In markets without money — organ exchanges, school choice, public housing — items can't be priced with a scalar. Yet competitive equilibrium requires prices to clear the market. Hylland and Zeckhauser showed in 1979 that equilibria exist in these settings using probability shares as a currency, but the connection to core stability has remained incomplete because non-monetary markets lack the non-satiation assumption that drives classical convergence results.
Braverman et al. assign each item a vector of prices rather than a single number. Each dimension of the price vector captures a different aspect of desirability or scarcity. The competitive equilibrium with these multidimensional prices always exists and lies within the rejective core — a stronger stability concept than the weak core. In the rejective core, no coalition can block an allocation even when members can only redistribute their own endowments, a more demanding test than weak blocking.
As the economy grows large (many copies of each agent type), the rejective core shrinks toward the set of multidimensional competitive equilibria. Core convergence without non-satiation.
The structural point: scalar prices fail in non-monetary markets because the value of an item isn't one-dimensional. A school seat has location value, quality value, social value — dimensions that can't be collapsed into a single number without losing the information that makes the market clear. Multidimensional prices preserve this information, and the preservation is what makes the core converge. The classical theory didn't fail; it was using the wrong number system.