Maximal revenue with multiple goods: Nonmonotonicity and other observations

Consider the problem of maximizing the revenue from selling a number of goods to a single buyer. We show that, unlike the case of one good, when the buyer's values for the goods increase, the seller's maximal revenue may well decrease. We then identify two circumstances where monotonicity does obtain: when optimal mechanisms are deterministic and symmetric, and when they have submodular prices. Next, through simple and transparent examples, we clarify the need for and the advantage of randomization when maximizing revenue in the multiple-good versus the one-good case. Finally, we consider seller-favorable mechanisms, the only ones that matter when maximizing revenue. They are essential for our positive monotonicity results, and they also circumvent well known nondifferentiability issues.