# Optimal Dynamic Auctions are Virtual Welfare Maximizers

## TL;DR

This paper characterizes optimal dynamic auctions as virtual welfare maximizers with monotone constraints and introduces an efficient approximation scheme through a novel ironing technique.

## Contribution

It provides a structural characterization of optimal dynamic auctions as virtual welfare maximizers and develops an efficient approximation method using a new ironing technique.

## Key findings

- Optimal dynamic auctions are virtual welfare maximizers with monotone constraints.
- A new ironing technique allows approximate and efficient handling of non-monotonic virtual values.
- The paper presents a Fully Polynomial Time Approximation Scheme for optimal dynamic auctions.

## Abstract

We are interested in the setting where a seller sells sequentially arriving items, one per period, via a dynamic auction. At the beginning of each period, each buyer draws a private valuation for the item to be sold in that period and this valuation is independent across buyers and periods. The auction can be dynamic in the sense that the auction at period $t$ can be conditional on the bids in that period and all previous periods, subject to certain appropriately defined incentive compatible and individually rational conditions. Perhaps not surprisingly, the revenue optimal dynamic auctions are computationally hard to find and existing literatures that aim to approximate the optimal auctions are all based on solving complex dynamic programs. It remains largely open on the structural interpretability of the optimal dynamic auctions. In this paper, we show that any optimal dynamic auction is a virtual welfare maximizer subject to some monotone allocation constraints. In particular, the explicit definition of the virtual value function above arises naturally from the primal-dual analysis by relaxing the monotone constraints. We further develop an ironing technique that gets rid of the monotone allocation constraints. Quite different from Myerson's ironing approach, our technique is more technically involved due to the interdependence of the virtual value functions across buyers. We nevertheless show that ironing can be done approximately and efficiently, which in turn leads to a Fully Polynomial Time Approximation Scheme of the optimal dynamic auction.