I'm not sure how mathematically rigorously you can analyze this game, given that
- the players are not playing under the same set of conditions (dictionaries)
- the conditions change as the game progresses (the players learn words from their opponents).
If everybody agrees to use the exact same dictionary to get their words from, then the game can be considered a heavily modified form of Grundy's game, which is a Nim variant where the players split an initial heap into smaller heaps of unequal size, until someone is given a position with only heaps of size 1 and 2, at which point that someone has lost. In this case, the initial heap is all the words in the dictionary, and the players then take turns splitting it into smaller heaps using the words.
The main modifications for this game, which in my opinion make it more difficult to analyze, are:
- there are potentially more than two players (Nim and Nim-variants assume only two players are playing)
- whenever one of the two existing heaps is used, the other one can no longer be acted upon.
So far, I haven't been able to find any existing research or informal work done on games with these restrictions; if you happen to stumble across some, let me know!
Without dictionaries, it becomes a lot more complex. I can see metagaming becoming more prevalent, which essentially turns the game into a subjective analysis of what you think others will do. I don't know how much rigorous game theory you can apply in this scenario other than what people already do to construct the "meta" for a game.
Given all this, I'm inclined to believe that there is no optimal strategy, but there are ways you can increase your chance of winning, such as
- having a very, very large vocabulary (or memorizing esoteric words from the dictionary)
- going first, if you're playing with a very large group
- always choosing a "midpoint" word between two "endpoint" words e.g. playing "nine" instead of "Nim" if you're given nil - nip. This ensures that your opponents get heaps that are of minimal possible size.
(Side note: There's also a potential loophole of people saying words that aren't valid but can't be challenged under the current rule set, and this could go on to infinity (a - ab $ \rightarrow $ aa - ab $ \rightarrow $ aaa - ab $ \rightarrow $ aaaa - ab $ \rightarrow \cdots $). Unless OP declares that that's a valid strategy, I believe there need to be more restrictions on what qualifies as a "word.")