A Simple Approximation Algorithm for Computing Arrow-Debreu Prices

We consider the Arrow-Debreu market with linear utilities in which there is a set G of divisible goods and a set B of buyers. Each buyer starts with an initial endowment of goods. The buyer's utility function is a linearly separable function of the goods that the buyer purchases. We develop a simple and efficient algorithm for determining an approximate market equilibrium. Our algorithm finds an E-approximate solution in O(n/epsilon(vertical bar B vertical bar vertical bar G vertical bar)) time, where n = vertical bar B vertical bar+ vertical bar G vertical bar. The running time can be further improved to O(n/epsilon(m+ vertical bar B vertical bar log vertical bar B vertical bar) where in is the number of pairs (i, j) such that buyer i has some utility for purchasing good j.