Online Market Equilibrium with Application to Fair Division (2103.12936v2)

Abstract: Computing market equilibria is a problem of both theoretical and applied interest. Much research to date focuses on the case of static Fisher markets with full information on buyers' utility functions and item supplies. Motivated by real-world markets, we consider an online setting: individuals have linear, additive utility functions; items arrive sequentially and must be allocated and priced irrevocably. We define the notion of an online market equilibrium in such a market as time-indexed allocations and prices which guarantee buyer optimality and market clearance in hindsight. We propose a simple, scalable and interpretable allocation and pricing dynamics termed as PACE. When items are drawn i.i.d. from an unknown distribution (with a possibly continuous support), we show that PACE leads to an online market equilibrium asymptotically. In particular, PACE ensures that buyers' time-averaged utilities converge to the equilibrium utilities w.r.t. a static market with item supplies being the unknown distribution and that buyers' time-averaged expenditures converge to their per-period budget. Hence, many desirable properties of market equilibrium-based fair division such as no envy, Pareto optimality, and the proportional-share guarantee are also attained asymptotically in the online setting. Next, we extend the dynamics to handle quasilinear buyer utilities, which gives the first online algorithm for computing first-price pacing equilibria. Finally, numerical experiments on real and synthetic datasets show that the dynamics converges quickly under various metrics.

• The paper introduces a new online market equilibrium for sequential item allocation, shifting from traditional static market models.
• It presents the novel PACE algorithm that ensures buyers’ time-averaged utilities and expenditures align with static market benchmarks.
• Empirical validations demonstrate rapid convergence and the achievement of fairness properties such as no envy and Pareto optimality.

The paper "Online Market Equilibrium with Application to Fair Division" addresses the problem of computing market equilibria in an online setting, which diverges from the traditional static Fisher markets. In the static case, full information about buyers' utility functions and item supplies is available. However, the authors expand this setting to scenarios where items arrive sequentially, requiring immediate and irrevocable allocation and pricing decisions.

Key Contributions

1. Online Market Equilibrium: The paper introduces a new concept of online market equilibrium. This equilibrium is defined using time-indexed allocations and prices that ensure buyer optimality and market clearance when evaluated ex post facto. Essentially, it extends the idea of market equilibrium into a dynamic environment where decisions are made continuously as new items become available.
2. PACE Algorithm:

The authors propose a novel allocation and pricing dynamics, termed as PACE (Proportional Allocation and Cumulative Expenditure). This approach is designed to be simple, scalable, and interpretable. It particularly ensures that: - Buyers’ time-averaged utilities align with the equilibrium utilities of a static market, where item supplies follow an unknown, possibly continuous distribution. - Buyers’ time-averaged expenditures are consistent with their per-period budgets.

1. Theoretical Guarantees: Under the framework where items are drawn i.i.d. from an unknown distribution, the PACE dynamics are shown to lead to an online market equilibrium asymptotically. Important properties of market equilibrium-based fair division like no envy, Pareto optimality, and proportional-share guarantee are achieved in the long run within the online context.
2. Extension to Quasilinear Utilities: The research extends the proposed dynamics to handle quasilinear utility functions, marking the development of the first online algorithm for computing first-price pacing equilibria. This extension broadens the applicability of the algorithm to a wider set of utility types, making it relevant even for settings involving quasi-linear preferences.
3. Empirical Validation: Numerical experiments conducted on both real and synthetic datasets demonstrate that the PACE dynamics converge quickly according to various metrics. This rapid convergence verifies the efficacy and practicality of the proposed model in real-world scenarios.

Implications

The findings of this paper have significant implications for real-world markets, particularly in domains where items must be allocated immediately upon arrival and can’t be revisited. The introduction of PACE provides a practical tool for achieving fairness and efficiency in such dynamic markets. Furthermore, the theoretical and empirical validations offer a solid foundation for future work in online algorithms for market equilibria, laying groundwork for more sophisticated applications in digital marketplaces and beyond.

This comprehensive paper thus bridges the gap between static market equilibrium theory and real-time allocation problems, contributing both novel concepts and practical tools for ongoing and future research in this field.