*Generalized Sprague-Grundy theory* has been used to analyze finite impartial loopy games with normal play. There is a nice short account by Mark S. in this answer. It was introduced by Cedric Smith in the 60s, and developed independently by Fraenkel and others.

The theory is usually described in the setting of finite directed graphs (see for example section IV.4 of Siegel's Combinatorial Game Theory).

I see that Fraenkl and Rahat (2001) looked at the case of infinite directed graphs which are *locally path-bounded*, i.e. where for any vertex $v$, the sup of the length of all paths starting at $v$ is finite.

I'm interested in the case of directed graphs which are allowed to contain infinite paths. Suppose we assume that every vertex has finite outdegree. It seems to me that much of the structure (e.g. all the results of Section IV.4 of Siegel) extends unchanged to this setting. In particular, every such game has a "loopy nim value" which is either a non-negative integer (in which case the game is equivalent to a finite nim heap) or of the form $\infty(\mathcal{A})$ for some subset $\mathcal{A}$ of $\mathbb{N}$ (see the answer by Mark S. linked above).

(Once we allow infinite paths, it's no longer important to allow cycles, since we can create multiple "copies" of each node. Without loss of generality one could consider "game trees", i.e. rooted trees directed away from the root, with all degrees finite.)

Can someone point me to a reference covering such games? Does the generalized Sprague-Grundy theory go through straightforwardly to that context as I imagine?