Stronger core results with multidimensional prices
This work addresses a foundational issue in matching theory for economists and computer scientists, offering a novel solution concept that ensures stability in resource allocation without money.
The paper tackles the problem of non-existence of competitive equilibria and empty strong core in one-sided matchings without money by proposing a generalization using multi-dimensional prices, showing that this solution always exists and resides within the rejective core, which is strictly stronger than the weak core.
We study one-sided matchings with endowments in the absence of money. It is well-known that a competitive equilibrium may not always exist and that the strong core may be empty in this setting [Hylland and Zeckhauser, 1979]. We propose a generalization of competitive equilibria that associates each item with a multi-dimensional price. We show that this solution concept always exists and resides within the rejective core [Konovalov, 2005]. Rejective core stability is strictly stronger than weak core stability: allocations in the rejective core are elements of the weak core, but the opposite is not true. Moreover, we show that the rejective core always converges to the set of competitive equilibria with multi-dimensional prices as the economy grows, demonstrating core convergence in a setting without non-satiation.