Stochastic algorithms, symmetric Markov perfect equilibrium, and the 'curse' of dimensionality

This paper introduces a stochastic algorithm for computing symmetric Markov perfect equilibria. The algorithm computes equilibrium policy and value fu nctions, and generates a transition kernel for the (stochastic) evolution o f the state of the system, It has two features that together imply that it need not be subject to the curse of dimensionality. First, the integral tha t determines continuation values is never calculated; rather it is approxim ated by a simple average of returns from past outcomes of the algorithm, an approximation whose computational burden is not tied to the dimension of t he state space. Second, iterations of the algorithm update value and policy functions at a single (rather than at all possible) points in the state sp ace. Random draws from a distribution set by the updated policies determine the location of the next iteration's updates. This selection only repeated ly hits the recurrent class of points, a subset whose cardinality is not di rectly tied to that of the state space. Numerical results for industrial or ganization problems show that our algorithm can increase speed and decrease memory requirements by several orders of magnitude.