Tuesday, November 30, 2010

Confusion over partiality

Perhaps the confusion over partiality is not just limited to myself... or perhaps I'm just teaching it poorly.

For our presentations, most students have chosen partisan games, but somehow many students have declared that their games are impartial during the presentations. The reasoning, it seems, is that these games are "impartialish" from the initial position, since both players have similar moves and the value is either in N or P. This has a bit of logic to it; the strategies for each side is independent of the player's identification (Left, Right, Blue, Red, etc). This "fake impartiality" only exists for this initial state, however. Once a move has been made, the game is nearly always partisan.

I feel like there is more to think about here, but perhaps I'm headed down the path of trying to trisect an angle...

When we consider partiality of a game, it does not seem that this is consistent through equivalence. For example:

{ | } is impartial and is equal to zero, but

{ -1 | 2} is not impartial, but is still equal to zero.

Perhaps I'm wrong and we can consider {-1 | 2} to be impartial, but it seems dirty somehow.

Having now studied partisan games to the point where I could teach them for a semester (as far as we got, anyways) I'm very ready to retreat back to my happy, impartial-only world. Nimbers are fairly easy to work with; Ups and Switches and Dyadic Rationals and 3+DoubleUp+* is a bit more frightening. My respect for the effort needed to get Aaron Siegel's CGT Suite to work properly is moon-bound. This stuff is crazy-interesting, but I'll be happy to resume needing only a knowledge of mex and XOR to get some research done.

(As a side note, our presenter won a game today, so the record is now: 6-1 for the audience.)

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