Partiality Continues to be Confusing, part 2

Common Trip-Ups:

Confusing Impartiality with Symmetry. Symmetric positions seem impartial because each of the moves for one player has an opposite in the set of moves for the other player. Those opposing positions have opposite values, however. While it is true that all impartial games are symmetric, the converse does not hold.

Separate Scores. Many games are nearly impartial, except the players keep track of different scores. The game board (not including the scores) may be changeable in the exact same ways by both players, except that then the players get different scores. 3,6,9 and Odd Scoring are good examples of this. These games are not impartial, because the scores are different.

All the positions are impartial games. { 0, *, *2, *3 | 0, * } may look impartial because all the positions are impartial positions. This is not impartial, however, because the players don't have all the same move options.

The Position is equivalent to an impartial game or 0. A game equivalent to * is not necessarily impartial. The game could be: { 0 | 0, *2 }, which is equivalent to * but does not have the same options for both players. Equivalence does not preserve impartiality. (Perhaps not everyone agrees with this!)

What are some other common problems I didn't list here?

Partiality Continues to be Confusing

Happy new semester & year! Let's dive right in!

I thought I had done a much better job explaining the difference between impartial and strictly partisan games last semester, but again far too many of my students claimed that their game was impartial during their presentations.

A few students presented symmetric games, and claimed they were impartial. A few made mistakes on games that were impartial except that each player had a separate score. These games, such as 3,6,9 and Odd Scoring, are not impartial because your turn affects your own score but does not affect your opponent's. Thus, the same moves are not allowed for both players.

For example, given the following Odd Scoring state: (parity of scores is given instead of actual numbers)

Left: Odd Right: Even
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _X_ ___ ...

One legal move for Left is to slide the marker one spot:

Left: Even Right: Even
___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ _X_ ___ ___ ...

Right does not have that move as one of its options. It can still slide the marker, but then the scores will both be odd instead of even. Thus, Odd Scoring is not impartial.

Last semester, I made a distinction to the class about impartial positions and impartial rulesets, defining each separately. A position is impartial if, recursively, all it's positions are impartial and if the set of Left positions is the SAME SET as the set of Right positions. (Not whether they are equivalent.) A ruleset is impartial if all positions available in that ruleset are impartial.

Before defining these, I defined symmetric positions as those equivalent to their opposite (G = -G means G is symmetric). I think this helped, because many students knew that their games were symmetric, but not impartial. This was a change from the previous year when most students claimed impartiality.

Perhaps it would be better next time to list common misconceptions about determining partiality.