Price Convergence Under a Probabilistic Double Auction

This paper studies a dual that applies to a class of quasilinear exchange economies with indivisible (or divisible) goods in search for an equilibrium. Our model aims at an economy with a large scale and an agent's revealed demand or supply is contaminated with stochastic errors (noises). Thus, it is computationally expensive, if not inapplicable, to run a Walrasian auction based on truthfully revealed total demand and supply. Instead, we study a probabilistic α-double auction and are interested in the convergence of a price process it generates. We allow the weight α to be a random variable with time-varying unknown distributions over [0,1] and show a convergence result when the two step sizes are either deterministically diminishing or probabilistically diminishing in the mean. An error bound is estimated when the two step sizes are constant, bounded away from zero, while α is at random. We provide conditions under which the double auction generates a price process that converges in probability to the set of Walrasian equilibria of the primal economy