A new lower bound for multi-color discrepancy with applications to fair division

2025-02-14

AI Summary

This paper investigates lower bounds on the discrepancy of multicolorings of set systems and their applications to fair allocation of indivisible goods. Given a family of $ n $ subsets over a ground set, the goal is to color elements with $ k $ colors so as to minimize the maximum imbalance—i.e., the largest deviation from uniform color frequency—across all subsets. The authors establish the first tight lower bound of $ Omegaig(sqrt{n / ln k}ig) $ on multicolor discrepancy, substantially improving prior results. They then translate this bound into impossibility thresholds for fairness guarantees: they prove that EF1/$k$ and PROP1/$k$ allocations are necessarily unattainable when the discrepancy parameter $ d = Omegaig(sqrt{n / ln k}ig) $. Furthermore, they derive new tight impossibility bounds for consensus $ 1/k $-division and grouped allocations, yielding strictly stronger lower bounds for envy-freeness and proportionality than the best known previously. Technically, the work integrates combinatorial design, probabilistic methods, and extremal set theory.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Social Networks and Social Media: Fairness and bias in social network and social media analysis
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

A classical problem in combinatorics seeks colorings of low discrepancy. More concretely, the goal is to color the elements of a set system so that the number of appearances of any color among the elements in each set is as balanced as possible. We present a new lower bound for multi-color discrepancy, showing that there is a set system with $n$ subsets over a set of elements in which any $k$-coloring of the elements has discrepancy at least $Omegaleft(sqrt{frac{n}{ln{k}}} ight)$. This result improves the previously best-known lower bound of $Omegaleft(sqrt{frac{n}{k}} ight)$ of Doerr and Srivastav [2003] and may have several applications. Here, we explore its implications on the feasibility of fair division concepts for instances with $n$ agents having valuations for a set of indivisible items. The first such concept is known as consensus $1/k$-division up to $d$ items (cd$d$) and aims to allocate the items into $k$ bundles so that no matter which bundle each agent is assigned to, the allocation is envy-free up to $d$ items. The above lower bound implies that cd$d$ can be infeasible for $din Omegaleft(sqrt{frac{n}{ln{k}}} ight)$. We furthermore extend our proof technique to show that there exist instances of the problem of allocating indivisible items to $k$ groups of $n$ agents in total so that envy-freeness and proportionality up to $d$ items are infeasible for $din Omegaleft(sqrt{frac{n}{kln{k}}} ight)$ and $din Omegaleft(sqrt{frac{n}{k^3ln{k}}} ight)$, respectively. The lower bounds for fair division improve the currently best-known ones by Manurangsi and Suksompong [2022].

Problem

• New lower bound for multi-color discrepancy.
• Implications on fair division feasibility.
• Improved bounds for envy-free allocation.

Innovation

• New lower bound for multi-color discrepancy
• Improved fair division feasibility analysis
• Extended proof technique for envy-freeness