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Fair Division with Social Impact

Michele Flammini, Gianluigi Greco, Giovanna Varricchio

TL;DR

This work extends fair division by incorporating a social impact function, proposing a two-valuation model with individual preferences $v_i$ and societal impact $s_i$. It analyzes the price of fairness (PoF) and proves impossibility results: no fair allocation can achieve a sublinear-factor approximation to the maximum utilitarian social welfare in general, under several fairness notions. The authors then develop multiple poly-time algorithms achieving near-optimal or near-linear approximations under ordered valuations, identical valuations, and weaker fairness criteria, including $\mathsf{EF}1$, $\mathsf{EFX}$, and $\mathsf{EF}2$, as well as novel socially aware notions $\mathsf{sEF}$ and $\mathsf{sEF}1$ that align fairness with societal welfare. They also show that a max-utilitarian, socially aware allocation exists and can be computed efficiently, offering a framework for allocating indivisible goods when allocations influence society, with several directions for future research such as alternative welfare functions and adjustable agent awareness.

Abstract

In this paper, we consider the problem of fair division of indivisible goods when the allocation of goods impacts society. Specifically, we introduce a second valuation function for each agent, determining the social impact of allocating a good to the agent. Such impact is considered desirable for the society -- the higher, the better. Our goal is to understand how to allocate goods fairly from the agents' perspective while maintaining society as happy as possible. To this end, we measure the impact on society using the utilitarian social welfare and provide both possibility and impossibility results. Our findings reveal that achieving good approximations, better than linear in the number of agents, is not possible while ensuring fairness to the agents. These impossibility results can be attributed to the fact that agents are completely unconscious of their social impact. Consequently, we explore scenarios where agents are socially aware, by introducing related fairness notions, and demonstrate that an appropriate definition of fairness aligns with the goal of maximizing the social objective.

Fair Division with Social Impact

TL;DR

This work extends fair division by incorporating a social impact function, proposing a two-valuation model with individual preferences and societal impact . It analyzes the price of fairness (PoF) and proves impossibility results: no fair allocation can achieve a sublinear-factor approximation to the maximum utilitarian social welfare in general, under several fairness notions. The authors then develop multiple poly-time algorithms achieving near-optimal or near-linear approximations under ordered valuations, identical valuations, and weaker fairness criteria, including , , and , as well as novel socially aware notions and that align fairness with societal welfare. They also show that a max-utilitarian, socially aware allocation exists and can be computed efficiently, offering a framework for allocating indivisible goods when allocations influence society, with several directions for future research such as alternative welfare functions and adjustable agent awareness.

Abstract

In this paper, we consider the problem of fair division of indivisible goods when the allocation of goods impacts society. Specifically, we introduce a second valuation function for each agent, determining the social impact of allocating a good to the agent. Such impact is considered desirable for the society -- the higher, the better. Our goal is to understand how to allocate goods fairly from the agents' perspective while maintaining society as happy as possible. To this end, we measure the impact on society using the utilitarian social welfare and provide both possibility and impossibility results. Our findings reveal that achieving good approximations, better than linear in the number of agents, is not possible while ensuring fairness to the agents. These impossibility results can be attributed to the fact that agents are completely unconscious of their social impact. Consequently, we explore scenarios where agents are socially aware, by introducing related fairness notions, and demonstrate that an appropriate definition of fairness aligns with the goal of maximizing the social objective.

Paper Structure

This paper contains 23 sections, 25 theorems, 8 equations, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution.
  3. Related Work.
  4. Preliminaries
  5. The Envy-Cycle Elimination.
  6. Sequential Allocation Algorithms.
  7. Social Impact.
  8. Impossibility Results
  9. Approximating maxUt subject to fairness
  10. A general approx. algorithm for ordered valuations
  11. An $n$-approx. subject to $\mathsf{EFX}$ for identical agents
  12. An $O(n^2)$-approximation subject to $\mathsf{EF}1$
  13. An $2n^2$-approximation for Case 1.
  14. An $2 n$-approximation for Case 2.
  15. An $n$-approx. subject to weaker fairness criteria
  16. ...and 8 more sections

Key Result

Theorem 1

An approximation better than $n-k+1$ to $\textsf{opt}$ is not possible when requiring $\mathsf{EF} k$, even for identical agents.

Theorems & Definitions (43)

  • Theorem 1
  • proof
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Lemma 1
  • Proposition 1
  • proof : Sketch
  • Theorem 3
  • proof : Sketch
  • ...and 33 more