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We consider combinatorial exchanges where agents have (possibly random) endowments and ordinal preferences over bundles of indivisible goods. For any market instance, we show that there exists an approximately feasible, individually rational and ordinally efficient lottery assignment that can be supported by prices from a novel competitive equilibrium foundation. Any such competitive allocation can be implemented as a lottery over deterministic allocations that are approximately feasible, individually rational, and efficient. When endowments are deterministic, it can be implemented over near-feasible weak core outcomes. Moreover, competitive allocations are ordinally envy-free and ex-post envy-free up to one good (where envy is only justified if another agent’s endowment is either smaller or worth less in equilibrium). A mechanism that selects such an allocation is strategyproof in the large. Our framework can be used in many real-world market design applications, such as organ exchanges, tuition exchanges, time bank sharing, shift exchanges, and resource reallocation.
