Selling Two Identical Objects

TL;DR

This paper identifies conditions under which deterministic selling mechanisms are optimal for two identical objects, analyzing scenarios with decreasing and increasing marginal values, and showing when bundling or deterministic sales are optimal.

Contribution

It provides new conditions under which deterministic mechanisms are optimal for selling two identical objects with correlated values, extending prior results to specific valuation models.

Findings

Deterministic sale of the first unit is optimal under DMV with McAfee-McMillan condition.

Bundling is optimal under IMV with the same distribution condition.

Stronger distribution conditions guarantee deterministic mechanisms are optimal in DMV.

Abstract

It is well-known that optimal (i.e., revenue-maximizing) selling mechanisms in multidimensional type spaces may involve randomization. We obtain conditions under which deterministic mechanisms are optimal for selling two identical, indivisible objects to a single buyer. We analyze two settings: (i) decreasing marginal values (DMV) and (ii) increasing marginal values (IMV). Thus, the values of the buyer for the two units are not independent. We show that under a well-known condition on distributions~(due to McAfee and McMillan (1988)), (a) it is optimal to sell the first unit deterministically in the DMV model and (b) it is optimal to bundle (which is a deterministic mechanism) in the IMV model. Under a stronger sufficient condition on distributions, a deterministic mechanism is optimal in the DMV model. Our results apply to heterogeneous objects when there is a specified sequence in which the two objects must be sold.