On the Complexity of Dynamic Mechanism Design

TL;DR

This paper studies the complexity of designing optimal dynamic auctions for selling two items to an agent, revealing NP-hardness for deterministic mechanisms and providing polynomial algorithms for randomized ones, with implications for auction design and communication.

Contribution

It proves NP-hardness of finding optimal deterministic mechanisms in simple dynamic settings and introduces polynomial algorithms for optimal randomized auctions.

Findings

Optimal deterministic mechanisms are NP-hard to compute.

Polynomial-time algorithms exist for optimal randomized auctions.

Significant revenue gaps between different auction types and information structures.

Abstract

We introduce a dynamic mechanism design problem in which the designer wants to offer for sale an item to an agent, and another item to the same agent at some point in the future. The agent's joint distribution of valuations for the two items is known, and the agent knows the valuation for the current item (but not for the one in the future). The designer seeks to maximize expected revenue, and the auction must be deterministic, truthful, and ex post individually rational. The optimum mechanism involves a protocol whereby the seller elicits the buyer's current valuation, and based on the bid makes two take-it-or-leave-it offers, one for now and one for the future. We show that finding the optimum deterministic mechanism in this situation - arguably the simplest meaningful dynamic mechanism design problem imaginable - is NP-hard. We also prove several positive results, among them a polynomial linear programming-based algorithm for the optimum randomized auction (even for many bidders and periods), and we show strong separations in revenue between non-adaptive, adaptive, and randomized auctions, even when the valuations in the two periods are uncorrelated. Finally, for the same problem in an environment in which contracts cannot be enforced, and thus perfection of equilibrium is necessary, we show that the optimum randomized mechanism requires multiple rounds of cheap talk-like interactions.