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Fast-Converging Tatonnement Algorithms for the Market Problem
Richard Cole, Lisa Fleischer
Dartmouth TR2007-602

Abstract:

Why might markets tend toward and remain near equilibrium prices? In an effort to shed light on this question from an algorithmic perspective, this paper defines and analyzes two simple tatonnement algorithms that differ from previous algorithms that have been subject to asymptotic analysis in three significant respects: the price update for a good depends only on the price, demand, and supply for that good, and on no other information; the price update for each good occurs distributively and asynchronously; the algorithms work (and the analyses hold) from an arbitrary starting point.

Our algorithm introduces a new and natural update rule. We show that this update rule leads to fast convergence toward equilibrium prices in a broad class of markets that satisfy the weak gross substitutes property. These are the first analyses for computationally and informationally distributed algorithms that demonstrate polynomial convergence.

Our analysis identifies three parameters characterizing the markets, which govern the rate of convergence of our protocols. These parameters are, broadly speaking:

1. A bound on the fractional rate of change of demand for each good with respect to fractional changes in its price.

2. A bound on the fractional rate of change of demand for each good with respect to fractional changes in wealth.

3. The relative demand for money at equilibrium prices.

We give two protocols. The first assumes global knowledge of only the first parameter. For this protocol, we also provide a matching lower bound in terms of these parameters. Our second protocol assumes no global knowledge whatsoever.


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   Richard Cole and Lisa Fleischer, "Fast-Converging Tatonnement Algorithms for the Market Problem." Dartmouth Computer Science Technical Report TR2007-602, August 2007.


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