In order to determine the values of some of these pieces, I have recorded the results of a series of computer chess games in which the computer played both sides, and in which all the normal pieces except one were used.
For example, in one series of games Black had all the normal pieces, but White had a piece that moved like Wazir or Alfil instead of the Knights; this new piece ( shall we call it the Waffle? ) has a mobility of 5.75 ( 3.5 from the Wazir, 2.25 from the Alfil ), but seems to be weaker than the Knight; after 5256 games, White's winning rate was 0.457002 ( for reference, White's winning rate in standard chess was 0.49227 after 13842 games ).
The large number of games played gives the results a degree of certainty; a result of 0.46 after 4000 games is different from one of 0.47, standard deviations, t-tests, and all.
The large number of games played was achieved by networking. One machine can crank out 36 games in four hours.
The number of games is always a multiple of 6 because the opening position is permuted 6 ways: that is, one game has Ra1, Nb1, Bc1, the next has Ra1, Bb1, Nc1, and so on. This is important, but the reason for it is several chapters ahead.
The results are reproducible, but their meaning is unclear. For example, King versus Knight has a win rate of 0.37: this seemed fine to me, it seemed like the results I got when I tried it by hand, with me playing both sides. Then I discovered a better strategy for the side with the King, and found that the real win rate should be over 50%!
I'm certain that you don't want to see more details about the methodolgy and the results.
The meaning is unclear, but sometimes it works. For example, it was the computer results that made the game of Different Augmented Knights become a reality. The game has been played by masters, and the computer results turned out to be correct.
Sometimes it works, but it doesn't help with the problem of finding a general solution to calculating the values of pieces.
Because I can't succeed at finding a way to calculate piece values, well, naturally, now I want to prove that it's impossible!