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MMS Approximations Under Additive Leveled Valuations

Mahyar Afshinmehr, Mehrafarin Kazemi, Kurt Mehlhorn

Abstract

We study the problem of fairly allocating indivisible goods to a set of agents with additive leveled valuations. A valuation function is called leveled if and only if bundles of larger size have larger value than bundles of smaller size. The economics literature has well studied such valuations. We use the maximin-share (MMS) and EFX as standard notions of fairness. We show that an algorithm introduced by Christodoulou et al. ([11]) constructs an allocation that is EFX and $\frac{\lfloor \frac{m}{n} \rfloor}{\lfloor \frac{m}{n} \rfloor + 1}\text{-MMS}$. In the paper, it was claimed that the allocation is EFX and $\frac{2}{3}\text{-MMS}$. However, the proof of the MMS-bound is incorrect. We give a counter-example to their proof and then prove a stronger approximation of MMS.

MMS Approximations Under Additive Leveled Valuations

Abstract

We study the problem of fairly allocating indivisible goods to a set of agents with additive leveled valuations. A valuation function is called leveled if and only if bundles of larger size have larger value than bundles of smaller size. The economics literature has well studied such valuations. We use the maximin-share (MMS) and EFX as standard notions of fairness. We show that an algorithm introduced by Christodoulou et al. ([11]) constructs an allocation that is EFX and . In the paper, it was claimed that the allocation is EFX and . However, the proof of the MMS-bound is incorrect. We give a counter-example to their proof and then prove a stronger approximation of MMS.
Paper Structure (8 sections, 2 theorems, 6 equations, 1 algorithm)

This paper contains 8 sections, 2 theorems, 6 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Models and Preliminaries
  5. EFX and Approximate MMS Guarantees
  6. Problem in Proof of Proposition 5.1 in christodoulou2024fairtruthfulallocationsleveled
  7. Existence of $\lfloor\frac{m}{n}\rfloor/\lfloor\frac{m}{n}\rfloor + 1\text{-MMS}$
  8. Conclusion and Discussion

Key Result

theorem 3.1

Algorithm alg constructs an EFX allocation for leveled valuations.

Theorems & Definitions (13)

  • definition 2.1
  • definition 2.2
  • definition 2.3
  • definition 2.4
  • definition 2.5
  • definition 2.6
  • definition 2.7
  • definition 2.8
  • theorem 3.1: christodoulou2024fairtruthfulallocationsleveled
  • proof
  • ...and 3 more