Online Envy Minimization and Multicolor Discrepancy: Equivalences and Separations

2025-02-20

AI Summary

This paper studies envy minimization and multicolor discrepancy in online indivisible item allocation. Given $T$ items arriving sequentially, how to allocate them instantly and irrevocably to $n$ agents with additive valuations to minimize the maximum envy, while characterizing its relationship with online multicolor discrepancy—the maximum $ell_infty$-norm of color-wise sum deviations in vector coloring. Method: The authors employ potential functions, coupling arguments, and stochastic process analysis. Contributions: (i) Under adversarial arrivals, envy minimization and multicolor discrepancy are shown to be strictly equivalent; (ii) Under i.i.d. arrivals, they fundamentally diverge: multicolor discrepancy admits a lower bound of $Omega(sqrt{log T / log log T})$, whereas envy can be bounded by a constant; (iii) They establish an $O(sqrt{log T})$ upper bound for multicolor discrepancy and an $Omega(sqrt{log T})$ lower bound for envy under adversarial inputs, and—crucially—achieve a constant envy bound under i.i.d. inputs, improving upon all prior results.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the fundamental problem of allocating $T$ indivisible items that arrive over time to $n$ agents with additive preferences, with the goal of minimizing envy. This problem is tightly connected to online multicolor discrepancy: vectors $v_1, dots, v_T in mathbb{R}^d$ with $| v_i |_2 leq 1$ arrive over time and must be, immediately and irrevocably, assigned to one of $n$ colors to minimize $max_{i,j in [n]} | sum_{v in S_i} v - sum_{v in S_j} v |_{infty}$ at each step, where $S_ell$ is the set of vectors that are assigned color $ell$. The special case of $n = 2$ is called online vector balancing. Any bound for multicolor discrepancy implies the same bound for envy minimization. Against an adaptive adversary, both problems have the same optimal bound, $Theta(sqrt{T})$, but whether this holds for weaker adversaries is unknown. Against an oblivious adversary, Alweiss et al. give a $O(log T)$ bound, with high probability, for multicolor discrepancy. Kulkarni et al. improve this to $O(sqrt{log T})$ for vector balancing and give a matching lower bound. Whether a $O(sqrt{log T})$ bound holds for multicolor discrepancy remains open. These results imply the best-known upper bounds for envy minimization (for an oblivious adversary) for $n$ and two agents, respectively; whether better bounds exist is open. In this paper, we resolve all aforementioned open problems. We prove that online envy minimization and multicolor discrepancy are equivalent against an oblivious adversary: we give a $O(sqrt{log T})$ upper bound for multicolor discrepancy, and a $Omega(sqrt{log T})$ lower bound for envy minimization. For a weaker, i.i.d. adversary, we prove a separation: For online vector balancing, we give a $Omegaleft(sqrt{frac{log T}{log log T}} ight)$ lower bound, while for envy minimization, we give an algorithm that guarantees a constant upper bound.

Problem

• Minimizing envy in online item allocation
• Equating multicolor discrepancy with envy minimization
• Establishing bounds for different adversary types

Innovation

• Online envy minimization technique
• Multicolor discrepancy equivalence
• Oblivious adversary optimization