Abstract
We present the first polynomial time algorithm for computing Walrasian equilibrium in an economy with indivisible goods and general buyer valuations having only access to an aggregate demand oracle, i.e., an oracle that given prices on all goods, returns the aggregated demand over the entire population of buyers. For the important special case of gross substitute valuations, our algorithm queries the aggregate demand oracle {widetilde{O}}(n) times and takes {widetilde{O}}(n^3) time, where n is the number of goods and the {widetilde{O}}(cdot ) notation denotes an asymptotic bound up to polylogarithmic factors. Both algorithms are randomized. At the heart of our solution is a method for exactly minimizing certain convex functions which cannot be evaluated but for which the subgradients can be computed. We also give the fastest known algorithm for computing Walrasian equilibrium for gross substitute valuations in the value oracle model. Our algorithm has running time {widetilde{O}}((mn + n^3) T_V) where T_V is the cost of querying the value oracle. A key technical ingredient is to regularize a convex programming formulation of the problem in a way that subgradients are cheap to compute. En route, we give necessary and sufficient conditions for the existence of robust Walrasian prices, i.e., prices for which each agent has a unique demanded bundle and the demanded bundles clear the market. When such prices exist, the market can be perfectly coordinated by solely using prices.
Highlights
1.1 A macroscopic view of the marketAs part of our everyday experience, prices reach equilibria in a wide range of economics settings
We show that it is possible to solve this problem with a linear dependence on the number of buyers and cubic dependence in the number of items: Theorem 2 (Informal) In a consumer market with n goods and m buyers whose valuation functions satisfy the gross substitute condition, we can find an equilibrium price and allocation using mn + O(n3) calls to the value oracle and O(n3) time
We show that: Theorem 3 (Informal) In a consumer market with n goods and m buyers whose valuation functions satisfy the gross substitute condition, robust Walrasian prices exist if and only if there is a unique Walrasian allocation
Summary
As part of our everyday experience, prices reach equilibria in a wide range of economics settings. We show that it is possible to compute market equilibrium by exploiting the very rudimentary information of aggregate demand, i.e. the quantity demanded for each item at a given price aggregated over the entire population of buyers. This result implies, among other things, that a market can be viewed as an aggregate entity. The market can only be accessed via an aggregate demand oracle: given prices for each item, what is the demand for each item aggregated over the entire population In this model, it is not possible to compute an allocation of items to buyers, since the oracle access model allows no access to buyer-specific information. The city of Berkeley, for example, has about 350 restaurants but 120,000+ people!
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