Question: In what way can I use maths to work out strategic advantages and disadvantages in a board game like chess?

Daniel Bearup answered on 9 Nov 2020:

The first step would typically be to assign values to each piece. It’s possible that you’ve encountered the values P = 1, N = 3, B = 3, R = 5, Q = 9 for example. These scores are a pretty good representation of the values of the pieces and connected to the mobility of those pieces.

However, it is clear that these static values miss something. A pawn on the 7th rank (about to queen) is worth a lot more than a pawn on the second rank! So you need a way to evaluate how the positions of pieces on the board affect their value, or, more generally, a way to evaluate how good your position is relative to your opponents.

There are natural measures you can consider to do such an evaluation (e.g. spaces controlled by your pieces, spaces attacked by your pieces, spaces where your pieces can be, differences in attack strength etc). in practice, this quickly becomes too complicated to do in your head so instead we rely on rules of thumb which are derived from computer analysis. e.g. the bishop pair is worth an extra pawn on an open board … .

Now the computer analysis programmes all take slightly different approaches, but I believe they are typically less “mechanistic” that the measures described above. Instead, programmers define some functions that evaluate a chess position. Then they let those programmes analyse positions, and evaluate which ones “do best” (typically by tacking on some more functionality to let them play chess and see who wins). They would eliminate the programmes that do badly, and “mutate” the programmes that do well, searching for the “best programme”. (Machine learning and genetic algorithms are the search terms for the basic theory.)