I'm not sure as to how the "classic" piece values were found (probably just estimates of what exchanges worked and which didn't).
I think this might be helpful to you: http://home.comcast.net/~danheisman/Articles/evaluation_of_material_imbalance.htm
mathematics of relative chess piece value
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Jul 14, 2010
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GM Larry Kaufman describes how he tried to figure out the value of the pieces in this article.
Why not try to find the geometric value of the pieces to determine their true strength?
Of course, a rigid adherence to those values is a folly. A knight is worth 3? That depends. A knight on the sixth rank in a good supported un-attackable outpost is worth more than that. A bishop is worth 3? In a closed position, it is worth less. In a pair in an open position, it is worth more. In some situations, a bishop as part of a pair may be worth as much as a rook, imo.
Trying to objectively quantify the value of the pieces is an attempt to avoid examining the position for its unique qualities, which is not a good idea.
I red that article, but I was curious to know if there are mathematical methods to estimate those values on a 8x8 chessboard.
For example what are the values of
a)the Fool of Omega Chess (The Fool assumes the identity of the last piece your opponent has moved.)
b)Capablanca's Archibishop (Bishop+Knight) and Chancellor(Rook+Knight)
c)The pieces in japanese and chinese chess (always on a 8x8 checkboard and no special rules as droppings)
Regards
Flamma_Aquila wrote:
Trying to objectively quantify the value of the pieces is an attempt to avoid examining the position for its unique qualities, which is not a good idea.
When did anyone suggest using the values to avoid examining the position? Read Kaufman's article, he talks about how the characteristics of the position change the value of the pieces, for example, rooks gaining in strength after a few pawns are traded.
Conquistador wrote:
Why not try to find the geometric value of the pieces to determine their true strength?
What do you mean exactly?
For example if you consider the endgame fool+king vs rook+king if you estimate that the opponent will move 50% the king and 50% the rook the fool strength should be about (4+5)/2=4,5
If you take the archibishop = (knight + bishop) its value should be 3+3=6 plus a little bonus like the queen so let's say its relative value will be 7
these of course are mere speculations
I mean by the pieces are as strong as their movements, their blind spots. and the type of position. For example, a rook can move orthogonally, but has weak points at the diagonals. A Bishop can move diagonally, but has weak points orthogonally. Ordinarily these pieces would be equal, but because of the nature of standard chess, the rook is generally more powerful. If the board was a different shape, it could favor the bishop. For example, instead of a square board, a diamond board generally would favor the bishops.
The queen would be as strong as both the rook and the bishop plus as additional point for having the powers of two pieces.
In my mind, a combination of pieces should be given a point to compensate for their clearly superior positional power. An Amazon (Rook+Bishop+Knight) would be given two points for combining three pieces.
Queen-Rook+Bishop-4.5+3.5+1=9
Amazon-Rook+Bishop+Knight-4.5+3.5+3+1+1=13
Rook-4.5
Bishop-3.5
Knight-3
The rook, bishop, knight, and other pieces are what I call base pieces because they are the basis of the power of the stronger pieces.
Conquistador wrote:
I mean by the pieces are as strong as their movements, their blind spots. and the type of position. For example, a rook can move orthogonally, but has weak points at the diagonals. A Bishop can move diagonally, but has weak points orthogonally. Ordinarily these pieces would be equal, but because of the nature of standard chess, the rook is generally more powerful. If the board was a different shape, it could favor the bishop. For example, instead of a square board, a diamond board generally would favor the bishops.
The queen would be as strong as both the rook and the bishop plus as additional point for having the powers of two pieces.
In my mind, a combination of pieces should be given a point to compensate for their clearly superior positional power. An Amazon (Rook+Bishop+Knight) would be given two points for combining three pieces.
Queen-Rook+Bishop-4.5+3.5+1=9
Amazon-Rook+Bishop+Knight-4.5+3.5+3+1+1=13
Rook-4.5
Bishop-3.5
Knight-3
The rook, bishop, knight, and other pieces are what I call base pieces because they are the basis of the power of the stronger pieces.
So everytime you create a piece which is a combo of two lesser pieces you gain +1 bonus (wonder why...) Ex:queen = rook(5) + bishop(3) + bonus = 9
If you take as an example the bishop pair : two same pieces that are not redundant give a bonus (+1/2 bonus for the bishop pair) Two bishops of the same color will never cross each other so they aren't redundant by default.
If you generalize you should consider in the evaluation the size and the shape of the board and how much it is crowded by pieces
In a 10 x 10 board for example the bishop value should be significantly higher than the knight value
Regards
I do not think the rook is strong enough to be given a 5, rather a 4.5. The bishop is geometrically similar to the rook so its value should be the same, minus a point to compensate for the position, so 3.5. The knight is a short range piece that is vulnerable to any point blank attack, yet it can move with an L shape over and around pieces. I give the knight a 3 to compensate for the weakness to the rook and bishop.
panderson2 wrote:
Conquistador wrote:
I mean by the pieces are as strong as their movements, their blind spots. and the type of position. For example, a rook can move orthogonally, but has weak points at the diagonals. A Bishop can move diagonally, but has weak points orthogonally. Ordinarily these pieces would be equal, but because of the nature of standard chess, the rook is generally more powerful. If the board was a different shape, it could favor the bishop. For example, instead of a square board, a diamond board generally would favor the bishops.
The queen would be as strong as both the rook and the bishop plus as additional point for having the powers of two pieces.
In my mind, a combination of pieces should be given a point to compensate for their clearly superior positional power. An Amazon (Rook+Bishop+Knight) would be given two points for combining three pieces.
Queen-Rook+Bishop-4.5+3.5+1=9
Amazon-Rook+Bishop+Knight-4.5+3.5+3+1+1=13
Rook-4.5
Bishop-3.5
Knight-3
The rook, bishop, knight, and other pieces are what I call base pieces because they are the basis of the power of the stronger pieces.
So everytime you create a piece which is a combo of two lesser pieces you gain +1 bonus (wonder why...) Ex:queen = rook(5) + bishop(3) + bonus = 9
If you take as an example the bishop pair : two same pieces that are not redundant give a bonus (+1/2 bonus for the bishop pair) Two bishops of the same color will never cross each other so they aren't redundant by default.
If you generalize you should consider in the evaluation the size and the shape of the board and how much it is crowded by pieces
In a 10 x 10 board for example the bishop value should be significantly higher than the knight value
Regards
I would consider that more of a relative position, rather than the relative piece value, so to consider a point value for that is a different matter.
Now if you take in account the mobility of a piece in a position, then you can calculate their relative piece value. What I was refering to earlier was an absolute piece value.
The Paladin by my absolute piece value is stronger than 6 points.
Paladin-Bishop+Knight-3.5+3+1=7.5
It has still to be addressed why 1 knight and/or bishop = 3/4 pawns
One fairy piece are the Berolina pawns who move diagonally and capture frontally.
I wonder what their value would be.
Interesting debate since so far.
Capa_Kaspa wrote:
good question... always wanted to know what was the basis in giving such 1-3-3-5-9 values.one explanation maybe the advantage or disadvantage one gets after trading(though this is ENTIRELY dependent on the position)
-2 rooks for a queen is almost an even trade(unless rook activity is hampered due to locked posns and absence of open files.So 2 rooks (10) are on par with a queen(9).
-pair of bishops are slight advantage over a rook so 2 bishops (6) are about par with a rook
-sometimes it is said that knights are slightly inferior to bishops, but again this is heavily dependent on position.knights get 3 simply because of their ability to jump over, thus being greatly beneficial in locked pawn formations.
-assigning a 4 to the king is gross injustice ! it is simply infinity.Collect as much material as u can, but u lose the game if yr king cant get out of a mating net ! It is this sinle fact that sometimes break all chess rules,no principle of chess is ultimate because of the fact that the king must be protected at all costs.
-as for Capablanca's Archbishop, in my book it gets 6, as it can play in both locked positions(as knight) and open positions (as bishop). hell it can switch colours as bishop too ! thus acheiving play on the entire board , nd not just on one half of it, as in case of a bishop.For the rest of the pieces, i havent got a clue!
The dynamic value of the king is 4 . The exchange value of the king is infinity.
In the endgame the king is as dangerous as a minor piece
Standard chess pawn-1
Berolina chess pawn-.5
The berolina pawn can only capture one direction while the standard pawn can capture in two directions, so it is slightly weaker. The extra direction of movement does not compensate for the extra blind spot.
I've always wondered what a Shogi Gold (General) is worth. For those of you not familiar, it moves like a king but no movement on the rear diagonals...
i.e.
A gold general can move one square orthogonally, or one square diagonally forward, giving it six possible destinations. It cannot move diagonally backward.
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Conquistador wrote:
Standard chess pawn-1
Berolina chess pawn-.5
The berolina pawn can only capture one direction while the standard pawn can capture in two directions, so it is slightly weaker. The extra direction of movement does not compensate for the extra blind spot.
Thanks
One speculation of mine.
The more powerful pieces are present in the initial setup the more White has the advantage because it has the initiative Ex: In a chess game with Amazons instead of queens white on average will win more.
The smaller the checkboard the more white wins for the same reason. Probably the 8x8 chessboard is not large enough to minimize the first move white advantage
Conquistador wrote:
Standard chess pawn-1
Berolina chess pawn-.5
The berolina pawn can only capture one direction while the standard pawn can capture in two directions, so it is slightly weaker. The extra direction of movement does not compensate for the extra blind spot.
Imo, the Berolina pawn should be at least = if not stronger to a normal pawn. I think by capturing forward, one needs to really be careful in defending. This type of pawn is very dangerous and, due to the nature of chess, it would completely change the game first of all and your opponent needs to be careful in not putting a random piece directly in front of it-I think blocking the pawn on both diagonals is harder than blocking it when it moves forward. Not to mention, the Berolina pawn can change its direction (although not in the same move) so it can be quite a beast. This also increases its mobility which should contribute to the value.
But to be honest, I think a game of chess with the Berolina pawns needs to be played first before coming up with a value-(This is how people like Kaufman came up with the value of normal chess pieces anyways-by taking a sample of a large number of games)
Everybody knows that the values of chess pieces are 1-3-3-5-9
the dynamic value of the king should be about 4
My question is: how these values were discovered? Just empirically?
If I design a fairy chess piece , given its properties can I estimate its relative value?