Comments/Ratings for a Single Item

'very close to being proportional to mobility squared'

I always thought that forking power should depend on number of
directions.

A recent comment made me wonder if I had, in effect, been underestimating
the
forking power of moves such as Bc4xf7+ (attacking both Ke8 and Ng8, for
example  1 e4 e5 2 Nf3 Nc6 3 Bc4 Bc5 4 b4 Bb4 5 Bb4 c3 6 Qb3 Na5 (in 1985,
in the Harding-Botterill book 'The Italian Game', this was ignored as a
simple error). 
After 8 Bxf7+ Kf8 9 Qa4 c6 , White needs to play Bxg8 to avoid losing a
piece.

In 2003, opponents on FICS will play 6...Na5. This discussion of
theoretical 
piece values ties directly into actual practical everyday playing of FIDE
Chess!

If there were a mathematical calculation for the Max Lange Attack or for
the Evans Gambit, it would make me unhappy; but if this calculation opened
the way to inventing a Max Lange equivalent in the Rookies versus
Colbberers game, I'd be happy overall.

In this discussion, we are asking questions that go far beyond the norm,
and if our findings ever allow one to answer 'what's the best move in
*this* position according to *these* rules, I think that none of us will
be happy with the result.

Basta Philosophy! 'Mobility squared'.

'Mobility squared' was always the sort of thing I felt iffy about. A
simple math, seems so attractive, as a chess master I doubted that things
were so clean.

In my early calcs, I know I tried to use something squared, maybe geom
dist,
and later I shied away from simple squared. Maybe something squared is
correct! 
If you prove I was wrong you may win the Nobel Prize for piece values
research.

(This is no joke, Do a web search, find how many professional
mathematicians link
to my values pages, and how few try to contribute.)

I always feared handwaving. 'How to Lie with Statistics' is a very good
book, and
it is very applicable to our field of endeavour.  I would always rather
miss a discovery rather than present a flawed arg for it. 

Thus I am prejudiced against anything squared. It seems too simple. 

However, I will listen; and my own personal judgment is far from final, as
I may be superceded.

What I am trying to say is that a good result may turn out to be a false
lead.

I mean, today you get numbers that look good, tomorrow raises doubts.

In order to feel this sort of pessimism, you need to be old enough to have

gone through a few cycles of Eureka! and Oops!.

Maybe you have something golden. I hope so.

It is late. I was thinking of deleting this whole comment and remaining
silent,

Instead, I will trust your judgment to take it for what it is worth.

'Or it could be something else entirely.'

Hooray! That is the sort of doubt that I feel!

I am so uncomfortable about having everybody take my primitive efforts as
golden.

It could be otherwise entirely.

Robert,

That is puzzling. Are there value gaps between the other augmented Knights
or do they test out fairly equal? Value of NF vs. R I could argue either
way as their moves are so unrelated.

I would think that the NF would be the strongest augmented Knight (even
though less mobile than NW) as it masks two Knight weaknesses:
colorswithching and inability to move a single square. 

NW masks one step inablility but isn't as forward as NF.

NA and ND mask colorswitching and give a a lot of coverage to the 2-square
distance.  These are very likely quite well mathced: NA more forward, ND
more mobile.

I really never had though of colorswitching as a major disadvantage, I
have even doubted it is worth considering. On the other hand one of the
nice things about Rooks is that they are neither colorbound nor
colorswitching.

Michael --

Here's the data on the augmented knights, obtained from Zillions vs.
Zillions using 5-ply fixed-depth searches.  The augmented knights are
placed in the rook positions and played against the orthodox army.

All the augmented knights give an advantage vs. rooks, probably in part
due to their ease of development and the rook's lack thereof.  The
following values are the various pieces' advantages over the rook, in
centipawns.

ND, NA = 27
NW = 38
NF = 82 (!)

The standard deviation for these measurements is about 10 cP.

More thoughts on augmented Knights:

Part of the advantage of the augmented Knights over the Rook may be a
Zillions artifact--Knights are strongest in the opening, Rooks in the
endgame. Zillions sometimes has trouble getting to an endgame, where human
masters would.  If my conjecture is correct, setting Zillions to deeper
plies would show the gap reducing or increasing much more slowly than
normal for repeating a Zillions calc at higher plies.

I suspect your results are not anomalous among the augmented Knights. The
NF has yet a third advantage--it cannot be driven from an outpost square
by an undefended pawn! All other augmented Knights can (as can the Rook,
but outposts are more important for short range pieces). This factor is
also almost certainly a part of why the Ferz is stonger than the Wazir.  

I would be curious to see what the numbers are for the various augmented
Knights vs Rook and each other if Berolina Pawn are used. I predict NW the
strongest but with a smaller gap, and Rook significantly better vs
augmented Knights (easier development as well as can't be attacked by an
undefended pawn).

My experience playing the game of Different Augmented Knights against a
strong chessplayer convinced me that the differences in value could safely
be ignored -- and it was this that gave me the confidence to proceed with
the next step, the Colorbound Clobberers.

My own computer-versus-computer simulations showed a slight advantage to
NF over the others. In addition to its ability to move to either color
square, the NF can escort a Pawn to promotion against a King, unaided by
its own King, and also very important in the endgame a NF can mark time:
NFe2-f3 keeps d4 defended! (Of course, ND can also sometimes do some of
this.)

NW has exceptional ability to draw by perpetual check (saving a lost
game).

When playing games using the NF, the most noticeable thing is that it
seems to have extreme flexibility. NFe2-g3 attacking Qh4 and defending the
K-side is great, but so is NFe2-f3 doing the same things! These choices
are powerful weapons in the hands of a strong player.

Numbers always underestimate the Rook. The new thought of 'King
interdiction' may play a large part in this. NF and Fibnif have some
interdiction ability, but not as much as WF....

Calculation would be so much easier if there were more known data points.
Instead, we begin with values known only for R, N, B, Q, and P; plus
as-Suli's estimates for A and F; plus recent chess variant experience
that NN is equal to Rook; plus old CV experience that on the cylindrical
board B == R. Developing a comprehensive theory of piece values based on
so few known data points is not easy.

Archangel worth only a Q? I'll buy that, based on the idea that the
displaced R move is worth a bit less, and the F move is duplicated. I fell
for a moment into the intuitive trap of envisioning the empty-board
mobility!, and of course in the late late endgame the Archangel must be
superior to Queen. 

I have the advantage of being a strong chessplayer, which means that not
only can I attempt to establish new known data points by playing both
sides of a game, but also once in a while I can talk some other strong
player into playing some wierd game against me. As a professional computer
programmer, I also had the chance to run comp-v-comp test series years
before anybody else. Over the course of 25 years, I have added a few known
and fairly trustworthy piece value data points: 

1. Ferz beats Wazir. I can't say by how much.

2. Augmented Knight equals Rook, or pretty close to equal. Of course, as a
competitive player, I'll choose NF every time, and try to win based on
its advantages! 

3. Commoner beats every other 2-atom piece because of its severe endgame
advantages. It has the largest absolute mobility, but its advantage is
much greater than the simple mobility numbers would indicate. A calc that
could 'predict' this would be wonderful!

4. What else? Is that all, in such a long time? Either I should have
worked harder or it is not so easy. 

Somebody reading this might have more money than math; if so, your
contribution to this developing science would be to pay grandmasters to
play with different armies (sponsor a tournament). This would create new
known data.

As you can see, developing known values by playtesting is extremely
expensive.
A good theory for calculating values would be such a help...

I have estimated the value of a Reaper and a Harvester and a Combine, but
I have fairly little faith in the correctness or exactness of these
estimates. These are simple and logical pieces, easy to estimate with the
current methods (easy to estimate though the estimate may be wrong!).

King Interdiction is a very promising new idea. Gryphon has double
interdiction! Is its practical value much greater than my estimate? Note
that until we can calculate the value of interdict, a NN ought to calc
higher than a Rook.

Another possible avenue of exploration is the interaction with Pawns.
There are many Pawns, and promotion is usually decisive. A Rook behind a
passed Pawn at a2 has value all the way down to a8 although current
calculations do not give it any credit for that. A Bishop supporting a
Berolina Pawn? Nobody knows..

In these few lines, I have pointed out what I think are promising
questions to explore. This is all pure speculation. Feel free to take a
different approach.

'ND, NA = 27
NW = 38
NF = 82 (!)'

NF is better, but not by that much. NF might be notably better when
Grandmasters play; but for normal masters, NF is barely better, hardly
notice it at all.

Your extreme results should be taken as a sign that you should distrust
the tool you used to take this measurement. Of course, lacking other tools
you will continue to use it! However, your faith in the value of its
measurements will be diminished.

The relative order is correct: NF is best, NW is second-best, NA and ND
are a bit behind. The quantification is way off. The 'quantum' of
advantage would be about 30 in your scale, and so NW is worth ND, close
enough. Eleven centipoints is nothing.

Therefore your tool has some value. If several different unreliable
measurements give the same result, one may have some degree of confidence
in the combined result.

'Part of the advantage of the augmented Knights over the Rook may be a
Zillions artifact--Knights are strongest in the opening, Rooks in the
endgame. Zillions sometimes has trouble getting to an endgame, where
human
masters would'

I seem to recall having written a short piece on the relationship between
opening values, endgame values, and the strength of the players, that is,
why the
Blackmar_Diemer Gambit is winning between 1800 players and losing between
GMs.

Opening advantages favor normal humans, endgame advantages favor GMs, in
both cases because of the precision of play required to overcome an
opening advantage and then profit from the endgame advantage. Exceptions:
Tal, RJF.

The fact that the Rook has just the one forward direction does not explain the lack of difference between Rook+Bishop and Rook+Knight as both wopuld gain from the non-Rook move. The more likely factors are (a) that adding the Knight move to a Rook with eight adjoining allies allows it to leap out of that space in a way that a Bishop move does not and (b) just adding the Rook move to a Bishop removes its colourbinding, adding it to a Knight rewmoves colourswitching, which is the Knight's property of always moving to the (not just an) other colour. Both compounds - and for that matter Bishop+Knight - can move to squares both of the same colour and of the opposite one. There are other kind of switching - the Silvergeneral is rankswitching (always moves an odd number of ranks), the Fibnif and Mushroom fileswitching (always move an odd number of files), and the Ferz and Camel both. Can everyone see why pieces that are both rankswitching and fileswitching are colourbound?

I don't see what is being explained here. The Kaufman values for solitary B and N are exactly equal with 2x5 Pawns on the board; with fewer Pawns the Bishop has a small edge, with more Pawns the Knight. As 5 Pawns is a quite typical middle-game case, that is about as equal as it can get. Only the _second_ Bishop (if it is on the opposite color, which of course it always is) is worth a lot more than a Knight, the so-called pair bonus, which amounts to half a Pawn.

Now the Chancellor (RN) is about half a Pawn weaker than a Queen (RB). So how come 'the Chancellor is doing so well'? Seems to me it is not doing well at all. Before adding R they (i.e. N and B) were equal, after adding it the B has gained appreciably more than the N. In fact about as much as the pair bonus, which could be interpreted as due to lifting the color-boundedness.

Such interpretations are a bit dangerous, though, at least when used quantitatively. The 'lifting of color-boundedness' argument could also be used when adding N to B or R. But there it would have to explain away nearly a full 2-Pawn difference between R and B, as RN is only marginally stronger than BN in practice. And it is a bit strange that lifting the color-boundedness in one case would buy you 0.5 Pawn, and in another case, combining even less valuable pieces, 1.75 Pawns.

Anyway, it seems to me that attempts are made to 'explain' something that is the reverse from what is true.

My first paragraph was a quote from the page itself that I was questioning. I was highlighting what the Rook gains from adding either Bishop or Knight move, and what both the Bishop and the Knight gain from adding the Rook move.
Could I also point out that '-boundedness' is not the right term here? Bounded is the past tense of the verb to bound, meaning to jump or leap or hop (in a general sense rather than the specific Chess ones), and as an adjective it means having a boundary so '-boundness' would be more correct - although so is the even briefer '-binding'. There are analogies with other verbs - you can ground an aeroplane and get a grounded aeroplane, but if you grind coffee you get ground coffee - not that I'm offering any.

This doesn't seem to be true. The only mating position for FAND unaided is K in a corner and FAND one A move away. But in order to force a K on b1 into a1, the FAND would need to threaten all of a2, b2, c2, and c1 (plus b1, if K's owner has other pieces), which it can only do from a0, which first of all is off the board, second of all is close enough for the K to capture it, and third of all is more than one move away from c3, where it needs to be to complete the mate.

WFND can mate unaided by threatening K with a D move and then duplicating K's moves until it is trapped on an edge. WFAN cannot, because K can move orthogonally to the threat.

I think that color-switching in the sense of true colors is a disadvantage, because it guarantees the piece has no mating potential with and against an orthodox King.

Your first definition seems the correct generalization: if there are no loops of odd length, the accessible squares should break up into two disjunct sets, one set being reachable only after an odd number of moves, the other after an even number. Of course there could be higher-order color switching schemes, like for a piece that only moves N, W and SE. Note that a Ferz is both color-bound and meta-color alternating within the set of accessible squares.

I agree about the FAND.

And Betza lists both can't-mate and color-switching as *separate* weaknesses in his list (IV&PV part 4).

So I still can't figure out why anyone believes that color-switching is a weakness, as opposed to merely an interesting trait.

Well, how about this then:

with color-switching pieces it is not possible to lose a tempo, to bring the opponent in zugzwang in an end-game. Of course there will also be color-bound pieces that suffer from the same problem, because the are meta-color switching. But compared to other non-color-bound pieces it is a weakness.

Is that a disadvantage specifically of switching pieces, though, or are they just the most extreme case on a sliding scale based on the length of the shortest odd cycle the piece can make? It seems intuitive that a hypothetical piece that takes, say, 9 moves to return to its starting square is much less likely to be able to lose a tempo in practice than a piece that can do it in 3 moves (though I could be wrong).

That would also be interesting because it suggests pieces that capture and move in different patterns should be affected if and only if their non-capturing move is switching, regardless of their capture pattern.

I largely agree with everything you say there. It would indeed be very interesting to see if there is an advantage on equiping a color-switching piece with an extra null-move. Unfortunately Fairy-Max cannot be configured to include null moves on pieces; I would have to program the capability to null-move separately.

If the right to null-move would be granted to a side irrespetive of the pieces it still has (equivalent to giving the King an extra null move), I am convinced it would be a very significant advantage, as Rooks lose their mating potential against such a King. And Rook endings are the most common endings in Chess. So it would bring in many draws in otherwise lost positions. This despite the fact that the King in itself is not even color-switching. (And in KPK it would also help a lot, both for the attacking and defending side!)

And those endgames show the null move at its strongest; averaged over the course of the whole game, is it worth even a quarter that much? A tenth?

On the reverse side, adding a null move to your King is obviously more than enough compensation for having your ENTIRE ARMY saddled with the 'cannot lose a tempo' weakness due to switching. That's regardless of the size of your army, which shows that giving the weakness to multiple pieces can't possibly be linear, but still, on a single piece it has to be worth only a tiny fraction of the null move.

So at a wild guess, we're talking about maybe a one centipawn penalty for the switching weakness, or even less? That's noise.

I am not sure I entirely follow your line of reasoning leading to the very small value of the 'triangulation' ability. And even if I would, it might not be valid, because piece values need not be strictly additive. (E.g. A Queen usually beats 2 Knights easily, but 3 Queens badly lose against 7 Knights, all the in presence of Pawns.)

The Wazir is indeed worth 130-140 centi-Pawn.

I should perhaps point out that I don't believe anything Betza claims here, unless empirical testing happens to show the same thing as he claims. For the simple reason that his statements are purely based on theoretical considerations that do not surpass the level of educated guessing, and his empirical testing, done in Human play, is not statistically significant. (As this would require tens of thousands of games.)

So it would not come as a big surprise to me if testing would show that color-switching has no significant impact on the value of a piece at all. An interesting test for this could for instance be to play augmented Knights against each other, which get a single extra King move, either orthogonal (preserving the color switching) or diagonal (breaking it), and then see which performs better, and by how much. (Or play Knights where one move is replaced by an Alfil or Dababba move against normal ones.)