TITLE: Name of a game : Remove two chips from a vertex or one chip from both ends of an edge
QUESTION [8 upvotes]: Consider a finite graph $\Gamma$ with a positive number $n_v\geq 0$ of chips stacked at each vertex $v$ of $\Gamma$. Two players play in turn with moves consisting either of removing two chips from a unique vertex $v$ such that $n_v\geq 2$ or of removing one chip from both endpoints $v,w$ of an edge $\{v,w\}$ such that $n_v,n_w\geq 1$. The first player with no move left loses (or wins when using the misere convention).
The game is trivial on complete graphs. Winning positions have a nice (not completely trivial) description on the graph $P_3$ (Dynkin graph of type $A_3$) with vertices $1,2,3$ and edges
$\{1,2\},\{2,3\}$: Given a position with $(a,b,c)$ chips
on vertices $1,2,3$, the existence of a winning strategy for the first player and $a+b+c$ even (the case $a+b+c$ odd is trivial) depends on parities of $a,b,c$ and
"coarsened" inequalities (coarsened in order to deal with two numbers of different parities) along edges after taking care of "fattened" boundaries.
A similar phenomenon seems to hold for the $P_4$ graph
and perhaps for all trees.
Has somebody a reference/name for this game?

REPLY [4 votes]: In their paper A generalization of Arc-Kayles (Intl. J. Game Theory 48 (2019), 491–511),     Antoine Dailly, Valentin Gledel, and Marc Heinrich define Weighted Arc Kayles to be a game played on graphs with counters on the vertices. The two players alternate choosing an edge and removing one counter on both endpoints. An edge can no longer be selected if any of its endpoints has no counter left.
In light of Richard Stanley's remark that removing two counters from a vertex $v$ may be thought of as removing a counter from both endpoints of a loop at $v$, your game seems to be equivalent to Weighted Arc Kayles.