An obstacle to using market mechanisms to allocate indivisible goods is the lack of competitive equilibria (CEs). Arrow and Hahn introduced social-approximate equilibria: price vectors with “small” excess demands. We define preferences called Δ-substitutes, where social-approximate equilibria exist with good-by-good excess demand bounded by 2(Δ−1), independent of economy size. For Δ=1, CEs exist even with income effects. A Δ greater than 1 allows for richer substitutability and complementarity patterns, broadening the scope for market mechanisms to allocate indivisible goods.