Quantifying the Control over Squares

Love this. Many comments suggest changes that reach beyond the scope of this concept. Tactical dynamics can be informed by a simple static control map. The control map doesn't need to reflect every element of gameplay. The only additional feature I would recommend is tension. Keep the subtractive 1 to -1 greyscale but add either a second map or just a second color scale so we can see contested squares. You might need to invert the control values back for the tension statistic though, since two queens staring each other down seems a lot more tense than two pawns. I suppose the tension would have to reflect the value of the piece under threat, since a bishop looking at my queen is even more tense than another queen, and certainly more tense than a bishop staring at a pawn.

It would also be awesome to have a tool to import a position to create the heatmaps for. Until then I might need to do a spreadsheet version for my correspondence games!

No offence to your effort, but you'd go a lot further by studying the source code of classical engines, something like SF11. They've had decades of experience of finding and fine-tuning analytical approaches to solving this exact problem.

Categorising square control can be broken down into Biased/Unbiased and Absolute/Relative threats.

Absolute Biased Control (Immediate-ply Threat): This is the direct, turn-dependent view of what squares are currently under attack by the player whose turn it is. It's a snapshot of the immediate threats.

Absolute Unbiased Control (Number of White and Black Attackers): This moves beyond the current turn and considers the potential number of attackers each side has on a given square. It's a more stable, long-term perspective on quantifying control over squares.

Relative Biased Control (Our Immediate Influence): Focuses on squares our pieces currently control, considering the value and activity of those pieces. Example: An active central knight controlling key squares vs. a passive back-rank rook.

Relative Unbiased Control (Our Potential Influence/Mobility): This focuses on squares our pieces could control in the future, considering their mobility and potential lines of attack. It's independent of immediate opponent threats. It's more about inherent strategic potential.

In conclusion, Chess as we well know it, changes with each ply move. What's constantly changing is the positional square control and piece safety. I assume for many the game is played with guiding chess principles, that often bend under the weight of probabilities. I try to remain vigilant, when I push pawns. It breaks the status-quo of a position. Our perspectives of square control intertwine, like two sides of a coin. White and Black exist in tandem. A move only has value once both sides have played. That's what defines the balance and complexity of the game.

Controlling important areas on the chessboard is the real key to getting ahead and staying ahead.

The player with the initiative is the one pushing the action and calling the shots, and the reason they can maintain that pressure is often because they control important key squares or areas of the chessboard. Those squares aren't just any random number of squares; they're the ones that give a player the advantage. Figuring out which squares are the important ones is the trick. It's like knowing which spots on the map give you the high ground or control key resources. Those are the "worth more" squares.

Even if we can't precisely measure control, players who hold key squares understand their value – it's what allows for creative and successful play.

In the endgame example, how are a6 and c6 not at ~ -0.92? (-1 from the black pawn +1/12 for the white king). The color scale does not appear to even allow for this value...

From a static cp eval point of view:
-3.40 nh8 gained value, yet it cannot move;
-1.51 b7 gained value and it never moved, but it is protecting the king;
+2.04 Bh5 lost value, jet it is temporarily preventing the knight from moving out of the corner, so the control value should be higher;
+0.02 a3 is assumed by the engine to be worthless in a static evaluation, but that pawn is the winning move and it's played next.
So the pawn that is controlling b4 is not important at all.

To prove the logic of the controlled key squares, we need to play the game out. If the logic is good, it will stand the test of time.

Mate in #19 Depth 63/99
3. a4 Kb8 4. a5 Kc8 5. Bf3 Ng6 6. Bxb7+ Kd7 7. a6 Ne5 8. a7 Ke6 9. a8=Q Nd7+ 10. Kb5 Nf6 ...

Let me know what you think about the square control score and if you have any suggestions on how to quantify piece coordination.

I'm not sure why my opinion matters on this... https://hxim.github.io/Stockfish-Evaluation-Guide/ (see "Mobility bonus" etc.)

Thanks for the link, your presence and opinions matter to me.

Recognition precedes control: You can’t properly assess the control of a square or piece until you understand its strategic importance in the current position. A master once told me: Try simplifying the position by ignoring a piece like it was off the chessboard and then reassess the position. Identifying what needs to be controlled or removed is crucial for winning games.

Not all squares have the same value. As plans evolve, so does the control of important squares. Controlling key squares, as well as maintaining mobility and activity (outposts, threats, promotions), is vital. Control often leads to more mobility for your pieces while limiting your opponent's space. It's important to ponder on which squares are worth controlling.

After looking at the link in the prior post I came to the conclusion that engines don’t assign fixed numerical values to "direct," "indirect," "potential," or "pawn" control. Instead, they evaluate the impact of control by looking at many things like:
King safety and activity;
Piece mobility and space;
The pawn structure’s strengths and weaknesses;
The quality of piece placement on key squares (often a result of control);
The number and value of pieces attacking or defending critical squares.

So to quantify control, we need to see the full picture.

@TotalNoob69 said in #7:

I believe that subtracting the attacking value of the two sides loses information. I would use red and blue for the two sides, ...

This idea could be expressed as a pair of numbers, written in a tiny size. For the oponent in red color in the upper right corner. For the player in blue in the bottom right of contested squares.

This representation is simpler and could work with several color schemes; has been used with the periodic table of the elements; don't need to print data on uncontested square and don't need to take in account the order of the battery.

I had used some times this scheme with correspondence games, because some times felt confused keeping in mind the value of pieces and counting moves.

I really enjoyed this! The idea of quantifying square control in such a structured, algorithmic way is brilliant — especially because it touches on something we all “feel” intuitively during a game, but rarely try to measure.

What really stood out to me is the potential for visualizing this as a heat map. That kind of representation would make positional control immediately digestible, even for less experienced players. Super inspiring — thanks for sharing this approach!