Approximate Competitive Equilibrium from Equal Incomes (A-CEEI) is a procedure for fair item assignment. It was developed by Eric Budish.
Background.
CEEI (Competitive Equilibrium from Equal Incomes) is a fundamental rule for fair division of divisible resources. It divides the resources according to the outcome of the following hypothetical process:
The equilibrium allocation is provably envy free and Pareto efficient. Moreover, when the agents have linear utility functions, the CEEI allocation can be computed efficiently.
Unfortunately, when there are indivisibilities, a CEEI does not always exist, so it cannot be used directly for fair item assignment. However, it can be approximated, and the approximation has good fairness, efficiency and strategic properties.
Assumptions.
A-CEEI only assumes that the agents know how to rank bundles of items. The ranking need not be weakly additive nor even monotone.
Procedure.
A-CEEI with parameters formula_1 divides the resources according to the outcome of the following hypothetical process:
Budish proves that, for any formula_4 , there exists formula_1-CEEI where formula_3 depends on the minimum between the number of different item-types and the number of different items that an agent may receive.
Guarantees.
The allocation satisfies the following properties:
Moreover, the A-CEEI mechanism is strategyproof "in the large": when there are many agents, each agent has only a small influence on the price, so the agents act as price takers. Then, it is optimal for each agent to report his true valuations, since it allows the mechanism to give him an optimal bundle given the prices.
Computation.
The A-CEEI allocation is hard to compute: it is PPAD complete.
However, in realistic-size problems, A-CEEI can be computed using a two-level search process: 
The agent-level program can be done in parallel for all agents, so this method scales near-optimally in the number of processors.
The mechanism has been considered for the task of assigning students to courses at the Wharton School of the University of Pennsylvania.
Comparison to maximum-Nash welfare.
The "Maximum-Nash-Welfare" (MNW) algorithm finds an allocation that maximizes the product of the agents' utilities. It is similar to A-CEEI in several respects:
However, A-CEEI has several advantages:
On the flip side, A-CEEI has several disadvantages:
The approximation error of A-CEEI grows with the number of distinct items, but not with the number of players or the number of copies of each item. Therefore, A-CEEI is better when there are many agents and many copies of each item. A typical application is when the agents are students and the items are positions in courses.
In contrast, MNW is better when there are few agents and many distinct items, such as in inheritance division.
Comparison to competitive equilibrium.
A-CEEI (and CEEI in general) is related, but not identical, to the concept of competitive equilibrium. 
References.
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