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Improved MMS Approximations for Few Agent Types

Jugal Garg, Parnian Shahkar

TL;DR

The paper addresses maximin-share (MMS) fairness for indivisible goods when agents fall into a small number of types. It introduces an approach that leverages the MMS partition of the majority type and an ONI$_\alpha$ reduction to achieve improved MMS guarantees, notably $\frac{4}{5}$-MMS for two types and $\frac{16}{21}$-MMS for three types, with polynomial-time algorithms. The key technical contributions are the SHV partition for same-type agents and the case-based algorithms that adapt partitions and bag-filling to multiple agent types, ensuring allocations meet their MMS targets. These results advance the state of the art for structured MMS instances and motivate further study on the trade-offs as the number of agent types grows, while providing practical methods for real-world scenarios with few distinct agent types.

Abstract

We study fair division of indivisible goods under the maximin share (MMS) fairness criterion in settings where agents are grouped into a small number of types, with agents within each type having identical valuations. For the special case of a single type, an exact MMS allocation is always guaranteed to exist. However, for two or more distinct agent types, exact MMS allocations do not always exist, shifting the focus to establishing the existence of approximate-MMS allocations. A series of works over the last decade has resulted in the best-known approximation guarantee of $\frac{3}{4} + \frac{3}{3836}$. In this paper, we improve the approximation guarantees for settings where agents are grouped into two or three types, a scenario that arises in many practical settings. Specifically, we present novel algorithms that guarantee a $\frac{4}{5}$-MMS allocation for two agent types and a $\frac{16}{21}$-MMS allocation for three agent types. Our approach leverages the MMS partition of the majority type and adapts it to provide improved fairness guarantees for all types.

Improved MMS Approximations for Few Agent Types

TL;DR

The paper addresses maximin-share (MMS) fairness for indivisible goods when agents fall into a small number of types. It introduces an approach that leverages the MMS partition of the majority type and an ONI reduction to achieve improved MMS guarantees, notably -MMS for two types and -MMS for three types, with polynomial-time algorithms. The key technical contributions are the SHV partition for same-type agents and the case-based algorithms that adapt partitions and bag-filling to multiple agent types, ensuring allocations meet their MMS targets. These results advance the state of the art for structured MMS instances and motivate further study on the trade-offs as the number of agent types grows, while providing practical methods for real-world scenarios with few distinct agent types.

Abstract

We study fair division of indivisible goods under the maximin share (MMS) fairness criterion in settings where agents are grouped into a small number of types, with agents within each type having identical valuations. For the special case of a single type, an exact MMS allocation is always guaranteed to exist. However, for two or more distinct agent types, exact MMS allocations do not always exist, shifting the focus to establishing the existence of approximate-MMS allocations. A series of works over the last decade has resulted in the best-known approximation guarantee of . In this paper, we improve the approximation guarantees for settings where agents are grouped into two or three types, a scenario that arises in many practical settings. Specifically, we present novel algorithms that guarantee a -MMS allocation for two agent types and a -MMS allocation for three agent types. Our approach leverages the MMS partition of the majority type and adapts it to provide improved fairness guarantees for all types.

Paper Structure

This paper contains 23 sections, 18 theorems, 27 equations, 2 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Additional Related Work
  3. Chores
  4. Preliminaries
  5. SHV $\frac{4}{5}$-MMS Partition of Same-type Agents
  6. Improved MMS Approximations
  7. $\frac{4}{5}$-MMS for $2$-type $\text{ONI}_\alpha$ Instance
  8. $\frac{16}{21}-$MMS for $3$-type $\text{ONI}_\alpha$ Instance
  9. Case 1: If $|C_2| > T_2$ and $|C_3|> T_3$
  10. Case 2: If $\exists i,i'\in\{2,3\} \text{ s.t. } i\neq i', |C_i| > T_i$ and $|C_{i'}|\leq T_{i'}$
  11. Case 3: If $|C_2| \leq T_2$, and $|C_3|\leq T_3$, $|C_1|\leq T_1$
  12. Case 4: If $|C_2| \leq T_2$, and $|C_3|\leq T_3$, $|C_1|> T_1$
  13. Conclusion
  14. Proof of Lemma \ref{['thm:case4']}
  15. $\bullet$
  16. ...and 8 more sections

Key Result

Theorem 1

Given an instance $\mathcal{I}$ and an $\alpha$-MMS allocation of $\mathtt{order}(\mathcal{I})$, one can compute an $\alpha$-MMS allocation of $\mathcal{I}$ in polynomial time.

Figures (2)

  • Figure 1: This figure illustrates Example \ref{['example1']}. The red bundles are saved in $F$, and the green bundles are allocated to type $2$ agents.
  • Figure 2: This figure represents the set of available HV and MV items when type $2$ gets inactive in \ref{['example1']}. Note that all crossed items, and items within red and green bundles are unavailable.

Theorems & Definitions (47)

  • Definition 1: Ordered instance
  • Theorem 1: Theorem 2 in barman2020approximation
  • Definition 2: Normalized instance
  • Lemma 1: Lemma 4 in simple
  • Definition 3: Valid reductions
  • Definition 4: Reduction rules
  • Definition 5
  • Lemma 2: Lemma 3.1 in garg2021improved
  • Lemma 3: Lemmas 2 and 3 in simple
  • Lemma 4: Lemma 6 in simple
  • ...and 37 more