Bidder Optimal Assignments for General Utilities

We study the problem of matching bidders to items where each bidder i has a general, strictly monotonic utility functions u_{i,j}(p_j) expressing her utility of being matched to item j at price p_j . 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.