Dividing Good and Better Items Among Agents with Bivalued Submodular Valuations

TL;DR

This paper introduces a simple algorithmic framework for fair allocation of indivisible goods among agents with bivalued submodular valuations, extending previous classes and analyzing solution guarantees and computational complexity.

Contribution

It develops a sequential algorithm based on Yankee Swap for computing various solution concepts in bivalued submodular valuations, including MNW and leximin, when parameters divide.

Findings

MNW and leximin allocations guarantee specific fractions of maximin share.

Computational intractability when parameters do not divide.

MNW and leximin are not necessarily envy free up to one good.

Abstract

We study the problem of fairly allocating a set of indivisible goods among agents with {\em bivalued submodular valuations} -- each good provides a marginal gain of either $a$ or $b$ ($a < b$) and goods have decreasing marginal gains. This is a natural generalization of two well-studied valuation classes -- bivalued additive valuations and binary submodular valuations. We present a simple sequential algorithmic framework, based on the recently introduced Yankee Swap mechanism, that can be adapted to compute a variety of solution concepts, including max Nash welfare (MNW), leximin and $p$-mean welfare maximizing allocations when $a$ divides $b$. This result is complemented by an existing result on the computational intractability of MNW and leximin allocations when $a$ does not divide $b$. We show that MNW and leximin allocations guarantee each agent at least $\frac25$ and $\frac{a}{b+2a}$ of their maximin share, respectively, when $a$ divides $b$. We also show that neither the leximin nor the MNW allocation is guaranteed to be envy free up to one good (EF1). This is surprising since for the simpler classes of bivalued additive valuations and binary submodular valuations, MNW allocations are known to be envy free up to any good (EFX).