The Price of EF1 for Few Agents with Additive Ternary Valuations

TL;DR

This paper investigates the efficiency loss, measured as the price of EF1, in resource allocation with additive ternary valuations, revealing bounds that depend on the number of agents, with specific results for small agent counts.

Contribution

It establishes lower bounds for the price of EF1 in large agent settings and provides exact bounds for instances with two and three agents.

Findings

Lower bound of (\u00a0 ext{n}) for large n

Price of EF1 is 12/11 for 2 agents

Price of EF1 is between 1.2 and 1.256 for 3 agents

Abstract

We consider a resource allocation problem with agents that have additive ternary valuations for a set of indivisible items, and bound the price of envy-free up to one item (EF1) allocations. For a large number $n$ of agents, we show a lower bound of $\Omega(\sqrt{n})$, implying that the price of EF1 is no better than when the agents have general subadditive valuations. We then focus on instances with few agents and show that the price of EF1 is $12/11$ for $n=2$, and between $1.2$ and $1.256$ for $n=3$.