Abstract
Consider a market with n unit demand buyers and m sellers, each selling one unit of an indivisible good. The buyers specify their preferences over items via utility functions u ij ( p j ), which is the utility of buyer i for item j when its price is p j . So far, this is the classic Shapley-Shubik assignment model [Shapley and Shubik 1971] which captures a variety of matching markets including housing markets and ad auctions [Edelman et al. 2007], except for the extension to general utility functions instead of the quasi-linear utilities in the original model. Shapley and Shubik show that a competitive equilibrium always exists in their model, and later work [Crawford and Knoer 1981, Quinnzi 1984, Gale 1984] shows that a competitive equilibrium must also exist for the model with general utility functions u ij (·), provided these u ij (·) are strictly decreasing and continuous everywhere.
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