# The Price of Connectivity in Fair Division

## TL;DR

This paper investigates how connectivity constraints in fair division of indivisible goods affect fairness, introducing the 'price of connectivity' to measure the fairness loss and providing bounds and characterizations for various graph classes.

## Contribution

It introduces the concept of the price of connectivity in fair division, derives tight bounds for different graph classes, and characterizes graphs that admit certain fairness guarantees.

## Key findings

- Connected allocations can guarantee at least 3/4 of the maximin share for biconnected graphs with two agents.
- For some graphs, the maximum guaranteed share drops to at most 1/2.
- Characterizes graphs that always admit EF1 allocations for three agents.

## Abstract

We study the allocation of indivisible goods that form an undirected graph and quantify the loss of fairness when we impose a constraint that each agent must receive a connected subgraph. Our focus is on well-studied fairness notions including envy-freeness and maximin share fairness. We introduce the price of connectivity to capture the largest gap between the graph-specific and the unconstrained maximin share, and derive bounds on this quantity which are tight for large classes of graphs in the case of two agents and for paths and stars in the general case. For instance, with two agents we show that for biconnected graphs it is possible to obtain at least $3/4$ of the maximin share with connected allocations, while for the remaining graphs the guarantee is at most $1/2$. In addition, we determine the optimal relaxation of envy-freeness that can be obtained with each graph for two agents, and characterize the set of trees and complete bipartite graphs that always admit an allocation satisfying envy-freeness up to one good (EF1) for three agents. Our work demonstrates several applications of graph-theoretic tools and concepts to fair division problems.