## Optimal Mechanisms for Selling Two Items to a Single Buyer

We extend the classical Myerson optimal auction and other related extensions of the work to design a revenue optimal mechanisms to allocate two heterogeneous items to a single buyer. We consider two different cases.

In the first case, the buyer’s valuation of the two items is assumed to be the uniform distribution over an arbitrary rectangle in the positive quadrant. We provide an explicit solution for any given rectangle, and prove that the optimal mechanism is to sell the items according to one of eight possible menus. We conjecture and experimentally validate that our methodology can be extended to a wider class of distributions. In the second case, we consider the above problem in a unit demand setting and explore two approaches to solving the problem: (a) the dual based approach (b) the virtual valuation based approach. We first show the difficulty in using the dual based approach and then use the virtual valuation based approach to provide a complete, explicit solution for our problem. In particular, we prove that the optimal mechanism is to sell the items according to one of five possible menus.

**References:**

Thirumulanathan, Rajesh Sundaresan, and Y. Narahari. Optimal mechanisms for selling two items to a single buyer having uniformly distributed valuations. **Journal of Mathematical Economics, **Elsevier, Volume 82, May 2019, pp. 1-30.

Thirumulanathan, Rajesh Sundaresan, and Y. Narahari. On optimal mechanisms in the two-item single-buyer unit-demand setting. **Journal of Mathematical Economics, **Elsevier, Volume 82, May 2019, pp. 31-60.