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Dividing Indivisible Items for the Benefit of All: It is Hard to Be Fair Without Social Awareness

Argyris Deligkas, Eduard Eiben, Tiger-Lily Goldsmith, Dušan Knop, Šimon Schierreich

TL;DR

The paper studies fair division of indivisible goods when each agent has a valuation v_i and a social impact s_i, aiming to maximize total social impact (SIM) while satisfying fairness constraints. It maps a comprehensive complexity landscape across seven fairness notions, showing NP-hardness for socially unaware agents in many cases, and establishing polynomial-time results when agents are socially aware. It demonstrates that SA-enabled allocations (e.g., SA-EF1, SA-swEF1, SA-EFL) exist and can be computed efficiently, but that relaxing social awareness or allowing partial/unawareness reintroduces hardness, including NP-hardness even with binary inputs or constant numbers of agents. The work highlights a nuanced trade-off between social awareness and computational feasibility, with implications for designing fair, socially aware allocation mechanisms in real-world settings and for identifying when such mechanisms can be computed efficiently. It also suggests future directions, including relaxations of SIM, alternative fairness notions, and extensions to chores, to develop a broader understanding of fair division under social considerations.

Abstract

In standard fair division models, we assume that all agents are selfish. However, in many scenarios, division of resources has a direct impact on the whole group or even society. Therefore, we study fair allocations of indivisible items that, at the same time, maximize social impact. In this model, each agent is associated with two additive functions that define their value and social impact for each item. The goal is to allocate items so that the social impact is maximized while maintaining some fairness criterion. We reveal that the complexity of the problem heavily depends on whether the agents are socially aware, i.e., they take into consideration the social impact functions. For socially unaware agents, we prove that the problem is NP-hard for a variety of fairness notions, and that it is tractable only for very restricted cases, e.g., if, for every agent, the valuation equals social impact and it is binary. On the other hand, social awareness allows for fair allocations that maximize social impact, and such allocations can be computed in polynomial time. Interestingly, the problem becomes again intractable as soon as the definition of social awareness is relaxed.

Dividing Indivisible Items for the Benefit of All: It is Hard to Be Fair Without Social Awareness

TL;DR

The paper studies fair division of indivisible goods when each agent has a valuation v_i and a social impact s_i, aiming to maximize total social impact (SIM) while satisfying fairness constraints. It maps a comprehensive complexity landscape across seven fairness notions, showing NP-hardness for socially unaware agents in many cases, and establishing polynomial-time results when agents are socially aware. It demonstrates that SA-enabled allocations (e.g., SA-EF1, SA-swEF1, SA-EFL) exist and can be computed efficiently, but that relaxing social awareness or allowing partial/unawareness reintroduces hardness, including NP-hardness even with binary inputs or constant numbers of agents. The work highlights a nuanced trade-off between social awareness and computational feasibility, with implications for designing fair, socially aware allocation mechanisms in real-world settings and for identifying when such mechanisms can be computed efficiently. It also suggests future directions, including relaxations of SIM, alternative fairness notions, and extensions to chores, to develop a broader understanding of fair division under social considerations.

Abstract

In standard fair division models, we assume that all agents are selfish. However, in many scenarios, division of resources has a direct impact on the whole group or even society. Therefore, we study fair allocations of indivisible items that, at the same time, maximize social impact. In this model, each agent is associated with two additive functions that define their value and social impact for each item. The goal is to allocate items so that the social impact is maximized while maintaining some fairness criterion. We reveal that the complexity of the problem heavily depends on whether the agents are socially aware, i.e., they take into consideration the social impact functions. For socially unaware agents, we prove that the problem is NP-hard for a variety of fairness notions, and that it is tractable only for very restricted cases, e.g., if, for every agent, the valuation equals social impact and it is binary. On the other hand, social awareness allows for fair allocations that maximize social impact, and such allocations can be computed in polynomial time. Interestingly, the problem becomes again intractable as soon as the definition of social awareness is relaxed.

Paper Structure

This paper contains 15 sections, 16 theorems, 26 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Related Work
  4. Preliminaries
  5. Solution Concepts
  6. Socially Unaware Agents
  7. Socially Aware Agents
  8. Limited Social Awareness
  9. Some Agents Socially Unaware
  10. Approximate Awareness
  11. Weak Social Awareness
  12. Discussion
  13. Relaxing SIM.
  14. Other Fairness Notions.
  15. Division of Chores.

Key Result

Lemma 1

For any $\mathcal{F}\in\{\text{tEF1},\text{EF1},\text{sEF1},\text{wEF1},\text{swEF1},\text{EFL},\text{EF}\}$, an instance $\mathcal{I} = ({{N}},{M},({v}_i)_{i\in{{N}}},(s_i)_{i\in{{N}}})$ of fair division with social impact admits a SIM and $\mathcal{F}$-fair allocation if and only if an instance $\

Figures (2)

  • Figure 1: A basic overview of the complexity landscape of deciding whether a SIM and $\mathcal{F}$-fair allocation exist.
  • Figure 2: An illustration of the social impact function used in the instance constructed in the proof of \ref{['thm:czechEnvy:NPh']}.

Theorems & Definitions (52)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Remark 1
  • Lemma 1
  • ...and 42 more