How can I figure out the relative piece values in variants I come up with?

I noticed that it can be hard to look up and find the values of some fairy pieces.  For instance I could not find the point values of knight-riders, camel-rider, alphil-riders, or dabbaba riders.  I understand that sometimes the values of pieces can vary from one game to another, meaning that even if I found the values of the given pieces, I couldn't necessarily assume that they are the same as they would be in some of the variants I create, especially given that some of the variants i create have starting positions very different from standard chess.  How can I figure out the relative piece values, in any variant I come up with?

I think for most variant pieces it's hard to determine an accurate value. In a few cases only, there might be ways to make an estimate:

1) If it's a compound piece (example chancellor = R + N) then you can add the value of the individual pieces. Sometimes there is a synergestic effect, which might add another point. So chancellor = 5 + 3 + 1 = 9.

3) HG Muller has made formula which seems to work for pure jumpers. The value is based on the number of squares the piece attacks (N). The formula is: value =33N + 0.69N^2 (centipawns). So for example:

knight (N=8): value = 33(8) + 0.69(8)^2 = 264+44 = 308 (3 points)
hawk (N=16): value = 33(16) + 0.69(16)^2 = 528+177= 705 (7 points)

Unfortunatelly I'm not aware of any way to estimate the value of other pieces.

In 'A World of Chess, Its Development and Variations through Centuries and Civilizations' (by Jean-Louis Cazaux & Rick Knowlton) values are given for variant pieces in a large number of different chess games. This is from page 4: "We are fortunate to have a program, provided by Zillions of Games, which analyses our diverse games to give us approximate values of the playing pieces."

Maybe somebody with more knowledge of Zillions than me can expand on this?

Zillions of Games is a commercial program that can be configured to play almost any chess variant. It uses a proprietry algorithm to guess the piece values. The values it comes up with are known to often suck big time, so people that are developing games for ZoG have to use tricks to make it guess better values (e.g. like specifying some moves of the piece multiple times, so that it thinks the piece has more moves.)

There is only one reliable way to determine piece values, and that is the empirical one. Exact mathematical calculations by a guessed method are still blind guesses. You will have to observe what the piece on average does in a real-life game. So the best method is just to configure a multi-variant engine for using the piece, and then let it play a couple of hundred games with that piece against some material of known value, to see which has the upper hand. E.g. if you would want to know the value of an Archbishop, you could replace the Queen of one player by an Archbishop in the FIDE setup, and have the program play, say, 400 games against itself starting from that position. (To average out the first-move advantage, you can alternately replace the white and the black Queen.) The Queen will win such a match with a score of around 62%. But if you then delete the f-Pawn for the Queen side in addition, the Archbishop will score about 54%. So Q > A, but Q < A+P, but the latter combinations are closer in value. Taking the scores seriously, it seems Q = A + 0.75P. You can do that for any piece.

Never believe piece values obtained by calculation, no matter how advanced. They are usually off by a large amount.

BTW, Nightriders are worth about half a Pawn more than Rooks. On an 8x8 board. Board size strongly affects the piece values. On 8x8 a lone Bishop and a lone Knight are about equal in value; on 10x8 they already differ by half a Pawn.

Camels are rather weak pieces on 8x8; against orthodox material they always seem to get lost without compensation in the end-game. Their opening value comes from the fact that their good forking power in a dense position allows you often to trade them for an orthodox minor. I am afraid a Camelrider would hardly be better; the chance that you can make the second step is quite small on 8x8.

I never measured Alfil- and Dababba-riders. But the pure riders would be very weak. The Alfil has only access to 8 board squares, and making a rider of it doesn't change that. It just means that it can sometimes move between those squares in somewhat fewer moves. Even if you would allow it to teleport to all of the 7 squares it is not on, it would still be a virtually worthless piece. The Dababba can visit 16 square, which is still pitiful. Of course when you combine these riders with a move that breaks their color binding, it suddenly becomes a lot more useful. Still not as much as the corresponding normal slide. On a crowded board the jumping moves are dangerous, but that advantage gets lost in the end-game, when the normal slides are also almost never blocked. And then they just attack fewer squares.

 You misunderstand. A nightrider has all the moves of a knight, plus the options of further movement. You're not forced to move as far as possible.

Do you know where I can find such an engine, and how to get it to play hundreds of games from a given starting position?

Some of the variants I've made up are on boards that are a different size than the standard chess board, which I understand a camel becomes more valuable on a larger board, and a non royal king becomes more valuable on a smaller board.  One variant I've come up with is 3d, and has pieces that can't exist in 2d such as the (2,1,1) leaper, the (2,2,1) leaper, and the (1,1,1) rider, and I'm wondering if an engine could play games using these pieces.

Take the squares the piece controls, add 1 to it, subtract 2 if it can only access 1 color, add 1.5 if it can go to any color, then divide it by 3

Really?

Here’s a rigorous explanation:

https://triceschess.com/articles_01.shtml

Not this crap again

PhDs in math nominated it for the Best Paper submitted to the ICJA that year. Only the feeble-minded criticize it. You just informed the world that you wallow in your own ignorance.

Just so everyone knows, a mathematician named Taylor wrote the first paper on chess pieces values in 1876. It was praised and considered noteworthy. His equations were for a board that was square in shape. The paper Ed Trice wrote in 2003 or 2004 was for a non-square, rectangular board. It was harder to derive. It was also extended to work for more pieces. It was a “Holy Grail” for computing values for any type of piece on any size board. People who are PhDs in math approved it for publication.

Based on this idea from @HGMuller I just got another idea about a possible approach:

Let's indeed write this multi-variant engine. But instead of playing entire games with it, we could have it run evaluations for a large number of random positions.

Assume that we have piecetypes [T1, T2, ..., Tn] with values (weights) [w1, w2, ..., wn]. We think we know some of these values (weights) but we also want to improve (tune) some of them. So we run the evaluations and that gives us a dataset of observations:
X: we had a position with n1 times T1, n2 times T2, ... nn times Tn
Y: Evaluation result
Parameters to be tuned: weights

Our engine does not need to be a state of the art top-engine. It will do the job with some limited minmax calculation, it should correctly detect mate/stalemate/King loss. But whatever piecetype we want to include must also be implemented.

At this point the question is, how good are the evaluations that our program produces? For this, we need to look at the mismatch between program evaluation, and plain material count, based on the very weights that we are trying to tune.

Let's imagine that we started with weights of 9 for pawn, 3 for Bishop and 1 for Queen. This deviates a lot from our intuitive knowledge.
And in the evaluation by our program, this should eventually push it towards a result, that deviates a lot from the plain material count.

This is where I think Machine Learning algorithms can be put to work:
https://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html

Today I finished some basic program that can do the evaluations with any vanilla or divergent-vanilla piecetype.
https://github.com/evert823/chesspython/

Work in progress...

The best litmus test is against strong humans at first, then other programs

The idea seems sound and logically based, however there is apparently one underlying assumption that may not be valid. Namely a particular w1 > w2 everywhere and always. I can think of a few examples.

white to move and play Bxf6 expecting Axf6 (the archbishop, knight + bishop) but instead black retakes with the chancellor (knight + rook)

So that leads to this position with white to move

Obviously he played Archbishop x Chancellor since C > A by all conventional wisdom.

This game has been analyzed on and off since July of 2005! This is known as "Winning the Light Exchange" in this variant because there's 3 types of non-chess exchanges : Heavy, Medium, and Light.

Usually the Light Exchange is made when the wiser player realizes his Archbishop can go on an immediate tactical rampage and/or the opponent's chancellor is somewhat out of play and just a spectator to the horror. But this is not the case here! I think the black Archbishop only makes 2 more moves in this game, yet black wins! So the distribution of the material long after the trade was more is important than the actual sum of the material, and such a concept is not easy to factor into any a.i. program.

Look for the move by move replay here and see if you can spot a fatal flaw

https://gothicchess.info/articles_04.shtml

Played July 7, 2005 or watch the entire game animated below

You can think of a few opportunities to promote your favorite chess variant, that's one thing I can notice.

Actually only the feeble minded would attach any value to the theory presented in that paper. Because it predicts a value of zero for pieces that cannot deliver any safe check, such as the non-royal King or the Shogi Gold and Silver Generals. And when it predicts pure nonsense for one piece, it is obviously not generally reliable for predicting the value of any piece; any relation it has to actual piece values would be pure coincidence. Since there are so few pieces with an uncontested value (basically only the orthodox Chess pieces Knight, Bishop, Rook and Queen) and an infinite number of mathematical formulas that could be used to calculate them, it is not much of an achievement to find one that gives a correct value for those 4 cases. And of course almost always it then gives total nonsense for unorthodox pieces.

If what you say here is true, the only thing it proves is that you don't have to be very smart tp get a Ph.D. in mathematics...

There is so much interesting information here.

I found this bot again...