Comments/Ratings for a Single Item

H.G.Muller:
| Larry Kaufman has applied the method on (pre-existing) Human
| Grand-Master games, to determine piece values for 8x8 Chess.

If I look at:
http://home.comcast.net/~danheisman/Articles/evaluation_of_material_imbalance.htm
he says things like:
  [...] an unpaired bishop and knight are of equal value [...], so
  positional considerations [...] will decide which piece is better.
and also see the section 'Applications'.

In other words, he is using a statistical approach merely as a point of departure for developing a theory which combines point values with other reasoning, such as positional judgement.

The values he gives though are interesting:
  P=1, N=3¼, B=3¼, BB=+½, R=5, Q=9¾
where BB is the bishop pair.

Hans Aberg:
| In other words, he is using a statistical approach merely as a point 
| of departure for developing a theory which combines point values with 
| other reasoning, such as positional judgement.
Of course. PIECE VALUES are only a point of departure. If you do a
principal-component analysis of the score percentage as a function of the
various evaluation characteristics, the piece values are the most
important components. But by no means the only ones.

If you read Larry Kaufman's paper, you see that he continues quantifying
the principal positional term responsible for the B vs N value, namely the
number of Pawns left on the board. And again, he uses the statistical
method to derive the numerical value, concluding that each own Pawn
affects the B-N difference by 1/16 Pawn. So his statistical analysis
doesn't stop at piece values. He also applies it to get the value of
positional terms.

The piece values Kaufman gets are very good. And, in so far I tested them,
they correspond exactly to what I get when I run the the piece combinations
from shuffled openings. For instance, I also find B=N and BB=0.5. Note that
Kaufman's Rook value is also dependent on the number of Pawns on the
board.

H.G.Muller:
| If you read Larry Kaufman's paper, you see that he continues
| quantifying the principal positional term ...

This is nothing new: this was done in classical theory. He just uses statistical input in an attempt to refine the classical theory.

| The piece values Kaufman gets are very good. And, in so far I tested
| them, they correspond exactly to what I get when I run the the piece
| combinations from shuffled openings.

You can sync your method against his values, to get piece value suggestions. But that is just about what you get out from it.

Hans Aberg:
| You can sync your method against his values, to get piece value 
| suggestions. But that is just about what you get out from it.

Define 'suggestions'. What I get are a set of piece values from which
you can accurately predict how good your winning chances are, all other
thing being equal or unknown. If other (positional) features are known,
you can factor those in in the same way.

This is by definition what piece values are meant to do. If the
'classical' system does 'something else', because it was obtained in
another way, they are simply no piece values.

H.G.Muller:
| Define 'suggestions'. What I get are a set of piece values from which
| you can accurately predict how good your winning chances are, all other
| thing being equal or unknown.

You do not get a theory that predicts winning chances, as chess isn't random. If the assumption is that opponents will have a random style similar in nature to the analyzed data, then it might be used for predictions.

It is clear that Larry Kaufman does not think of his theory in terms of 'x pawns ahead leads to a winning chance p'. You can analyze your data and make such statements, but it is an incorrect conclusion it will be a valid chess theory predicting future games - it only refers to the data of past games you have analyzed.

My take on the value of the pieces is for the following purposes:
1. If you want to do a universal build your own army variant, this allow you to see if the sides would be balanced.
2. When pitting one side against another, if the sides are unbalanced, this system should allow a balancing in points for handicapping reasons of the forces.

Rich Hutnik:
| 2. When pitting one side against another, if the sides are unbalanced,
| this system should allow a balancing in points for handicapping reasons
| of the forces.

Games where both sides have equal material are also unbalanced, as in general, there is an advantage to play the first move.

Hans Aberg:
| You do not get a theory that predicts winning chances, as chess 
| isn't random. If the assumption is that opponents will have a 
| random style similar in nature to the analyzed data, then it might 
| be used for predictions.
This is where we fundamentally differ, and it makes it completely
pointless to discuss anything in detail as long as we argue based on such
mutually excusive axioms. Chess as we play it is a game of chance, as
players don't have perfect knowledge, and thus randomly choose between
positions they cannot distinguish based on the knowledge they have. And
there is no logical necessity for the condition of similar randomness that
you impose. It is well known that the eventual distribution of  random
walker does not depend on the details of the steps he can make. Only on
the variance. (The central limit theorem of probability theory!) In
particular, it is an empirical fact that statistical analysis of
computer-computer games, as I did, produces the same winning probabilities
as analyzing GM games (as Kaufman did). So even if you were in principle
right (which I doubt) that sufficiently different nature of the randomness
would produce different overall statistics, observations then apparently
show that computers and Human GMs are not 'sufficiently different'.

| It is clear that Larry Kaufman does not think of his theory in 
| terms of 'x pawns ahead leads to a winning chance p'. You can 
| analyze your data and make such statements, but it is an incorrect 
| conclusion it will be a valid chess theory predicting future games 
| - it only refers to the data of past games you have analyzed.
For one, this is not clear at all, and very unlikely to be true. Anyway,
you cannot know what Kaufman thinks or doesn't. Fact is that he
translates the statistics into piece values in exactly the same way I do.
The rest of you statement is totally at odds with standard statistical
theory, which maintains that probabilities of stochastic events can be
measured to any desired accuracy / confidence by sampling. This is really
to ridiculous for words.

To Rich:
For balancing armies, piece values the way I derive them are exactly what
one needs. The pice values represent winning robabilities, and balancing
them will creae a game where the winning chances will be equal.

H.G.Muller:
| Chess as we play it is a game of chance...

The main point is that such a statistic analysis is only valid with respect to a certain group of games, as long as as the players stick to a similar strategy. The situation is like with pseudo-random numbers, where in one case one discovered that if the successive numbers generated were plotted in triples, they fell into a series of sloped planes. Such a thing can be exploited. So there results a circle of making better pseudo-random generators and methods to detect flaws, without ever resulting in true random numbers. A similar situation results in cryptography.

In chess, the best strategy is trying to beat whatever underlying statistical theory the opponent is playing against. When playing against computer programs this is not so difficult, because one tries to figure what material exchanges the opposing program favors and shuns, and then tries playing into situations where that is not valid. Now, this requires that the human player gets the chance of fiddling around with the program interactively for some time in order to discover such flaws - learning this through tournament practice is a slow process - plus a way to beat the computers superior combinatorial skills if the latter is allowed to do a deeper search by brute force.

| Anyway, you cannot know what Kaufman thinks or doesn't.

His stuff looks like all the other chess theory I have seen, only that he uses a statistic analysis as an input, attempting to fine-tune it. By contrast, you are the only guy I have seen that thinks of it as a method to predict the average outcome of games. You might benefit from asking him, or others, about their theories - this is how it looks to me.

You might still do values and percentages and display them as your analysis of past games in a certain category, but there is gap in the reasoning claiming this will be true as a prediction for general games.

I still can't see the point you are tring to make. If piece values cannot
be used to predict outcomes of games, they would be useless in the first
place. Why would you want to be an exchange or a piece ahead, if it might
as frequently mean you are losing as that you are winning? Fact is that I
OBSERVE that the piece values I have given below do statistically predict
the outcome of games with good precision.

If, according to you, that should not be the case, than apparently you are
wrong. If, according to you, that is not what 'good' piece values should
do, than that satisfactorily explains why you would consider your set of
piece values, which would cause whatever entity that uses them to play by
to lose consistently, in your eyes can still be 'good', and further
discussion could add nothing to that.

What you say about opponent modelling doesn't have anything to do with
piece values. Precisely knowing the limitations of your opponent allows
you to play a theoretically losing strategy (e.g. doing bad trades) in
order to set a trap. In general, this is a losing strategy, as in practice
one cannot be sufficiently sure about where the opponent's horizon will
be. So it will backfire more often than not.

And the datasets that are analyzed by me or Kaufman are not dominated by
players engaging in such opponent modelling. I can be certain of that for
my own data, as I wrote the engine(s) generating it myself. So I know they
only attempt theoretically optimal play that would work against any
opponent.

H.G.Muller:
| If piece values cannot be used to predict outcomes of games, they
| would be useless in the first place.

If one is materially behind, one knows one should better win the middle game, or win material back before coming into the end-game, unless the latter is a special case.

| Why would you want to be an exchange or a piece ahead, if it might
| as frequently mean you are losing as that you are winning?

This is indeed what happens if with programs focusing too much on material, or weak players starting with a piece ahead.

| Precisely knowing the limitations of your opponent allows
| you to play a theoretically losing strategy (e.g. doing bad trades) in
| order to set a trap.

Sure, this seems to be essentially to be the effect of a brute force search on a very fast computer.

| In general, this is a losing strategy, as in practice
| one cannot be sufficiently sure about where the opponent's horizon will
| be.

In human tournament playing, a top player either plays against opponents of lower horizon, or against well known opponents whose playing style has been well analyzed. In the first case, there is not much need to adapt playing as one can see deeper, but in the latter case, one certainly does choose strategies adapted against the opponent. Now, with computer programs, at least in the past, the GMs were pitted against program which they did not know that well, the latter which ran in special versions and on very fast computers when tried on humans. So humans do not get much of  chance to develop better strategies. But this may not matter if the strategy does not allow them to handle the theoretical faulty combinations used by computer by relying on a somewhat deeper search.

| Fact is that I OBSERVE that the piece values I have given below
| do statistically predict the outcome of games with good precision.

You only observe past events, not the future, and a statistical prediction is only valid for for a true stochastic variable, or in situations that continue behave as such. But don't worry about this:

If you find methods to duplicate the analysis by Larry Kaufman, and use that to compute values for various pieces on boards like perhaps 8x8, 10x8, 12x8, then it seems simple enough to modify engines to play with different chess variants (if protocols like UCI are extended to cope with it).

I think though the real test will be when humans play against those programs.

Fairy-Max is already able to play most Chess variants, and WinBoard
protocol already supports those variants. Many engines are now able to
play Capablanca-type variants under WinBoard protocol, some of them quite
strong. But as I already have accurate piece values, and my engines seem
to be significantly stronger than the competition in any Chess variant I
have bothered to configure them for, there is no incentive whatsoever to
do as you say. First build an engine that beats mine, then I might worry
when apparent misevaluations of a position correlate with the presence or
absence of certain pieces.

I have no interst in convincing anyone to use my empirically derived piece
values. On the contrary, if engine builders want to insist on using guessed
piece values that make their engines play losing Chess, I only applaud it,
as it means my engines will remain the best forever. If someone wants to
prove that there exists a set of piece values that works better, I
encourage them to do it (and then I mean by play-testing, rather than idle
talk or fanciful numerology). I am not going to waste my time on such a
wild goose chase.

H.G.Muller:
| Indeed, I plan to submit a paper to the ICGA Journal discussing the
| piece values and the empirical statistical method used to obtain them.

You might have a look at things like:
  http://en.wikipedia.org/wiki/Perfect_information
  http://en.wikipedia.org/wiki/Complete_information
  http://en.wikipedia.org/wiki/Nash_equilibrium
  http://en.wikipedia.org/wiki/Prisoner's_dilemma
Your claims are similar to the idea that chess players under some circumstances get a Nash equilibrium. This might happen, say, if the players focus on only simple playing strategies where piece vales have an important role, and they are unable to switch to a different one. Note that the prisoner's dilemma leads to such an equilibrium when repeated, because players can punish for past defections. In chess, this might happen if chess players are unable to develop a more powerful playing theory, say due to the complexity. - Just an input, to give an idea of what reasoning one might expect to support claims of predictions.

H.G.Muller:
| Note that a Nash equilibrium in a symmetric zero-sum game must be the
| globally optimum strategy.

Chess isn't entirely symmetric, since there is in general a small advantage of making the first move. But for players (or games) adhering to a piece value theory throughout as a main deciding factor, perhaps such balance may occur. The only world champion that was able to do that, winning by playing with very small positional and material advantages, was perhaps Karpov. Kasparov learned to break through that heavily positional playing, in part by training against the Swedish GM Andersson who specialized in a similar hyper-defensive playing. A more normal way of winning is at some point making material sacrifices in exchange for strong initiative particularly combined with a mating attack, and then winning by either succeeding by a mate or via some material gains neutralizing to a winning end-game. Perhaps when determining piece values, such games should be excepted. And since computers are not very good at such strategies, perhaps such game exclusion occurs naturally when letting computers playing against themselves.

H.G.Muller:
| As piece values are only useful as strategic guidelines for quiet
| positions, they cannot be sensitive to who has the move.

At the beginning of the game, white is thought to have a slight advantage, and the first task of black will be attempting to neutralize that. And it might be possible set a picee value to that positional advantage, just as when reasoning in terms of getting positional compensation for a sacrifice. Somewhat less han a pawn, perhaps. If one knows about the black/white winning statistics, one might be able to set a value on it that way. It may not be usable for a computer program as it does not change sides, but only computes the relative values of moves.

Sure, this is what people do and have done for ages. It is well known that the advantage of having the move is worth 1/6 of a Pawn, (corresponding in normal Chess to a white score of 53-54%) and that, by inference, wasting a full move is equivalent to 1/3 of a Pawn.

But the point is that this does not alter the piece values. It just adds to them, like every positional advantage adds to them. In my test the advantage of having the lead move is neutralized by playing every position both with white to move and black to move.

Muller:

You have my best regards toward your worthwhile effort to publish your
empirical, statistical method for obtaining the material values of pieces
in the ICGA Journal.  My assessment is that it will surely be a much
better paper than the junk [name removed] published in the same journal
regarding piece values.

[The above has been edited to remove a name and/or site reference. It is
the policy of cv.org to avoid mention of that particular name and site to
remove any threat of lawsuits. Sorry to have to do that, but we must
protect ourselves. -D. Howe]

H.G.Muller:
| But the point is that this does not alter the piece values.

Right, though that might be just a preferred way to structure theory because it suits human thinking. Essentially, define contexts, and attach values to them. First define piece values in nutralk settings. Then observe that the bishop pair gets an added value. Then try to figure out values for good and poor bishops. And so on. By contrast, computers tend to be very poor at handling such contexts, so other methods might be suitable for programs.