What is the name of the following combinatorial game:
Two players, moving in turn.
Positions: $0,1,2,\ldots$.
Moves: $n\longmapsto n-1$ or $n\longmapsto \lfloor n/2\rfloor$ if $n>0$.
No move for $0$ which loses.
The determination of the winning strategy is easy: $n=2^k(2m+1)>0$ wins if and only if $k$ is even.
Determinating the winning strategy for the misere convention ($0$ wins) is also easy:
$2^k$ wins for $k$ odd and loses for $k$ even,
$2^k(2m+1)$ with $m\geq 1$ wins for $k$ even and loses for $k$ odd.
This game has certainly been described somewhere. Does it has a name? Does somebody have a reference?
