A remoteness formula for generalized Wythoff games

Abstract

Smith's remoteness function records how long an impartial game lasts when the winner wins as quickly as possible and the loser delays defeat. For the generalized Wythoff game $\mathrm{WYT}(a,b)$, in which two positive integer parameters $a$ and $b$ determine which simultaneous removals from the two piles are allowed, we give a formula for the remoteness of every position, building on the known values at losing positions. The formula depends on two indices: a local index determined by the coordinates of the losing positions in a window of length $b$ ending at the smaller pile, and the integer part of the pile difference divided by $a$. When $b=1$, complementary Beatty sequences make the local index explicit, which gives an explicit formula for $\mathrm{WYT}(a,1)$, including classical Wythoff's game. We also show that if each attempted move in a finite impartial game independently fails with a fixed probability and passes the turn, the probability that the player to move wins under optimal play is an explicit exponential function of remoteness. For every success probability strictly between $0$ and $1$, the optimal attempts are exactly the moves that reduce remoteness by one.