When it comes to chess variants, theres quite a few concepts I've thought about and would like to at some point share on this website, a lot of them are vague and essentially none have I thought through to an actual meaningful conclusion, if its even possible to do so in the first place with these kind of things. I have no idea if people would even be interested in hearing them, but for those that are I'll post them regardless. This topic seems about as good a place to start as any other, so to get the ball rolling...

Introduction

This is just an outline I have on how to calculate Ultima piece values, not sure if anyones tried something like this (someone probably has), but thought I'd share and outline some of my (incomplete) work...

Ultima pieces are generally characterised by homogenous movement and differing capture technique. For a chess piece, its power is essentially just a matter of mobility. How effective an Ultima piece however is not as immediately obvious as with a chess piece. What we require is a brute force technique of calculating its power...

Statistical capture probability

This essentially involves calculating the probability of a piece making a capture across all situations. The pieces value comes out as an equation involving up to 3 variables-number of friendly pieces on the board=f, number of enemies on the board=e, board length=l. I'll spare people the actual maths, but the idea is to get an equation that can then be plotted across 3 axis, for all values of each piece.

Basic Method

So to carry out the calculation, you basically have to place the piece on every part of the board, and calculate the possibility of it making a capture from there. Anyone who knows anything about probability theory knows that it will come down to two basic words; AND (multiplying), OR (adding), with the net result being a value between 0 and 1 of an event happening.

So for a piece on square X the process becomes basically; probability of capturing in this direction, added to probability of then capturing in THIS direction if you couldn't previously capture and so on till you have a sum of variables.

The ultimate point is to get an average of all positions to get the final equation. You can simplify it by the symmetry of the board to only work it out for squares in one quarter and then half that again as only the central diagonal from the corner out has no mirror squares in that quarter, then you get the final polynomial.

Simple example; Remover

Probably the simplest to work with would be a "remover" piece (rifle captures any adjacent piece). So for this, taking e=no. of enemies pieces on board, f=no. of friendlies, l=length of board, A=area of board/number of squares on board, P=capture probability, the equation will be 4*(P for corner squares)+24*(P for edge squares)+36*(P for inner squares), as there are 4 corner squares, 24 edge squares and 36 inner squares, with no more differentiation needed between squares the remover can be on, as the remover only attacks adjacent squares. This is then divided by the total number of squares, 64 in this case, to get the average per square-the pieces actual value at a point in a game.

So the equation for P for a corner square is e/A+[1-(e/A)][(e-1)/A]+[1-[1-(e/A)]][(e-2/A)], which is about as simple as calculations get with this. Here the idea is P for a particular direction for a remover is (no. of enemies)/(number of spaces on the board), with for the next you using the unitary compliment (1-X) to multiply, in other words, probability of capturing in this square IF you didn't capture in the previous.

I won't bother doing any further calculations or finishing the equation for the remover, needless to say it gets drastically more complicated. I myself gave up for more complex pieces like the coordinator and pincer pawn, having only been able to slug it out for one night...

Open question; how to calculate for noncapturing Ultima pieces?

Its much more difficult to calculate any clean values for noncapturing pieces, by these methods at least, as for one thing there are so many types, each of what of which would require their own approach, one might reduce enemy mobility (immobiliser) and have a value proportional to enemy mobility, the other may force enemy movement and/or increase own mobility (pusher/swapper) with a value inversely proportional to piece mobility, or maybe a piece may do something else entirely eg a protector preventing friendly pieces from being captured, which would have a value proportional to the capture threat of the enemy. Then you could have pieces more abstract again...