KAAK is a general draw (only 0.2% of the black-to-move positions is lost).
After reading this, I tried to get a "Mate in __" position with KAA vs K. Here's the resulting position:
Indeed, that is the technique for executing the final mate. The longest forced mate according to fairygen is this mate in 16:
[Event "Alibaba end-game"]
[Site "ONTWIKKELLAPTOP"]
[Date "2014.12.26"]
[Round "-"]
[White "fairygen"]
[Black "hgm"]
[Result "1-0"]
[Variant "fairy"]
[FEN "8/k3A3/3K4/8/6A1/8/8/8 w - - 0 1"]
[SetUp "1"]
{--------------
. . . . . . . .
k . . . A . . .
. . . K . . . .
. . . . . . . .
. . . . . . A .
. . . . . . . .
. . . . . . . .
. . . . . . . .
white to play
--------------}
1. Kc5 Ka6 2. Ag4e4 Ka5
(2... Kb7 3. Ae4e6 Ka6 4. Ae6c6 Ka7 5. Ae7e5+ Kb7 6. Kb5 Ka7 7. Ac6e6
Kb7 8. Ae5c5 Kb8 9. Kb6 Ka8 10. Ae3 Kb8 11. Ae3c3 Ka8 12. Kc7
Ka7 13. Ac4 Ka8 14. Ac4c6 Ka7 15. Ac3c5#)
3. Ac2 Ka6 4. Ac2c4+ Kb7
(4... Ka7 5. Kc6 Kb8 6. Ae6 Ka7 7. A7c5+ Kb8 8. Ae3 Ka7 9. Ae3c3
Kb8 10. Kb6 Ka8 11. Kc7 Ka7 12. Ac4 Ka8 13. Ac4c6 Ka7 14. Ac3c5#)
5. Ac4c6 Ka7 6. Ae7e5 Kb7 7. Kb5+ Ka7 8. Ac6e6 Kb7 9. Ae5c5 Kb8 10. Kb6
Ka8 11. Ae3 Kb8 12. Ae3c3 Ka8 13. Kc7 Ka7 14. Ac4 Ka8 15. Ac4c6 Ka7 16.
Ac3c5# 1-0 {checkmate}
I see the only practical use of Jester in sacrificing it for a pawn in attack against the king or to organize a pawn breakethrough. In ending it isn't worth even a single pawn, because a pawn can be sufficient for a win, but with Jester it is hard even to hold one pawn. Althougn, in ancient chess Bishops were even weaker - it was able to leap over one square and only diagonally.
Still as in a real game it is hard to avoid exchanging Jester for a pawn, its value must be equal to 1 point.
The exception is a pair a Jesters, when the opponents king is at the side of the board, as it's seen at the diagram above, but is it possible to push the king to a such position from the center of the board with a king and Jesters?
In ending it isn't worth even a single pawn, because a pawn can be sufficient for a win, but with Jester it is hard even to hold one pawn.
That is flawed logic, as by that samekind of logic Bishops and Knights would also be worth less than a Pawn. And we all agree (I suppose) that they are worth significantly more, and that trading Bishops and Knights for Pawns at the earliest opportunity is a (badly) losing strategy. Two Knights don't fare any better than two Alibabas/Jesters.
The value of a Chess piece is mainly determined how well it is able to support Pawns or prevent their advance, and hardly by whether it has mating potential by itself. There exist pieces that are worth ~7 Pawns in the presence of Pawns that have no mating potential on their own. (E.g. Team-Mate Chess has 4 pieces worth than a Rook that cannot inflict checkmate on a bare King.)
Agreed. 960 variants, huh?
That is flawed logic, as by that samekind of logic Bishops and Knights would also be worth less than a Pawn. And we all agree (I suppose) that they are worth significantly more, and that trading Bishops and Knights for Pawns at the earliest opportunity is a (badly) losing strategy. Two Knights don't fare any better than two Alibabas/Jesters.
...
I think that with a Bishop or a Knight it's much easier to play against pawns in the ending, than with Jester. But it was just an example and not a particular way of determining the value of chess pieces. Maybe I underestimated the power of Alibaba, especially a pair of them, but is it really possible to mate with a King+2 A when the opponents King is in the center of the board?
Pawn = 1
Bishop = 3
Knight = 3
Rook = 5
Queen = 9 or 10
King = the game
Maybe I underestimated the power of Alibaba, especially a pair of them, but is it really possible to mate with a King+2 A when the opponents King is in the center of the board?
KAAK is a general draw (only 0.2% of the black-to-move positions is lost).
So the answer is no: only in the exceptional case where the bare King is already trapped in the the right corner (one that can be reached by one of the Alibabas) mate can be forced.
But of course this is even worse for a pair of Knights, where you can never force mate no matter how unfavorable the position of the bare King, (unless it is mate in one, of course). So in itself this fact doesn't indicate an Alibaba is worth less than a Knight (which for other reasons it of course is).
To HGMuller: but two Knights can always push a King into a corner of the board and cannot force a mate due to a stalemate situation. This still shows power of Knights against Jesters. K+2N vs K is a rare and rather a theoretical situation, but in the game may occur a position where Knight will be preferable to Bishop.
To establish the value of Jester studying of the following situations could be worthy:
1. (in an ending with several pawns from the both sides) Knight vs Jester + extra pawn - here I would prefer Knight (or Bishop instead of Knight);
2. (the same) a minor piece vs Jester + 2 extra pawns (but not linked passers) - if the side with Jester is winning, it means that its value is about 1.5 points;
3. (the same) 2 minor pieces vs 2 Jesters + 3 extra pawns - if this is a draw, it proves that the value of Jester = 1.5;
4. (the same) 2 minor pieces vs 2 Jesters + 2 extra pawns - and if this is about a draw, it would show that value of a pair of Jesters is closer to 4 points.
Only a practical game can determine the real value, but in the ending it is seen more clearly.
Indeed, these are good methods to get information on the end-game value of a piece. What I usually do is put Pawns on 2nd and 7th rank starting from the edge, 2 or 3 on each wing, in various combinations, and then hide a King behind them on one side, and the pieces under test on the other. That way you can easily create some 16 different starting positions for each material combination, which is very helpful for driving up game diversity from what the normal randomization of moves would give you when starting all games from the same position.
I will try to run some games like that, starting with Knight + 4/5 Pawns against Alibaba + 5/6 Pawns.
It might also be interesting to play Alibaba + 2-4 Pawns against 4-6 Pawns.
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.
A bishop is not valued because it can only move on one color. I would say it is worth 2.
2 and a half.
I tried 400 games of Alibaba + 6 Pawns vs Knight + 5 Pawns. The Knights won this by 56.5% (127+, 198=, 75-). This is not a very large score, though; even a Pawn in the opening is worth an extra 18%, and in the late end-game it is much more. So the end-game value of the Alibaba seems just slightly less than Knight minus Pawn. (i.e ~200, when N = 325.)
I also tried about a dozen games of Alibaba + 4 Pawns vs 6 Pawns. There both the Pawns and the Alibaba won a few times. So it seems reasonable to assume the end-game value of the Alibaba is about 2 Pawn.
Well, the opening value of the Alibaba seems to be a bit higher than I had guessed. An opening position with 2 Alibabas instead of 2 Knights loses by 74% (measured over 400 games, i.e. with statistical error 2%). En extra Pawn usually means ~18% extra score, so the 74% represents an advantage of about 1.5 Pawn (perhaps a bit more, because the score is so extreme that it gets in the region where it might start to saturate).
Two Alibabas do beat a Knight + Pawn by 57%, however, swhich translates to slightly less than half a Pawn. When we put the usual P=100 and N=325 (Kaufman values) that makes about 470 for two Alibabas, i.e. 235 per Alibaba.
I guess the opening value of the Alibaba is boosted because of the Elephantiasis effect, which depresses the value of all the opponent minors a bit. And the opponent has 4 such minors, which now all have to play a trade-avoiding strategy w.r.t. the Alibabas, which are worth significantly less, limiting the usefulness of these minors.