CREDIBLE, SEALED-BID, OPTIMAL REPEATED AUC-TIONS WITH DIFFERENTIABLE ECONOMICS

Abstract

Online advertisement auctions happen billions of times per day. Bidders in auctions strategize to improve their own utility, subject to published auctions' rules. Yet, bidders may not know that an auction has been run as promised. A credible auction is one in which bidders can trust the auctioneer to run its allocation and pricing mechanisms as promised. It is known that, assuming no communication between bidders, no credible, sealed-bid, and incentive compatible (aka "truthtelling" or otherwise truthful-participation-incentivizing) mechanism can exist. In reality, bidders can certainly communicate, so what happens if we relax this (typically unrealistic) constraint? In this work, we propose a framework incorporating cryptography to allow computationally-efficient, credible, revenue-maximizing (aka "optimal") auctions in a repeated auction setting. Our contribution is two-fold: first, we introduce a protocol for running repeated auctions with a verification scheme, and we show such a protocol can eliminate the auctioneer's incentive to deviate while costing negligible additional computation. Secondly, we provide a method for training optimal auctions under uncertain bidder participation profiles, which generalizes our protocol to a much wider class of auctions. Our empirical results show strong support for both the theory and competency of the proposed method.

1. INTRODUCTION

The problem of designing optimal, or revenue-maximizing, auctions bears significant theoretical and practical importance in economics: every Google search involves a sponsored search auctionfoot_0 , webpage views involve real time auctions for ads, and online platforms like Ebay and Amazon have created markets ran by auctions. This problem is non-trivial: the auctioneer's revenue is dependent on the "best response" strategy of each bidder, which can each be dependent on each other. In his Nobel-prize-winning work, Myerson showed the n-bidder, 1-item optimal auction can be solved by essentially computing a virtual bid for each bidder, then maximizing welfare Myerson (1981) ; Daskalakis (2015) . What about multi-item auctions? This has been shown to be no easy task, one clear reason for this difficulty is the size of the bundling space which grows exponentially. Additionally, an auctioneer may set reserve prices or draw lotteries to earn additional revenue. In essence, the optimal auction can be weird and "defying intuition" Daskalakis (2015). Given no analytical solution have been found in designing the optimal multi-item auction, Daskalakis et al. ( 2014) have turned towards the complexity of this problem. They demonstrated that, under reasonable assumptions, finding the optimal multi-item auction is #P-hard. This has motivated the line of work called "differentiable economics" that focus on using machine learning to find desirable solutions to mechanism design problems Dütting et al. (2019) , which includes auction design. Differentiable economics approaches consider an auction as a function that takes bids as inputs and returns what item is allocated to who and how much each bidder pays. This function is usually encoded as a neural network, which can be backpropagated on given a differentiable loss function. The loss function is parameterized by the revenue, incentive compatibility-which we will provide a definition and discuss in more detail in later sections-or other desirable properties



A sponsored search auction is one where the website owner auctions different ad spots on the webpage when a certain keyword is searched.

