Structure and complexity of ex post efficient random assignments

TL;DR

This paper investigates the structural and computational properties of ex post efficiency in random assignment problems, revealing complexity results and proposing a new, weaker efficiency concept called robust ex post efficiency.

Contribution

It introduces the concept of robust ex post efficiency, characterizes it, and analyzes its computational complexity, contrasting it with traditional ex post efficiency.

Findings

Checking ex post efficiency is NP-complete.

Robust ex post efficiency can be tested in polynomial time with few agent types.

The random serial dictatorship rule is not robust ex post efficient.

Abstract

In the random assignment problem, objects are randomly assigned to agents keeping in view the agents' preferences over objects. A random assignment specifies the probability of an agent getting an object. We examine the structural and computational aspects of ex post efficiency of random assignments. We first show that whereas an ex post efficient assignment can be computed easily, checking whether a given random assignment is ex post efficient is NP-complete. Hence implementing a given random assignment via deterministic Pareto optimal assignments is NP-hard. We then formalize another concept of efficiency called robust ex post efficiency that is weaker than stochastic dominance efficiency but stronger than ex post efficiency. We present a characterization of robust ex post efficiency and show that it can be tested in polynomial time if there are a constant number of agent types. It is shown that the well-known random serial dictatorship rule is not robust ex post efficient. Finally, we show that whereas robust ex post efficiency depends solely on which entries of the assignment matrix are zero/non-zero, ex post efficiency of an assignment depends on the actual values.