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The problem of designing a profit-maximizing, Bayesian incentive compatible and individually rational mechanism with flexible consumers and costly heterogeneous supply is considered. In our setup, each consumer is associated with a flexibility set that describes the subset of goods the consumer is equally interested in. Each consumer wants to consume one good from its flexibility set. The flexibility set of a consumer and the utility it gets from consuming a good from its flexibility set are its private information. We adopt the flexibility model of [1] and focus on the case of nested flexibility sets - each consumer's flexibility set can be one of k nested sets. Examples of settings with this inherent nested structure are provided. On the supply side, we assume that the seller has an initial stock of free supply but it can purchase more goods for each of the nested sets at fixed exogenous prices. We characterize the allocation and purchase rules for a profit-maximizing, Bayesian incentive compatible and individually rational mechanism as the solution to an integer program. The optimal payment function is pinned down by the optimal allocation rule in the form of an integral equation. We show that the nestedness of flexibility sets can be exploited to obtain a simple description of the optimal allocations, purchases and payments in terms of thresholds that can be computed through a straightforward iterative procedure.

Optimal Mechanism Design with Flexible Consumers and Costly Supply