mathematics of relative chess piece value

 Those values IMO are useful because it's true that the position determines the piece value, but Chess is dynamic so positions change and you don't know how the game will be after 20 moves so you have to have an idea of what is their average value.

The value of the pieces is meant to use only as a rough guide in determining which exchanges are benificial and which are not ... if you let such guides/rules become like unbreakable dogma it will hurt you more than help you in this game of kings and king of games !   Yes, there is dogma in chess too !

 So one definition of pieces value is how they would handle the endgame if they were present after all trades had been done. Maybe some value adjustments have to be done to take in account their activity in the earlier more crowded phases of the game (were knights excel for example) .

It's weird that the ArchBishop is so synergetic and almost as strong as a Queen: there's something going on.

In Chess the Bishop pair is worth 1/2 pawn because the Bishops complete each other. I wonder what kind of synergies you could have with more than a couple of pieces of the same type (Ex: three Rooks) .

About the King value: are you saying that Lasker was wrong ?!

I'd politely ask if you have a list of values of the various fairy pieces even considering different topology or rules Ex: Circle Chess, Absorption Chess etc...

There is often reason to sacrifice rook for knight or bishop. Dulling an attack(He is attacking with his knight and other pieces, you take it), building an attack(If the knight is in a strong defensive position), control of the center(To prepare an attack or so, a slightly uncommon thing.), simplifying an obviously won position(2R + 3P vs 1 N + 1B is better to trade off the pieces and queen a pawn), ruining his pawn structure(Making a triple or double pawn, bringing his pawn onto a worse file) and ruining your opponents position(Bringing his king into the center, the knight is a backbone of his position) are all good reasons for the sacrifice in certain positions. 

True, but all roads lead the player to the endgame and there what the pieces are capable of doing on an open board matters a lot. If you have material superiority don't you try to force exchanges (expecially the Queens) on your opponent?

what are the cannon, soldier, and horse worth in chinese chess

oh oh oh I'vew got this one, the pieces are worth e=mc2 where m is the mass of the piece.  Good rule of thumb, buy heavy chessmen...

 or if you put them in a 8x8 chessboard

In Chinese Chess, piece values are a bit tricky, because an (unpassed) Pawn is often worth less than a tempo. But the rule of thumb is Rook=10, Cannon=5, Horse=4.5, unpassed Pawn=1, passed Pawn=2-3 (depending on ts distance to the enemy Palace). It is Difficult to put a value on Advisor and Elephant, as they can only be used in defense. Initially 2 is a reasonable approximation, but when the opponent does not have any attacking material left, they obviously are worth nothing. (Well, A could still be worth something as Cannon mount. But if the opponent has Cannons and no Rooks, A often become a liability, as he will use them as Cannon mount, and you cannot get them easily out of the way.) In KAAEEKP you are NOT ahead, despite your '8' vs his 2-3.

In a FIDE-Chess context, opening value of the Cannon is a little below the Knight. Two Horses almost exactly balance a single Knight. I never tested a Soldier. But the Wazir, which moves one step orthogonal in all directions, and is thus superior to even a passed Soldier, is only about 1.4 Pawn.

Some other remarks:

That you can sacrifice your advantage to speed things up if you have enough to spare has nothing to do with piece values. It is just a general strategic principle. It is still a sacrifice. If you were not enough ahead when making it, you would draw or even lose because of it. Note further that, although the ratio of the piece values in Chess is surprisingly constant, the conversion of Reinfeld points to winning probability is not, and increases nearly threefold from opening to late end-game. So by greatly advancing the game stage, (e.g. trading Queens) you might spend some Reinfeld points (say, give an additional Pawn), and your remaining Reinfeld points advantage might translate to a larger winning probability (e.g. because you re now in a Pawn ending, one passer ahead...).

The attackers pieces won't be any more valuable, and the opponent's pieces would not be any less valuable if you have no compensation...

The given example is irrelevant to this discussion anyway. This is NOT an exchange sacrifice. It is not a sacrifice at all. You trade R+P for Q+B by plain tactics. Which is a good deal, exactly because the piece value of Q+B is far greater than R+P. If you did not know that, why would you trade? If I can play 1.RxB, QxR 2.RxQ I would not call it an exchange sacrifice either. I would just call it collecting an insufficiently defended Bishop. Piece values are a strategical concept. In a tactical situation strategic concepts are meaningless. Who cares if you are a Queen ahead, when it is hanging, and it is the opponent's turn? One only counts material in tactically quiet positions.

What I said pertains to the case where you have R+3P against B+3P, and trade R for B+P, without any of the Pawns promoting within your tactical horizon.

I had a game here where I was up two exchanges. During the game I tried to keep both of them, really got into trouble, and then did end up winning. Later analysis with the computer taught me something. The bishops can be extremely powerful, sometimes equal or even more powerful than a rook. But there's one thing that the rooks can nearly always do that the bishops cannot, and that is sacrifice the exchange. The computer kept recommending all sorts of exchange sacrifices, which still left me with a +1.5 pawn advantage (roughly).

Here is the game: http://www.chess.com/echess/game.html?id=6720436

EDIT: Annotating this game might make a good blog... (-:

Well, let me put it in another way then:

Suppose you could type a material composition into a computer, which would then go through its collection of a million stored PGM games, and randomly pick a position from it with the material composition you requested. And that your life then depended on winning that game, starting from the retreived position...

Would you ask for a position where you had R+3P, and the opponent B+3P? Or would you rather ask if you could get the B+3P in stead, because "piece values are meaningless outside the context on the board", and the computer refused to let you see the board before you picked?

I am struggling with the notion that I can "win" an exchange yet the net result is my spending the rest of the game trying to recover a decent position. As someone noted, it's more of a sacrifice by my opponent to my detriment. And, I thought I was being so clever.

In a similar situation, the computer analysis called me an idiot for not grabbing the material.

 This is a very important and very true observation. It is known as the "Elephantiasis Effect". In short, it comes to this:

Stronger pieces lose value in the presence of lower-valued pieces of the opponent, because the latter can easily interdict their access to part of the board. By acting on the assumption that a piece is better, you have to handle it such that it is protected from exchange with the inferior piece, which makes it less usefull.

This, for instance, explains why Q clearly has the upper hand against R+B, but 3Q on average lose against 2Q+R+B. (Or, more realistically, Q+C+A against C+A+R+B in a Capablanca game, as C and A are nearly as valuable as Q.) If in normal Chess you sacrifice your Queen against B+R, you are in dire straits. But early in a Capablanca game, you get the compensation that your remaining two 'Queens' now are bothered by fewer opponent lower pieces, and thus gain in power. This more than offsets the intrinsic loss

An extreme case of this effect can be seen in the starting position of the "Charge of the Light Brigade" tournament (left). White has 7 Knights, which totally crush black's 3 Queens.

Let's say this ... It's impossible to play good chess if you don't know the long term value of your pieces. when I have at least  +2p advantage the first thing I try to do usually is to exchange the Queens to cut my opponent legs  , don't you do the same?

Regards

Ozzie, that's a pretty amazing game from your opponent -- two bishops vs two rooks and I think he had decent drawing chances until 35...Qxe5?!

If there are not combinations available I simplify. Why do I have to take risks?

If my opponent is behind in material he has to seek complications not me.

Regards

What about the pre-computer grandmasters ... what did they think about the topic?

I believe that in empirical (or inductive) reasoning, you start by analyzing concrete data, for example the results of actual games, and look for patterns that may help to explain it.  Larry Kaufman or John Watson take what I would describe as an empirical approach to coming up with piece values by examining the outcome of games with certain balances of pieces.

In theoretical (or deductive) reasoning, you start with a bunch of logical assumptions about the nature of what you are looking at it, and try to build a model that you can then apply to specific cases.  It strikes me that older theorists like Peter Pratt (1803) or Paul Rudolf von Bilguier (1843) worked from a set of assumptions about what attributes of each piece were important rather than trying to induce the values from a set of games.

It's quite common even now for thread starters to argue that knights are stronger than bishops, say, based on their claimed attributes.  Perhaps the recent spate of waffle pictures was related to the lack of empirical evidence cited for such claims.

Let's say that if you have a mathematical way to deduct those avg pieces values, when you're a playing a chess variant like loop/crazyhouse chess, you can calculate them once , you don't need to play countless games to find them empirically.

Regards