Varius, multiplex, multiformis: Mechanism design in multi-dimensional environments.

This dissertation consists of three essays in mechanism design theory. The underlying theme is the design of efficient mechanisms in complex, multi-dimensional environments. These essays are theoretical in nature and attempt to expand our knowledge of the workings of mechanisms in general quasi-linear settings. The first essay considers a mechanism design environment with private valuations, multi-dimensional types and an arbitrary allocation set. In this setting, it is known from Holmstrom (1979) that if agents' valuation functions are smooth with respect to individual types, then any efficient, dominant strategy direct mechanism is in the class of Groves mechanisms. In this essay, I extend this result to include non-smooth valuations, and present a necessary and sufficient condition for the uniqueness of Groves mechanisms among all efficient, dominant strategy mechanisms. This condition imposes restrictions on the behavior of the one-sided directional derivatives of the valuation functions with respect to individual types: at the efficient allocations, valuations are allowed to have upward, but not downward, kinks. I argue that non-smooth valuation functions consistent with the Upward Kink Property arise naturally in many important economic settings, as in situations where agents take private actions that lie outside the scope of the mechanism design problem. The main contribution of the first essay can be stated as a necessary and sufficient condition for the so called Payoff Equivalence Principle for all dominant strategy implementable allocation rules that are affine maximizers. In the second essay of this dissertation, I demonstrate that the Upward Kink Property is sufficient for Payoff Equivalence for all dominant strategy implementable allocation rules, not just affine maximizers. In the final essay of this dissertation, I depart from the private value environment considered previously, and report some positive results for the ex post implementation of efficient allocation rules in a social choice framework with interdependent valuations and multi-dimensional types. I first provide a complete characterization of ex post incentive compatible allocation rules for general linear settings, where valuations for the social alternative are linear in individual types. Next, I show that in the one-dimensional case a surplus extraction tax scheme implements the efficient allocation rule ex post. Extending this insight, I show by construction the existence of efficient, ex post incentive compatible mechanisms in two economically important classes of multi-dimensional models: in separable environments, where valuations are separable and thus each dimension of the type space interacts with one dimension of the allocation set; and in quasi-separable environments, where the interaction of several dimensions of the type space with each dimension of the allocation set can be aggregated into a single component.