Session: Pricing problems and variance reduction in a world of uncertainty
Chair: Yu-Ching Le
Cluster:
Talk 1: Optimal Pricing for Probabilistic Selling with Pity Time Mechanism
Speaker: Kwei-Long Huang
Abstract: Blind boxes have grown increasingly popular among consumers, prompting many companies to adopt probabilistic selling to segment markets and boost profits. To ease concerns about uncertainty, some sellers disclose product probabilities, while others implement a “pity time” mechanism, which guarantees the desired product obtained by consumers after repeated unsuccessful attempts. This study investigates the use of pity time mechanisms in probabilistic selling involving vertically differentiated products (high vs. low quality) and diverse consumer types (surplus-oriented vs. quality-oriented). We derive optimal prices and profits across different selling formats and find that, whether offering only probabilistic products or multiple product options, implementing a pity time mechanism consistently benefits sellers, consumers, and society — achieving a win-win-win outcome for all stakeholders.
Talk 2: Competitive Demand Learning: A Non-cooperative Pricing Algorithm with Coordinated Price Experimentation
Speaker: Yu-Ching Le
Abstract: We consider a periodical equilibrium pricing problem for multiple firms over a planning horizon of $T$ periods. At each period, firms set their selling prices and receive stochastic demand from consumers. Firms do not know their underlying demand curve, but they wish to determine the selling prices to maximize total revenue under competition. Hence, they have to do some price experiments such that the observed demand data are informative to make price decisions. However, uncoordinated price updating can render the demand information gathered by price experimentation less informative or inaccurate. We design a nonparametric learning algorithm to facilitate coordinated dynamic pricing, in which competitive firms estimate their demand functions based on observations and adjust their pricing strategies in a prescribed manner. We show that the pricing decisions, determined by estimated demand functions, converge to underlying equilibrium as time progresses. {We obtain a bound of the revenue difference that has an order of $\mathcal{O}(F^2T^{3/4})$ and a regret bound that has an order of $\mathcal{O}(F\sqrt{T})$ with respect to the number of the competitive firms~$F$ and $T$.} We also develop a modified algorithm to handle the situation where some firms may have the knowledge of the demand curve.
Talk 3: An Accelerated Variance Reduced Extra-Point Approach to Finite-Sum Hemivariational Inequality Problem
Speaker: Kevin Huang
Abstract: In this paper, we develop stochastic variance reduced algorithms for solving a class of finite-sum hemivariational inequality (HVI) problem. In this HVI problem, the associated function is assumed to be differentiable, and both the vector mapping and the function are of finite-sum structure. We propose two algorithms to solve the cases when the vector mapping is either merely monotone or strongly monotone, while the function is assumed to be convex. We show how to apply variance reduction in the proposed algorithms when such an HVI problem has a finite-sum structure, and the resulting accelerated gradient complexities can match the best bound established for finite-sum VI problem, as well as the bound given by the direct Katyusha for finite-sum optimization respectively, in terms of the corresponding parameters such as (gradient) Lipschitz constants and the sizes of the finite-sums. We demonstrate the application of our algorithms through solving a finite-sum constrained finite-sum optimization problem and provide preliminary numerical results. Archival version: https://doi.org/10.48550/arXiv.2211.03269
