Comments/Ratings for a Single Item

Hans Aberg:
| If it is doubled, then in a 7-ply search, if the positions are 
| independent, a search for all would require 2^7 = 128 more positions 
| to search for. If there are 10 times more average moves, then 10^7 
| more positions need to be searched.
Well, actually alpha-beta pruning makes it such that in a 7-ply search you
would only need 2^4 = 16 times as many positions, or 10^4 times. (And, as
the number of transpositions would likely go up, you would save a lot of
that with a hash table.) But these are still respectable numbers, so you
would perhaps only be able to do a 5-ply or a 4-ply search. 

But it is still not clear to me that a Human would not suffer as bad from
this. The game of Arimaa was a deliberate attempt to defeat computers
through this strategy, by creating branchin ratios of thousands. But I am
not sure if this has been succesful. The prevailing opinion under Chess
programmers is that computers are poor at Arimaa not because it is too
challenging, but mainly because no serious programmer cares...

Furthermore, computers are not really totally ignorant on strategical
matters either. But they cannot found by search, and must be programmed in
the evaluation. So it would also depend on the difficulty to recgnize the
strategical patterns for a computer as opposed to a Human. And until the
game strongly simplifies, the main strategic goal is usually to gain
material, using piece values as an objective. Unless the opponent really
ignores his King safety. Then the strategic goal will become to start a
mating attack.

H.G.Muller:
| But it is still not clear to me that a Human would not suffer as bad
| from [increasing the average number of moves].

This is clearly the difficulty.

| Furthermore, computers are not really totally ignorant on strategical
| matters either. But they cannot found by search, and must be programmed
| in the evaluation.

The strength in orthodox chess derives much from implementing human heuristics. So if the game is changeable or rich in this respect, it will be more difficult for computers.

| So it would also depend on the difficulty to recgnize the
| strategical patterns for a computer as opposed to a Human.

Computers have difficulty in understanding that certain positions are generally toast. They make up by being very good at defending themselves, so a good program can hold up positions that to humans may look undefendable. So a either a variant should have better convergence between empirical reasoning and practice, or one should let the humans have access to a cmoputer that do combinatorial checking.

| And until the game strongly simplifies, the main strategic goal is
| usually to gain material, using piece values as an objective.

That seems optimistic :-). A GM advice for becoming good in learning end-games.

| Unless the opponent really ignores his King safety. Then the 
| strategic goal will become to start a mating attack.

There are all sorts of tactics possible.

One thing one can try against a computer program is giving it material in exchange for initiative. Then, the human may need some assistance on the combinatorial side.

If the game is highly combinatorial the computer is favored. So one can try to close it.

So a chess variant should admit stalling. If it is always possible to break into highly combinatorial positions, then the computer will be favored.

Me:
| As I said, the outcome is decided by the best playing from both sides. 
| So if one starts to play poorly in the face of a material advantage, 
| that is inviting a loss.
Someone:
| We still don't seem to connect. What gave you the impression I
| advocated to play poorly? Problem is that even with your best play, it
| might be a loss. And as it is an end leaf of your search tree, which
| is limited by the time control, you have no time to analyze it until
| checkmate, or in fact analyze it at all. You have to judge in under a
| second if you are prepared to take your chances in this position,

The original context was what how I think that the classical piece value system is constructed:

By experience, certain generic types of endgames will empirically classified by this system. Those that aspiring becoming GMs study hard to refine it, so that exceptions are covered. Once learned, it can be used to instantly evaluate a position.

This is then not a statistical system.

Reinhard:
| Thus there are no trades caused by different views of pieces values, 
| which could make a trade attractive for BOTH sides. Thus your model 
| is never verified at such critical moments. 
If there is a best strategy (i.e. best set of piece values) it must be
shared by both sides. The theoretically best play is amongst the set of
self-play games. And it is only the score that results from theoretically
best play that determines the piece value.

Hans Aberg:
| By experience, certain generic types of endgames will empirically 
| classified by this system. Those that aspiring becoming GMs study 
| hard to refine it, so that exceptions are covered. Once learned, 
| it can be used to instantly evaluate a position.
What do you mean by this? The result of end-games is not determinedby the
material present. There are KRKBPP end games that are won for the Rook,
and that are won for the Bishop. Similar for KRKBP and KRKBPPP. So how
would you derive a piece-value system of this?

Me:
| By experience, certain generic types of endgames will empirically 
| classified by this system. Those that aspiring becoming GMs study 
| hard to refine it, so that exceptions are covered. Once learned, 
| it can be used to instantly evaluate a position.
Someone:
| What do you mean by this? The result of end-games is not determinedby
| the material present. There are KRKBPP end games that are won for the
| Rook, and that are won for the Bishop. Similar for KRKBP and KRKBPPP.
| So how would you derive a piece-value system of this?

It hangs on the word 'generic', meaning some general empiric cases. Then the cases you cover would be special cases studied separately as a refinement. These are excepted when defining the piece value system.

H.G.Muller:
| It is too vague for me. Could you give a specific example?

Take K+P against K. Then it can only be won if the king is well positioned against the pawn and the pawn isn't on the sides if the opponent king is in good position. So in general, one P ahead does not win, but in special circumstances it can be, So a player that isn't very good at end-games will probably try to get more material, but a better one will know, and an even better player will be able trying to play for the most favored end-game. A weak player may loose big in the middle game because they don't know how to keep the pieces together, but a GM might do that because knowing that the alternatives will lead to a lost end-game, so trying something wilder may be a better try. But a GM may not have much use of such long-range strategic skills against a computer that can search all positions deeper, becoming reduced, relative the computer, not being to keep the pieces together.

I have been following this discussion a bit, am pretty confused, but think I have a few things down.  I had a few questions here regarding this entire discussion:
1. I know when they try to evaluate American football for fantasy league play, and are evaluating players, they will tend to treat the defense as a single entity.  The reason for this is that it is next to impossible to be able to tell the value of a single player in defense.  Can one argue that chess is similar?  Pieces work with one another.
2. Similar to this last question, are the value of pieces in Chess960 the same as they are in FIDE Chess?  Also, if players had free set up in the back row with their pieces, would they have the same value then?  Are rooks worth more if they start out in the middle, instead of the outside?

I ask this question, because of some things I am interested in doing here.  First, I would be interested in pitting Chess against Shogi or XiangQi in some way that would have the sides roughly equal to one another, by either point handicapping or something else.  

Secondarily, if you take a look at Near Chess, I have found that the game takes on entirely different dynamics if you move the rooks back or not.  This also relates to doing Near Chess vs Normal (FIDE Chess).  Playing the Near Chess side is a different animal than Normal Chess, and I found, even without castling, Near Chess holds its own.  For FYI, Near Chess is here:
http://www.chessvariants.org/index/msdisplay.php?itemid=MSnearchess

(I just bring up Near Chess now, because it fits a concern I have, and it seems like the conversation).

So, in a nutshell, would anyone want to focus on the value of a set of chess variant RULES vs another one, rather than single pieces without a context they fit into?  How about a test, for example of the value of a Queen in Chess960 depending on where it starts, and which pieces are next to it?  Maybe even work out a bidding system for configurations. 

I hold out hope that this may be a reality one day, to effectively balance a range of chess variants, so a player could play with their favorite set up, against another player's set up, and it be reasonably close.

Reinhard: Within the boundary condition that the game should be played wth a strategy belonging to a sub-class of all possible strategies, (in this case a strategy with an evaluation based on additive piece values that are constant throughout the game), there is a best strategy. And the strategy will not lead to deterministic play, as there is often choice between moves that are evaluated equal, and the choice between them is based on random factors outside the strategy. So the outcome of a position under such a strategy is not fixed, but probabilistic.

That the strategy is not globally optimal, and that there exist strategies (not based on piece value but, for example, material tables), is not a problem if the playing strength of the piece-value-based strategy is not very much below the global optimum. And, considering the success that current engines hae in being the best Chess entities on the planet, I would say the piece-value-based strategies are doing a pretty good job. In particular, Joker80 is doing a pretty good job for 10x8 Chess. The strategies of our best engines are such that the variability of the result obtained from the set of positions with a certain material composition is not dominated by errors that the engines make in playing out such positions, but by the W/D/L statistics within the set. At that point, there would be very little change on improving the quality of play any further.

To give ane example: without knowledge of other details, it is better to have two minors than a Rook in a position which otherwise contains only equal numbers of Pawns (plus Kings). This despite the fact that zillions of KRPPPBNPPP positions are won for the Rook side. It is just that there are far more that are won for the B+N side. That the engines you use to decide which positions are won and which are lost (for a sample of the positions) now and then bungle a won game that was close to the limit, on the average cancels out, provided it is only the games close to the win/loss boundaries that can be bungled by the average playing inaccuracy. Because the narrow strips on either side of the boundary have equal area. Only if the accuracy gets very low, so that games very far from the boundary can still be bungled, the exact shape of the boundary (e.g. its curvature) becomes important.

Hans Aberg:
| So in general, one P ahead does not win, but in special circumstances 
| it can be,  ...
This is already not true. In general, one Pawn ahead does win in a Pawn ending. KPK is an exception (or at least some positions in it are).

But is is still completely unclear to me how this has any bearing on piece values. KPK is a solved end-game (i.e. tablebases exist), so the concept of piece value is completely useless there. In solved end-games it only matters if the position is won, and having KPK in a won position is better than having KQK in a drawn position. I don't see how you could draw a conclusion from that that a Pawn has a higher value than a Queen.

Piece values is a heuristic to be used in unsolved positions, to determine who has likely the better winning chances. Like in an early middle-game position, where you just lost the exchange, and it will make you feel like you are losing the game because of that fact...

Me:
| So in general, one P ahead does not win, but in special circumstances 
| it can be,  ...
H.G.Muller:
| This is already not true. In general, one Pawn ahead does win in a Pawn ending. KPK
| is an exception (or at least some positions in it are).

My statement referred to this particular ending. For other endings, I indicated it depends on the playing strength, noting that a GM would make sure to win whenever possible, but a weaker player may prefer more material. It is classification for developing playing strategies, not the theoretically best one.

| But is is still completely unclear to me how this has any bearing on piece values.

Change the values radically, and see what happens...

| KPK is a solved end-game (i.e. tablebases exist), so the concept of piece value is
| completely useless there. In solved end-games it only matters if the position is
| won, ...

It is true of all chess positions, not only end-game.

| ...and having KPK in a won position is better than having KQK in a drawn position.

So here one the point-system would be useless, if one faces the possibility of having to choose between those two cases. But the point system will say that KQ will win over KP unless there are some special circumstances, not that it will win in a certain percentage if players of the same strength are making some random changes in their play.

| I don't see how you could draw a conclusion from that that a Pawn has a higher value
| than a Queen.

I have no idea what this refers to.

| Piece values is a heuristic to be used in unsolved positions, to determine who has
| likely the better winning chances.

Only that the 'winning chances' does not refer to a percentage of won games of equal strength players making random variations, which is what you are testing. It refers to something else, which can be hard to capture giving its development history.

Hans Aberg:
| Change the values radically, and see what happens...
What do you mean? Nothing happens, of course. KPK positions that are won,
will remain won. What did you expect? That they become draws when I reduce
the Pawn value, or become lost positions when I say that a Pawn has a
negative value???
| It is true of all chess positions, not only end-game.
Except for the unfortunate detail that threredo not exist anyChess
positions other than some late end-games and a few mate-in-N problems that
are known to be won...
| But the point system will say that KQ will win over KP unless there 
| are some special circumstances, not that it will win in a certain 
| percentage if players of the same strength are making some random 
| changes in their play.
Only a quantitative measure can establish if the circumstances are
special, or not. If they would occur 90% of the time, I would call them
common, not special.
| Only that the 'winning chances' does not refer to a percentage of 
| won games of equal strength players making random variations, which 
| is what you are testing. It refers to something else, which can be 
| hard to capture giving its development history.
So in your believe system, if a certain position, when played by expert
players to the best of their ability, is won in 90% of the cases by white,
it might still be that black has 'the better chances' in this position?
If it is, this would be a very good conclusion to end this discussion
with. I don't see how going for such positions with 'better chances'
(guided by piece values) in play would win you many games, though...

H.G.Muller:
|| Change the values radically, and see what happens...
| What do you mean? Nothing happens, of course.

You say that a pawn ahead is always a win with only some exceptions. So is a rook. So set values of these the same - does not work to predict generic rook against pawn end-games.

| If they would occur 90% of the time, I would call them common, not special.

90 % with respect to what: all games, those that GMs, newbies, or a certain computer program play?

| So in your believe system, if a certain position, when played by expert
| players to the best of their ability, is won in 90% of the cases by white,
| it might still be that black has 'the better chances' in this position?

The traditional piece value system does not refer to a statement like: 'this material advantage leads to a win in 90 % of the cases'.

Hans Aberg:
| You say that a pawn ahead is always a win with only some 
| exceptions. So is a rook. So set values of these the same - 
| does not work to predict generic rook against pawn end-games.
You seem to attach a variable meaning to the phrase 'a pawn ahead', so
that I no longer know if you are referring to KPK, or just any position.
The rule of thump amongst Chess players is that in Pawn endings that you
cannot recognize as obvious theoretical wins/draws/losses (like all KPK
positons, positions with passers outside the King's square etc.) a Pawn
advantage makes it 90% likely that you will win. In a Rook ending, OTOH, a
single Pawn advantage will be a draw in 90% of the cases.
| 90 % with respect to what: all games, those that GMs, newbies, or a 
| certain computer program play?
You will always be able to find players that will bungle any advantage,
amongst Humans as well as computers. 90% is the limit to which you
converge with increasing quality play. And you converge to it rather
quickly, because Pawn endings aren't that complicated. Players at a level
where 'opposition' and the 'square rule' are familiar concepts will
probably achieve close to this percentage.
| The traditional piece value system does not refer to a statement 
| like: 'this material advantage leads to a win in 90 % of the cases'.
This is because even the weakest players get to learn the piece-value
system, and the weaker they are, the more faithfully they usually adhere
to it. Even 6 year olds get to learn this before they know about castling
and e.p. capture. And to people that can only look 2 ply ahead a Pawn
advantage obviously means a lot less (in terms of score benifit they will
achieve because of this) than to 1900-rated players.

So the win percentage derived from a certain advantage according to the
piece-value system is not fixed. But that is not the same as saying that
for a given level of play, a larger advantage does not result in a higher
score. In fact it does. At any level of play, giving Knight odds between
equal players leads to a much larger win percentage than giving Pawn odds,
while Rook odds would give even more (if by then it would not already be
100%).

But if a certain position (or set of positions characterized by the same
material imbalance, as I use in my test matches) would be won in 90% of
the cases between equal players at any level above 1800 Elo by white (but
tail off below that, when you get into the regions of the real patzers),
would it, according to your definition, still be possible that black
actually had the better 'winning chances'? And that a piece-value system
that would predict these better 'winning chances' for black thus was a
correct and viable system?

I would prefer (and advice anyone else) to use a piece-value system that
would actually predict that white had better winning chances in that
position! As, by the time you will have gone through the trouble of
forcing the opponent to this position, selected on basis of the system,
you will be subjected to these very same winning chances.

I missed this one:

Hans Aberg:
| You say that a pawn ahead is always a win with only some 
| exceptions. So is a rook. So set values of these the same - 
| does not work to predict generic rook against pawn end-games.

Why would you want to set the values the same? Because both a Pawn and a
Rook advantage in the end-game is 100% won?

In the first place they aren't. An extra Pawn will win 90%, an extra Rook
99.9-100%. But, for the sake of argument, change the example to an extra
Rook or an extra Queen, and say that both now win 100%. Does that prove
the piece value of a Rook and a Queen are the same?

Of course NOT. 100% is far outside the linear regime, and obviously the
score becomes meaningless there. I never take into account the results of
thest that are more than 70% biased. I don't have to, as a full Pawn
advantage in the opening in the linear range only adds 12%, (so I get a
62% score for pure Pawn odds), and in principle I can always balance the
material such that I get scores below that by adding a Pawn.

So what would NOT work, and what I thus do NOT do, is first giving full
Archbishop odds, and then giving Chancellor odds. Because both these odds
would lead to ~95% scores, and trying to peel a piece value out of the
fact if they are 95% or 96% falls far below the resolution (determined by
statistical noise), and would be extremely sensitive to the exact initial
position (as the 5% non-wins would be due to tactical freak accidents like
quick mates or perpetuals).

What I do is play C vs A or CC vs AA (nearly equal) to see who wins. Or Q
vs A+P and Q vs C+P (to see who does better). Or A vs R+R or A vs R+B or A
vs R+N+P, to see if A typically beats those combinations (which it does,
but never by more than 60% or so, or I would give stronger material to the
weak side).

But if Q beats A by a larger score (say 59%) than A+P beats Q (say 53%), I
consider it evidence that Q-A > 0.5 Pawn. Because I know (from observation)
that adding another 50cP advantage to the imbalance in favor of the weak
side (like replacing an unpaired Bishop by a Knight) will in general
simply adjust the existing score by 6% at the expense of the side who you
give the weakness. And that duplicating a weakness (like giving one side
two unpaired Bishops in stead of the other's two Knights) will simply
double the score excess to 12%, the same as for Pawn ods, and can be
exactly compensated by giving the Knight-side an extra Pawn. So Q+N will
still beat A+B(unpaired), consistent with Q-A in the point system being
more poinst than the B-N difference of 0.5 Pawn, and A+P-Q being less.

H.G.Muller:
| You seem to attach a variable meaning to the phrase 'a pawn ahead', so
| that I no longer know if you are referring to KPK, or just any position.

I tried to explain how explain the idea of a general, or 'generic' situation by restricting to this example. In a general case, one could not classify the games as exactly.

| The rule of thump amongst Chess players is that in Pawn endings that
| you cannot recognize as obvious theoretical wins/draws/losses (like all
| KPK positons, positions with passers outside the King's square etc.) a
| Pawn advantage makes it 90% likely that you will win.

It haven't seen anything like that in end-game books, or players from the time whan I was active, before the days of computer chess. I think that nowadays, people learn much by playing against strong computer programs, rather than learning classical theory. So I do no think such percentages relate to any classical chess theory, and possibly no formal mathematical statistics. Such a statistical approach, even if formalized, may still work well in a computer that by brute force can do a deeper lookahead than a human, but may fail otherwise.

H.G.Muller:
| Why would you want to set the values the same? Because both a Pawn and
| a Rook advantage in the end-game is 100% won?

I describe how a piece value theory might be developed without statistics. Since one P or R ahead generically wins, set them to the same value. Now this does not work in P against R; so set the value higher than P. Then continue this process in order to refine it, comparing different endings that may appear in play, taking away special cases, always with respect to tournament practice, using postmortem game analysis.

Hans Aberg:
| I describe how a piece value theory might be developed without 
| statistics. Since one P or R ahead generically wins, set them to 
| the same value. Now this does not work in P against R; so set the 
| value higher than P. Then continue this process in order to refine 
| it, comparing different endings that may appear in play, taking away 
| special cases, always with respect to tournament practice, using 
| postmortem game analysis.

It seems to me that in the end this would produce exactly the same
results, at the expense of hundred times as much work. You would still
have to play the games to see which piece combinations dominantly win. But
now you would throw away the information on how much, making the outcome of
65% equivalent to one of 95%.

While in fact there is very much information in this, as tsmall advantages
turmn out to be additive to a high degree of accuracy. If Pawn odds is 62%,
and Q vs A is 59%, I can be pretty sure that A+P vs Q will be 53%. In the
system you propose, you would have to actually play A+P vs Q before you
would know if A is worthe more or less than Q minus P.

This in particular applies to the fractional advantages. If a B-pair wins
56% from a B anti-pair, and you would only interpret that as 'B-pair is
good', you would have to explicitly test the B-pair against any other
fractional advantage (e.g. Q+BB(unpaired) vs C+BB(paired), Q+BB(unpaired)
vs A+BB(paired), etc to know that the pair advantage was half a Pawn,
rather than 3/4 or 1/4.

Why do you think this is an attractive method, and why would you expect it
to give different results in the first place? Can you construct a
hypothetical example (of score percentages) where your analysis method
would produce different results from mine?

H.G.Muller:
| It seems to me that in the end this would produce exactly the same
| results, at the expense of hundred times as much work. You would still
| have to play the games to see which piece combinations dominantly win.

Perhaps the statistical method is only successful because it is possible by a brute-force search to seek out positions not covered by the classical theory.

Hans Aberg:
| Perhaps the statistical method is only successful because it is 
| possible by a brute-force search to seek out positions not covered 
| by the classical theory.

I am not sure what 'brute force' you are referring to. Do you consider
Monte-Carlo sampling of a quantity you want to measure a 'brute force'
approach? I would associate that more with exhaustive search. Playing a
few thousand games can hardly be called 'exhaustive', considering the
size of the game tree for Chess.

What do you mean by 'classical theory'? Are you referring to the piece
value system 1,3,3,5,9, or some more elaborate point system including
positional advantages, or very complex methods of proving a certain
position can be won against any play?

What does it matter anyway how the piece-value system for normal Chess was
historically constructed anyway? Don't you agree that the best way to play
the game through a piece-value heuristic is to make sure that positions
with better piece values (in your favor, i.e. yours minus those of the
opponent) will have a better probability to be won? (It will be understood
that the heuristic will only be applied to 'quiet positions': the players
are supposed to be smart enough to search low-depth tactical trees that
makes them recognize that the are not a Pawn ahead but a Queen behind when
the opponent has a passer that they cannot stop, or has the move and can
capture their hanging Queen.)

H.G.Muller:
| I am not sure what 'brute force' you are referring to.

See
  http://en.wikipedia.org/wiki/Computer_chess

| What do you mean by 'classical theory'?

What was used before the days of computers. A better term might be 'deductive theory' as opposed to a 'statistical theory', i.e., which has the aim of finding the best by reasoning, though limited in scope due to the complexity of the problem.

| What does it matter anyway how the piece-value system for normal Chess
| was historically constructed anyway?

It is designed to merge with the other empiric reason developed to be used by humans.

You might have a look at a program like Eliza:
  http://en.wikipedia.org/wiki/ELIZA
It does not have any true understanding, but it takes while for humans to discover that. The computer chess programs are similar in nature, but they can outdo a human by searching through many more positions. If a human is compensated for that somehow (perhaps: allowed to use computers, take back at will, or making a new variant), then I think it will not be so difficult for humans to beat the programs. In such a setting, a statistical approach will fail sorely, since the human will merely play towards special cases not covered by the statistical analysis. The deductive theory will be successively strengthened until in principle the ideal best theory will emerge. This latter approach seems to have been cut short, though by the mergence of very fast computers that can exploit the weakness of human thinking, which is the inability of making large numbers of very fast but simple computations.