Online Envy Minimization and Multicolor Discrepancy: Equivalences and Separations

TL;DR

This work resolves the relationship between online envy minimization and online multicolor discrepancy across adversary models. It proves an equivalence under oblivious adversaries, achieving the optimal $O(\sqrt{\log T})$ bound for both problems and matching $\Omega(\sqrt{\log T})$ lower bounds, while also establishing a separation under i.i.d. adversaries: online vector balancing incurs a $\Omega\left(\sqrt{\frac{\log T}{\log\log T}}\right)$ bound, whereas online envy minimization admits a constant upper bound. The key technical contribution is a versatile tree-reduction framework that extends Banaszczyk-style discrepancy methods to sets of vectors on edges and yields subgaussian prefix-sums via a carefully constructed distribution over signs, enabling optimal oblivious-adversary guarantees. For stochastic inputs, the paper introduces a two-phase welfare-plus-envy strategy with pre-sampling and quantile-based coupling, plus a novel distribution-agnostic concentration tool to rule out heavy envy-cycles, yielding tight performance under i.i.d. models. Overall, the results close several open questions and illuminate the precise impact of adversary strength on online fairness problems, with broad implications for online fair division and discrepancy theory.

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, $Θ(\sqrt{T})$, 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 $Ω(\sqrt{\log T})$ lower bound for envy minimization. For a weaker, i.i.d. adversary, we prove a separation: For online vector balancing, we give a $Ω\left(\sqrt{\frac{\log T}{\log \log T}}\right)$ lower bound, while for envy minimization, we give an algorithm that guarantees a constant upper bound.

Figures (1)

• Figure 1: Adversary strategy for the two-agent lower bound. In state $L_d$, an item valued $(1,v_d)$ arrives, while in state $R_d$, an item valued $(v_d,1)$ arrives. The arrows indicate whether agent $L$ or agent $R$ is given the item in each state. The arrows are labeled by the amount envy changes after that item is allocated.

Theorems & Definitions (39)

• Proposition 1
• proof
• Proposition 2
• proof
• Definition 1: Subgaussian norm
• Theorem 1: kulkarni2024optimal
• Theorem 2
• proof
• Theorem 3
• Theorem 4