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 show that it may be the most promising method for approximating optimal multi-item auctions. Besides optimality, credibility of the auction is another major consideration. Consider a sealed-bid, one-item, second-price auctionfoot_1 being run between bidder 1 and bidder 2, whereas bidder 1 has valuation of $2 and bidder 2 has valuation of $3. Acting in their best strategy, bidder 1 and bidder 2 bid $2 and $3, and the auctioneer should then allocate the item to bidder 2 charging them $2. However, since the auction is sealed-bid, the auctioneer can tell bidder 2 that they won the auction and bider 1 bid $2.99, which would increase the auctioneer's revenue by $0.99. An auction is said to be credible if the auctioneer has no incentive not to stick to their proposed auction. The significance of auction credibility was brought to light when Google was called out for gaming their proposed second-price online ad auction Schiff (2022). It is known that there exists no sealed-bid, incentive-compatible, and credible auction if communication between bidders is precluded Akbarpour & Li (2020). The authors of this work admit that modern cryptography along with bidder communication can potentially break the trilemma, but they consider the costs, in terms of computing resources and latency, of cryptographic constrictions too high. In this paper, we propose an approach for running repeated auctions that greatly reduces the cost of a credible multi-item auction when verified either by revealing bids (i.e. we greatly reduce the number of bids that must be revealed) or using a cryptographic mechanism such as zero-knowledge proofs. First, we show that in a sequence of repeated auctions, we need not run the verification mechanism for every round. Instead, we can punish deviations with a penalty that when high enough, can prevent the auctioneer from being untruthful even when only a random set of auctions are audited. The repetition of auctions naturally brings up an issue regarding bidder participation which previous works in differentiable economics did not have to deal with, as it is unrealistic to assume that the same set of bidders participates from start to end in repeated auctions. We address this issue by proposing a model that takes account of bidder participation uncertainty, and we provide a method to extend previous works in differentiable economics to this model, which we support with experimental results. Verification tools. To prevent the auctioneer from deviating, we need some verification method that does not reveal additional information. Angel & Walfish (2013) proposed a cryptographic verification system called VEX that can be efficiently applied to second price auctions. In VEX, the auctioneer acts as the prover and the bidders act as the queriers. Under some given algorithm, the queriers can verify what the prover proposed is correct without information leakage in a considerable amount of time. More generally, Liu et al. (2021) has proposed zero-knowledge proof structures that can work for neural networks, and Mishra et al. (2020) has described a cryptography system which can also be applied to neural networks. With all that in mind, we can be confident that it is realistic to introduce verification tools during auction design.

2. RELATED WORK

There also exist concrete work on granting credibility in auction design for specific scenarios. Ferreira & Weinberg (2020) finds a credible and optimal auction for MHR valuations with commitment,



A sponsored search auction is one where the website owner auctions different ad spots on the webpage when a certain keyword is searched. In a second price auction, the auctioneer allocates the item to the highest bidder, and charges them the bid of the second highest bidder. The best strategy for any bidder in the second price auction is to bid exactly how much they value the item.



of the auction Peri et al. (2021); Kuo et al. (2020). Although differentiable economics is a newlyemerged field, recent progress Dütting et al. (2019); Rahme et al. (2020; 2021a); Curry et al. (2021)

Neural networks. RegretNet Dütting et al. (2019) was the first work to train incentive compatible auctions to maximize revenue using deep learning. RegretNet has two components: the allocation net and the payment net. Each network treats the corresponding part of the auction as a function, taking the bids from the bidders and outputting the allocation/payment. There are various version of other networks developed on the basis of RegretNet to cover specific needs. Peri et al. (2021) considers possible human preference in the allocation process. Kuo et al. (2020) focuses on improving the fairness of the auction mechanisms. There are also works focusing on improving the accuracy and efficiency of RegretNet. Rahme et al. (2020) proposed ALGNet as a more efficient version of RegretNet, which considers auction design as a auctioneer versus bidders adversarial model.

