What is the correct value of the pieces?

[Carotino]

On this subject, there are dozens of different evaluations, but no one was able to give a definitive answer.
The classic evaluation is:

P=100
N=A=300
R=500
Q=900

Steinitz (and Fischer!) believed that the bishop was higher than the knight, and gave it a bonus of 1/2 Pawn. Capablanca, however, gave the horse a penalty of half pawn, then:

N=300 and B=350 (Steinitz) or
N=250 and B=300 (Capablanca)

Kasparov, in a his book, gives to the rook a value equal to a bishop + 1 1/2 pawns...

In computer-chess but we have to be more precise, using the centipawns. Houdini uses the following evaluation which is, obviously, the result of a patient work of tuning (Among other things, the value assigned to the knight has a "strange" resemblance to the greek pi.):

P=93
N=314
B=321
R=491
Q=947 (995)

Other authors however, did not care about this aspect, and have included the traditional values ​​in their engines.
Theoretically, the functions of assessment should be able to assign the right value to each piece (relative to the position), but I think you have to start from values​​, as much as possible corrected.

I believe that these "axioms" of Steinitz are a good starting point:

- The bishop is slightly greater than 3 pawns.
- The knight is slightly less than 3 pawns.
- Bishop + 2 pawns are equal to a Rook.
- Knight + 2 pawns are slightly less to the Rook.
- The Queen is equal to approximately 3 pieces.
- 2 rooks are slightly higher than the queen.

Anyone have an idea to transform these rules correctly in numbers? For example, in the first rule, what is meant by "slightly greater"? How can you turn this into a number?
Half a pawn is a value that is too approximate, but then what is the correct one? 1/4? 1/8? 1/10?

[kbhearn]

The problem is you have to look at a lot more context.

There are for instance positions where due to lack of secure outposts, a R is better than B+N despite nominally being a bit more than a pawn worse in an average position.

RRvQP is another 'equality' that isn't really equal and can go either way (are the Rs coordinated with an open file? does the Q have a passed pawn? are kings exposed?).

In the context of a chess engine, the value of the pieces that turns out to be optimal will depend in large part as to how well the 'other factors' are handled by the rest of the evaluation. i.e. if in a RvNB position the R gets bonuses for an open board, and the minors get penalised for insecure structure, then the base value of the R can be on average lower than it would need to be without this knowledge, which will make it more accurate in the more typical positions where NB > RP.

This would also be noted in BvN (or BBvBN) where a bishop pair bonus combined with context-specific information about bad bishops, knights in closed positions, useful knight outposts, dominated knight, etc could help determine which is the better minor, and therefore bishop and knight would not need to be differentiated significantly on base value of the piece.

[ZirconiumX]

Ummm... CLOP anyone?

Taking a guess:

P=100
N=295
B=305
R=505
Q=915

Matthew:out

[velmarin]

Humans or Grand Masters give a material of the parts valuation, but valuation dynamic is in your brain, it is difficult to explain this assessment.

The math assessment following the guidelines given Roberto is not difficult.
But then comes the static judging the position and dynamics,
Everything has to be outfitted to be very effective.

Horses usually has a static valuation difference if they are centralized or cornered.
Torres is almost indifferent to his column.

[Houdini]

Three comments:
1) The value of the pieces evolve with the phase of the game. The values you give above for Houdini are not "piece values", they're just multipliers for a more complex function. They could very well have been 100, 50, 87, 26, 151...
2) Piece values are influenced by the PST (piece-square tables). If you increase all the PST values by 10, the "piece values" will decrease by the same amount.
3) Piece values very much depend on the dynamic factors in the evaluation. If you increase the mobility score for of a piece, its static piece value will have to be decreased to maintain overall balance.

All this means that there is no such thing as a simple "piece value" in Houdini (or in any other engine, for that matter). Comparing the multiplier values you've found in the Houdini executable to classical, "human" piece values is completely meaningless.

Robert

[hgm]

On this subject, there are dozens of different evaluations, but no one was able to give a definitive answer.

You never heard of Larry Kaufman, then?

Even discarding positional factors, piece values strongly depend on other material that is on the board. Otherwise it would be hard to explain why 3 Queens lose so badly from 7 Knights...

[Don]

Three comments:
1) The value of the pieces evolve with the phase of the game. The values you give above for Houdini are not "piece values", they're just multipliers for a more complex function. They could very well have been 100, 50, 87, 26, 151...
2) Piece values are influenced by the PST (piece-square tables). If you increase all the PST values by 10, the "piece values" will decrease by the same amount.
3) Piece values very much depend on the dynamic factors in the evaluation. If you increase the mobility score for of a piece, its static piece value will have to be decreased to maintain overall balance.

All this means that there is no such thing as a simple "piece value" in Houdini (or in any other engine, for that matter). Comparing the multiplier values you've found in the Houdini executable to classical, "human" piece values is completely meaningless.

Robert

Komodo, as well as other programs and probably Houdini too have different values for different game stages.

In Komodo we "center" everything so that the values are actually meaningful to us without too much interpretation. For example our bishop mobility is negative if the bishop is not very mobile and is close to zero when it has "average" mobility. Of course what is average is a judgement call but we try to make that judgement. So for mobility we have defined N squares of mobility to be the centering point - and that is different for each piece. Our piece square tables all have negative and positive entries also in order to capture this same concept. Whenever we implement a new evaluation concept we don't want to have to "lower" or "raise" the value of something else to compensate so we take care to "center" anything that has anything more than a trivial consequence on the evaluation function.

Our values for the opening are:

600, 3100, 3225, 4350, 9100

ending:

925, 3100, 3225, 5475, 9700

We also have other modifiers which have the effect of changing these such as the bishop pair, rook pair, "major piece pair involving queen" and a bunch of other stuff which also considers the effect of the pawns.

I doubt these values will work for just any program but I present them for what they are worth.

I do have a hypothesis on this too. If a program is terrible at handling some specific piece, should the value of that piece be the same as in a program that handled that piece well? Traditionally, programs did not handle their rooks well for example and the programs chances improved if there were no rooks on the board. Of course programs do well with queens too, due to tactics. In the chess club I once attended some weak kid there specializing in the use of knights because it works against weaker plays and he had a knack for playing them. For HIM, the knights were worth more than the bishops - and his entire play was based on deploying and attacking with the knights. He was not hard to beat, but you had to watch his knights and if you could win them his game fell completely apart. If he were a computer program I would say his knights NEEDED to be valued much more than the bishops.

The short version of this is that I believe the values of pieces should be somewhat based on the other heuristics - what really matters is maximizing your strengths and minimizing your weaknesses and for computer chess you have to literally assign numbers to those things.

[lkaufman]

There is a difference between ideal values for a human and ideal values for a computer. Don answered the question for a computer (Komodo). For humans, the best middlegame values in my opinion are given in my recent book ("the Kaufman Repertoire for Black and White"). They are N = 3.5, B = 3.5, R= 5.25, two bishops = 7.5 (i.e. half a pawn bonus), Q = 10. Probably a single bishop is a tad better than a single knight on average, so for more precision I could have said B = 3.55, N = 3.45, but I wanted to keep everything as a multiple of 1/4 pawn.

[Carotino]

Of course the "final value" of pieces is a dynamic process and depends on the evaluation function, PST, etc.. But I believe that a correct estimation of the initial value is still important.
The values ​​provided by Larry are good and have sufficient accuracy for humans, but for computer but for a computer I believe that we must use a finer tolerance of 1/4 of pawn. What? that is the question!

Another consideration: usually, we tend to give to the bishop a value slightly higher than the knight. In fact, in my opinion, should be the evaluation functions of the engine to change this (depending on the mobility, PST, controlled square, pawn structure, etc.. etc ...). I think the starting value should be equal... Or not?

[lkaufman]

Of course the "final value" of pieces is a dynamic process and depends on the evaluation function, PST, etc.. But I believe that a correct estimation of the initial value is still important.
The values ​​provided by Larry are good and have sufficient accuracy for humans, but for computer but for a computer I believe that we must use a finer tolerance of 1/4 of pawn. What? that is the question!

Another consideration: usually, we tend to give to the bishop a value slightly higher than the knight. In fact, in my opinion, should be the evaluation functions of the engine to change this (depending on the mobility, PST, controlled square, pawn structure, etc.. etc ...). I think the starting value should be equal... Or not?

As Robert explained, it doesn't matter what the "starting values" are, as long as all the other terms properly adjust for it. In Komodo we try to set the starting values to be the average values, just so that when we make changes we don't get misled. The average value of an unpaired bishop is slightly higher than the average value of a knight; currently we are using one-eighth of a pawn for this difference.