Beauty is a Beast, Frog is a Prince -- Assortative Matching with Nontransferabilities

We present sufficient conditions for monotone matching in environments where utility is not fully transferable between partners these conditions involve not only complementarity in types of the total payoff to a match as in the transferable utility case but also monotonicity in type of the degree of transferability between partners we apply our conditions to study some models of risk sharing and incentive problems deriving new results for predicted matching patterns in those contexts.