Computing Walrasian Equilibria: Fast Algorithms and Economic Insights

We give the first polynomial algorithm to compute a Walrasian equilibrium in an economy with indivisible goods and general valuations having only access to an aggregate demand oracle, i.e., an oracle that given a price vector, returns the aggregated demand over the entire population of buyers. Our algorithm queries the aggregate demand oracle $\tilde{O}(n)$ times and takes $\tilde{O}(n^3)$ time, where n is the number of items. 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 $\tilde{O}((mn+n^3 )T_V)$ where $T_V$ is the cost of querying the value oracle. En route, we give necessary and sufficient conditions for the existence of robust Walrasian prices, i.e., price vectors such that 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.