We study the pricing of three-part tariffs (3PTs), where service providers charge a fixed fee with an allowance of free units, and a per-unit fee for additional units above the allowance. This is a pricing strategy that has been widely used in a variety of industries, including telecommunications and internet services. Many papers study the properties of the optimal solution of 3PTs. However, the existing solution approaches proposed in prior papers can only be used to solve 3PT problems under certain restricted conditions. Solving 3PT problems with multiple consumer types has been considered as “intractable” in the literature. In this paper, we provide an optimal solution for 3PT problem with multiple consumer types. We first provide a solution approach to solve 3PT problem with discrete units and multiple discrete consumer types whose willingness-to-pays for different units follow Spence-Mirrlees single-crossing property. The cost of our solution approach is low-rank polynomial to the problem size. We then extend the discrete solution to solve 3PT problems when the consumer types and unit amounts are not required to be discrete. The solution values for the continuous problems are within arbitrary prespecified error tolerances.

我们研究了三部曲 (3PT) 的定价，其中服务提供商收取固定费用并允许免费单位，并对超过允许额的额外单位收取每单位费用。这是一种广泛应用于各种行业的定价策略，包括电信和互联网服务。许多论文研究了 3PT 最优解的性质。然而，现有论文中提出的现有解决方案只能用于在某些受限条件下解决 3PT 问题。解决具有多种消费者类型的 3PT 问题在文献中被认为是“棘手的”。在本文中，我们为具有多种消费者类型的 3PT 问题提供了最优解。我们首先提供解决离散单元和多个离散消费者类型的 3PT 问题的解决方法，这些消费者类型的不同单元的支付意愿遵循 Spence-Mirrlees 单交叉属性。我们解决方案的成本是问题规模的低阶多项式。然后，当消费者类型和单位数量不需要离散时，我们扩展离散解决方案来解决 3PT 问题。连续问题的解值在任意预先指定的误差容限内。