1. Introduction

Most research in combinatorial game theory assumes normal play, where the first player unable to move loses, as opposed to misere play, where the first player unable to move wins. It is rather remarkable how much changes when we simply switch the goal from getting the last move to avoiding the last move. At first glance one might think misere play is merely the “opposite” of normal play, but this is not at all the case. There is actually no relationship between normal outcome and misère outcome: for every pair of (not necessarily distinct) outcomes , there is a game with normal outcomeand misère outcome [11]. Likewise, strategies from normal play are in general neither the same nor reversed for misère play. For example, in normal play, Left would always choose a move to 1 = ﹛0|·﹜ over a move to 0 = ﹛·|·﹜, but in misère play there are situations in which Left should choose 1 over 0 and others where Left prefers 0 over 1. This means that 0 and 1 are incomparable in misere play, which goes against our intuition that Left is trying to run out of moves before Right.

So we really are in a fog in misere play. We look to the elegant algebra of normal-play games and hope for some semblance of structure, but we are dismayed at every turn:

Zero is trivial. In normal play we have the wonderful property that every previous-win game is equal to zero. In misere, the zero game is next-win, but our hopes that perhaps every next-win game is equal to zero are more than dashed; in fact, only the game ﹛·|·﹜ is equal to zero [11].