# Computing large market equilibria using abstractions

## TL;DR

This paper introduces a method for approximating market equilibria using abstractions, reducing computational complexity while maintaining accuracy, applicable to fair division and market design.

## Contribution

It proposes a novel abstraction-based approach for computing market equilibria, including bounds on key economic quantities and practical methods for valuation estimation and problem size reduction.

## Key findings

- Coarse abstractions yield near-equilibrium solutions in real data.
- Bounding techniques ensure economic properties are preserved in abstractions.
- Abstraction methods improve computational efficiency for large markets.

## Abstract

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 (typically the valuation of every buyer for every item) and computing power. We consider ameliorating these issues by applying a method used for solving complex games: constructing a coarsened abstraction of a given market, solving for the equilibrium in the abstraction, and lifting the prices and allocations back to the original market. We show how to bound important quantities such as regret, envy, Nash social welfare, Pareto optimality, and maximin share/proportionality when the abstracted prices and allocations are used in place of the real equilibrium. We then study two abstraction methods of interest for practitioners: (1) filling in unknown valuations using techniques from matrix completion, (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. We find that in real data allocations/prices that are relatively close to equilibria can be computed from even very coarse abstractions.