(Almost Full) EFX for Three (and More) Types of Agents
Pratik Ghosal, Vishwa Prakash HV, Prajakta Nimbhorkar, Nithin Varma
TL;DR
This work extends the existence of envy-freeness up to any good (EFX) to settings with multiple valuation types. By grouping agents into $k$ types and using a leading-agent and minimally envied-subset framework, the authors show that EFX allocations exist with at most $k-2$ unallocated goods for $n$ agents when valuations come from $k$ distinct types, and achieve complete EFX under MMS-feasible valuations when all but two agents share valuations. The approach combines Pareto-dominating improvements, the MES concept, and the Plaut-Roughgarden algorithm, alongside leveraging BergerCFF22 and ChaudhuryKMS21 to guarantee progress. They also prove a complete EFX allocation for the two-outlier case under MMS-feasibility, broadening prior results for small numbers of agents. Overall, the results broaden the practical reach of EFX in heterogeneous, multi-type valuation environments and outline clear open questions about minimizing unallocated items and outlier counts.
Abstract
We study the problem of determining an envy-free allocation of indivisible goods among multiple agents with additive valuations. EFX, which stands for envy-freeness up to any good, is a well-studied relaxation of the envy-free allocation problem and has been shown to exist for specific scenarios. EFX is known to exist for three agents, and for any number of agents when there are only two types of valuations. EFX allocations are also known to exist for four agents with at most one good unallocated. In this paper, we show that EFX exists with at most k-2 goods unallocated for any number of agents having k distinct valuations. Additionally, we show that complete EFX allocations exist when all but two agents have identical valuations.