mathematics of relative chess piece value

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?

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

Edit: Sorry, I seem to have mistook post #2 for the OP. I think most of what I've written is still cogent. Let's leave it there.

I'm in broad agreement with most of what you've said. Can't comment about "Archbishop"s as I don't know that piece. I think the piece values are not far from the truth, except (always position will have final say):

1. I prefer two Rooks to a Queen.

2. I tack a little (indefinable) value onto the Bishop pair in my calculations.

Generally, I use: 1 3 3 5 9

It seems sufficient for my level of play.

I said dynamic value of the king

In my point system, the absolute piece values of two rooks and the queen are equal.

Two rooks-4.5+4.5=9

Queen-Rook+Bishop-4.5+3.5+1=9

But the relative piece value is different when they are lined up.  A point is added to compensate for the combined strength.

Two rooks-4.5+4.5+1=10

Quen-Rook+Bishop-4.5+3.5+1=9

I continue the post at:

http://www.chess.com/forum/view/general/mathematics-of-relative-chess-piece-value

Regards

Why do that?

Why not post a subject once?

This is an interesting question that I have thought about a bit.  Notice that a pawn, not a rook pawn, on an open board attacks, can capture on two squares,  The king can attack eight squares.giving him a value of four. The knight hits eight, giving him a value of four also; and the bishop, in the center, attacks 13 squares, seemingly giving him a strength of six plus. 

However the number of squares covered by these luminaries decreases rapidly near the sides and corners of the board,  Also our friend the bishop only moves on squares of one color.

The rook always has the same number of squares, 14, anywhere he is placed; and the queen equals the power of the rook plus the bishop, yielding 27, which is also divided and trimmed by the edge of the board.  These numbers give one roughly the same values, 1  3 3 5 9.  Specific properties of an individual piece, whether it is active or impeded, or again the rules of checkmate, are irrelevant to the values of the pieces, 

It is interesting to apply this appoach to Fairy Chess values and to chess and other varients as in Games Ancient and Oriental.  Good discussion,

Because the argument covers both forums I suppose .

You're missing my point.

Why make two forums the same? I call that spam.

I think that the value of pieces should be estimated using as an indicator the number of moves that can be made legally with that piece (or, more realistically, the number of moves that can be made without hanging it). For a better evaluation we must take into account the fact that even if now a piece can't move (0 value) it can move soon if, let's say, we move a pawn.

To exemplify, in this position the white Bishop is worth 0 points (pawns) because it has no moves and white can't do anything to change that.

In the second diagram the Queen is certainly not worth 9 pawns because it now has 1 move and it's a bad one. And there's no chance to chang

I'd put the Arch bishop at a 7/8. It is capable of checkmating without a supporting piece! That's pretty cool. For this diagram  pretend the Bishop is the Archbishop

As for relative value of pieces, I'll take the four points of this pawn and knight rather than the 5 points of a rook during the middle game almost every time

but the bishop still defends 2 squares like a pawn so its value IMO is 1

The queen in the corner controls 2 squares too (defending the king is meaningless)

so she should be 1 too in that diagram

About the archibishop we were discussing in the other post that a piece who sums the properties of two lesser pieces receive a +1 bonus value per sum because of the positional advantage

Ex: Queen = Rook(5) + Bishop(3) + bonus(1)=9

     Archbishop = Knight(3)+ Bishop(3) + bonus(1)=7

     Amazon= Rook(5) + Bishop(3) + Knight(3) + bonus(2)=13

Read this article: http://home.comcast.net/~danheisman/Articles/evaluation_of_material_imbalance.htm

 I have made a systematic investigation of the relationship between the number of squares covered by a piece, and its empirical value. (For my method, see the tread on this same subject in the General section of this forum.) Especially for sliders (e.g. R, B, Q) the relation is still somewhat illusive. But fow short-range leapers (e.g. N), a reasonably coherent picture emerged:

The value of a piece, that (away from any edge) covers n squares (in an unblockable manner, like for a Knight n=8) is about (30 + 5/8*n)*n (in centiPawn). So for n=8 this would be around 280. Pieces with only 4 targets (the Shatranj Queen) are worth ~130, with 12 targets around 450, with 16 targets around 640... It worked pretty well upto 24 targets (the 'Lion', jumping directly to every square in a 5x5 area), which is around 1150.

This can be corrected a little bit for some global properties of the particular combination of target squares: slowness (K and N both have n=8, but the dynamic value of K is suppressed by the fact that its largest step is 1), color-boundedness, mating potential, irreversibility. These are minor effects, though.

Forward moves turn out to be worth about twice as much as backward or sideway moves. For divergent pieces, (which capture different from the way they move, such as the FIDE Pawn), captures turn out to be worth about twice as much as non-captures. (Note that this theory does not apply to the Pawn, which derives most of its value from the fact that it promotes to a super piece, not from its tactical abilities.)

This gives an indication why the situation with sliders is so complicated: a Rook might have a lot of moves, but only 25% of those go forward (against 50% for the Bishop). Well, a bit more in practice, because you tend to operate from yur own half, during most of the game. But what is even more important, is that most of these moves are non-captures. Only at the end of each ray there can be a capture. A Knight can have 8 captures, but R and B at most 4. And the captures count the most! This explains why a Nightrider (a multi-step Knight) is stronger than a Rook (by ~0.5 Pawn), despite the fact that its maximum coverage is smaller than that of the Rook (12 vs 14), and it has no mating potential: it can attack along 8 rays, 4 of them looking forward.

I have a book by Hans Berliner, I can't think of the title, that gives a very detailed explanation of the value of the pieces.

Sure, explanations are plentiful. You can dream uphundreds that get the values of the orthodox pieces, which were given in advance, exactly right. And then, when you apply them to onorthodox pieces, of which the originator of the explanation did not know the value in advance, they are totally off. (If they can be applied at all, and do not lack essential values for fudge factors that were magically chosen to give exactly the right value for the known cases.)

So they really do not explain much. They just observe, and try to express the observation into a formula without any predictive power outside the observations.