The following game is based on a somewhat "stupid" version of the Euclidean algorithm (where we allow only subtractions).
Positions are given by finite non-empty multisets (repeated elements allowed) of strictly positive integers.
An allowed move picks two elements $a\geq b$ in the multiset and replaces the larger element $a$ by the difference $a-b$ if $a>b$, respectively removes one of the two (identical) elements from the multiset if $a=b$.
Both player have the same moves and the first player which cannot move (which happens if and only if the multiset is reduced to a singleton defining the gcd of all numbers in the initial multiset) loses.
Is there an infinite subset $\mathcal S$ of $\mathbb N$ such that the second player has a winning strategy on all non-empty finite subsets of $\mathcal S$? (Distinct powers of $2$ looked very good to my computer: The second player has a winning strategy for all non-empty subsets of $\{1,2,4,8,16,32\}$. The game has however first-player winning strategies for some subsets of powers of $2$ involving $64$, e.g. $\{1,2,8,16,64\}$.)
Final remark: The standard Conway theory can be applied but seems a bit useless: The number of possible moves gets large for large multisets.
