Basic theories of loopy (normal-play) games which may go on forever under the usual disjunctive sum (the game ends when there are no moves available for you in any component on your turn) were introduced by Smith in the 60s for impartial games and in Winning Ways for well-behaved partizan games. At times, the best some player can do is force the game to go on forever.

In On Numbers and Games, a theory for a diminished disjunctive compound (where the game ends as soon as a single component has run out of moves) of finitely many loopfree impartial games was given, based on the idea of foreclosing a game near the end. I suspect that something very similar could be said in the case of normal-play compounds of partizan games, although I haven't been able to find a paper mentioning even that; Compound Node-Kayles on Paths is the only resource I have found aside from ONAG that seems to mention diminished disjunctive compounds at all.

Example and Question

However, some weird things definitely happen when dealing with a disjunctive compound of an infinite collection of partizan games: Let $0$ be the exact position $\left\{\mid\right\}$ (as opposed to the disjunctive-sum equivalence class thereof), $1$ be the exact position $\left\{\left. 0\right\vert\right\}$, and $-1$ be the exact position $\left\{\left\vert 0\right.\right\}$. Then let $G$ be $\left\{ \left\{ 0\left|0\right.\right\} \left|\left\{ 0\left|0\right.\right\} \right.\right\}=\left\{0\left|0\right.\left\Vert0\left|0\right.\right.\right\} $ and let $H$ be $\left\{ \left\{ 0\left|\left\{ -1\left|1\right.\right\} \right.\right\} \left|\left\{ \left.\left\{ -1\left|1\right.\right\} \right|0\right\} \right.\right\} =\left\{ 0\left|\left\{ -1\left|1\right.\right\} \right.\left\Vert \left.\left\{ -1\left|1\right.\right\} \right|0\right.\right\} $.

The disjunctive compound of infinitely many copies of $G$ is a $\mathcal{P}$-position: any move can be responded to by ending the game. However, the diminished compound of infinitely many copies of $H$ is drawn! A move in one copy of $H$ must be responded to in that component, but that leaves $\left\{\left.-1\right\vert1\right\}$, not $0$, which can always be left alone to make a threat in a new component.

Is there some other terminology for this type of game I should be looking for? Where can I read about this sort of phenomena? Or, at least, are there any related references I should be looking at?


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