Comparing the Fairness of Recursively Balanced Picking Sequences

TL;DR

This paper compares the fairness of recursively balanced picking sequences in allocating indivisible goods, showing they have similar welfare prices and identifying the best sequence for maximin share guarantees.

Contribution

It provides a comparative analysis of recursively balanced picking sequences, highlighting their welfare equivalence and optimality for MMS guarantees.

Findings

All such sequences have the same egalitarian welfare price.

The sequence where the last first picker is compensated yields the best MMS guarantee.

Recursively balanced sequences are fairer and more predictable than other methods.

Abstract

Picking sequences are well-established methods for allocating indivisible goods. Among the various picking sequences, recursively balanced picking sequences -- whereby each agent picks one good in every round -- are notable for guaranteeing allocations that satisfy envy-freeness up to one good. In this paper, we compare the fairness of different recursively balanced picking sequences using two key measures. Firstly, we demonstrate that all such sequences have the same price in terms of egalitarian welfare relative to other picking sequences. Secondly, we characterize the approximate maximin share (MMS) guarantees of these sequences. In particular, we show that compensating the agent who picks last in the first round by letting her pick first in every subsequent round yields the best MMS guarantee.