There have been many attempts to write mathematical formulas or create tables of piece values, but I don't think any have gained widespread acceptance. Authorities can't really even agree on the values of the orthodox pieces in FIDE Chess, so there's no accepted way to determine whether any given valuation is "correct" (or even precisely what it would mean if it were). Some people have tried computer tests to determine values empirically, which I think is a promising direction, but results don't always agree with other computer tests or with the accepted values (such as they are) of the orthodox pieces, so it's not always clear what they mean, and they're not much help in predicting the value of any piece that wasn't specifically tested.
Most people agree that value is primarily related to a piece's "mobility", or how many different ways it can typically move or capture (after somehow accounting for the fact that certain moves are more likely to be possible than others; e.g. a Bishop can move 7 squares, but only in rare situations).
But then there's a bunch of other factors that we're pretty sure are real but that no one really knows the true value of, like mating potential, development speed, colorbound pairs, stealth, how they cooperate with allied pieces, and so forth. Notice that many of these aren't even intrinsic to the piece, but relate to the other pieces on the board, which means that they change from variant to variant (and even over the course of a single game, as pieces get captured and removed from play).
I found Ralph Betza's About the Values of Chess Pieces to be helpful when I started researching piece values for For the Crown, so that might be a good place to start reading if you want to know more.
There have been many attempts to write mathematical formulas or create tables of piece values, but I don't think any have gained widespread acceptance. Authorities can't really even agree on the values of the orthodox pieces in FIDE Chess, so there's no accepted way to determine whether any given valuation is "correct" (or even precisely what it would mean if it were). Some people have tried computer tests to determine values empirically, which I think is a promising direction, but results don't always agree with other computer tests or with the accepted values (such as they are) of the orthodox pieces, so it's not always clear what they mean, and they're not much help in predicting the value of any piece that wasn't specifically tested.
Most people agree that value is primarily related to a piece's "mobility", or how many different ways it can typically move or capture (after somehow accounting for the fact that certain moves are more likely to be possible than others; e.g. a Bishop can move 7 squares, but only in rare situations).
But then there's a bunch of other factors that we're pretty sure are real but that no one really knows the true value of, like mating potential, development speed, colorbound pairs, stealth, how they cooperate with allied pieces, and so forth. Notice that many of these aren't even intrinsic to the piece, but relate to the other pieces on the board, which means that they change from variant to variant (and even over the course of a single game, as pieces get captured and removed from play).
I found Ralph Betza's About the Values of Chess Pieces to be helpful when I started researching piece values for For the Crown, so that might be a good place to start reading if you want to know more.