# On the Value of Chess Squares

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## Abstract

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## 1. Introduction

Chess is not a game. Chess is a well-defined form of computation. You may not be able to work out the answers, but in theory, there must be a solution, a right procedure in any position. —John von Neumann

#### Connections with Previous Work

## 2. Value of Chess Squares and Pieces

- (1)
- A comprehensive dataset $\mathcal{D}$ is systematically created, encompassing a diverse range of chessboard states s. This dataset is derived from an extensive chess database, as described in Section 2.4.
- (2)
- Utilizing the state-of-the-art chess engine, Stockfish 16 [6], an evaluation is derived for each state s in $\mathcal{D}$. This evaluation yields the true value $c\left(s\right)$, indicative of the strategic strength of the corresponding chessboard state.
- (3)
- A neural network architecture is designed and implemented. This network is specifically tailored to approximate the function $V:s\mapsto c\left(s\right)$, where $V\left(s\right)$ represents the predicted value of state s as determined by the neural network.
- (4)
- In the applications section, the constructed neural network is rigorously tested by employing it to predict $V\left(s\right)$ for a myriad of states s, thereby assessing its results in relation to anecdotal chess maxims.

#### 2.1. Centipawn Evaluation and Optimal Play

Bellman’s principle of optimality: An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decision. (Bellman, 1957)

#### 2.2. Q-Values

#### 2.3. Neural Network Architecture

#### 2.4. Data

## 3. Piece Evaluation: Knights and Bishops

#### Magnus Carlsen

## 4. Pawn Valuation

No pawn exchanges, no file-opening, no attack—Aron Nimzowitsch

- Pawns gain value as they cross the fourth rank: This point highlights an important principle in chess, where advancing pawns beyond the fourth rank often leads to increased positional strength and potential threats. As pawns move forward, they gain control over more squares, restrict the opponent’s piece mobility, and open up lines for their own pieces. Crossing the fourth rank is a significant milestone that can significantly impact the dynamics of the game.
- Pawns on the
**h**and**a**files are very good on the fifth rank: This point emphasizes the strategic importance of pawns positioned on the h and a files when they reach the fifth rank. Pawns on these files can have a powerful influence on the game, particularly in the endgame. Placing pawns on the fifth rank provides support for the central pawns, helps control key central squares, and may facilitate piece activity and potential attacks on the opponent’s position. - Pawns on the sixth rank are deadly, especially when supported by a pawn on the fifth rank: This point highlights the strength of pawns on the sixth rank, which is just two steps away from promotion. Pawns advanced to this rank become highly dangerous as they pose a direct threat concerning promotion to a more powerful piece. When supported by a pawn on the fifth rank, these pawns can create a formidable pawn duo, exerting significant pressure on the opponent’s position and potentially leading to advantageous tactical opportunities.
- Edge pawns tend to be weaker than central pawns: This point draws attention to the relative weakness of pawns placed on the edges of the board (such as the a and h files) compared to pawns in central positions. Edge pawns have fewer potential squares to advance or support other pieces, limiting their mobility and influence. In contrast, central pawns control more critical squares, contribute to a stronger pawn structure, and have a greater impact on the overall game dynamics.

**g**and

**h**files for White) can have a more immediate and aggressive impact compared to advancing pawns on the queenside (

**a**and

**b**files for White). Advanced kingside pawns can create open lines, potentially exposing the opponent’s king to attacks or weakening their pawn structure. Understanding this distinction helps players assess the strategic implications of pawn advances on different sides of the board.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Turing, A.M. Proposed Electronic Calculator (1945). In Alan Turing’s Automatic Computing Engine: The Master Codebreaker’s Struggle to Build the Modern Computer; Oxford University Press: Oxford, UK, 2005. [Google Scholar] [CrossRef]
- Shannon, C.E. XXII. Programming a computer For Playing Chess. Lond. Edinb. Dublin Philos. Mag. J. Sci.
**1950**, 41, 256–275. [Google Scholar] [CrossRef] - Bronowski, J.; Bragg, M.; Gower, N. The Ascent of Man; The Folio Society: London, UK, 2012. [Google Scholar]
- Botvinnik, M.M. Computers, Chess and Long-Range Planning; Springer: New York, NY, USA, 1970. [Google Scholar]
- Good, J. A Five Year Plan for Automatic Chess (excerpts). In Computer Chess Compendium; Springer: New York, NY, USA, 1988; pp. 118–121. [Google Scholar]
- Romstad, T. Stockfish-Open Source Chess Engine. 2011. Available online: http://www.stock.shchess.com/ (accessed on 15 June 2023).
- Dalgaard, M.; Motzoi, F.; Sørensen, J.J.; Sherson, J. Global Optimization of Quantum Dynamics with AlphaZero Deep Exploration. NPJ Quantum Inf.
**2020**, 6, 6. [Google Scholar] [CrossRef] - Silver, D.; Huang, A.; Maddison, C.J.; Guez, A.; Sifre, L.; Van Den Driessche, G.; Schrittwieser, J.; Antonoglou, I.; Panneershelvam, V.; Lanctot, M.; et al. Mastering the game of Go with deep neural networks and tree search. Nature
**2016**, 529, 484–489. [Google Scholar] [CrossRef] [PubMed] - Maharaj, S.; Polson, N.; Turk, A. Chess AI: Competing paradigms for machine intelligence. Entropy
**2022**, 24, 550. [Google Scholar] [CrossRef] - Dean, J.; Corrado, G.; Monga, R.; Chen, K.; Devin, M.; Mao, M.; Ranzato, M.; Senior, A.; Tucker, P.; Yang, K.; et al. Large scale distributed deep networks. Adv. Neural Inf. Process. Syst.
**2012**, 25, 1–9. [Google Scholar] - Silver, D.; Schrittwieser, J.; Simonyan, K.; Antonoglou, I.; Huang, A.; Guez, A.; Hubert, T.; Baker, L.; Lai, M.; Bolton, A.; et al. Mastering the game of go without human knowledge. Nature
**2017**, 550, 354–359. [Google Scholar] [CrossRef] [PubMed] - Kapicioglu, B.; Iqbal, R.; Koc, T.; Andre, L.N.; Volz, K.S. Chess2vec: Learning vector representations for chess. arXiv
**2020**, arXiv:2011.01014. [Google Scholar] - Maharaj, S.; Polson, N. Karpov’s Queen Sacrifices and AI. arXiv
**2021**, arXiv:2109.08149. [Google Scholar] - Maesumi, A. Playing Chess With Limited Look Ahead. arXiv
**2020**, arXiv:2007.02130. [Google Scholar] - Gupta, A.; Grattoni, C.; Gupta, A. Determining Chess Piece Values Using Machine Learning. J. Stud. Res.
**2023**, 12, 1–20. [Google Scholar] [CrossRef] - Ubdip. Finding the Value of Pieces. Lichess Blog. 2022. Available online: https://lichess.org/@/ubdip/blog/finding-the-value-of-pieces/PByOBlNB (accessed on 1 June 2023).
- Ubdip. Comments on Piece Values. Lichess Blog. 2022. Available online: https://lichess.org/@/ubdip/blog/comments-on-piece-values/Ps9kghhO (accessed on 1 June 2023).
- Watkins, C.J.; Dayan, P. Q-learning. Mach. Learn.
**1992**, 8, 279–292. [Google Scholar] [CrossRef] - O’Donoghue, B.; Osband, I.; Munos, R.; Mnih, V. The Uncertainty Bellman Equation and Exploration. In Proceedings of the International Conference on Machine Learning, Stockholm, Sweden, 10–15 July 2018; pp. 3836–3845. [Google Scholar]
- Clifton, J.; Laber, E. Q-learning: Theory and Applications. Annu. Rev. Stat. Its Appl.
**2020**, 7, 279–301. [Google Scholar] [CrossRef] - Priest, G. The logic of backwards inductions. Econ. Philos.
**2000**, 16, 267–285. [Google Scholar] [CrossRef] - LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] [PubMed] - Saadat, M.N.; Shuaib, M. Advancements in Deep Learning Theory and Applications: Perspective in 2020 and Beyond. Adv. Appl. Deep Learn.
**2020**, 3, 3–22. [Google Scholar] - Vargas, R.; Mosavi, A.; Ruiz, R. Deep Learning: A Review; Queensland University of Technology: Brisbane, Australia, 2017. [Google Scholar]
- Gupta, A.; Sharma, B.; Chingtham, P. Forecast of Earthquake Magnitude for North-West (NW) Indian Region Using Machine Learning Techniques. EarthArxiv
**2023**. [Google Scholar] [CrossRef] - Gupta, A.; Tayal, V.K. Analysis of Twitter Sentiment to Predict Financial Trends. In Proceedings of the 2023 International Conference on Artificial Intelligence and Smart Communication (AISC), Greater Noida, India, 27–29 January 2023; pp. 1027–1031. [Google Scholar]
- Gupta, A.; Tayal, V.K. Using Monte Carlo Methods for Retirement Simulations. arXiv
**2023**, arXiv:2306.16563. [Google Scholar] - Korsos, L.; Polson, N.G. Analyzing Risky Choices: Q-learning for Deal-No-Deal. Appl. Stoch. Model. Bus. Ind.
**2014**, 30, 258–270. [Google Scholar] [CrossRef] - Polson, N.; Witte, J.H. A Bellman View of Jesse Livermore. Chance
**2015**, 28, 27–31. [Google Scholar] [CrossRef] - Nimzowitsch, A. My System; Open Road Media: New York, NY, USA, 2022. [Google Scholar]

**Figure 1.**Each square on the chess board can be defined by the letter of its column and the number of its row. For example, the White knight on the highlighted square in this diagram is located at $f5$.

**Figure 2.**Predicted CPL advantage $c\left(s\right)$ offered by the White knight in grandmaster games.

**Figure 3.**Predicted CPL advantage $c\left(s\right)$ offered by the White bishop in grandmaster games.

**Figure 4.**Predicted winning chance $w\left(s\right)$ offered by the White knight in grandmaster games.

**Figure 5.**Predicted winning chance $w\left(s\right)$ offered by the White bishop in grandmaster games.

**Figure 6.**Predicted CPL advantage $c\left(s\right)$ offered by the Black knight in grandmaster games.

**Figure 7.**Predicted CPL advantage $c\left(s\right)$ offered by the Black bishop in grandmaster games.

**Figure 8.**Predicted winning chance $w\left(s\right)$ offered by the Black knight in grandmaster games.

**Figure 9.**Predicted winning chance $w\left(s\right)$ offered by the Black bishop in grandmaster games.

**Figure 12.**Predicted CPL advantage $c\left(s\right)$ offered by the White knight in Carlsen’s games.

**Figure 13.**Predicted CPL advantage $c\left(s\right)$ offered by the White bishop in Carlsen’s games.

**Figure 14.**Predicted winning chance $w\left(s\right)$ offered by the White knight in Carlsen’s games.

**Figure 15.**Predicted winning chance $w\left(s\right)$ offered by the White bishop in Carlsen’s games.

**Figure 16.**Predicted CPL advantage $c\left(s\right)$ offered by the Black knight in Carlsen’s games.

**Figure 17.**Predicted CPL advantage $c\left(s\right)$ offered by the Black bishop in Carlsen’s games.

**Figure 18.**Predicted winning chance $w\left(s\right)$ offered by the Black knight in Carlsen’s games.

**Figure 19.**Predicted winning chance $w\left(s\right)$ offered by the Black bishop in Carlsen’s games.

**Figure 21.**Predicted CPL advantage $c\left(s\right)$ offered by the White pawn in grandmaster games.

**Figure 22.**Predicted winning chance $w\left(s\right)$ offered by the White pawn in grandmaster games.

**Figure 23.**Predicted CPL advantage $c\left(s\right)$ offered by the Black pawn in grandmaster games.

**Figure 24.**Predicted winning chance $w\left(s\right)$ offered by the Black pawn in grandmaster games.

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**MDPI and ACS Style**

Gupta, A.; Maharaj, S.; Polson, N.; Sokolov, V.
On the Value of Chess Squares. *Entropy* **2023**, *25*, 1374.
https://doi.org/10.3390/e25101374

**AMA Style**

Gupta A, Maharaj S, Polson N, Sokolov V.
On the Value of Chess Squares. *Entropy*. 2023; 25(10):1374.
https://doi.org/10.3390/e25101374

**Chicago/Turabian Style**

Gupta, Aditya, Shiva Maharaj, Nicholas Polson, and Vadim Sokolov.
2023. "On the Value of Chess Squares" *Entropy* 25, no. 10: 1374.
https://doi.org/10.3390/e25101374