Dead ends in misère play: The misère monoid of canonical numbers

Abstract

We find the misère monoids of normal-play canonical-form integer and non-integer numbers. These come as consequences of more general results for the universe of dead-ending games. Left and right ends have previously been defined as games in which Left or Right, respectively, have no moves; here we define a dead left (right) end to be a left (right) end whose options are all left (right) ends, and we define a dead-ending game to be one in which all end followers are dead. We find the monoids and partial orders of dead ends, integers, and all numbers, and construct an infinite family of games that are equivalent to zero in the dead-ending universe.