Three essays in the interface of optimization with mechanism design, nonexclusive competition, and prophet inequalities

Abstract

Mechanism Design. We consider the mechanism design problem of a principal allocating a single good to one of several agents without monetary transfers. Each agent desires the good and uses it to create value for the principal. We designate this value as the agent's private type. Even though the principal does not know the agents' types, she can verify them at a cost. The allocation of the good thus depends on the agents' self-declared types and the results of any verification performed, and the principal's payoff matches her allocation value minus the verification costs. It is known that when the agents' types are independent, a favored-agent mechanism maximizes her expected payoff. However, this result relies on the unrealistic assumptions that the agents' types follow known independent probability distributions. We assume that the agents' types are governed by an ambiguous joint probability distribution belonging to a commonly known ambiguity set and that the principal maximizes her worst-case expected payoff. We consider three types of ambiguity sets: (i) support-only ambiguity sets, which contain all distributions supported on a rectangle, (ii) Markov ambiguity sets, characterized through first-order moment bounds, and (iii) Markov with independence ambiguity sets. For each of these ambiguity sets, we show that a favored-agent mechanism, which we characterize implicitly, is optimal and also Pareto-robustly optimal. The optimal choices of the favored agent and the threshold do not depend on the verification costs in all three cases. Nonexclusive Competition. A freelancer with a time constraint faces offers from multiple identical parties. The quality of the service provided by the freelancer can be high or low and is only known by the freelancer. The freelancer's time cost is strictly increasing and convex. We show that a pure-strategy equilibrium exists if and only if the preferences of the high-type freelancer satisfy one of the following two distinct conditions: (i) the high-type freelancer does {not} prefer providing his services for a price equal to the expected quality at the no-trade point; (ii) the high-type freelancer prefers providing his services for a price equal to the expected quality at any feasible trade point. If (i) holds, then in equilibrium, the high-type freelancer does not trade, whereas the low-type may not trade, trade efficiently, or exhaust all of his capacity. Moreover, the buyers make zero profit from each of their traded contracts. If (ii) holds, then both types of the freelancer trade at the capacity in equilibrium. Furthermore, the buyers make zero expected profit with cross-subsidization. In any equilibrium, the aggregate equilibrium trades are unique. Prophet Inequalities. Prophet inequalities bound the expected reward obtained in a class of stopping problems by the optimal reward of the corresponding offline problem. We show how to obtain prophet inequalities for a large class of stopping problems associated with selecting a point in a polyhedron. Our approach utilizes linear programming tools and is based on a reduced form representation of the stopping problem. We illustrate the usefulness of our approach by re-establishing three different prophet inequality results from the literature. (i) For polymatroids with nonnegative coefficients in their unique Minkowski sum of simplices, we prove the 1/2-prophet inequality. (ii) We prove the 1/n-prophet inequality when there are n stages, the stages have dependently distributed rewards, and we are restricted to choosing a strategy from an arbitrary polyhedron. (iii) When the feasible set of strategies can be described via K different constraints, we obtain the 1/(K+1)-prophet inequality.