The notion of a p-best-response (p-BR) set was developed in order to relax some of the stringent epistemic requirements for the emergence of equilibrium play. Although this concept has clear conceptual advantages, one drawback of non-fixed point approaches is the loss of computational tractability in all but the simplest of settings. We show that when best-response correspondences are monotone, "exact" p- best response sets (exact p-BR) and p-minimal bestresponse (p-MBR) sets, as developed in Tercieux (2006), can be solved for as the Nash equilibria of an auxiliary game, so that traditional optimization methods may still be applied. As a result, we are able to state necessary and sufficient conditions for a set to be a p-MBR set. We also show that in the case of GSC, the set of exact p-BR sets is a complete lattice, but is completely unordered if at least one player's payoffs exhibit strict strategic substitutes. Examples are given.