Optimal Dynamic Auctions Are Virtual Welfare Maximizers

19 Pages Posted: 5 Dec 2018

See all articles by Vahab Mirrokni

Vahab Mirrokni

Google Inc.

Renato Paes Leme

Google Inc.

Pingzhong Tang

Tsinghua University

Song Zuo

Google Inc., New York

Date Written: November 11, 2018

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.

Keywords: Dynamic Auctions; Bank Account Mechanisms; Revenue Maximization; Virtual Values; Ironing; Virtual Welfare Maximizers

JEL Classification: D44; C73; D82

Suggested Citation

Mirrokni, Vahab and Paes Leme, Renato and Tang, Pingzhong and Zuo, Song, Optimal Dynamic Auctions Are Virtual Welfare Maximizers (November 11, 2018). Available at SSRN: https://ssrn.com/abstract=3282600 or http://dx.doi.org/10.2139/ssrn.3282600

Vahab Mirrokni

Google Inc. ( email )

1600 Amphitheatre Parkway
Second Floor
Mountain View, CA 94043
United States

Renato Paes Leme

Google Inc. ( email )

1600 Amphitheatre Parkway
Second Floor
Mountain View, CA 94043
United States

Pingzhong Tang

Tsinghua University ( email )

Beijing, 100084
China

Song Zuo (Contact Author)

Google Inc., New York ( email )

111 8th Ave
New York, NY 10011
United States

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