Market Equilibria in Polynomial Time for Fixed Number of Goods or Agents

We consider markets in the classical Arrow-Debreu model. There are n agents and m goods. Each buyer has a concave utility function (of the bundle of goods he/she buys) and an initial bundle. At an “equilibrium” set of prices for goods, if each individual buyer separately exchanges the initial bundle for an optimal bundle at the set prices, the market clears, i.e., all goods are exactly consumed. Classical theorems guarantee the existence of equilibria, but computing them has been the subject of much recent research. In the related area of Multi-Agent Games, much attention has been paid to the complexity as well as algorithms. While most general problems are hard, polynomial time algorithms have been developed for restricted classes of games, when one assumes the number of strategies is constant [20, 11]. For the Market Equilibrium problem, several important special cases of utility functions have been tackled. Here we begin a program for this problem similar to that for multiagent games, where general utilities are considered. We begin by showing that if the utilities are separable piece-wise linear concave (PLC) functions, and the number of goods (or alternatively the number of buyers) is constant, then we can compute an exact equilibrium in polynomial time. Our technique for the constant number of goods is to decompose the space of price vectors into cells using certain hyperplanes, so that in each cell, each buyer’s threshold marginal utility is known. Still, one needs to solve a linear optimization problem in each cell. We then show the main result –   that for general (non-separable) PLC utilities, an exact equilibrium can be found in polynomial time provided the number of goods is constant. The starting point of the algorithm is a “cell-decomposition” of the space of price vectors using polynomial surfaces (instead of hyperplanes). We use results from computational algebraic geometry to bound the number of such cells. For solving the problem inside each cell, we introduce and use a novel LP-duality based method. We note that if the number of buyers and Part of the work done while the author was visiting Microsoft Research, India. agents both can vary, the problem is PPAD hard even for the very special case of PLC utilities – namely Leontief utilities [8].