Spatial Market Equilibria on Networks

Computing market equilibria is an important practical problem for market design, for example, in fair division of items. However, computing equilibria requires large amounts of information, often the valuation of every buyer for every item, and computing power. In “Computing Large Market Equilibria Using Abstractions,” the authors study abstraction methods for ameliorating these issues. The basic abstraction idea is as follows. First, construct a coarsened abstraction of a given market, then solve for the equilibrium in the abstraction, and finally, lift the prices and allocations back to the original market. The authors show theoretical guarantees on the solution quality obtained via this approach. Then, two abstraction methods of interest for practitioners are introduced: (1) filling in unknown valuations using techniques from matrix completion and (2) reducing the problem size by aggregating groups of buyers/items into smaller numbers of representative buyers/items and solving for equilibrium in this coarsened market.

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Market equilibria and money

Fixed Point Theory and Algorithms for Sciences and Engineering ◽

10.1186/s13663-021-00705-4 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sjur Didrik Flåm

Keyword(s):

Fixed Point ◽

Steady State ◽

Value Added ◽

Pareto Optimal ◽

Generalized Gradients ◽

Market Mechanisms ◽

Mathematical Arguments ◽

Second Welfare Theorem ◽

Market Equilibria ◽

Welfare Theorem

AbstractBy the first welfare theorem, competitive market equilibria belong to the core and hence are Pareto optimal. Letting money be a commodity, this paper turns these two inclusions around. More precisely, by generalizing the second welfare theorem we show that the said solutions may coincide as a common fixed point for one and the same system.Mathematical arguments invoke conjugation, convolution, and generalized gradients. Convexity is merely needed via subdifferentiablity of aggregate “cost”, and at one point only.Economic arguments hinge on idealized market mechanisms. Construed as algorithms, each stops, and a steady state prevails if and only if price-taking markets clear and value added is nil.

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