In two-player combinatorial games, the last player to move either wins (normal play) or loses (misère play). Traditionally, normal play games have garnered more attention due to the group structure which arises on such games. Less work has been done with games played under the misère play convention, Just as in normal play, misère games can be placed in equivalence classes, where two games G and H are equivalent if the outcome class of G + K is the same as the outcome class of H+K for all games K. However, Conway showed that, unlike in normal play, these misère equivalence classes are sparsely populated, making the analysis of misère games under such equivalence classes far less useful than their normal play counterparts [ONAG]. Even though these equivalence classes are sparse, Conway developed a method, called genus theory, for analyzing impartial games played under the misère play convention [Allen 2006; WW; ONAG]. For years, this was the only universal tool available for those studying misère games.

In [Plambeck 2009; 2005; Plambeck and Siegel 2008; Siegel 2006; 2015b], many results regarding impartial misère games have been achieved. These results were obtained by taking a game, restricting the universe in which that game was played, and obtaining its misère quotient. However, while, as Siegel [2015a] says “a partizan generalization exists”, few results have been obtained regardingthe structure of the misère quotients which arise from partizan games.

For a game G ={GL|GR}, we define Ḡ = {GR|GL}. Those familiar with normal play will notice that under the normal play convention rather than Ḡ, we would generally write̶G. In normal play, this nomenclature is quite sensible as G+(-G)=0 [Albert et al. 2007], giving us the Tweedledum–Tweedledee principle; the second player can always win the game G+(-G) by mimicking the move of the first player, but in the other component. However, in misère play, not only does the Tweedledum–Tweedledee strategy often fail, G + Ḡ is not necessarily equivalent to 0. For example, *2+*2=*2+*2 is not equivalent to 0 [Allen 2006; WW]. However, having the property that G+Ḡ is equivalent to 0 is much desired, as it gives a link to which partizan misère games may behave like their normal counterparts.