2007-08-27 Sam Trenholme CVPages Comments

This discussion inspired me to look a little more closely at the short-range project. Here is what I have come up with so far:

31 short-range pieces

I will briefly look at 31 possible 1-2 square short range pieces that are on a square board. Basically, here is an ASCII diagram of the five possible places where a given piece may or may not be allowed to move:

`3 2 1 2 3
2 5 4 5 2
1 4 . 4 1
2 5 4 5 2
3 2 1 2 3`

1 is the Dababa; 2 is the Knight; 3 is the Alfil (Interesting fact: 'Alfil' is the Spanish word for what we call a Bishop); 4 is the wazir; and 5 is the ferz.

Here is a look at all 31 combinations of these possible moves, or, if you will, 'atom' pieces:

1 | 2 | 3 | 4 | 5 | Name | Colorbound |
---|---|---|---|---|---|---|

N | N | N | N | Y | Ferz | 2-way |

N | N | N | Y | N | Wazir | No |

N | N | N | Y | Y | Guard; Commoner | No |

N | N | Y | N | N | Alfil | 8-way |

N | N | Y | N | Y | Alfil-Ferz | 2-way |

N | N | Y | Y | N | Waffle; Phoenix | No |

N | N | Y | Y | Y | Guard + Alfil | No |

N | Y | N | N | N | Knight | No |

N | Y | N | N | Y | Knight + Ferz (Augmented knight) | No |

N | Y | N | Y | N | Knight + Wazir (Augmented knight); Vicar | No |

N | Y | N | Y | Y | Crowned Knight; Centaur | No |

N | Y | Y | N | N | Knight + Alfil (Augmented knight); Kangaroo; Newton | No |

N | Y | Y | N | Y | High Priestess | No |

N | Y | Y | Y | N | Alfil Knight Wazir | No |

N | Y | Y | Y | Y | Crowned Knight + Alfil | No |

Y | N | N | N | N | Dababa | 4-way |

Y | N | N | N | Y | Ferz + Dababa | 2-way |

Y | N | N | Y | N | Woody rook | No |

Y | N | N | Y | Y | Guard + Dababa | No |

Y | N | Y | N | N | Alibaba; Deacon | 4-way |

Y | N | Y | N | Y | Alibaba + Ferz | 2-way |

Y | N | Y | Y | N | Alibaba + Wazir | No |

Y | N | Y | Y | Y | Mastodon; Jumping General | No |

Y | Y | N | N | N | Knight + Dababa (Augmented knight) | No |

Y | Y | N | N | Y | Knight + Dababa + Ferz | No |

Y | Y | N | Y | N | Minister | No |

Y | Y | N | Y | Y | Minister + Ferz | No |

Y | Y | Y | N | N | Squirrel; Tower | No |

Y | Y | Y | N | Y | Squirrel + Ferz | No |

Y | Y | Y | Y | N | Squirrel + Wazir | No |

Y | Y | Y | Y | Y | Lion | No |

The power of these pieces depends on the size of board we are using. The Alfil is clearly the weakest piece; probably worth less than a pawn. The Lion is clearly the most powerful piece; probably worth about two queens. Seven of the pieces are colorbound; an 8-way colorbound piece needs eight of the piece to cover the entire board, a 4-way colorbound piece four pieces to cover the board, and a 2-way colorbound piece needs two pieces to cover the board (such as the Bishop in FIDE chess). There is one 8-way colorbound piece (the Alfil), two 4-way colorbound pieces, and four 2-way colorbound pieces in our mix. The other 24 possible pieces are not colorbound.

The pinwheel piece

You may observe that the knight is a unique 'atom'; all other four 'atoms' can move four squares; the knight can move eight squares. One possible way to divide up the knight in to two 'subatomic' pieces is to make what I call 'pinwheel' pieces. There are two possible pinwheel pieces; the two pinwheel pieces combined make a knight. Here is a diagram of the 'left-handed pinwheel':

`. X . . .
. . . . X
. . * . .
X . . . .
. . . X .`

And the 'right-handed pinwheel':

`. . . X .
X . . . .
. . * . .
. . . . X
. X . . .`

In other words, the 'left-handed pinwheel' can, from e4, go to d6, g5, f2, and c3. The 'right-handed pinwheel' can, from e4, go to f6, g3, d2, and c5.

Each 'pinwheel' piece is 5-way colorbound; you need 5 pinwheel pieces to cover every square on the board. However, its colorboundness is unusual and a pinwheel combined with any one of the other atoms (Ferz, Wazir, Dababa, Alfil, or even the other Pinwheel) becomes a non-colorbound piece.

Dividing up the knight in to the two pinwheels, we now have 64 possible short range pieces, 54 of which are not colorbound.

Other ways of dividing up the Knight

The pinwheel is a very unusual piece. It does not preserve left-right symmetry, which means it will not be as popular with chess variant inventors. The only widely known Chess Variant I know of with left-right asymmetrical pieces is Tori Shogi. However, it is far more common to have pieces that do not preserve forwards-backwards symmetry, including Chess' pawn, and Shogi's lance, silver and gold generals.

There is one way of breaking up a knight in to two moves-to-4-squares atoms that preserves both left-right and forwards-backwards symmetry, and two ways to break up the knight in to 4-square atoms that only preserve left-right symmetry.

All three ways of breaking up a knight have already been discussed by Betza. To summarize:

Sub-knight atoms #1: Narrow and wide knights.

Narrow knight:

`. X . X .
. . . . .
. . * . .
. . . . .
. X . X .`

Wide knight:

`. . . . .
X . . . X
. . * . .
X . . . X
. . . . .`

Both of these pieces are 4-way colorbound.

Sub-knight atoms #2: Crab and Barc

Crab:

`. X . X .
. . . . .
. . * . .
X . . . X
. . . . .`

Barc:

`. . . . .
X . . . X
. . * . .
. . . . .
. X . X .`

Betza liked these sub-Knight atoms the most; they are unique in that, unlike other symmetrical sub-Knight atoms, they are not colorbound. He preferred the Crab over the Barc, since it encourages one to attack the other player.

Sub-knight atoms #3: Forward knight and Backwards knight

Forward knight:

`. X . X .
X . . . X
. . * . .
. . . . .
. . . . .`

Backwards knight:

`. . . . .
. . . . .
. . * . .
X . . . X
. X . X .`

These pieces are not very useful by themselves until combined with other atoms; the forward knight is probably the more useful atom to add to other pieces.

The eight fold path

^{1}of chess is intriguing but limits the number of possibilities. This limitation is not necessarily a bad thing, it allow all the combinations to be explored.However starting with shogi instead of chess would informed only a bilateral symmetry and a stipulation that the piece can go forwad as well as back

^{2}. These conditions leads to 15,876 different basic pieces, which if all utilized require a game board of 252x252.I have some ideas on using a sliver of these possibilities on a much smaller board that I will post when I have the materials written up. However like many such ideas an effective dynamic piece valuation system need to be implemented.

Note 1: fold the ascii diagram in half top to bottom, then left to right, then diagonally. Now unfold. There is now eight sectors, each sector, including the creases, has numbers 1 thru 5.

Note 2: there are exceptions in shogi and its variants to these two constraints, however vast majority of shogi piece types obey them.

ReplyOptionsYou seem to grossly overestimate the strength of a Lion. Not even an Amazon (Q+N) is worth two Queens. In fact there is practically no edge in having an Amazon against Queen+Knight for the opponent (on a 10x8 board). Perhaps onlty 1/5 of a Pawn or so. And the Amazon covers all squares a Lion covers, plus a lot more thorough its slider moves.

It is true that the Amazon is not completely upward-compatible to the Lion, as the latter can make the move to the second square on diagonals and orthogonals as a jump. But it is hard to believe that that would be worth more than all distant slider moves together.

So I would estimate the Lion more as Queen + 2 Pawns. I will run some Capablanca matches where one side hashas Lion, and the other has Queen+Knight, to conclusively prove that Lion is weaker than Q+N.

ReplyOptionsI completed the test, and it turns out in the beginning the Lion was even lucky, when it was ahaed against Queen (as a Queen replacement in a full capablanca opening setup) by 74% after 41 games. I let the test run until 128 games, and at that point the lead of Lion vs Queen dropped to 66%. I also tried 160 games where I gave additional Pawn odds to the Lion side (so an imbalance of Lion vs Queen + Pawn), and that ended 51.6% in favor of the Lion.

So the extra Pawn made a score difference of 14.4%, in accordance with the 12% I usually find for pure Pawn odds. The statistical error in so few games is3-3.5%, so this can be called perfect agreement. Lion is hardly stronger than Q+P, though, and not even close to Q+2P like I guessed based on the first 41 games. So with Q=950, this would make Lion = ~1075. When I tried Lion vs Queen + Knight, the Lion was totally crushed.

Despite the Lion not being that much stronger than a Queen, there still is a marked synergy effect between the moves. All the pieces I measured so far that had 8 moves (A+F, D+W, D+F, N, K=F+W) did have piece values around 285 (The Knight, with 300, being the stongest). Those with 16 moves (A+F+N and D+W+N) did have piece values around 640. This is abut0.5 Pawn stronger than the sum of its component pieces. But the Lion has 24 target squares, and could be seen as the sum of three 8-move pieces (e.g. (A+F)+(D+W)+N). The values of those add up to ~875. So it seems the synergy here is a full 2 Pawns. There obviously is a contribution that is non-linear in the number of moves a short-range piece has. If N is the number of moves, the approximate piece value seems to be around 30*N+5/8*N*N. (For N=8, 16, 24 this formula produces 280, 640 and 1080, respectively.) For N=4 it would produce 130, and indeed the value of a pair of Ferzes was found to be about a quarter Pawn below that of a Knight (but this was on 8x8, so I really should redo it on 10x8).

This formula gives a very rough estimate of the value of a short-range piece, completely ignoring the details of the move pattern. It thus most certainly will have to be refined, for instance to take account of the different value of forward moves compared to similar backward or lateral moves. (All pieces the formula was based on were 8-fold symmetric, and thus averaged out such effects.) Color-boundedness might also play an important role, distributing the piece value over a base value nd a bonus, where the bonus would very quickly drop to a negligible value if you do not posses one piece of every 'color' class.

ReplyOptionsOne factor not yet considered that has to have some bearing on the subject is the pattern of movement and attack for pieces that attack the same number or even exactly the same squares. The lion steps to any square 1 away or leaps to any square 2 away; it's unblockable. Some versions may make a null move, move off then back on their start square, even capturing while doing so. I use a "double-commoner" piece, which steps 1 square in any direction, and then steps 1 square in any direction again, except back to its start square. It moves to and attacks the same 24 squares as the lion, but it cannot leap, so can be blocked. It must be worth less.

In the 16 squares category, Charles Daniel and I both use a DWAF, which steps one or leaps 2 squares orthogonally or diagonally. His may capture 2 adjacent pieces by jumping over one to land on the other. It must be worth more.

And either of them is probably worth more than the minister or high priestess, as they are less approachable than the minister or priestess - a DWAF guards all 8 adjacent squares, and can only be freely attacked by a knight at a distance of 2. Approachability and attack fraction - the number of squares a piece attacks in its neighborhood - must affect the value of pieces, but how much? The indications you've given so far seem to say "not much".

Indeed, much remains to be done. I am already very happy with the semi-empirical formula I got for the average of the value of a set of pure short-range leaper with the same number of targets. I am inclined to interpret the formula as (30+5/8*N)*N, where the second factor (N) gives the direct effect of the moves (most likely through the capture aspect of it), while the 30+5/8*N describes the contribution of extra moves to the average mobility, which helps you to manouevre the piece in position to exert its attack.

My plan is to use the Lion as reference, and measure the contribution of each individual move (or pairs of moves related by lateral symmetry, as they should have identical contribution by symmetry arguments) to its value, by disabling them one at a time. Disabling a move will have a relatively small effect on the manouevrability, as the Lion has so many moves that it hardly suffers by missing one. But is should contribute proportionally to the second factor. So by measuring the drop in strength by disabling the move (by playing such a handicapped Lion against an unimpedent one), the second factor can be replaced by a sum of weights of all enabled moves, rather than just their total number. This way, each of the 14 possible move types can be assigned a weight. In fact I will measure the contributions of the capture and non-capture aspects of each move separately, and for the leaps will measure the effect of the leap being a 'lame' one, for capture and non-capture separately.

The captures will probably contribute almost nothing to the manouevrability, as Chess is not Pac-Man, and pieces are not manouevred in place by captures: the opponent will defend its pieces and recapture the first time we capture, so pieces usually do not survive their first capture. Unless it is a recapture itself, in which case we end up in a place by choice of the opponent, so usually not where we want to be. (Capture followed by 'recapture' on another square is far less common, as it is bad strategy to reply to a threat with a counter-threat,, as it gives the opponent the opportunituy to sac his threatened piece for anything of value.) So in a refined formula I expect the factor 30+5/8*N to be refined by replacing the N by a weigted sums over non-capture moves, while in the second term N will be replaced by a sum over capture moves.

It will take long enough to figure all that out, as there are 14 moves, whichhave to be tried as capture and non-capture, while 9 of them have also to be tested both ways in one or more lame versions. The double-Commoner you describe would furthermore introduce a kind of semi-lame moves, where a target an be reached through more than one lame path, so that the move can be blocked, but only by occupying several squares. (My guess is that such a move is practically as good as a direct leap.) And the lame Knight jumps are kid of double lame, as occupying one square can block two moves (of the Chinese Horse).

But after this project of disabling moves or move aspects one at the time in the Lion, whe should look how well certain special cases are predicted by the formula. I expect that for the weaker pieces (with only a small fraction of the Lion's move) the formula will not work, but will need some reduction factor for certain disadvantages that the moves it can do still have as a group. For instance, the Commoner is not merely an 8-target subset of the Lion, but a very special sub-set, where all distance-2 moves have been taken out. This is an especially detrimental combination of handicaps, as it reduces the speed of the piece. Similarly, taking away all color-changing moves will produce a global handicap of being color-bound, while taking away all color-conserving moves will cause some Zugzwang problems (the cause that KNNK is draw).

Another bad one would be to only allow moves in a certain direction, making the piece an irreversible one where every move would contract its 'event horizon' to cover a progressiely smaller part of the board. So I expect a piece that moves like the Pawn (but all moves capt+noncapt) to be far less valuable than a piece that moves diagonally forward and straight backward (fFbW), despite the fact that a forward move in general is worth much more than a backward move. So all these special conditions resulting from a fatal lack of cooperativity of the moves would have to be identified and quantified by measurements on the pieces that exhibit them.

And then I am not even talking about pieces that can do multiple captures in one move, or capture by jumping. These I don't consider proper Chess pieces, as one of the defining characteristics of a CV is that capture is made by replacement of the enemy piece on the target square. Checkers is not a CV! Fairy-Max does not support such pieces.

As this will be a very computationally intensive project, I am considering making a Joker80 derivative that supports other Leapers than Knight (plus a single King). Joker80 is far stroonger than Fairy-Max, and beats it at Gothic Chess even against a time odds of a factor 90. So it really would speed up things if I could use Joker80 to play the required matches. To achieve an accuracy of 1/10 of a Pawn (1.2% score) will require 1500 games…

Today I encountered a surprising effect, subverting the validity of my tests. The engine I use for self-play, Fairy-Max, only applies a positional bonus for centralizing pieces, if the pieces are worth less than a Rook. This wrks in normal Chess, as Rooks and Queens are better not centralized, as they will not benifit from it (at least not the Rook), and be much to vulnarable to attach by lower pieces.

Now this strategy backfires on very valuable short-range pieces, like the Lion. Fairy-Max is not able to exploit its capabilities well, because it does not centralize it. This leads to passive play, the main asset of the army often spending the entire game on or near the back rank.

To my surprise this causes an arifact if I play an unimpeded Lion against one that lacks certain moves. If I, for instance, disable the backward Alfil jups, and play the thus handicapped Lion against a normal one, it turns out the handicapped Lion beats the normal one by a sizable margin !? The reason is that, because the handicapped Lion has more forward moves than backward moves, it does not aimlessly wander around during periods where the engine gets out of ideas (and starts to do randm non-losing moves), but on the average moves forward, increasing pressure on the opponent, and often running into a winning tactic because of it.

So there is an unfortunate coupling between the way a piece moves, and strategy, which can have a sizable impact on strongly suboptimal strategies. Before I can meaningfully execute the test program I had in mind, I will have to learn Fairy-Max a better strategy for handling short-range pieces. Making them subject to the centralizing drive, would probably do it. For strong pieces like Q, A or C this centralizing has little to no benifit, because all these pieces are sliders. So they can be put into play easily also from the back rank. For short-range pieces, this is not the case, however.

I would suspect any piece that leaps 2 would do fairly well to stay near the knight's "optimum squares". I know nothing about the theory behind all this, but the knights have "optimum" squares in the neighborhood of the center. These squares [with maybe some tweaking after experience] should also be the better and best locations for any pieces that move 2, 3, or 4 squares maximum in a turn. They are the squares that the short range pieces can get to in 1 or 2 moves max, that give the piece [almost?] full movement range, and attack the center into the opponent's side of the board. These squares are not, in the beginning up to midgame, the exact center squares generally, but near/next to them. This makes the piece easier to guard as that placement keeps it out of the full crossfire of the enemy [unless you're losing badly ;-) ]. In midgame, those pieces are good anywhere around the center, or even along the midline, either side of the board, but may get hurt if they cross the board completely. In the end, a pair of stronger shortrange leapers in a breakaway attack against the enemy king can finish a game in short order. One common defense tactic is keeping 1 or 2 shortrange pieces back, often a pair of weak ones, to guard the pawns and approaches to the king. The king, also, especially in entirely shortrange games, is a valuable defensive piece. This has been my experience playing games against human opponents with piece mixes from pure 2-square shortrange to roughly 50:50 mixes of short and long range pieces, either predominating, across a range of board sizes.

Yes, something like that should do it. In Fairy-Max I even have a very simplified approach to centralizing SR and light pieces (like anything is extremely minimalist there; one cannot make the smallest engine in the world by paying attention to too many details…). Each piece gets a penalty equal to the square of the distance to the point between e4 and e5 (on 8x8; on 10x8 that is f4 and f5). This makes such pieces migrate towards the center.

I now changed Fairy-Max such that it decides to apply this penalty to a piece not based on the programmed value, but by an independent parameter for each piece type in the .ini file. So I now can play with a Lion that is penalized for staying asleep on the back rank. Problem is that this version is 1.8 times slower, as I had to compile it myself. The previous version was made for me by a compilation wizzard, using a better compiler, which I dobn't have.

I hope this helps. A second problem is that Fairy-Max might not be equally strong with black and with white (even if I give black the first move). This because of the direction in which it scans the board for generating moves. For moves with equal scores it picks the first it encounters. And that gives it a bias for piece moves with black, and for pawn moves with white… Or, for a piece like the Lion, which has a veryl long list of moves, the move listed first might get a larger probability to be chosen than moves at the end of the list that defines it. And that same move might be a forward move for black, but a backward move for white, causing black and white on the average to handle the piece differently! This effect should be a lot larger if all moves tend to score equal, because the piece is not attracted to the center. A centralizing drive hopefully eliminates most of it.

So I have to be very careful. I will play each setup equally often with black and white having the handicapped Lion to eliminate the directional bias, and equal number of times with white and black having the first move (to eliminate the bias of the leading move). I am testing if there is already a black vs white bias in pure Capablanca Chess, but there seems to be none: If I alternately let black and white start, the result is not significantly different from 50%. I will play 400 games to establish this. Then I will play 400 games (200 with black, 200 with white) of a Lion that is attracted towards the center, versus a Lion that diffuses neutrally over the board, to make sure that centralizing a Lion is not an inferior strategy. After that, I am ready to try handicapped Lions vs normal ones.

ReplyOptionsOk, I completed several tests. In standard Capablanca Chess (i.e. without Short-Range leapers in stead of A and C) Fairy-Max did not have any bias towards white or black. That is no guarantee, however, that the Lions will not introduce such a bias by themselves, because thire moves being considered in an order that is the same for white and black. To eliminate such a possible effect, I will make sure that all subsequents tests will be done with each test position also played after color-reversal.

A second test I did was measure the effect of centralization drive on the Lions. This indeed is fairly large. If I play two versions of Fairy-Max against each other, from an opening position that is Capablanca Chess with A and C replaced by Lions, the version that tries to centralize its Lion scores 57.5% (measured over 1000 games) against a version that doesn't. So leaving the Lion freely diffusing over the board was indeed a very inferior strategy. Backward moves (the same as moves away from the center, as a Lion in enemy territory doesn't survive long) are usually bad strategy. If the engine knows this, there is no harm in adding such moves to a piece, as it will only do them in case of an emergency. But otherwise, adding the moves will seduce the engine into using them too often, and the upward compatible (and thus better) piece will cause a worse result, because of the clumsy handling.

I have a fast compile now of the version that allows me to program the centralization drive of a piece through the .ini file, 1.5 times as fast as my own compile. This means I can switch my testing again to 40/1', speeding things up. So I am currently running games where one side has two handicapped Lions, lacking one of the moves (or a pair of symmetry-equivalent moves). When I have done all 14 of those, I will report the results here.

ReplyOptions