Bidder optimal assignments for general utilities

We study the problem of matching bidders to items where each bidder i has general, strictly monotonic utility functions ui,j(pj) expressing her utility of being matched to item j at price pj. For this setting we prove that a bidder optimal outcome always exists, even when the utility functions are non-linear and non-continuous. Furthermore, we give an algorithm to find such a solution. Although the running time of this algorithm is exponential in the number of items, it is polynomial in the number of bidders.