I was tired of digging up pieces of Ralph Betza's musing about chess piece values on chessvariants and related pages. So I'm mirroring the content here as I need them.
(I'm still grumpy that website so fucking adamant about ai generated images that add nothing on the piece articles.)
All of these are not a science, just hunches and estimates from some random chess player dude. But hey they're something at least. I'm not experienced enough to make my own estimates.
Furthermore the "true value" of a chess piece is already a vaporous concept. I'm mostly using it to fill the bot's minimax on the chess roguelike.
Written 08/09/2025.
Original here with a corrected chart here.
A Short Rook is a Rook but whose maximum range is limited. In Betza notation they're Rx with X being the max range.
These articles had estimation for the strength of those limited Rooks on a normal 8x8 board. How accurate they are? I don't know but they're the best I got. I'm not experienced enough in variant chess to make my own judgement.
The average mobility on the charts was assuming the probability that a square is empty is 0.69 (nice).
This is a metric to evaluate a piece. It doesn't map directly to piece value, but it's a something to toy with.
Consider a w by h board and a leaper movement, denoted as (x,y) (or in Betza notation Lxy), as in "move x spaces vertically and y horizontally (or vice versa)". For example a Knight is (1,2) (or (2,1)).
The Empty-Board Average Random Mobility is the average number of moves that movement provide when the piece is placed on a random position on the board.
For a single part of the move it is (w-x)*(h-y)/(w*h). If the move is orthogonal (x = 0 or y = 0) or diagonal (x = y) multiply by 4 and you're done. If the move is neither multiply by 8 if the board is square.
If the board is not square, then add the value of the corresponding (y,x) pair then multiply by 2 or 4 instead. This simplifies back to the original formula for diagonal moves.
If a piece has multiple such moves and they don't overlap, you can just add them together.
Examples on an 8x8 board (might have gotten carried with the extreme leapers but I was curious):
And on an 8x10:
Riders like Rooks, Bishops, Queens or Nightriders (eew) can have their movement blocked. They are the infinite sum of their "unit vector" but on blockable leapers, bounded by the board size.
So for example for a Rook to do its (0,2) move, it needs its (0,1) move to be an empty square. Betza as an arbitrary numbers chooses 0.69 as the probability that a given square is empty.
Consdering a (x,y) rider, the mobility of its (N*x,N*y) move is thus the empty board mobility of that move times 0.69^(N-1) (with that constant adjustable as you see fit, it's arbitrary after all).
There is a paragraph on the value of the different directions of a move. It's a "quick and dirty rule of thumb" that's not elaborated on, but that's something to start on at least.
Betza considers that for an orthogonal move (Wazir, Dabbaba, Rook), front move = both sideways = 3~4 x backwards (so a ~43/43~/14 or ~44/44/12 split).
For a Bishop, both forwards = 3~4 x both backards (so a 75/25 or 80/20 split).
"The four forward moves of the Knight, about like the Bishop" idk what that means, but then the forwardest/long moves are "a bunch more" than the less forward/wide ones.