Maximin Shares with Lower Quotas

Hirota Kinoshita, Ayumi Igarashi

Keywords: Fair division, Maximin shares, Approximation, Constraints, Quotas

TL;DR: We develop polynomial-time algorithms with provable fairness guarantees for resource allocation under quota constraints.

Abstract:

We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to *lower* and *upper quotas* on the number of items allocated to each agent.
Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution.
This paper focuses on the fairness criterion known as *maximin shares (MMS)* and its approximations.
Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores.
Furthermore, we consider the generalized scenario where items are partitioned into multiple *categories*, each with its own lower and upper quotas.
In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time.
These results extend previous work on the *cardinality constraints*, i.e., the special case where only upper quotas are imposed.

Area: Game Theory and Economic Paradigms (GTEP)

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Submission Number: 320