Recursive comparison tests for dicot and dead-ending games under misère play

In partizan games, where players Left and Right may have different options, there is a partial order defined as preference by Left: G ⩾ H if Left wins G + X whenever she wins H + X for any game position X. In normal play, there is an easy test for comparison: G ⩾ H if and only if Left wins G−H playing second. In misère play, where the last player to move loses, the same test does not apply-for one thing, there are no additive inverses-and very few games are comparable. If we restrict the arbitrary game X to a subset of games u, then we may have G ⩾ H “modulo U”; but without the easy test from normal play, we must give a general argument about the outcomes of G + X and H + X for all X ∈ U. In this paper, we use the novel theory of absolute combinatorial games to develop recursive comparison tests for the well-studied universes of dicots and dead-ending games. This is the first constructive test for comparison of dead-ending games under misère play using a new family of end-games called perfect murders.