Multi-parameter mechanism design and sequential posted pricing

Abstract

We study the classic mathematical economics problem of Bayesian optimal mechanism design where a principal aims to optimize expected revenue when allocating resources to self-interested agents with preferences drawn from a known distribution. In single parameter settings (i.e., where each agent's preference is given by a single private value for being served and zero for not being served) this problem is solved [20]. Unfortunately, these single parameter optimal mechanisms are impractical and rarely employed [1], and furthermore the underlying economic theory fails to generalize to the important, relevant, and unsolved multi-dimensional setting (i.e., where each agent's preference is given by multiple values for each of the multiple services available) [25].In contrast to the theory of optimal mechanisms we develop a theory of sequential posted price mechanisms, where agents in sequence are offered take-it-or-leave-it prices. We prove that these mechanisms are approximately optimal in single-dimensional settings. These posted-price mechanisms avoid many of the properties of optimal mechanisms that make the latter impractical. Furthermore, these mechanisms generalize naturally to multi-dimensional settings where they give the first known approximations to the elusive optimal multi-dimensional mechanism design problem. In particular, we solve multi-dimensional multi-unit auction problems and generalizations to matroid feasibility constraints. The constant approximations we obtain range from 1.5 to 8. For all but one case, our posted price sequences can be computed in polynomial time.This work can be viewed as an extension and improvement of the single-agent algorithmic pricing work of [9] to the setting of multiple agents where the designer has combinatorial feasibility constraints on which agents can simultaneously obtain each service.

Extreme value theorems for optimal multidimensional pricing

“…Our approach is different than that of [5,6] in that we study directly the optimal revenue (as a random variable,) rather than only relating its expectation to a benchmark that may be off by a constant factor. Clearly, the optimal revenue is a function of the values (which are random) and the optimal price vector (which is unknown).…”

“…Previous work on this problem by Chawla et al [5,6] provides factor 2 approximation to the revenue achieved by the optimal price vector. The elegant observation enabling this result is to consider the following mental experiment: suppose that the unit-demand buyer is split into n "copies" t 1 , .…”

“…So we can go back to our original setting and design a mechanism whose revenue comes close to Myerson's revenue in the hypothetical scenario. Using this approach [6] obtains a 2-approximation to the optimal revenue. Moreover, if the distributions {F i } i are regular (this is a commonly studied class of distributions in Economics,) the corresponding price vector can be computed efficiently.…”

Equilibrium in Combinatorial Public Projects

“…5 A generalization of submodular valuations which we will use in this work is that of fractionally subadditive (or XOS ) valuations (see [9]): a valuation v is fractionally subadditive if and only if there exist a set of additive valuations…”

“…Some examples include posted price mechanisms that approximate revenueoptimal auctions [5], and simultaneous single item auctions that have good social efficiency relative to the fully efficient combinatorial auctions [6,1,13,10].…”