  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.
