Piececlopedia: Crowned Knight

Value speculation

This piece "may" be weaker than archbishop. The archbishop is assessed to be worth about 7 points (6.75 averaged)

The Archbishop is actually worth 8.75, on a scale where Q=9.5. Replace Q by A on one side, and delete the f-Pawn for the other, and the Archbishop has the upper hand. In the end-game A beats R+N+P pretty heavily, in the presence of more Pawns.

Centaur is a strong piece, but weaker than Archbishop. More like 8.

HGMuller thanks for replying my thread.

Make sure to check out this links:

http://chessvariants.wikidot.com/assigning-piece-values-with-cast

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.95.326&rep=rep1&type=pdf

Well, the presented methods obviously suck when they do not get the right values. The problem is that it is very easy to sit down for some numerological hocus pocus that is tuned to reproduce the classical values of the orthodox Chess pieces, and then take as gospel what it produces for fairy pieces. Many people have done it, in addition to the two examples you give. Often what comes out is complete nonsense, as might be expected for things that spring purely from the imagination.

The point is that Chess has a 'reality', albeit in its own, very restricted universe. Theories about piece value are thus comparable to scientific theories, which have to be validated by empirical data. The classical values of the orthodox Chess pieces are based on hundreds of years of experience with the game, rather than part of the game rules. They are confirmed with good precision by statistical analysis of GM games, as shown by Larry Kaufman.

In practice an Archbishop totally crushes any combination of orthodox material that is worth about 7 (like two minors + Pawn, Rook + 2 Pawns), i.e. it scores >80% from a set of quiet starting postions containing such an imbalance amidst a lot of other material. (Of course there have to be enough Pawns in these starting positions, as it is well known that you need a huge advantage to win in Chess without Pawns, and even Queen vs Bishop + Knight and nothing else can be a dead draw.) And an Archbishop scores >60% against combinations of orthodox material that are worth about 8 (like Rook + minor).

So it seems the theory that it is worth about 7 just doesn't 'cut any wood', and should be considered strongly falsified, no matter the amount of fancy calculation that people forward as 'evidence' for it. Chess and arithmetic are only weakly related. Even educated guessing cannot change reality.

As to the value of the Man/Commoner:

I don't see the logic of assigning a different value to a royal and non-royal version of the same piece. If anything, being royal should a handicap, as far as the tactical abilities of the piece are concerned, as it cannot move to check. And if you incorporate the 'royalty value' the King should be infinite, as losing it immediately finishes the game, no matter how much material you got in compensation.

The value of a second King in Spartan Chess (where you have to capture both in order to win) is around 4.5, so the 'spare royalty' aspect is apparently worth 1.25-1.5 Pawn. (I.e. the results improve as much by replacing the second King with a Rook, as they deteriorate when you replace King + Pawn by Rook.)

The quoted value, Commoner = Knight + 0.25 Pawn, sounds about right as end-game value, though. (One often encounters the claim that the value is ~4, but this is totally off.) Replacing Knights by Commoners for one side in the initial position suppresses its score to significantly below 50%, though.

I see your point and reasoning.

My question is how much is the bonus point alligned for the Centaur.

Knight + Man combined

If Rook + Bishop equals 8 + 1 = 9 (Queen)

Rook + Knight equals ≈ 8 + 0.5 = 8.5 (Chancellor)

The Queen is estimated to be 0.5 points stronger than Chancellor.

Do the restricted mobility influence the combined bonus value.

In theory the Stock Centaur is worth 6.00 points.

Knight 3.00

Mann 3.00

Plausible speculations

• 6.60
• 6.50
• 6.35

It turns out to be difficult to predict the synergy bonus. It is generally true that a compound piece is worth more than the sum of its components, however. For short-range leapers with N moves the average formula

value = 1.05 * (30 + 5/8 * N)*N centiPawn

holds reasonably well (as average value of all such leapers with N moves). For N=4, 8, 12, 16, 20, 24 this predicts 136, 294, 472, 672, 892 and 1134, respectively. This clearly shows the non-linear relation, and predicts a bonus of 81 for combining two 8-movers to one 16-mover. These are just the 'multiplet averages', though; not all pieces with N moves are equally valuable. Forward moves are worth about twice as much as sideway or backward moves. Concentration of the moves also seems valuable. The Centaur is one of the best 16-movers.

Chancellor is indeed half a Pawn weaker than Queen, in a FIDE context. This might depend on more or less coincidental end-game facts, like Queen + minor vs Queen being a draw, but Queen + Bishop vs Chancellor being a win, and Q vs R being a win, and C vs R often being a draw.

I am currently running a test where one side has a Centaur instead of a Rook and a Knight. (The latter deleted in all possible combinations of Queen-side and King-side, and the Centaur than put in one of the emptied spots.) The R+N are in the lead by 70-51 so far, i.e. ~58%. This is about half as much as classical Pawn odds, which causes a 65% score between equal opponents. This would suggest Centaur = R + N - 0.5P = 7.5. But it is a bit of a tricky test, as deleting one of the Rooks interferes with castling, which is an extra handicap for the Centaur over the normal piece value. And it is quite normal that a second super-piece is effectively worth less (because the compensating lighter material also hinders the existing Queen). E.g. a second Queen loses to R + B, while the first Queen is about a full Pawn stronger than R+B. So perhaps it would be better to determine the Centaur value by substituting Q for Centaur, and deleting a Knight on one side, and substituting Q for a third Rook for the other side. Then you would not have to interfere with castling either.

BTW, the 'safe check' valuation method used by Ed Trice doesn't seem to make much sense. It would predict value zero for the non-royal King, and value the Centaur the same as Knight!

If you was to count all the squares the queen can travel to in the above position you would end up with 27 squares.

Queen = 27

If you was to do the exact same thing to every chess piece

Rook = 14

Bishop = 13

Knight = 8

Pawn = 1

Now look at your archbishop. It can move to 21 squares

ArchBishop = 21

Pawns have a value of 1

Knights have value of 3

Bishop have value of 3

Rook have value of 5

Queen have value of 9

From 8-13 the value is 3 however, their is some restrictions.

The knight is given a value of 3 becuase even though it can only control 8 squares it has the chance to go on both colors

The bishop on other hand is restricted to only 1 color all game. So even though The bishop has 13 squares of influence. The bishop is bound by a color complex.

The Arch Bishop on the other hand will have features of the knight so it will not be bound by any one single color.

In regular chess

3 minor pieces = queen.

1 rook + 1 minor +1 pawn = queen.

In some cases 1 rook + 1 minor piece = queen. It can also be consider very acceptable.

If the minor pieces have a complex of 8-13 squares and 3 of them are considered good for a queen.

8x3= 24

13x3 = 39

Anything between 24 and 39 is considered a value of a queen.

1 rook + 1 minor + 1 pawn = queen

(14) + {8-13} + 1 = queen

14+8+1 = 23

14+13+1 = 28

So the range is 23-28 = queen

1 rook + 1 minor piece = queen

14 + {8-13}

14 + 8 = 22

14 + 13 = 27

So in practical terms anything with a square activity of 22-39 can be equal to a queen.

ArchBishop = 21

So maybe you should considerably change the following value.

The archbishop is assessed to be worth about 7 points (6.75 averaged)

I personally think the value should be more in the 8 to 9 range. Some people may even just consider it a 9.

"The archbishop is assessed to be worth about 7 points, intermediate between the rook and queen and can be considered as a week queen. The archbishop is closer in strenght between the rook than the queen in that regard."

So that means that the archbishop is an intermediate piece between the rook and queen, that leans toward the rook's strenght.

The Centaur give me the impression that the piece feels like an upgraded rook.

Rook = 14 squares 

Centaur = 16 squares 

16 / 14 * 1.05 = 1.2 * 5 = 6.00.

1. Queen = 9
2. Chancellor 8.5
3. Arcbishop 6.75 to 7.00
4. Centaur 6.35 to 6.50?

The knight and the king both are slow moving pieces, that can influence the synergy bonus.

Apart from that it seems that the knight suffers an huge inflation penality on wider board variations such as 10x8 (Gothic / Capablanca) and 24x24.

I disagree the archbishop is not bound by any color complex. It is far superior than a rook. It is more equal to a queen.

Good catch. I had not noticed that the Archbishop was listed in Wikipedia under 'Princess'. I corrected it now. (It is not clear to me what "closer in strength between ..." means anyway.)

I completed the test with the Centaur vs Rook + Knight as Queen substitutes now, and indeed the result is much closer (presumably because it never spoils castling): 54% in favor of R+N, which amounts to 0.25 Pawn. On the Kaufman scale, where B=N=3.25, R=5 and Q=9.5 this would suggest a value of 8. But I also tested it against a Bishop pair (Centaur scores 53%) and B-pair + Pawn (Centaur scores only 37%). The B-pair is worth 7.0 (due to the pair bonus of 0.5), so that suggests 7.25. Perhaps it should not be surprising that the B-pair performs well here, as a Centaur is quite vulnarable to Bishop attacks. As Rook-like pieces in the opening position often underperform 0.25 Pawn, due to the difficulty of developing them, a reasonable estimate for the Centaur value on the Kaufman scale seems 7.5.

I do not understand how you get this assessment.

All you are doing is reciting information you saw on Wikipedia. Than you are posting the Wikipedia link.

How on earth did they come up with the value of 7.

About the same way as Jules Verne got its detailed description of the Lunar surface: sitting behind a desk doing some educated guessing.

" The Chancellor is roughly equivalent to the Queen even though the ideal value of N is presumably less than Bishop: the Bishop is colorbound and its practical value is ever so slightly more than a Knight, combining it with R removes the colorboundness, and therefore is a classical case of "combining pieces to mask their weaknesses and thus allow their practical values to be fully expressed"; and therefore one might expect the Q to be worth notably more than the Chancellor. "

Commoner or the Man

Commoner is notably stronger in practice than a minor piece.

This is a very short-range and very flexible piece that is much weaker than a Knight in the opening, very strong in the middlegame if it can occupy the center, and almost always wins against a Knight or Bishop in the endgame.

Yes, that's right. An endgame such as K + WF + Pawns versus K + B + Pawns is almost always a win for the WF. Not only that, but the Pawnless endgame of K plus WF versus King is a forced win.

The weakness of this piece is that it takes a long time to get from one section of the board to another; for example, in the opening, it takes 2 or 4 moves to get a WF properly developed. Its strength is that it concentrates a lot of striking power in a small area.

These are no doubt quotations from Ralph Betza. Like many of the concepts he introduced in his theories on piece values, the "masking of weaknesses" is a sound (albeit qualitative) concept. Pieces have 'global' properties that follow from the way their moves cooperate, and thus cannot just be considered as an addition of independent moves. So the Bishop would have a value based on its number of moves, weighted by the ease with which they can be blocked, but because of its color binding, would still be less valuable than a piece that has the same number of moves and blockability that does not suffer from color binding. Combining it with a Rook or Knight breaks the color binding, so that the associated penalty disappears, and thus contributes to the synergy.

A Knight has a similar global deficiency, in the 'speed department', and it disappears when you combine it with Rook or Bishop, but not when you combine it with King/Man. Bishop and Knight also lack 'concentration' of their attacks (a property from which mating potential often follows). So it is understandable that the BN compound, combining two pieces that both have a lot of deficiencies, gains the most in terms of synergy.

Note, however, that even in absense of deficiencies of the components, combining their moves would still provide some synergy. Because the moves of one can be used to aim the attacks of the other, and vice versa. Another way of saying this is that when you combine (non-overlapping)  move sets X and Y, the piece can reach all destinations of X and Y in one move. But in two moves it cannot only reach destinations of two X-steps or two Y-steps, which X or Y could have reached in two steps, but also destinations reached by one X and one Y step.

The problem with Betza's theories is that all these effects are hard to quantify.

As to his remarks on the Commoner, it seems to me Betza is highly exaggerating. Most of the end-games of Commoner vs Bishop with equal Pawns end in a draw, and when the Bishop side has an extra Pawn, most non-drawn games end in a win for the Bishop.

The problem of course usually being that what you 'feel' and what is true are entirely different things, not necessarily related to each other in any way. This is why there is so much nonsense around concerning piece values. Everyone feels something different, and thinks his feeling is just as good as anybody else's.

In this particular case you seem to be quite off, for a piece that nearly balances Rook + Knight.