Author Topic: How to calculate the value of a piece?  (Read 118 times)

ubersketch

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How to calculate the value of a piece?
« on: March 30, 2018, 08:38:07 pm »
This question has been on my mind for a while and it perplexes me. It would be interesting if we had a systematic way of assigning pieces values.

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Martin0

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Re: How to calculate the value of a piece?
« Reply #1 on: March 31, 2018, 02:03:23 am »
First you take one piece (preferably a weak one) and use it as a base to value the other pieces. You can set its value to whatever you want, but numbers like 1 or 100 are nice numbers. Then you estimate how much other pieces are worth compared to that piece and set their values accordingly.

When it comes to estimating those values, I believe the best approach is to use statistics from a large sample of games between players of equal strength. Naturally this is easiest done with an engine playing against itself to get that large sample size.

Trying to find a formula for piece values that would work for any variant on any board size is put simply not possible, but some methods are clearly better than others. No matter what you do it is not something that can accurately assign the values, it will always be an estimate. And the value for a piece in practice will always be relative anyway.
« Last Edit: March 31, 2018, 02:06:51 am by Martin0 »

HGMuller

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Re: How to calculate the value of a piece?
« Reply #2 on: March 31, 2018, 08:04:27 am »
Piece values are empirical quantities, which represent who has the advantage in a large collection of positions with a material imbalance (differing in the placement of the pieces). To measure them you can play a large number of games starting from an opening-like position (i.e. pieces behind a closed rank of Pawns), and observe how much the result differs from 50%.

As a caveat I want to mention the whole concept of 'piece value' is an approximation. In practice, how  much a piece is worth depends on what other material you have, and what material the opponent has. Usually the corrections are small compared to the 'base value' of the piece (i.e. the value averaged over many total-material combinations with the same imbalance). But in extreme situations, the effect can be large. Seven Knights consistently beat three Queens (when they all start behind a rank of 8 Pawns, and there are no other pieces (except Kings)). This is totally at odds with the value you would observe in games between more similar armies, where a Queen on one side can be almost perfectly balanced by 3 Knights on the other side. If you take the latter as evidence that a Queen is worth 3 times as much as a Knight, you would expect the Knight side in the 3Q-7N game (which I named 'Charge of the Light Brigade') to be two Knights short of equality. But in fact the Queens stand no chance at all. A more mundane example of cooperative material effects is the Bishop pair: a Bishop is worth about half a Pawn more when you already have a Bishop on the other square shade.

Then there is an entirely different question: "can the empirical piece values be predicted from how the piece moves by a comparatively simple calculation (i.e. without playing games)". This turns out to be very difficult. Lots of methods to calculate this, which sound very plausible, turn out to give values that differ very much from the empirical value. E.g. lots of guestimates are around for the 'Capablanca pieces', the RN and BN compounds (Chancellor and Archbishop). Because every Chess player knows that a Rook is worth two Pawns more than a Bishop, these guestimates almost always have the Archbishop value ~2 Pawns below the Chancellor value. While the empirical value difference is only a quarter Pawn. (If one side has two Archbishops and a Pawn, instead of the opponent's two Chancellors, all other pieces being equal, the Archbishops usually win.) The explanation for this that I currently favor is that moves to orthogonally adjacent squares provide some extra value to the piece. (This also explains why the Rook is worth more than the Bishop, even on a cylindrical board where both have the same number of moves.)
« Last Edit: March 31, 2018, 09:27:03 am by HGMuller »

GothicChessInventor

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Re: How to calculate the value of a piece?
« Reply #3 on: March 31, 2018, 09:18:04 am »
This question has been on my mind for a while and it perplexes me. It would be interesting if we had a systematic way of assigning pieces values.

I published a paper on this years ago. Google for "80 square chess" or download the paper here:

https://pdfs.semanticscholar.org/330e/6cada5af2191248e09b5910527744592e10d.pdf

ubersketch

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Re: How to calculate the value of a piece?
« Reply #4 on: March 31, 2018, 11:01:07 am »
I think it would be a good idea to create a piece with a value of 1.
It cannot move and cannot capture.
« Last Edit: March 31, 2018, 12:37:45 pm by ubersketch »

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HGMuller

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Re: How to calculate the value of a piece?
« Reply #5 on: March 31, 2018, 03:33:13 pm »
1 what? What is your unit? And why do you think such a piece would have value 1 on that scale? I would be inclined to think that such a piece would have negative value. It is worse than having a 'hole' in that place (i.e. a square inaccessible to both players). Because you cannot go there, but your opponent can, by capturing it. And I would expect a hole to be sort of neutral. Although that might depend on where exactly on the board such a hole is. E.g. a hole on the second rank might provide a good shelter for the white King, and would just represent an obstacle for black.

GothicChessInventor

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Re: How to calculate the value of a piece?
« Reply #6 on: March 31, 2018, 04:11:49 pm »
I think it would be a good idea to create a piece with a value of 1.
It cannot move and cannot capture.

I think it should have a value of i, where i squared = -1, and e raised to the power of 2 x pi x i = 1.

Asher Hurowitz

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Re: How to calculate the value of a piece?
« Reply #7 on: March 31, 2018, 04:57:36 pm »
If it had a value of one, and it cannot move, then a piece that can move therefore can technically move an infinite amount of more spaces than the base piece, and thus is infinitely more powerful, no matter the move, as 1 unit of movement is infinitely more than 0 units of movement.
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ubersketch

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Re: How to calculate the value of a piece?
« Reply #8 on: March 31, 2018, 05:55:56 pm »
If it had a value of one, and it cannot move, then a piece that can move therefore can technically move an infinite amount of more spaces than the base piece, and thus is infinitely more powerful, no matter the move, as 1 unit of movement is infinitely more than 0 units of movement.
Assigning it 0 seems to make more sense. I'll call this piece the zero piece or stone.
Also, we have to mention that the zero piece still may be useful as it can block attackers.

"Okay now I can't stop staring at that alien crotch." ubersketch 2k17

Martin0

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Re: How to calculate the value of a piece?
« Reply #9 on: April 01, 2018, 03:23:58 am »
Here is a stupid example trying to decide how much light squared bishops are worth compared to dark squared bishops.


In this specific case it would make sense to value the light squared bishops as 0. Black will also be able to checkmate white. We could try with a lot of different positions for the dark squared bishop and kings where the bishop is not directly lost and white is not directly stalemated and we would get the same result.

I don't think anyone would be stupid enough to try to make conclusions with piece values on this example, such as 1 dark squared bishop is worth more than 32 light squared bishops, but I sort of like it to show off how piece values really are relative.

GothicChessInventor

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Re: How to calculate the value of a piece?
« Reply #10 on: April 02, 2018, 12:47:07 am »
[Deleted by moderator]
« Last Edit: April 02, 2018, 01:05:07 am by Asher Hurowitz »

Asher Hurowitz

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Re: How to calculate the value of a piece?
« Reply #11 on: April 02, 2018, 01:06:37 am »
[Deleted by moderator]

GothicChessInventor

Thank you for your contribution but please try to keep criticism constructive. Every addition adds something to the forum!
Thanks!
Asher
Indeed it is certain that Chess Variants make me happy.

My Youtube Channel on Chess Variants
https://www.youtube.com/channel/UCb-lm6a8XZo0eqAhfR4QWOg

My Lecture at Yale University, aged 13
https://www.youtube.com/watch?v=smDO2Dpx5tg

My Youtube Channel on Rubik's Cube
Search"Polyhedral Paradise"

GothicChessInventor

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Re: How to calculate the value of a piece?
« Reply #12 on: April 02, 2018, 09:05:21 am »
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GothicChessInventor

Thank you for your contribution but please try to keep criticism constructive. Every addition adds something to the forum!
Thanks!
Asher

That was as plain as I could state it. See you later. Good luck with the forum.

chilipepper

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Re: How to calculate the value of a piece?
« Reply #13 on: April 06, 2018, 06:17:31 pm »
Here is a stupid example trying to decide how much light squared bishops are worth compared to dark squared bishops.

...(image removed)...

In this specific case it would make sense to value the light squared bishops as 0. Black will also be able to checkmate white. We could try with a lot of different positions for the dark squared bishop and kings where the bishop is not directly lost and white is not directly stalemated and we would get the same result.

I don't think anyone would be stupid enough to try to make conclusions with piece values on this example, such as 1 dark squared bishop is worth more than 32 light squared bishops, but I sort of like it to show off how piece values really are relative.
I like this example with all the white bishops, illustrating how piece value can be highly affected by what other pieces are on the board. Those white bishops might actually be a liability (negative value?) because they prevent White's king from moving onto a light square, where it would be more safe.

fyi#1: the diagram is a little bigger than necessary. I need to use ctrl + mousewheel to shrink the image, which also reduces the text size. But I'm a big fan of diagrams, illustrations, graphs, etc., so thanks for spending the time to show it. :)

fy1#2: not sure what comment had to be deleted, but I'm glad we have an attentive moderator. I've seen plenty of other forums where spam, rude comments, and insults can be very unpleasant for people interested in a good conversation. :)
« Last Edit: April 06, 2018, 06:20:16 pm by chilipepper »
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Martin0

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Re: How to calculate the value of a piece?
« Reply #14 on: April 07, 2018, 07:56:26 am »
I'm not exactly sure why, but I can't change the image size with something like

[img=500x500]https://i.imgur.com/Bzuk3TJ.png[/img]

I can change the size of the uploaded image (something I just forgot to do this time), but that would require some extra work.