Reachability of Fair Allocations via Sequential Exchanges

TL;DR

This paper explores the conditions under which one EF1 fair allocation can be transformed into another through a sequence of EF1-preserving exchanges, revealing complexity results and special cases where reachability is guaranteed.

Contribution

It introduces the reachability problem in EF1 allocations, proves its PSPACE-completeness in general, and identifies cases with guaranteed reachability and efficient exchange sequences.

Findings

Reachability of EF1 allocations is not always possible, even with two agents.

Deciding reachability is PSPACE-complete in general.

Reachability is guaranteed for two agents with identical or binary utilities.

Abstract

In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.