# The Price of Fairness for Indivisible Goods

## TL;DR

This paper analyzes the efficiency loss in fair allocations of indivisible goods using the concepts of price of fairness and introduces the strong price of fairness to measure worst-case efficiency loss.

## Contribution

It provides tight bounds on the worst-case efficiency loss for various fair allocation notions, including EF1, balancedness, MNW, and leximin, and introduces the strong price of fairness.

## Key findings

- Tight bounds on efficiency loss for EF1, balancedness, MNW, and leximin.
- Introduction of the strong price of fairness concept.
- Analysis of worst-case efficiency loss in fair allocations.

## Abstract

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We also introduce the concept of strong price of fairness, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation as in the price of fairness. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions, for both the price of fairness and the strong price of fairness.