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Chess variation entropy and engine relevance for humans

Marc Barthelemy

TL;DR

The paper addresses the mismatch between engine-based positional evaluations $E$ and human cognitive limits in chess. It introduces the entropy of the principal variation, denoted $S$, as a metric of move-sequence complexity, and uses Stockfish evaluations on about 9,500 positions from the World Rapid Championship to quantify how $S$ correlates with skill level and with $|E|$. The main finding is that high entropy (=$S>4$) variations are common for beginners and intermediates when $|E|<100$, while experts exhibit lower $S$ overall except near balanced positions where $|E|\approx 0$; results are robust across openings. The study argues for AI-generated evaluations that convey the underlying complexity via skill-adaptive guidance, translating analysis into actionable, human-accessible recommendations.

Abstract

Modern chess engines significantly outperform human players and are essential for evaluating positions and move quality. These engines assign a numerical evaluation $E$ to positions, indicating an advantage for either white or black, but similar evaluations can mask varying levels of move complexity. While some move sequences are straightforward, others demand near-perfect play, limiting the practical value of these evaluations for most players. To quantify this problem, we use entropy to measure the complexity of the principal variation (the sequence of best moves). Variations with forced moves have low entropy, while those with multiple viable alternatives have high entropy. Our results show that, except for experts, most human players struggle with high-entropy variations, especially when $|E|<100$ centipawns, which accounts for about $2/3$ of positions. This underscores the need for AI-generated evaluations to convey the complexity of underlying move sequences, as they often exceed typical human cognitive capabilities, reducing their practical utility.

Chess variation entropy and engine relevance for humans

TL;DR

The paper addresses the mismatch between engine-based positional evaluations and human cognitive limits in chess. It introduces the entropy of the principal variation, denoted , as a metric of move-sequence complexity, and uses Stockfish evaluations on about 9,500 positions from the World Rapid Championship to quantify how correlates with skill level and with . The main finding is that high entropy (=) variations are common for beginners and intermediates when , while experts exhibit lower overall except near balanced positions where ; results are robust across openings. The study argues for AI-generated evaluations that convey the underlying complexity via skill-adaptive guidance, translating analysis into actionable, human-accessible recommendations.

Abstract

Modern chess engines significantly outperform human players and are essential for evaluating positions and move quality. These engines assign a numerical evaluation to positions, indicating an advantage for either white or black, but similar evaluations can mask varying levels of move complexity. While some move sequences are straightforward, others demand near-perfect play, limiting the practical value of these evaluations for most players. To quantify this problem, we use entropy to measure the complexity of the principal variation (the sequence of best moves). Variations with forced moves have low entropy, while those with multiple viable alternatives have high entropy. Our results show that, except for experts, most human players struggle with high-entropy variations, especially when centipawns, which accounts for about of positions. This underscores the need for AI-generated evaluations to convey the complexity of underlying move sequences, as they often exceed typical human cognitive capabilities, reducing their practical utility.
Paper Structure (4 sections, 5 equations, 4 figures)

This paper contains 4 sections, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Entropy of a variation
  3. Engine evaluation and human players
  4. Conclusion

Figures (4)

  • Figure 1: From a given position, the path towards the best (winning) position necessitates to find the best moves consecutively. The probability to reach the ideal position after $n$ half-moves is $P(n)$ (Eq. \ref{['eq:Pn']}).
  • Figure 2: (Top) Evolution of the average entropy $S$ versus ply (or half-move) for different levels of players (beginner, intermediate, expert). (Bottom) Evolution of the probability to observe an entropy larger than $4$ during the game, computed for different levels. These results are obtained for an average of 100 games played at the World Rapid Chess Tournament 2023 (Samarkand).
  • Figure 3: (Top) Probability to have a Stockfish $E$ (in centipawns). (Bottom) Cumulative distribution of the absolute value $|E|$. We observe here that positions with evaluation $|E|<100$ correspond to about $\approx 64\%$ of all positions (results computed for all positions for 100 games during the World Rapid chess championship 2023).
  • Figure 4: Probability to have a position with $S>4$ versus the engine evaluation $E$ for different levels. Computed for 100 games during the World Rapid chess championship 2023 (Samarkand).