Multi-Apartment Rent Division

2024-03-12

arXiv.org

AI Summary

This paper addresses the practical challenge where tenants must jointly select among multiple candidate apartments, extending the classical rent-division problem to a multi-apartment setting that simultaneously optimizes apartment selection, room assignment, and rent division. We propose “negotiated envy-freeness”—a novel fairness criterion capturing inter-apartment coordination—and prove that a solution always exists and is computable in polynomial time. We further introduce the stronger notion of “global envy-freeness” and characterize its existence threshold under random valuations via probabilistic analysis. Our approach integrates fair division theory, linear programming, and combinatorial optimization, relaxing the conventional single-apartment assumption. Theoretically, we establish existence and computational tractability guarantees; algorithmically, we provide efficient methods for computing fair allocations in this generalized setting—achieving simultaneous advances in both theoretical foundations and practical feasibility.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Security and Privacy: Large-scale security measurements

Abstract

Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called negotiated envy-freeness. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called universal envy-freeness and study its existence when values are drawn randomly.

Problem

• Generalizing rent division to multiple apartment choices
• Fairly selecting apartment and assigning rooms/rent
• Extending envy-freeness to multi-apartment setting

Innovation

• Generalized rent division to multi-apartment selection
• Introduced negotiated envy-freeness fairness guarantee
• Polynomial-time optimization over envy-free solutions