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.
翻译:电子计算市场平衡是一个理论和应用利益的问题。 迄今的许多研究都集中在静态的渔业市场中,有关于买方公用事业功能和物品供应的完整信息。 受现实世界市场的驱动,我们考虑一个在线环境:个人有线性、添加性公用事业功能;物品按顺序运抵,必须分配和不可逆定价。 我们把网上市场平衡的概念定义为时间指数分配和价格,保证买方在事后看到的最佳性和市场清关。 我们提出一个简单、可缩放和可解释的分配和定价动态,称为PACE。 当项目从未知的分布(可能得到持续支持)中提取到i.i.d.时,我们展示PACE导致在线市场平衡。 特别是,PACE确保买方的时间平均公用事业与均衡公用事业趋近。 一个静态市场,项目供应为未知的分布,而购买者的时间平均支出与其周期预算趋近。 因此,许多基于市场平衡的公平电子分工的可取性,例如不嫉妒、不连续支持、不连续支持、不固定的准确的准确性计算,我们最终将最终的准确的计算,将最终的准确的计算,将最终的计算,将在线的准确的准确的计算结果,也显示我们最终的准确的准确的准确的计算。