I don't know much about Nash-like game-theory, except that - at this stage - I don't want to know more about it.
The way I see it, when considering the 4-element game, Greg has one continuous strategy parameter (is that even a word?), namely the probability p with which to choose the (plausibly more profitable) group Z4. Kathleen has considerably more leeway (I think) varying from using the correct algorithm outlined before, via calling out a group at probability q_i at any of the branching points in her algorithm to just saying Z4 or Z2xZ2 at random (read: at a fixed probability q_0).
So I think I should just fix Greg's choice to a given distribution of groups for a given size (let's say uniform over isomorphism type) and let Kathleen do her best. Of course, this is evading difficult questions! What if Greg says 4096? Reasonable number, right? Nobody has any precise idea how many different isomorphism types of groups there are for that size of group, but I can tell you one thing: it is huge!
This leads us to the next question: when is a given distribution of groups fair?