Further generalizations of the Wythoff game and the minimum excludant

For any non-negative integers a and b, we consider the following game WY T (a, b). Given two piles that consist of x and y matches, two players 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 < x′ + y′, and [min(x′, y′) < b or |x′ − y′| < a]. The player who takes the last match is the winner in the normal version of the game and (s)he is the loser in its misere version. It is easy to verify that the cases (a = 0, b = 1), (a = b = 1), and (b = 1,∀ a) correspond to the two-pile NIM, the Wythoff, and Fraenkel games, respectively. The concept of the minimum excludant mex is known to be instrumental in solving the last two games. We generalize this concept by introducing a function mexb (such that mex = mex1) to solve the normal and misere versions of the game WY T (a, b).