In this talk we will present a general duality-theory framework for revenue
maximization in additive Bayesian auctions involving many bidders, multiple items
and arbitrary joint value distributions. Although the single-item case has been
resolved in a very elegant way by the seminal work of Myerson [1981], optimal
solutions involving more items still remain elusive. The framework extends linear
programming duality and complementarity to constraints with partial derivatives.
The dual system reveals the geometric nature of the problem and highlights its
connection with the theory of bipartite graph matchings. We will demonstrate the
power of the framework by applying it to special single-bidder settings with
independent item valuations drawn from various distributions of interest, to design
both exact and approximately optimal auctions.
We will also briefly discuss the standard Bayesian auction setting, where multiple
bidders have i.i.d. valuations for a single item, showing that for the natural class
of Monotone Hazard Rate (MHR) distributions, offering the same, take-it-or-leave-it
price to all bidders achieves an (asymptotically) optimal revenue.
Some of the related papers can be found in the following links:
https://arxiv.org/abs/1404.2329
https://arxiv.org/abs/1510.03399
https://arxiv.org/abs/1404.2832
https://arxiv.org/abs/1810.00800