Establishing the value of a chess piece

Assume that this was a standard piece:

Yes, this piece is colorbound, and I believe it's value in pawns to be at most 3. Also, it should be noted that on the 10x10 board, the two jesters are located between the bishop and knight on both sides of the king. So, just as the bishop pair is worth slightly more than the value of two induvidual bishops combined (supposedly 6 pawns), I'd assume the "jester pair" to behave similarly?

Good question. LIM's estimate looks pretty close, I'd say. It's "fighting value" would be close to that of a knight (3) due to its jumping ability, close to a bishop (3.25) due to being stuck on one color, close to a king (4) due to having either diagonal or rectangular motion, so that averages out to a little over three.

P.S.--The board size should be irrelevant.

The problem with this piece (commonly called 'Alibaba', because it is a compound of the Shatranj Alfil and the Dababba) in that it is not just color bound, but suffers a higher-order of meta-color binding that confines it to a quarter of the board. I would be surprised if its intrinsic value was worth more than 1.5 Pawn.For comparison, in Shatranj the Alfil (which jumps 2 diagonally) is worth about the same as a Shatranj Pawn, but because Shatranj Pawns promote to worthless pieces they are worth only about half as much as a FIDE Pawn. (E.g. a Rook is worth 8 of them.) Perhaps due to its forking power the Alibaba's opening value could be 2, because there is a fair chance to trade it against an opponent minor, if you are quick.

The only way to know for sure is to play a couple of hundred materially unbalanced games, between equally strong opponents (i.e. self-play of a Chess engine), where one side has Alibabas in stead of Knights or Bishops, and see how he fares. (Probably the sides that has the Bishops or Knights should be given additional Pawn odds, to make that it not completely crushes the side with Alibabas, and will get you in the score range where result score is approximately linearly dependent on material value.)

Such tests could for instance be done with Fairy-Max.

It is worse than a knight because it is reduced to ¼ of the board. I would give its value at 2-2.5 approximately.

The piece combines the defects of the bishop (colorbound) with those of a knight (limited scope) - and more (as it can never reach half of the square of its own color). 

Intuitively you don't want to give it a very low value, as it does leap about... it may have some tactical power... but I would guess that if you got a knight and an alibaba for your rook, it's more or less an equal trade. It's probably worth the exchange in itself. 

Of course its value is lessened the bigger the board is. For those of you who think that it doesn't matter 8*8 or 10*10, try to imagine what would be the value ratio of a bishop vs. a knight on a 100*100 chessboard. 

There is a software, Zillions of Games, which does that. Learn how to write a simple zrf file with all the pieces you like, place them in the center of the table and Zillions will oblige and tell you how much it thinks the piece is worth. Finnaly put a pawn on the home row, get its value and divide them. (30 minutes work for the first piece, 5 minutes for every additional one; you may even find a ready made zrf file for it on the web - just download it).

Good observations from everyone, and a facinating topic (to me). Wouldn't it be great if there were a formula where you could plug in the values of the piece's motion (jumping ability, colorboundness, range, directionality, size of board, etc.) and get a single, outputted, fighting value for that piece?

Well, it's also relative to all the other pieces, so you'd have to plug in all the different pieces and the size of the board.

Ratio of max to min controlled squares (mid board vs corner), whether it could leap, and % of all squares possible to reach would be the big ones.

It attacks 8 squares in the middle of board, same as Knight.

It attacks more squares in the center, and less when placed on the flank, same as Knight.

It can only attack one color complex, which is disadvantageous.

Can never reach half the squares of it's own color complex.

The way it moves lacks flexibility, unlike the Knight. Once it moves, 2 of the squares available were already available last turn, so it gains few new attacking squares after moving.

My estimation is between 2 pawns, on an 8x8 board, but this depends. Do you add it to the already existing pieces in chess? Then it may be a bit better, because the jumping pieces are better the more cluttered the board is, or it may be worse, just because it's low value means it's stealing squares from more powerful pieces.

Probably about 2. It combines the worst of the knight (short-range) with the worst of the bishop (one-colour).

In fact it's worse than even 1 colour, it can only access 16 squares!

I think you would need (or at least it would be helpful) to change the colour scheme of the board too .

The "black" squares should become different (and easily distinguishable dark shades) and similarly the "white" squares should become different (and easily distinguishable) shades of light in order to make it easier to distinguish the colour complexes. Although the Alibabas are trapped on the colour complexes they begin on there might be some endgames where the only way to avoid stalemate, inability to mate or threefold repetition was to promote a Pawn to a third, fourth etc. Alibaba.

On a board size that was divisible by 4 (8x8,12x12 etc.) an Alibaba of any colour complex (remembering that there are four colour complexes on the board) would have the same number of squares theoretically available to it.

On a board size that was divisible by 2 but not 4 an Alibaba whose colour complex includes a corner square would have more squares available to it than one that didn't.

Another way to describe a Knight's move is to the closest square of opposite color non-adjacent.  In this way, a Knight (or Jester) controls small pieces of diagonals (almost like a Bishop).  And like a Bishop, a Knight moves freely on ranks and files whereas the one-directional path of a pawn makes a pawn inferior.  And, better than a mono-color Bishop, the Knight can alternate square colors and is stronger than a Bishop in tight, crammped conditions with many pawns on the board.  Some variations of the "Knight's Tour" involve the Knight covering one quadrant of the 64 square board at a time.  For this, we can see how the Knight can access every square on a 16 square board.  I don't even want to speculate about an 8 or 10 square board unless it involves some kind of puzzle unique to that board size.  I hope these observations fuel your further progress on the question.

The problem with Zillions of Games is that it has a heuristic for determining piece values which is very inaccurate, and often totally off. It has for instance no idea that the Archbishop (BN compound) is worth nearly as much as a Queen. So I would not consider using Zillion's static evaluation as a reliable way getting piece values. You really have to play games to get the empirical value of a piece. And it is better to do that with Fairy-Max than with Zillions, as it is in general a lot stronger, so that you can use faster time control to get the same quality of play. And you will need several hundreds of games to get the statistical error acceptably small.

The general formula for the piece value of short-range leapers is

1.1*(30 + 5/8*N)*N centi-Pawn

where N is the number of moves it would have when not hindered by the board edge. But that doesn't take into account 'global' properties implied by the specifics of the move patters, like color-binding, forwardness, speed, mating potential. The Alibabba really takes a huge hit on color binding.

Here's some more "important" questions:

• Can two Jesters (Alibabas) and a king force checkmate against a lone king?
• Is the Rook vs. Jester endgame usually a draw, or a win for the side with the rook?
• Is there a fortress position(s) in the Queen vs. two Jesters endgame?
• Can a Jester(s) and a king force stalemate against a lone king?

Any help is appreciated, as always. Also, if anyone thinks up another question regarding this piece, please do post it here! Thank you.

<Togex> I like these questions! Definitely mind opening and instrumental in acquiring a real feel for evaluting the ali-baba. 

KAAK is a general draw (only 0.2% of the black-to-move positions is lost).

KBAK with unlike B and A is 80% won with white to move, and 43% lost with black to move, which is far more than for a general draw, but far less than for a general win. E.g. KNAK is 27% won wtm, and 1% lost btm. Both KBAK and KNAK have lengty wins (35 or 33 moves to conversion, max).

KRKA is a general win (17 moves to conversion max).

KQKAA is a general win, no matter which meta-colors the Alibabas are on.

The same holds for KQBA and KQNA. For the latter there are some perpetual check draws, though. (Just from the top of my head, e.g.  KQ6/8/2k5/1n6/2a5/8/8/8 w - - 0 1 .) These are so few in number that they are not visible from the tablebase statistics (99.044% won with white to move, most non-won positions being forks on K+Q by a protected Alibaba).

The KAK endgame even is a dead draw under rules where stalemate is a win. (Not a sinle lost position with black to move.)

Wow! How did you come up with all this information?

By running fairygen ( http://kirill-kryukov.com/chess/discussion-board/viewtopic.php?f=6&t=6258 ), after defining the Alibaba in its pieces.ini file.

Btw, the data I gave is all for 8x8.