The study of market equilibria is central to economic theory, particularly in efficiently allocating scarce resources. However, the computation of equilibrium prices at which the supply of goods matches their demand typically relies on having access to complete information on private attributes of agents, e.g., suppliers' cost functions, which are often unavailable in practice. Motivated by this practical consideration, we consider the problem of setting equilibrium prices in the incomplete information setting wherein a market operator seeks to satisfy the customer demand for a commodity by purchasing the required amount from competing suppliers with privately known cost functions unknown to the market operator. In this incomplete information setting, we consider the online learning problem of learning equilibrium prices over time while jointly optimizing three performance metrics -- unmet demand, cost regret, and payment regret -- pertinent in the context of equilibrium pricing over a horizon of $T$ periods. We first consider the setting when suppliers' cost functions are fixed and develop algorithms that achieve a regret of $O(\log \log T)$ when the customer demand is constant over time, or $O(\sqrt{T} \log \log T)$ when the demand is variable over time. Next, we consider the setting when the suppliers' cost functions can vary over time and illustrate that no online algorithm can achieve sublinear regret on all three metrics when the market operator has no information about how the cost functions change over time. Thus, we consider an augmented setting wherein the operator has access to hints/contexts that, without revealing the complete specification of the cost functions, reflect the variation in the cost functions over time and propose an algorithm with sublinear regret in this augmented setting.

翻译：市场均衡研究是经济理论的核心，尤其在高效配置稀缺资源方面。然而，计算商品供需匹配的均衡价格通常需要获取市场主体私人属性的完整信息（例如供应商成本函数），而这些信息在实践中往往难以获得。基于这一实际考量，我们研究不完全信息环境下的均衡定价问题：市场运营商需从具有私有成本函数（且运营商未知）的竞争供应商处采购商品以满足客户需求。在不完全信息环境下，我们考虑随时间学习均衡价格的在线学习问题，同时联合优化三个与$T$周期均衡定价相关的性能指标：未满足需求、成本遗憾和支付遗憾。首先考虑供应商成本函数固定且客户需求恒定的情形，提出遗憾为$O(\log \log T)$的算法；当需求随时间变化时，遗憾为$O(\sqrt{T} \log \log T)$。其次考虑供应商成本函数可随时间变化的情形，证明若市场运营商对成本函数变化模式完全未知，则任何在线算法无法在所有三个指标上实现次线性遗憾。因此，我们扩展设置，允许运营商获取反映成本函数时间变化但未完全揭示函数规范的线索/上下文，并在该增强设置中提出具有次线性遗憾的算法。