A recursive process related to a partizan variation of wythoff

Wythoff queens is a classical combinatorial game related to very interesting mathematical results. An amazing one is the fact that the P-positions are given by $(\lfloor \phi n, \phi^2 n \rfloor)$ and $(\lfloor \phi^2 n, \phi n \rfloor)$ where $\phi = \frac{1 + \sqrt{5}}{2}$. In this paper, we analyze a different version where one player (Left) plays with a chess bishop and the other (Right) plays with a chess knight. The new game (call it chessfights) lacks a Beatty sequence structure in the P-positions as in wythoff queens. However, it is possible to formulate and prove some general results of a general recursive law which is a particular case of a partizan subtraction game