Do Prices Coordinate Markets?

A Walrasian equilibrium outcome has a remarkable property: the induced allocation maximizes social welfare while each buyer receives a bundle that maximizes her individual surplus at the given prices. There are, however, two caveats. First, minimal Walrasian prices necessarily induce indifferences. Thus, without coordination, buyers may choose surplus maximizing bundles that conflict with each other. Accordingly, buyers may need to coordinate with one another to arrive at a socially optimal outcome---the prices alone are not sufficient to coordinate the market. Second, although natural auctions converge to Walrasian equilibrium prices on a fixed population, in practice buyers typically observe prices without participating in a price computation process. These prices are not perfect Walrasian equilibrium prices, but we may hope that they still encode distributional information about the market. To better understand the performance of Walrasian prices in light of these two problems, we give two results. First, we propose a mild genericity condition on valuations so that the minimal Walrasian equilibrium prices induce allocations resulting in low overdemand, no matter how the buyers break ties. In fact, under our condition the overdemand of any good can be bounded by 1, which is the best possible at the minimal prices. Second, we use techniques from learning theory to argue that the overdemand and welfare induced by a price vector converge to their expectations uniformly over the class of all price vectors, with sample complexity linear and quadratic in the number of goods in the market respectively. The latter results make no assumption on the form of the valuation functions.