Lasker Piece Values

Starting with more Pawns is to the advantage of the Knights, because the Pawns offer support points. This makes it easier to keep all Knights protected twice at all times, to remain immune for 2N-for-Q trades, and still leave enough freedom to actually move some of the Knights. Furthermore, in the face of the numeric majority of the Knights, the white Pawns become a liability, as the cannot be protected with Queens and King against attack by 2 Knights.

Below you see how QueeNy beats Stockfish with only 6 Knights vs 3 Queens:

This of course doesn't prove that it is actually won with only 6 Knights. It only proves that Stockfish is stupid. It is not prepared to trade Q for 2N while it still can, which would be the route to a draw or even win. QueeNy would win this with white every time, even against itself.

I'd certainly agree that actual play or simulatiions are a better method of determining piece value, now that the resources  are available.

Short of that, I think that coming up with a mathematical method and seeing if the results make any sense is one way to begin.

Sure, it would be great to have a theory behind piece values that would make it possible to predict those values with high accuracy just by seeing how they move. The (30+5/8*N)*N formula for short-range leapers is a step in that direction. But it is exactly the "seeing if the results make any sense" where virtually everything you read on the internet about values of unorthodox pieces fails. Originators of such theories just calculate values based on rules that according to them make sense, and then 'test' the values that come out against the same 'sense' that provided the rules in the first place. This is circular reasoning.

The mentioned paper is a particularly bad example of this, as the method used (counting 'safe checks'), although it leads after some tweaking to correct values for the othodox pieces and 'intuitively correct' (but alas, empirically wrong) values for Archbishop and Chancellor, produces 'obviously non-sensical' values for other pieces. No one could seriously believe a non-royal piece moving as King is worth absolutely nothing just because it cannot deliver 'safe checks'. And if the method so obviously sucks for some not-so-unreasonable pieces, why would it magically produce correct results for pieces that happen to occur in Capablanca Chess?

The problem is that the orthodox pieces are so similar that they do not even offer a reliable starting point for extrapolation. It is like fitting a line through a single point, you can pretty much make of it what you want. For example, all orthodox pieces are totally symmetric and have the same captures and non-captures. Except Pawns, but most of their value is due to the ability to promote, and due to their tactical agility. So whether captures are worth 2 times as much as non-captures, ratherthan 1.5 or 2.5 times will always remain a blind guess, if you have no empirical data on divergent pieces to guide you.