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 his 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. We give sufficient conditions under which every mechanism that finds a bidder optimal outcome is incentive compatible. We also give a mechanism that finds a bidder optimal outcome if the conditions for incentive compatibility are satisfied. The running time of this mechanism is exponential in the number of items, but polynomial in the number of bidders.