We study the design of mechanisms in combinatorial auction domains. We focus on settings where the auction is repeated, motivated by auctions for licenses or advertising space. We consider models of agent behaviour in which they either apply common learning techniques to minimize the regret of their bidding strategies, or apply short-sighted best-response strategies. We ask: when can a black-box approximation algorithm for the base auction problem be converted into a mechanism that approximately preserves the original algorithm's approximation factor on average over many iterations? We present a general reduction for a broad class of algorithms when agents minimize external regret. We also present a new mechanism for the combinatorial auction problem that attains an @math approximation on average when agents apply best-response dynamics.
 We study mechanisms for utilitarian combinatorial allocation problems, where agents are not assumed to be single-minded. This class of problems includes combinatorial auctions, multi-unit auctions, unsplittable flow problems, and others. We focus on the problem of designing mechanisms that approximately optimize social welfare at every Bayes-Nash equilibrium (BNE), which is the standard notion of equilibrium in settings of incomplete information. For a broad class of greedy approximation algorithms, we give a general black-box reduction to deterministic mechanisms with almost no loss to the approximation ratio at any BNE. We also consider the special case of Nash equilibria in full-information games, where we obtain tightened results. This solution concept is closely related to the well-studied price of anarchy. Furthermore, for a rich subclass of allocation problems, pure Nash equilibria are guaranteed to exist for our mechanisms. For many problems, the approximation factors we obtain at equilibrium improve upon the best known results for deterministic truthful mechanisms. In particular, we exhibit a simple deterministic mechanism for general combinatorial auctions that obtains an O(√m) approximation at every BNE. This paper is meant to familiarize the audience with some of the fundamental results in the theory of implementation and provide a quick progression to some open questions in the literature.