A Generalization of the AL method for Fair Allocation of Indivisible Objects

TL;DR

This paper extends the AL method for fair allocation of indivisible objects to handle agents with indifferences, providing theoretical guarantees, improved efficiency, and complexity limitations.

Contribution

It generalizes the AL method to include indifferences in preferences and proves its axiomatic properties, also improving computational efficiency.

Findings

Generalized AL method for indifferences

Proved axiomatic properties of the generalized method

Achieved O(m) speedup in checking envy-free assignments

Abstract

We consider the assignment problem in which agents express ordinal preferences over $m$ objects and the objects are allocated to the agents based on the preferences. In a recent paper, Brams, Kilgour, and Klamler (2014) presented the AL method to compute an envy-free assignment for two agents. The AL method crucially depends on the assumption that agents have strict preferences over objects. We generalize the AL method to the case where agents may express indifferences and prove the axiomatic properties satisfied by the algorithm. As a result of the generalization, we also get a $O(m)$ speedup on previous algorithms to check whether a complete envy-free assignment exists or not. Finally, we show that unless P=NP, there can be no polynomial-time extension of GAL to the case of arbitrary number of agents.