mathematics of relative chess piece value

Actually the smaller chess board would weaken an Amazon as their powers cannot be used effectively.  I do not believe the first move advantage should change with more pieces.  It would only change the complexity of a position.

It's a powered pawn with 6 capture squares vs 2 capture squares of the ordinary pawn so The gold general value should be 3.

AFAIK even if  pawn-like pieces cannot go backward this liability is compensated by the promotion so no special maluses are applied

I would give the gold general 2 points because it is weaker than the Bishop or the knight.

A consort or a commmodore piece can move just like a king, yet is still weaker than the bishop as it is only a short range piece.  I would give it 2.5 points.

The consort/ commodore is worth more

from wikipedia:

"It is not meaningful to assign a value to the king relative to the other pieces, as it cannot be captured or exchanged. In this sense, its value could be considered infinite. As an assessment of the king's capability as an offensive piece in the endgame, it is often considered to be slightly stronger than a bishop or knight – Emanuel Lasker gave it the value of a knight plus a pawn (i.e. four points on the scale of Chess piece relative value) (Lasker 1934:73). It is better at defending nearby pawns than the knight is, and it is better at attacking them than the bishop is (Ward 1996:13)."

It would be curious to know the values of Baroque/ultima chess.

they all move like the queen except the pawns who move like a rook but their

capture methods are different:

The Long leaper captures like checkers but in all directions not diagonally only: there is the multi capture variant and the variant were there can be only one capture

The Withdrawer  captures by moving directly away from an adjacent piece.

The Coordinator  captures any opposing piece that is on either of the two squares found at a) the intersection of its own file and the King's rank, and b) the intersection of the King's file and its own rank

In another variant the Coordinator captures passively when the king moves

The Chamaleon captures as the piece it's going to capture

The Immobilizer does not capture anything, but immobilizes all adjacent enemy pieces.

The pawns - or pincers, as it were - move like standard chess Rooks. A pawn captures any opposing piece horizontally or vertically between the square to which the pawn moved and a friendly piece

In other variants the pawns can capture passively

Anther cool piece is the Swapper which simply swap location with any other piece friendly or enemy without capturing but only if an orthodox queen could capture it

Probably the only way to figure out which are their exchange dynamic values is to empirically run countless simulations in the computer changing the piece evaluation numbers and see what is the optimal arrangement.

No the consort or commodore should not be worth more than a bishop.  When you are talking about dynamic potential, yes they would be stronger.  But in my absolute geometric value system, they are somewhat limited in their movements which results in their lower score.

What's the difference between dynamic potential and g.value system ?

Dynamic potential is the strength of a piece at a particular position while my geometric value system measures the absolute value of a piece in general based on its movement abilities.  So if you are looking at a particular position and say the bishop is stronger than the knight, then that is the dynamic potential.  My system would say that in general the bishop is stronger than a knight.

Now if you wanted to measure the relative geometric value at a particular position, than you can calculate the mobility of a piece.  More mobility=stronger piece.

Well if you're talking about the average mobility  here there are the values.

knight 2.4    
bishop 4    
rook 6.4    
queen 10.4    
king 3

of course AM changes with the size and shape of the chessboard. I tried the cylinder chess variant where pieces on one side can reenter on the other side of the board (left/right) and there knights are crap compared to bishops and other heavy pieces

Still It's interesting how the knights manage to be so useful despite their substantial inferior mobility.

panderson, how exactly did you calculate average mobility? I'm guessing it's something related to how many squares a piece can get to in one move averaged over all starting squares.

Why not how many squares a piece can get to in two moves? Often this is important. A piece goes from starting square to destination square to attack something on a third square.

I think it doesn't do the knight justice compared to the bishop to come out with such less mobility. The bishop is severely limited in mobility since it can only cover half the board.

I got it from wikipedia

http://en.wikipedia.org/wiki/Chess_piece_relative_value

 Probably you obtain it by calculating how many squares in average a piece can move  on an empty board and then you make a proportion to the AM of the pawn but I'm not sure about it.

Of course since the board is empty these are endgame values I suppose.

For a real pedestrian calculation with as much value as any of the others (i.e. none) you can simply count the number of squares the piece can influence when positioned on one of the four centre squares (excluding the one it's currently on) and then baseline that against a value of 1 for a pawn:

Just like any other methodology, completely worthless without the context of the position being taken into consideration.

The evaluation functions are the core of Chess . Afaik Larry Kaufman  is in primary charge of the evaluation function of rybka. But the pieces values alone are not sufficient. For example how much worth are two connected pass pawns? A black square domination? And a Rook on 7? I saw in one kingcrusher youtube video Topalov vs Aronian  exchanged two rook vs two knights and obtained as a compensation two connected pawns + bishop pair winning easily

regards

Just a small correction, the pawn only controls 3 squares.

Yes, a good point.

One interesting thing I've noticed playing some chess variants is that the piece value change.

Ex: In Chess variant with the Shogi drop rule

a) you really never reach the endgame because the pieces are recycled

b)knights are deadly

c)pawn can be dropped to the seven rank so they are deadly too

So the values are about Q>K>P>B>R

In the cylinder variant where column a and h are connected, bishops kick ass and knights suck so the values are about

Q>B=R>K>P

One can do a lot of (very) educated guessing about the value of Chess pieces. To my surprise, when I actually started determining those values empirically, it turns out that even the most educated guesses are more often than not completely wrong!

My method for determining piece values is conceptually quite simple: I wrote a computer program where the way the various pieces move is configurable by the user. I then let this computer program play against itself, starting from positions with a certain material imbalance. E.g. to find the value of the Amazon (Q+N), I replace the Queen of one side by an Amazon, and delete one of its Knights in compensation. I then play several hundred games. To average out particular positional effects, and get more game variety, I shuffle the opening positions in a Chess960-like way. And I eliminate the white advantage by playing each setup twice, with mirrored colors.

In the end the match result tells me who had the advantage. For Z (=Amazon) versus Q+N the game is almost exactly balanced (50% result). While an advantage of a Pawn (pawn odds) leads to a 68% score. I can verify that by playing Z vs Q+B, which indeed gives the expected result (about 59% for the Q+B side if the Bishop was part of a pair, equality otherwise). If the result is very unbalanced (e.g. when I play Z vs Q+R), I can compensate with extra Pawn odds in favor of the disadvantaged side (i.e. make it Z+P vs Q+R).

So consider this debunked: combining two pieces does not always produce a synergy, as it so clearly does for Q=R+B, where the synergy is nearly 1.5 Pawn. When I delete these pieces from the opening position, indeed the side with Q wins quite heavily.

It can work the other way to. Replace the Queen by an Archbishop (=B+N compound), and let the Queen side face additional Pawn odds (so Q vs A+P). Turns out A+P have the upper hand. The difference between Q and A value is less than a Pawn! I verified this in almost all possible material combinations (e.g. Q vs R+R and A+P vs R+R, Q vs R+B+P and A vs R+B) and it stands all tests: in a context of normal Chess pieces, A+P are better than Q. So if B=N=3 and Q=9, A>8 (about 8.25, to be exact). Debunk 7... The synergy is enormous here, nearly 2.5 Pawn. I was at a loss to explain why. Many programs that can play Capablanca Chess do not know this, and squander their Archbishops by tradng them for inferior material, counting themselves rich! And then it is really funny to see how the Archbishop rips them apart!

Debunk #3: the effective value of K=4. Not true! A non-royal piece, moving as King, (known as Commoner or Man, M), clearly is inferior to Knight. (Despite its ability to mate. Such an ability seems to count for very little.) If N=3, M~2.75.

For those who want to try for themselves: The program (Fairy-Max) is available as part of the WinBoard package. (WinBoard is a popular Chess user interface, which, amongst others, allows you to pit two Chess programs against each other.)

You misunderstand the meaning of the concept "piece value". It is defined as the _average_  over all positions. So by definition it is not dependent on the particular position just like the average weight of an American is the same for you as it is for your neighbor. Positions like the one above, where the Bishop had no moves, do hardly anything to suppress the average, as the are even statistically unlikely when you would randomly set up the pieces. Let alone in real games, where people would try their best to avoid them. So it is of little importance that a Knight is extremely poor mobility in a corner. You just don't go there...

Similarly, positions where you can checkmate in one are too rare amongst all Chess positions to have any impact on the average.

I think the piece values are a bit more than just academic. For instance, if you did not know a Rook was more vauable than a Bishop, you would probably do unsound exchange sacrifices in the majority of games. Just because in the opening / early middle-game a Bishop is almost always more active than a Rook. And you would not perform very well.

The point of the piece values is that they allow you to say something about your chances, which is true more often than not, without having detailed knowledge on the exact position. Which is very convenient, if that position is so far ahead that it could be almost anything, depending on what your opponent will play.

Of course I am not saying that piece values are the only term that should go into evaluation of a Chess position. King Safety, mobility, Pawn structure etc. are all known to be very important as well. But the piece values are a very useful aid in what trades to strve for, and which to frown upon (i.e. only consider them when they have clearly identifiable massive compensation in positional terms).

E.g. thinking that an ArchBishop is worth 6, will get you toasted in any real game quite quickly, because you tend to trade it for B+N plus some relatively minor other compensation (e.g. to break the B-pair)...