Computation of Optimal Dynamic Mechanism with Participation Requirements

We consider dynamic mechanism design problems in which a principal procures up to one unit of a product/service in every period from an agent who is privately informed about its marginal production cost in each period, which are i.i.d. random variables. The model incorporates participation requirements: continued participation by the agent is costly (e.g., the agent consumes a fixed amount in each period to continue the partnership), and in each period the agent’s production cost must be fully paid for. In this setting, “static mechanisms” are no longer optimal and closed-form solutions are not available, which makes the development of tractable computational methods crucial. Our main results provide regularity conditions on the distribution of the private information such that the optimal contracts demonstrate “state-dependent-threshold” structures. Under our conditions, in any period, the optimal procurement level is the whole unit if the marginal production cost is below a threshold, and is a fraction between 0 and 1 if the marginal production cost is above this threshold. The threshold and the fractional procurement amount depend on the state of the system, which characterize the agent’s total expected future utility. These results allow dramatic reduction of the action space in the dynamic optimization model for the mechanism design problem. We demonstrate the impact of our structural results in computation time in a numerical study. Our results are based on the theory of infinite-dimensional conic optimization and show that “ironing” can occur even with well behaved distribution, such as uniform, in this dynamic setting. The analysis focuses on what we call the “dynamic virtual valuation,” a generalization of the Myersonian virtual valuation from the static setting, and is potentially relevant to other dynamic mechanism design problems. To illustrate the applicability of our framework in other settings, we apply the methodology to a model where no promise of future payment is allowed but the agent can accumulate gains to cover its future participation costs. This problem is non-concave, due to strategic termination of the partnership from the principal. Despite this difficulty we establish similar structural properties for the optimal contract, which imply tractable computational methods for this model as well.