Further generalizations of the Wythoff game and the minimum excludant

Given non-negative integers a and b, we consider the following game WYT(a,b). Given two piles that consist of x and y matches, and two players having alternate turns; a single move consists of a player choosing x′ matches from one pile and y′ from the other such that 0≤x′≤x,0≤y′≤y,0