Approximations in bayesian mechanism design for multi-parameter settings

We study the classic mathematical economics problem of Bayesian mechanism design where a principal aims to optimize an objective, typically expected revenue or welfare, when allocating resources to self-interested agents with preferences drawn from a known distribution. In single-parameter settings where each agent's preference is given by a single private value, this problem is solved (Myerson 1981). Unfortunately, these single-parameter optimal mechanisms are impractical and rarely employed (Ausubel and Milgrom, 2006), and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-parameter setting, where each agent's preferences require multiple values to describe (Manelli and Vincent, 2007). The main theme of this work is that we can solve multi-parameter mechanism design problems by analogy to properly chosen single-parameter ones. In order to implement this approach, however, the mechanisms we design must be robust to changes in the setting they are used in. Thus, in contrast to the theory of optimal mechanisms, we develop a theory of sequential posted-price mechanisms, where agents are offered take-it-or-leave-it prices in sequence. We prove that these mechanisms are approximately optimal in single-parameter settings. Further, these posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical, allowing them to generalize naturally to multi-parameter settings. We consider two types of multi-parameter settings in this thesis. First, we consider settings with multiple services and unit-demand agents. In such settings agents value each service differently, but desire at most one. In particular, we achieve constant-factor approximations for the multi-parameter multi-unit revenue-maximizing auction problem, and bound the economic benefit of randomization in such settings. Second, we consider settings where agents face budget constraints. An agent with a hard budget constraint cannot make payments exceeding it, even when the agent's value for service greatly exceeds the budget. We show how unconstrained Bayesian mechanism design can guide the design of mechanisms for budget-constrained agents, and achieve approximations to both revenue and welfare. Some of our results extend to settings where budgets are private information of the agents.