source of rich mathematical problems, chess continues to captivate minds. With its finite rules but near-infinite possibilities, chess serves as an ideal environment for developing and testing algorithms while also inspiring tools and concepts valuable for the study of complex systems. In a recent study (Barthelemy, 2025), we introduce a metric—the "fragility score"—to quantify the tension in chess positions, providing a fresh perspective on the critical moments that shape games.

The Science of Chess: A Brief History

The intersection of chess and science is not new. Famous scientists such as Alan Turing (Turing, 1953) and Claude Shannon (Shannon, 1950), among others (Levy, 2013), recognized in the 50s the potential of chess as a playground for exploring artificial intelligence. This culminated in historic milestones such as IBM’s Deep Blue defeating Garry Kasparov in 1997 and, more recently, Google DeepMind’s AlphaZero (Sadler & Regan, 2019) redefining chess strategy. 

While computational advancements have transformed chess analysis, the scientific exploration of its deeper dynamics has lagged. Indeed, most of the scientific effort in chess has focused on developing AI and engines capable of surpassing human play, yet the quantitative understanding of chess dynamics has remained largely unexplored. With the advent of powerful engines and large game databases, the door is however now open to a rigorous, data-driven exploration of chess as a complex system (Blasius & Tonjes, 2009; Chacoma & Billoni, 2024; Chassy & Gobet, 2011; De Marzo & Servedio, 2023; Schaigorodsky, Perotti, & Billoni, 2016; Sigman, Etchemendy, Slezak, & Cecchi, 2010). In the spirit of complex systems approaches, chess can be viewed as an evolving network of interacting pieces (Barthelemy, 2025), with its structure providing insights into the dynamics of positions and potential advantages for one side. 

Interaction Graph and the Fragility of Positions

A “fragile” position can be understood as one where a single change—like the loss of a key piece—can dramatically alter the structure of the position and the balance of the game. In order to do that, we represent each position by an interaction graph (see Fig. 1 for an example), where nodes correspond to pieces and edges represent attacks or defenses. Directed edges capture the asymmetry of chess: a piece attacking another is not necessarily defended in return. A key metric for characterizing the importance of a node in a graph is the betweenness centrality (BC) which measures (Freeman, 1977) how often a node lies on the shortest paths between other nodes. For the chess interaction graph, a piece with a high BC value is pivotal, as its capture could initiate a cascade of exchanges and deeply alter the structure of the position. If such a piece is under attack, the position could undergo significant changes in the next moves, making it ‘fragile’.  By summing the BC values of attacked pieces, we can then define the “fragility score” of a position (Barthelemy, 2025). A high score signals a position on the brink of transformation.

Figure 1: Example of a position interaction graph for a position from Mchedlishvili - Van Foreest (44th FIDE Chess Olympiad Chennai 2022 Open). On the left, we show the position at the ply 49 after 25. Nxd4 (which happened after 24…Rd4!!). On the right, we show the corresponding interaction graph (we denote the corresponding pieces of nodes with uppercase letters for white pieces and lowercase for black pieces and the position of each piece is shown on each node). Arrows indicate defense (green and blue arrows) and attacks (red arrows). The key piece is here the white knight and has the largest betweenness centrality. 

Universality, Critical Moments and Tipping Points

By analyzing over 20,000 games from chess champions like Carlsen, Kasparov, and Capablanca, we observed a universality of fragility patterns, with a peak occurring typically around the 15th move. This corresponds to the middlegame—a phase known for its rich tactical and strategic opportunities. Interestingly, we also found that certain pieces, like pawns and knights, are disproportionately involved in high-tension positions. This aligns with their roles as key drivers of attacks and defenses. 

The fragility score doesn’t just offer a retrospective view; it identifies pivotal moments in real-time, and we observed that in many famous games, the peak fragility coincides with brilliant, game-changing moves. We analyzed the top 10 games ever played, as ranked by Chess.com (Chess.com, 2022), and computed the fragility score for each. For each game, we identified the move corresponding to the maximum fragility, and in many cases, the maximum fragility coincides with brilliant and decisive moves (with minor variations of the order of a few moves), suggesting that during these high tension phases, creativity and skill are crucial in shaping the outcome of the game. This phase of maximum fragility often dictates the fate of the game and naturally corresponds to critical and noteworthy moves.

Broad implications and the future of Chess Science

The fragility score invites us to view chess not merely as a game of calculation but as a dynamic interplay of structure and tension, offering a novel lens for analyzing decision-making in complex systems. By identifying tipping points, it reveals how small changes can cascade into large-scale effects—a principle with applications in fields as diverse as economics, biology, and social networks. While chess experts possess practical knowledge about position strength and move quality, quantitative studies could uncover guiding principles and descriptors that distill the intricate web of interactions between pieces into insights comprehensible and actionable by humans.

This approach is still in its infancy, and significant work remains to fully understand the dynamics of chess games and provide practical recommendations for anticipating critical moments and making informed decisions. While processing vast amounts of data and encoding patterns into countless parameters, engines identify hidden patterns without truly understanding them. Humans may never surpass engines again, but complex systems science offers a path to decode chess’s complexity in ways that align with Réti’s vision of advancing chess to the third stage of scientific understanding (Réti, 1924) by establishing scientific laws for the game. This, ultimately, is one of the ambitions of the science of chess.

References

Barthelemy, M. (2025). Fragility of chess positions: Measure, universality and tipping points. Physical Review E, 111, 014314. https://journals.aps.org/pre/abstract/10.1103/PhysRevE.111.014314

Blasius, B., & Tonjes, R. (2009). Zipf’s law in the popularity distribution of chess openings. Physical Review Letters, 103(21), 218701.

Chacoma, A., & Billoni, O. (2024). Emergent Complexity in the Decision-Making Process of Chess Players. arXiv preprint arXiv:2406.15463. DOI: 10.48550/arXiv.2406.15463

Chassy, P., & Gobet, F. (2011). Measuring chess experts’ single-use sequence knowledge: An archival study of departure from ‘theoretical’ openings. PLoS One, 16, e26692. https://doi.org/10.1371/journal.pone.0026692

Chess.com. (2022). The best chess games of all time. Retrieved from The Chess.com website: https://www.chess.com/article/view/the-best-chess-games-of-all-time

De Marzo, G., & Servedio, V. (2023). Quantifying the complexity and similarity of chess openings using online chess community data. Scientific Reports, 13(1), 5327. https://rdcu.be/d7GeW

Freeman, L. (1977). A set of measures of centrality based on betweenness. Sociometry, 40, 35-41. https://www.jstor.org/stable/3033543

Levy, D. (2013). Computer chess compendium. Springer Science & Business Media.

Maslov, S. (2009). Power laws in chess. Physics, 2, 97.

Réti, R. (1924). Curso superior de ajedrez. Four conferences given in Buenos Aires in 1924. Buenos Aires.

Sadler, M., & Regan, N. (2019). Game changer: AlphaZero's groundbreaking chess strategies and the promise of AI. Alkmaar, The Netherlands: New in Chess.

Schaigorodsky, A., Perotti, J., & Billoni, O. (2016). A study of memory effects in a chess database. PLoS One, 11, e0168213. https://doi.org/10.1371/journal.pone.0168213

Shannon, C. (1950). Programming a computer for playing chess. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(314), 256-275. https://vision.unipv.it/IA1/ProgrammingaComputerforPlayingChess.pdf

Sigman, M., Etchemendy, P., Slezak, D., & Cecchi, G. (2010). Response time distributions in rapid chess: A large-scale decision making experiment. Frontiers in Decision Neuroscience 4(60). https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2010.00060/full 

Turing, A. (1953). Chess. In B. Bowden, Faster than thought. London: Pitman.

Announcement

Chessable is accepting applications for the 2025 cycle of the Chessable Research Awards. Applications are open until May 15, 2025. For more information, please visit the Chessable Research Awards site.