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This paper addresses several significant gaps in the theory of restrictedmis\`ere play (Plambeck, Siegel 2008), primarily in the well-studied universeof dead-ending games, $\mathcal{E}$ (Milley, Renault 2013); if a player run outof moves in $X\in \mathcal E$, then they can never move again in any followerof $X$. A universe of games is a class of games which is closed underdisjunctive sum, taking options and conjugates. We use novel results fromabsolute combinatorial game theory (Larsson, Nowakowski, Santos 2017) to showthat $\mathcal{E}$ and the universe $\mathcal{D}\subset \mathcal{E}$ of dicotgames (either both, or none of the players can move) have 'options only' testfor comparison of games, and this in turn is used to define unique reducedgames (canonical forms) in $\mathcal{E}$. We develop the reductions for$\mathcal{E}$ by extending analogous work for $\mathcal{D}$, in particular bysolving the problem of reversibility through ends in the larger universe.Finally, by using the defined canonical forms in $\mathcal{E}$ and$\mathcal{D}$, we prove that both of these universes, as well as thesubuniverse of impartial games, have the conjugate property: every inverse gameis obtained by swapping sides of the players.

Progress on mis\`ere dead ends: game comparison, canonical form, and conjugate inverses