The Price of EF1 for Few Agents with Additive Ternary Valuations

2025-08-13

AI Summary

This paper studies the price of EF1—the efficiency loss incurred in achieving envy-freeness up to one good (EF1)—in the allocation of indivisible goods under additive valuations. Using techniques from combinatorial optimization and game theory, combined with asymptotic analysis and explicit constructions, we establish the first tight bounds for small numbers of agents: the exact price of EF1 is 12/11 for two agents and lies in [1.2, 1.256] for three agents. Moreover, we prove a lower bound of Ω(√n) on the price of EF1 for large n. These results characterize the fundamental trade-off between EF1 fairness and social welfare: fairness imposes only a modest efficiency cost in small-scale settings, but the loss grows significantly with system scale. Our work thus provides foundational theoretical benchmarks for designing and evaluating fair allocation mechanisms.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements

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 $Ω(sqrt{n})$, 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 $n=3$.

Problem

• Study EF1 allocation price for additive ternary valuations
• Establish lower bound Ω(√n) for many agents
• Determine EF1 price for 2 and 3 agents

Innovation

• Additive ternary valuations for agents
• Lower bound of Ω(√n) for EF1
• EF1 price bounds for few agents