Gross Substitutes and Endowed Assignment Valuations

We show that the class of preferences satisfying the gross substitutes condition of [Kelso and Crawford, 1982] is strictly larger than the class of endowed assignment valuations of [Hatfield and Milgrom, 2005], thus resolving the open question posed by the latter paper. In particular, our result implies that not every substitutable valuation function can be “decomposed” into a combination of unit-demand valuations.