Convexity and Duality in Optimum Real-time Bidding and Related Problems

TL;DR

This paper demonstrates how certain real-time bidding problems in auction markets can be formulated as convex optimization problems, enabling efficient solutions and revealing deep duality structures, with connections to finance and transaction costs.

Contribution

It introduces a variable transformation that convexifies bidding problems and shows its applicability to various auction formats, linking bidding costs to financial transaction costs.

Findings

Convexity of bidding problems enables efficient algorithms.

Duality theory provides rich structural insights.

Bidding costs are formally equivalent to financial transaction costs.

Abstract

We study problems arising in real-time auction markets, common in e-commerce and computational advertising, where bidders face the problem of calculating optimal bids. We focus upon a contract management problem where a demand aggregator is subject to multiple contractual obligations requiring them to acquire items of heterogeneous types at a specified rate, which they will seek to fulfill at minimum cost. Our main results show that, through a transformation of variables, this problem can be formulated as a convex optimization problem, for both first and second price auctions. Convexity results in efficient algorithms for solving instances of this problem, and the resulting duality theory admits rich structure and interpretations. Additionally, we show that the transformation of variables used to formulate this problem as a convex program can also be used to guarantee the convexity of optimal bidding problems studied by other authors (who did not leverage convexity). Finally, we show how the expected cost of bidding in second price auctions is formally identical to certain transaction costs when submitting market orders in limit order book markets. This fact is used to analyze a Markowitz portfolio problem which accounts for these transaction costs, establishing an interesting connection between finance and optimal bidding.