Comments by Hans Aberg

In initial position setups, impose the condition that in the start position the pawns are protected by a piece. Then in N B and B N setups, the B pawn is unprotected, unless next to the B, there is a piece that can move diagonally.

Keeping the notation G = Q + N for 'general' (as 'amazon' abbreviated to 'A' would cause confusion with the 'archbishop' - and there is little convergence of names between variants), I get the following 10x8 board combinations (for white):
  R N B A Q K G B N R
and
  R C N B Q K B N G R
Here, I have required: N and B next to each other, the two B on different square colors, the two new pieces different and the more powerful piece on the kings side.

The intent of the chess variants I posted is to design playable variations, where the pieces are used. It would be interesting to know which ones generate more exiting (including with respect to opening developments) or shorter games.

Here is yet another one: Enhanced 8x8 Orthodox chess:
  G C A Q K A C G
i.e., gotten form Orthodox chess by the piece replacements R -> G, N -> C, B -> A. So there are no minor pieces here.

H.G.Muller:
| I think the following array would be more logical:
|    C H A M K A H C
|  where M=Amazon and H=Nightrider.

Yes. I was thinking about putting a Q+N piece besides the K, too. The nightrider may be too powerful though, and may be hard to learn for humans.

| ...but my guess is that it would make the
| game too tactical, without quiet positions, almost reversi-like...

One function that the minor pieces have in orthodox chess, is being suitable
for sacrifice. So having only powerful pieces could loose out on the tactical side too.

| Only the poor King has no enhancement. You could replace it by a Centaur,
| but this might make checkmating it too difficult.

The king, as piece for becoming mated, is just fine. I thought about the K+N piece a bit too.

One variation I am thinking about is 'Spartan kings': two kings, just as the Spartans, where 
When two kings remains, one can be taken; game ends when the last remaining king has become mated.

But one must think carefully about the design objectives: orthodox chess has a particularly good blend of strategy and tactics. With the Capa variation I gave, the idea is to preserve as much as possible of that, while lessening the amount of draws by adding some material.

H.G.Muller:
| I set up a tactically dead A+5P vs R+N+6P position (
| http://home.hccnet.nl/h.g.muller/BotG08G/KA5PKRN6P.gif ), and let it
| play a couple of hundred times to see who had the advantage. Turns out
| the position was well balanced.

This might be the way to go, because the standard exchange is equal pieces. The next step is a refinement: 'if I exchanged my R for a B, what do I need to keep a balance?' - something like 2 Pawns. The piece value system probably cannot predict well the balance in more complicated unequal exchange positions, but humans would probably try to avoid them, and they do not arise naturally, at least in human play.

So if an A is exchanged for a C, what is needed to keep the generic end game balance? And so forth. With a list of balanced combination, perhaps a reliable piece value system can be constructed. It would then only apply to the examined exchange combinations.

H.G.Muller:
| Well, this is basically how I got the empirical values I quoted. Except
| that so far I only did it for opening positions, so the values are all
| opening values. But, like I said, they don't seem to change a lot during
| the game. For the complete list of exchanges that I tried, see
| http://z13.invisionfree.com/Gothic_Chess_Forum/index.php?showtopic=389&st=1

There might be some problems here: For the classical piece value system, basically, what one gets is determining balance when the position is positionally equal and when both players play 'correctly'. By contrast, you take a statistical approach, and as you say for opening positions. The classical piece value system depends on a deeper analysis; perhaps the statistical approach can serve as a first approximation.

So the approach I suggested is to find a set of end-game positions where the outcome is always a draw or always a win, with extreme positions excepted. This might produce different values. In this setup, it is not possible to set values as exactly, as equal material means 'always result in draw for the positions considered'; perhaps it leads to only integral values, as a fraction pawn value cannot be used to decide the outcome of games. The fractions might be introduced as position compensations, like in orthodox chess reasoning, for example 'the sacrificed pawn is compensated by positional development'.

|| So the approach I suggested is to find a set of end-game positions
|| where the outcome is always a draw or always a win, with extreme
|| positions excepted. This might produce different values.'
H.G.Muller:
| The problem is that such positions do not exist, except for some very
| sterile end-games.

There is a difference between finding formally proved positions, and those that have such an outcome by human experience. If one in middle game exchanges ones queen for a rook and temporary very strong initiative, suppose this initiative does not lead to an immediate mate combination - it might be very effective to threaten a mate, which the opponent can only avoid by giving back some extra material - how much more material is needed in order to ensure a draw? A bishop perhaps, if the pawns are otherwise favorable, otherwise at least some more material, a pawn or two. This is a different judgement than a statistical one: a purely statistical judgment can probably be quite easily beaten by a human. It goes into a difficult AI problem: computers are not very good at recognizing patterns and contexts. But the classical piece value system probably builds on some such information.

H.G.Muller:
| It seems to me that the example you sketch is exactly what the piece-value
| system cannot solve and is not intended to solve. You want to have an
| estimate of how much material it wil cost the opponent to solve a certain
| mobility or King-safety advantage. Those are questions about the
| corresponding positional evaluation, not about piece values.

The situation I depicted is the other way around: I see a good queen for rook exchange that looks promising. Can I at least ensure a draw? How much additional material would I need to get to ensure. Next, have a look at the likely pawn development. So the traditional system will tell me whether I need 1, 2 or 3 pawns.

The statistical system does not say anything - I am not interested in how players on the average would handle this position.

H.G.Muller:
| I suspect you misunderstand what quantity is analyzed. In any case not how players handle | the position. But the very question you do ask, 'what are my chances for a draw with 1, 2 | or 3 Pawns in compensation', can only be answered in a statistical sense. The answer will | never be 'with 1 Pawn I will lose, with 2 Pawns I will draw, and with 3 Pawns, I will
| win'. It will be something like: 'With one Pawn I will have 5% chance on a win and 10% on | a draw, (and thus 85% for a loss) with 2 Pawns this will be 20-30-50, and with 3 Pawns 50-| 30-20. And I can count a Passer as 1.5, so if my 2 Pawns include a passer it will be 35-
| 30-35).'

No, this is the flaw of your method (but try to refine it): 

Chess is not played against probabilities, as in say poker. There is really thought to be a determined outcome in practical playing, just as the theory says. I can have a look at my opponent and ask 'what are the chances my opponent will not see my faked position' - but that would lead to poor playing. Much better is trying to play in positions that your opponent for some reason is not so good at, but it does not mean that one takes a statistical approach to playing.

Playing strength is dependent roughly on how deep on can look - the one that looks the furthest. There are two methods of looking deeper - compute more positions. Or to find a method by which positions need not be computed, because they are unlikely to win. 'Unlikely' here does not refer to a probability of position, but past experience, including analysis. With a good theory in hand, one can try to play into positions where it applies -- this is called a plan.

If I have an when the position values applies, I can try to play into such situation, and try to avoid the others.

When I looked at your statistical values, I realized I could not use them for playing, because they do not tell me what I want to know. A computer that does not care about such position evaluations may be able to use them. But I think the program will not be very strong against an experienced human.

But in the end, it is the method that produce the wins that is the best.

H.G.Muller:
| Well, let us take one particular position then: the opening position of
| FIDE Chess. You maintain that the outcome of any game starting from this
| position is fixed?

Since there is only a finite number of positions, there is theoretically an optimal strategy.

| A quick peek in the FIDE database of Grand Master games
| should be sufficient to convince you that you are very, very wrong about
| this. Games starting from this position are lost, won and drawn in
| enormous numbers. There is no pre-determined outcome at all.

They probably haven't searched through all positions to find the optimal strategy.

| Computers that use the statistical approach, such as Rybka, are
| incomparably strong. Humans, using the way you describe, simply cannot
| compete. These are the facts of life.

I suspect that humans are not allowed to search through zillions of positions like computers do. Specifically, if strong human players are allowed to experiment to search out of the weaknesses of the computer programs and using other computer programs to check against computational mistakes, I think that these computer programs will not be as strong.

But as for the question of piece values, humans and computers may prefer different values - it depends on how they should be used.

Reinhard Scharnagl:
| It is pure luck, that chess computer program strengths accompanied our masters for a time.
| Nobody will try to win a sprint against a Porsche, because the difference is even more obvious. So 
| it does not tell anything about how to handle a subject, when different species are competing.

The computer programs for orthodox chess succeeds in part because the number of positions required for achieving a good lookahead is fairly small relative to the capacity of the most powerful computers.

If one designs a chess variant with more material in a way that it is still very strategic to humans, then it will be harder for computers to be good on that variant, as the number of positions that must be searched for will be much larger.

GMs only search 6-8 ply deep, looking through at most a few hundred searches per position. That is feasible for a brute force computer (with pruning and noting that position evaluation still is needed), give the rather low average number of moves per position in orthodox chess.

But suppose that the average number of moves per position is made a couple of times larger. If the game is still such that humans can play it strategically, then that might improve for humans in the competition with computers.

H.G.Muller:
| Chess is a chaotic system, and a innocuous difference between two
| apparently completely similar positions [...] can make the difference
| between win and loss.

This is only true if positions are viewed out of context.

Humans overcome this by assigning a plan to the game. Not all position may be analyzable by such a method. The human analyzing method does not apply to all positions: only some. For effective human playing, one needs to link into the positions to which the theory applies, and avoid the others. If one does not succeed in that, a loss is likely.

The subset of positions where such a theory applies may not be chaotic, then.

H.G.Muller:
| Computers have no insight what to prune, and most attempts to make
| them do so have weakened their play. But now hardware is so fast that
| they can afford to search everything, and this bypasses the problem.

So it seems one should design chess variants where the average number of moves per position is so large that one has to prune.

| Making the branching ratio of a game larger merely means the search
| depth gets lower. If this helps the Human or the computer entirely
| depends on if the fraction of PLAUSIBLE moves, that even a Human
| cannot avoid considering, increases less than proportional. Otherwise
| the search depth of the Human might suffer even more than that of the
| computer. So it is not as simple as you make it appear below.

I already said that: the variant must be designed so that it is still very strategic to human. - It is exactly as complicated as I already indicated :-).

Therefore, I tend to think that perhaps a 12x8 board might be better, with a Q+N piece, and perhaps an extra R+N piece added.

H.G.Muller:
| It never happened to you that early in the game you had to step out of
| check, and because of the choice you made the opponent now promotes with
| check, being able to stop your passer on the 7th?

Early in the game, most things happen by opening theory. And if one is getting an advantage like a passer, one should be careful to not let down the defense of the king, including computing checks.

With those computer programs, a tactic that may work is to let down the defenses of the king enough that the opponent thinks it is worth going after it, and then exploit that in a counterattack.

| I think that if you are not willing to consider arguments like 'here I
| have a Knight against two Pawns (in addition to the Queen, Rook,
| Bishop and 3 Pawns for each), so it is likely, although not certain, that
| I will win from there', the number of positions that remains acceptable
| to you is so small that the opponent (not suffering from such scruples) will
| quickly drive you into positions where you indeed have 100% certainty....
| That you have lost!

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. So a material advantage of one pawn must happen in circumstances of where one can keep the initiative, otherwise, it might be better to returning that material for getting the initiative hopefully.

| What is your rating, if I may ask?

I have not been active since the 1970s, just playing computers sometimes. About expert, I think.

H.G.Muller:
| Why do you think the bigger board and the stronge piece make the game
| more strategical?

I said: if one increases the average number of moves in each position, then a full search may fail, as there will be too many of them. Then a different strategy is needed for success. 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.

Strategic positions is another matter: indeed, in orthodox chess, trying to settle for positions were advantage depends on long term development is a good choice against computers, the latter which tend to be good in what humans find 'chaotic' positions.

The design of a variant must be so that it admits what humans find strategic, and so it is possible to play towards them from the initial position. I am not sure exactly what factors should be there. Just putting in more material may indeed favor the computer. In orthodox chess, one can stall by building a pawn chain, and then use the minor pieces for sacrifices to create breakthrough. The chess variant must contains some such factors as well.

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.

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.

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.

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'.

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.

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.