My interesting point system for value of pieces

I have no idea if this has been done before, but I thought the end result of this thought experiment had some pretty math.
Let’s say we count the number of squares a certain piece controls, and then do that for every square. (For example the knight would control 2 squares in the corner and 8 in the center). Over all 64 squares of the chess board, the average control for the knight would be 5.625 squares. We then have:
Pawn: 1.75
Bishop: 8.75
Rook: 14
Queen: 22.75
I made the pawn worth one point, so we divide by 1.75 and get the following:

Pawn = 1
Knight = 3.2 ~3
Bishop = 5
Rook = 8
Queen = 13

And look at that! By some luck all of the values are also in the fibonacci Sequence! Because of this, they get the interesting property that: knight + bishop = rook, Bishop + rook = queen. Now obviously the value of the pieces are not constant and a knight can be worth more than a rook in some positions. Just thought I’d share this all with you.

The common idea is different:

Pawn = 1, Bishop and Night = 3, Rook = 5, Queen = 9.

Yep, it is different than the widely accepted values. This is just deriving values based on average control over 64 squares

The original values are already based on the number of squares a piece can control on the opponents side of the board divided by two, with +1 for the Queen on stalemate potential.

I don't get the point.

Bishop and Knight* It's N and not K because the K is for king.

I mean when writing the moves.

Very interesting

I love it!
And the point is that the rule-of-thumb values are an extreme generalization.

In my experience, the truth is somewhere in between. A knight can be worth more than a bishop, especially in closed positions.

You should also do an analysis of how many accessible squares each piece has in different accepted openings (at the end of piece development) - I think we'll have another interesting average plus a distinction between LSB and DSB, queen's knight vs. king's knight, etc. Just an idea.

I also wonder where a knight is worth more than a rook (unless it's passivated).
And I figure a pawn is not 1.75 but 2 since there is never a pawn on the backrank (it ceases to be a pawn unless you prefer for it to underpromote to itself - because it loves itself or some other brainless reason).
But ok, this is mosquito talk.
Can you also include the king? It becomes an attacker later in the game..

@1
Your averaging over the whole board is not representative. You want your pieces in the center, where they are more active. That is why pawn, knight, bishop are worth more and rook, queen are worth less than in your system

Pawn: 3 -> 1 = 1
Knight: 8 -> 2.67 ~ 3
Bishop: 13 -> 4.33 ~ 3
Rook: 14 -> 4.66 ~ 5
Queen: 27 -> 9 = 9

This paper arrived in Table 6 at
https://arxiv.org/pdf/2009.04374.pdf 
P = 1, N = 3.05, B = 3.33, R = 5.63, Q = 9.5

Your Fibonacci coincidence is just that: a coincidence.

All values in 1, 3, 3, 5, 9 are odd.
That means that only positions with an even number of men can be materially balanced:
all positions with an odd number of men have a material advantage for one side.

@10
True, every piece (besides rook being the exception in that it always controls 14 squares regardless of where it is) is always more active in the center. I feel that some pieces just get worse than others once they reach a corner. For example, a Knight will control 2 squares in a corner and 8 in the center… 4 times weaker. It would also take that knight at least two moves in order to reach an 8 square control position from the corner. However a bishop will control 7 in a corner and 13 in the center… only about 46% weaker and get to its most active square in one move. How much this applies to actual gameplay is up to debate… I doubt anyone would actually willingly kill their knight and put it in a corner if they didn’t have to

Some pretty cool math there.

I don't really consider mathematical piece values when playing too much (except when considering exchanges between two different pieces), but either way, I still find the math here quite fun and thought-provoking.

@9
I should’ve clarified, but actually for the 1.75 pawn value, I disregarded the 1st and 8th rank and averaged over 48 squares. I didn’t include en passant because I didn’t want to think too hard about it haha. And as for the king, it does in fact become an attacker in the end game.

If we apply the same rules as before, the king has an average of 6.5625 squares over the entire board. Divide this by 1.75 and we get:

King = 3.75 ~ 4

Where as the the knight is ~3.2. Meaning that the King, according to average control, is roughly half a square stronger than the knight!

To show everything visually, I created a “heat map” for each piece, based on its power or control at each square. I then assigned a color ranging from 2 squares=purple and 27 squares=red. Link are included below:

Rook:
https://drive.google.com/file/d/1n_bT0x86ar6rcGIjpgn_Y4AviSTEAbKs/view?usp=drivesdk

Queen:
https://drive.google.com/file/d/1ED0_dP-HINwwzjQdbiPFMhJzcd007x5c/view?usp=drivesdk

King:
https://drive.google.com/file/d/1uSAhU2RTHHp5JghZoQPcL2muH1lxNGVO/view?usp=drivesdk

Knight:
https://drive.google.com/file/d/1dl307spJPc0TNDACMSHuLIMDRkRXU9ji/view?usp=drivesdk

Bishop:
https://drive.google.com/file/d/1G5tdP0-gm5nJcwvPyvVKaCzSYrHuqOal/view?usp=drivesdk

Thanks for the discussion everyone👍🏻

Edit: I miscalculated the average knight value. It’s actually 5.25. Dividing by 1.75 brings it to 3 points flat

So the knight is 3 and the bishop is 5?

This will result in some changes to my repertoire ...

Yup, it definitely makes it clearer why a bishop is generally more valuable than a knight.