Misere nim variant

Question

Is there a name (and strategy) for this nim variant?

There are $n$ lists of objects, say $L_1,\ldots,L_n$ where $L_i = \{a_{i,1},a_{i,2},\ldots,a_{i,n_i}\}$. Players take turns choosing a list and any number of sequential elements to remove. If they do not remove from one of the endpoints of the list, the list becomes effectively partitioned into two lists.

Of course, there is a "regular" version of the game where the last player to remove an element wins, and the "misere" version where the last player to remove an element loses.

Answer 1

Sebastian Goette and Max Alekseyev together answered this question almost completely.

The game you describe is equivalent to the Octal Game "$0.7777\ldots$". I do not believe it has another name.

Under normal play, it is easy to see that a single list of $n$ objects is equivalent (in the sense of the Sprague-Grundy theory) to a Nim heap of size $n$: By induction all the moves that don't split up the list are to the Nim heaps of size $<n$. And as Sebastian Goette points out, if $b+c<a$ then $b\oplus c<a$, so that a Nim heap of size $n$ won't appear among the options.

Since the same is so similar to Nim, the misère play should be similar as well. You should play as in normal-play Nim until your prescribed move would leave only lists of size 1, and then make a move which leaves your opponent an odd number of such lists.