Ex-post optimal knapsack procurement

We consider a budget-constrained mechanism designer who selects an optimal set of projects to maximize her utility. A project's cost is private information and its value for the designer may vary. In this allocation problem, the selection of projects - both which and how many - is endogenously determined by the mechanism. The designer faces ex-post constraints: The participation and budget constraints must hold for each possible outcome while the mechanism must be implementable in dominant strategies. We derive the class of optimal mechanisms and show that it has a deferred acceptance auction representation. This feature guarantees an implementation with a descending clock auction. Only in the case of symmetric projects do price clocks descend synchronously such that the cheapest projects are executed. The case in which values or costs are asymmetrically distributed features a novel tradeoff between quantity and quality. Interestingly, this tradeoff mitigates the distortion due to the informational asymmetry compared to environments where quantity is exogenous.