We study when pure bundling, i.e., offering only the grand bundle of all products, is optimal for a multi-product monopolist. Pure bundling is optimal if consumers with higher values for the grand bundle have higher relative values for smaller bundles compared to the grand bundle. Conversely, pure bundling is not optimal if consumers with higher values for the grand bundle have lower relative values. We prove the results by decomposing the problem into simpler ones in which types can be ranked according to their values for the grand bundle. ∗We thank Nageeb Ali, Mark Armstrong, Ben Brooks, Gabriel Carroll, Robert Kleinberg, Vijay Krishna, Alexey Kushnir, Preston McAfee, Stephen Morris, Henrique Oliveira, Marco Scarcini, Ilya Segal, Dan Vincent, Rakesh Vohra, Jidong Zhou, and various seminar participants. We thank Berk Idem, Xiao Lin, and Garima Singal for excellent research assistance. We are particularly grateful to Ron Siegel for numerous discussions about the paper. This paper reinterprets and improves on some of the results that were previously in manuscripts titled “Reverse Mechanism Design” and “Multi-dimensional Virtual Values and Second-degree Price Discrimination”. †Department of Economics, Penn State University, nima.haghpanah@gmail.com. ‡EECS Department, Northwestern University, hartline@eecs.northwestern.edu. What is a multi-product monopolist’s optimal selling strategy? This is a classical economic question of importance for both positive and normative analysis, dating back to Stigler (1963) and Adams and Yellen (1976). We characterize when pure bundling, i.e., offering only the grand bundle of all products, is the optimal selling strategy. Our characterization is easy to state and has a straightforward intuition. Consider a monopolistic seller of products 1 to n, and a buyer who needs at most one unit of each product. Assume that production costs are zero. The buyer’s privately known type t identifies a value v(b, t) for each bundle of products b ⊆ {1, . . . , n}, and is drawn from a distribution. To maximize expected profit, should the seller use pure bundling and offer only the grand bundle of all products b∗ = {1, . . . , n}? Or should she use more complex strategies such as offering a menu that includes multiple bundles at possibly different prices? The optimality of pure bundling depends on how the buyer’s relative values, the ratio v(b, t)/v(b∗, t) for each bundle b, change with the buyer’s value for the grand bundle v(b∗, t). Pure bundling is optimal if relative values are first-order stochastically non-decreasing in the value for the grand bundle; i.e., types with higher values for the grand bundle are more likely to have higher relative values. Conversely, pure bundling is not optimal if relative values are first-order stochastically decreasing in the value for the grand bundle. The characterization has a straightforward economic intuition. Let us compare the profit from selling only the grand bundle at some price p, to the profit from a “mixed bundling” strategy of selling a smaller bundle at a discounted price in addition to the grand bundle at the full price p. Mixed bundling has a gain and a loss compared to pure bundling. The gain is from selling to more types, i.e., types who are unwilling to pay the full price for the grand bundle but are willing take the discounted offer. The loss is from types whose demand is diverted from the full price to the discounted offer. These types have high value for the grand bundle and high relative value for the smaller bundle (so that they find the discounted offer attractive). The loss is larger than the gain if types with higher value for the grand bundle are more likely to have high relative values, and is smaller if such types are more likely to have low relative values. To see the interpretation of our results, consider selling a car that can be customized with an audio system. Mapped into our model, there are two products, the “basic” car and the audio system, and three non-empty bundles. The value of the bundle that contains only the audio system is zero. Suppose that consumers differ across two dimensions. First, whereas some consumers need cars to get to work, others can take public transit. The former group have higher values for all bundles than the latter. Second, some consumers care more

中文翻译：

---

什么时候是纯捆绑最佳？

我们研究什么时候纯粹捆绑，即只提供所有产品的大捆绑，对于多产品垄断者来说是最佳的。如果与大捆绑相比，对大捆绑具有较高价值的消费者对较小捆绑具有更高的相对价值，则纯捆绑是最佳的。相反，如果消费者对大捆绑的价值较高，而相对价值较低，则纯捆绑不是最佳的。我们通过将问题分解为更简单的问题来证明结果，其中类型可以根据它们的值对大束进行排序。∗我们感谢 Nageeb Ali、Mark Armstrong、Ben Brooks、Gabriel Carroll、Robert Kleinberg、Vijay Krishna、Alexey Kushnir、Preston McAfee、Stephen Morris、Henrique Oliveira、Marco Scarcini、Ilya Segal、Dan Vincent、Rakesh Vohra、Jidong Zhou 和各种研讨会参与者。我们感谢 Berk Idem, Xiao Lin, 和 Garima Singal 提供出色的研究帮助。我们特别感谢 Ron Siegel 对论文进行了多次讨论。本文对之前在题为“逆向机制设计”和“多维虚拟价值和二级价格歧视”的手稿中的一些结果进行了重新解读和改进。†宾夕法尼亚州立大学经济系，nima.haghpanah@gmail.com。‡西北大学 EECS 系，hartline@eecs.northwestern.edu。多产品垄断者的最优销售策略是什么？这是一个对实证分析和规范分析都很重要的经典经济问题，其历史可以追溯到 Stigler (1963) 和 Adams and Yellen (1976)。我们描述纯捆绑销售，即只提供所有产品的大捆绑销售，是最佳销售策略。我们的特征很容易表述并且具有直接的直觉。考虑产品 1 到 n 的垄断卖方，以及每个产品至多需要一个单位的买方。假设生产成本为零。买方的私人已知类型 t 为每组产品 b ⊆ {1, . . . ，n}，并且是从分布中提取的。为了最大化预期利润，卖家是否应该使用纯捆绑销售并只提供所有产品的大捆绑销售 b* = {1, . . . ，n}？或者她应该使用更复杂的策略，例如提供包含多个套餐的菜单，价格可能不同？纯捆绑的最优性取决于买方的相对价值，即每个捆绑 b 的比率 v(b, t)/v(b∗, t)，随着买方对大捆绑 v(b∗, t) 的价值的变化. 如果大捆绑的相对值是一阶随机不减少的，则纯捆绑是最佳的；即，具有较高值的​​总包的类型更有可能具有较高的相对值。相反，如果大捆绑的相对值是一阶随机递减的，则纯捆绑不是最佳的。这种表征具有直接的经济直觉。让我们比较以某个价格 p 只销售大捆绑包的利润，与以折扣价出售小捆绑包和全价 p 的大捆绑包的“混合捆绑”策略的利润。与纯捆绑相比，混合捆绑有利有弊。收益是从销售到更多类型，即，不愿意为大礼包支付全价但愿意接受折扣优惠的类型。损失来自需求从全价转移到折扣报价的类型。这些类型对于大捆绑包具有较高的价值，对于较小的捆绑包具有较高的相对价值（因此他们发现折扣优惠很有吸引力）。如果总包具有较高值的​​类型更有可能具有较高的相对值，则损失大于增益，如果此类类型更有可能具有较低的相对值，则损失较小。要查看对我们结果的解释，请考虑销售可以使用音频系统进行定制的汽车。映射到我们的模型中，有两种产品，“基本”汽车和音响系统，以及三个非空包。仅包含音频系统的包的值为零。假设消费者在两个维度上有所不同。首先，一些消费者需要开车上班，而另一些消费者则可以乘坐公共交通工具。前一组的所有捆绑包的值都高于后者。二、部分消费者更关心