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Fair Division Under Inaccurate Preferences

Trung Dang, Daniel Halpern, Anuran Makur, Alexandros Psomas, Japneet Singh, Paritosh Verma

TL;DR

This paper explores the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy, and gives an efficient online algorithm that guarantees logarithmic maximum envy with high probability.

Abstract

The fair allocation of scarce resources is a central problem in mathematics, computer science, operations research, and economics. While much of the fair-division literature assumes that individuals have underlying cardinal preferences, eliciting exact numerical values is often cognitively burdensome and prone to inaccuracies. A growing body of work in fair division addresses this challenge by assuming access only to ordinal preferences. However, the restricted expressiveness of ordinal preferences makes it challenging to quantify and optimize cardinal fairness objectives such as envy. In this paper, we explore the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy. We consider various settings based on whether the true preferences of the agents are stochastic or worst-case, and whether the inaccuracies, modeled as additive noise, are stochastic or worst-case. When the true preferences are stochastic, we show that envy-free allocations can be computed with high probability; this is achieved both in the setting with stochastic and worst-case noise. This generalizes a notable result in stochastic fair division, which establishes a similar guarantee, albeit in the absence of any noise. When the true preferences are worst-case, and the noise is bounded, we analyze the maximum envy achieved by the Round-Robin algorithm. This bound is shown to be tight for deterministic algorithms, and applications of this bound are provided. Lastly, we consider a setting with worst-case preferences and noise, where the true preferences for each item are revealed upon its allocation. Here, we give an efficient online algorithm that guarantees logarithmic maximum envy with high probability. This result generalizes a known result from algorithmic discrepancy to a setting with noisy input data.

Fair Division Under Inaccurate Preferences

TL;DR

This paper explores the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy, and gives an efficient online algorithm that guarantees logarithmic maximum envy with high probability.

Abstract

The fair allocation of scarce resources is a central problem in mathematics, computer science, operations research, and economics. While much of the fair-division literature assumes that individuals have underlying cardinal preferences, eliciting exact numerical values is often cognitively burdensome and prone to inaccuracies. A growing body of work in fair division addresses this challenge by assuming access only to ordinal preferences. However, the restricted expressiveness of ordinal preferences makes it challenging to quantify and optimize cardinal fairness objectives such as envy. In this paper, we explore the broad landscape of fair division of indivisible items given inaccurate cardinal preferences, with a focus on minimizing envy. We consider various settings based on whether the true preferences of the agents are stochastic or worst-case, and whether the inaccuracies, modeled as additive noise, are stochastic or worst-case. When the true preferences are stochastic, we show that envy-free allocations can be computed with high probability; this is achieved both in the setting with stochastic and worst-case noise. This generalizes a notable result in stochastic fair division, which establishes a similar guarantee, albeit in the absence of any noise. When the true preferences are worst-case, and the noise is bounded, we analyze the maximum envy achieved by the Round-Robin algorithm. This bound is shown to be tight for deterministic algorithms, and applications of this bound are provided. Lastly, we consider a setting with worst-case preferences and noise, where the true preferences for each item are revealed upon its allocation. Here, we give an efficient online algorithm that guarantees logarithmic maximum envy with high probability. This result generalizes a known result from algorithmic discrepancy to a setting with noisy input data.
Paper Structure (22 sections, 24 theorems, 61 equations, 1 algorithm)

This paper contains 22 sections, 24 theorems, 61 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Additional Related Work
  4. Preliminaries
  5. Allocations.
  6. Fairness Notions.
  7. Inaccurate Preference Model.
  8. Stochastic True Valuations
  9. Stochastic Values and Stochastic Noise
  10. (1)
  11. (2)
  12. Stochastic Values and Worst-case Noise
  13. Worst-Case True Valuations
  14. Worst-Case Values and Worst-case Noise
  15. Applications of \ref{['thm: worst-case valuations and worst-case error upper bound']}
  16. ...and 7 more sections

Key Result

Lemma 1

conc_ineq_book Given any two real-valued, nondecreasing functions $f$, $g$ and a real-valued random variable $X$, ${\mathbb{E}}[f(X)\cdot g(X)] \geq {\mathbb{E}}[f(X)] \cdot {\mathbb{E}}[g(X)]$.

Theorems & Definitions (44)

  • Definition 1: Envy-freeness
  • Lemma 1: Chebyshev's Association Inequality
  • Theorem 1
  • Lemma 2
  • proof
  • Theorem 2
  • Theorem 3
  • Lemma 3
  • Theorem 4: Theorem 5.5 (b) of barbanel2005geometry
  • proof : Proof of \ref{['lem: barbanel shit']}
  • ...and 34 more