Robust Auction Design in the Auto-bidding World

1 Introduction . As auto-bidding—the practice of using optimization algorithms to procure advertising slots—is becoming the prevalent option in online advertising , a growing body of work is revisiting auction theory from the lens of the auto-bidding world [ Aggarwal et al. , 2019 , Babaioff et al. , 2021 , Balseiro et al. , 2021a , Deng et al. , 2021b , Golrezaei et al. , 2021c ] . A benefit of auto-bidding is that it simplifies advertisers ’ bidding process by asking advertisers for their high-level goals and then bidding on behalf of the advertisers.1 The main difference with classic auction theory stems from the model adopted for agent behavior . Unlike the classic utility maximization model , where each agent aims to maximize its own quasi-linear utility given by the difference between value and payment , the behavior of an auto-bidding agent is determined by its underlying optimization algorithm . In particular , the prevalent adopted model for the behavior of auto-bidding agents is that of value maximization , where each agent aims to maximize total values subject to an ROS ( return on spend ) constraint on the average spend per opportunity . For example , two common auto-bidding strategies are target CPA ( cost per acquisition ) and target ROAS ( return on ad spend ) auto-bidding , in which an algorithm optimizes the 1See https : //support.google.com/google-ads/answer/6268637 and https : //www.facebook.com/business/m/onesheeters/value-optimization-with-roas-bidding ; and Aggarwal et al . [ 2019 ] , Balseiro et al . [ 2021a ] , Deng et al . [ 2021b ] for more background introductions . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . total value ( e.g. , the number of conversions ) subject to an ROS constraint specified by the advertiser ( e.g. , the average spend for each conversion should not exceed a pre-specified target ) . One surprising observation when agents are value maximizers is that the Vickrey-Clarke-Groves ( VCG ) auction , which is truthful and efficient for utility maximizers , is no longer truthful nor efficient . Value maximizers have incentives to strategize their bids and , moreover , there exist auction instances where the social welfare at equilibria under the VCG auction is only 1/2 of the optimal social welfare [ Aggarwal et al. , 2019 , Deng et al. , 2021b ] . This is because the ratio between bids and values can be very different for each value-maximizing agent , and an agent with a low value may end up outbidding other bidders with high values in some auctions , creating allocation inefficiency . While there have been recent work addressing this problem , most previous studies have several shortcomings , e.g. , they reduce such efficiency loss by introducing boosts that explicitly or implicitly rely on the accurate knowledge of the advertiser values [ Deng et al. , 2021b ] , or they aim to optimize for either value-maximizing buyers or utility maximizing buyers ( and not a mixture of auto-bidders ) [ Balseiro et al. , 2021a ] , or they aim to optimize only welfare and not revenue [ Deng et al. , 2021b ] . In this paper , we aim to address the above shortcomings and propose simple auctions taking inaccurate value signals as additional inputs and we prove that the auctions have robust approximation guarantees in terms of both social welfare and revenue . Such inaccurate value signals could be the outcomes of some machine learning models that have bounded multiplicative errors on advertiser values.2 Our key theoretical result is a generic lemma that transfers the accuracy of the value signals into approximation guarantees on both welfare and revenue for various auctions . Moreover , our main theorems not only work in the pure auto-bidding world , but also in mixed environments in which both value maximizers and utility maximizers coexist . Furthermore , the approximation guarantees apply to a very general set of market outcomes , i.e. , they hold as long as no bidder is using a bidding profile that is always dominated under any competing bids . Such a broad solution concept includes classical notions such as complete-information Nash equilibrium as refinements and , as we discuss in our empirical results , additionally provides guarantees that hold along the convergence path when agents individually optimize their strategies using simple update rules . Our auctions are based on VCG or generalized second-price auctions ( GSP ) and use the inaccurate signals as reserve prices and/or as ( additive ) boosts . Reserve prices are minimum prices the agents should clear to win the auction . Boosts are additive transformations on top of bids , i.e. , agents are ranked based on the sum of their bids and boosts . Moreover , boosts are subtracted from standard payments to maintain incentive compatibility for utility maximizers . Intuitively , adding properly chosen boosts can push the auction to a more efficient outcome [ Deng et al. , 2021b ] . On the other hand , in contrast to the classic utility maximization setup in which reserve prices usually do not help improve social welfare , modest reserve prices in fact may improve social welfare when there are value-maximizing agents in the market . The key difference is that in truthful auctions like the VCG auction , utility-maximizers never react to reserve prices , but value-maximizers do react to reserve prices to satisfy their ROS constraints . It turns out that , with properly chosen reserve prices , one can eliminate bad bidding strategies by reducing the set of bidding strategies that could be best responses to other bidders ’ strategies . As a result , the remaining good bidding strategies lead to an outcome with better social welfare guarantees . 1.1 Our results . We propose a set of simple auctions and prove approximation guarantees both in terms of social welfare and revenue . In particular , these approximation guarantees for the proposed auctions are all tight except for the social welfare approximation for GSP with reserve ( Appendix E ) . The conclusions are robust against : i ) signal inaccuracy , ii ) agent behavior models ( value/utility maximization or intermediate ) , iii ) system status beyond equilibrium . The following Table 1 summarizes most of our approximation results , where 2 [ 0 , 1 ] describes the approximation accuracy of the signals and 2 [ 0 , 1 ] indicates a hybrid behavior model between pure value maximization and pure utility maximization ( e.g. , = 0 for pure value maximization and = 1 for pure utility maximization ) . All formal definitions will be given later in Section 2 . 2The learning task of the signals can be a complex problem given the potential interaction with advertiser incentives . In this paper , we are agnostic about how the signals are learned and hence the learning problem is out of the scope . Nevertheless , the line of work on incentive-aware learning [ Epasto et al. , 2018 , Golrezaei et al. , 2021a ] could be relevant to avoid or mitigate inappropriate incentives . GSP with reserve and boost = 0 ( 1 + ) /2 Corollary 4.6 1 = 0 : value maximization ; = 1 : utility maximization ; 2 ( 0 , 1 ) : intermediate models ( see Program ( 1 ) ) . 2 For 0 < < 1 , ( 1 + ) /2 > 1/ ( 2 ) > . Observe that in a world with value maximizers only ( i.e. , = 0 ) , if all the value maximizers can hit their targets ( see Program ( 1 ) ) , the revenue equals to the social welfare . In such an environment , our social welfare guarantees can directly imply the same revenue guarantees under the condition that value maximizers all hit their targets . We provide a general framework for proving approximation results in the presence of value maximizers through a novel technical lemma ( Lemma 3.1 ) . In fact , our approximation results are mostly derived from this lemma . Finally , we conduct empirical analyses with semi-synthetic data and validate our theoretical findings for the performance of our mechanisms in VCG auctions . 1.2 Related Work . As a central topic in economic study , since the seminal work of Vickrey–Clarke–Groves ( VCG ) auctions [ Clarke , 1971 , Groves , 1973 , Vickrey , 1961 ] and Myerson ’ s auction [ Myerson , 1981 ] , auction design has been successfully deployed in many different fields . Examples includes combinatorial auctions for reallocating radio frequencies [ Cramton et al. , 2006 ] and generalized second-price auctions as well as dynamic auctions for online advertising [ Aggarwal et al. , 2006 , Edelman et al. , 2007 , Mirrokni et al. , 2020 , Varian , 2007 ] . In contrast to works assuming utility-maximizing agents , a growing body of work has recently focused on auto-bidders such as target CPA bidders and target ROAS bidders . Aggarwal et al . [ 2019 ] find the optimal bidding strategies for a general class of auto-bidding objectives and prove the existence of pure-strategy equilibrium . Deng et al . [ 2021b ] show how boosts can be used to improve the efficiency guarantees when target CPA and target ROAS auto-bidders coexist . Balseiro et al . [ 2021a ] characterize the revenue-optimal single-stage auctions with either value-maximizers or utility-maximizers with ROS constraints under various information structure . Golrezaei et al . [ 2021c ] study the auction design for utility-maximizers with ROI constraints , where the behavior model is equivalent to the special case of ours with = 1/ ( 1 +minimal ROI ) . Besides the model of constrained optimization , there is another generalization of the utility models with ROI constraints adopted by Babaioff et al . [ 2021 ] , Goel et al . [ 2014 , 2019 ] . In particular , the negative payment term in the utility function is replaced by a general cost , which is often a convex function of the payment . Independent of the above , the term of value-maximizer has been used in some other works but with quite different mathematical models Fadaei and Bichler [ 2016 ] , Wilkens et al . [ 2016 , 2017 ] . The main distinction is that in these models , the target ROS constraints are imposed on a single auction rather than across a set of auctions , hence the bidders become insensitive to the marginal tradeoff between values and payments , leading to different behavior patterns . Prior to these works , the simplest auto-bidding model is budget optimization with utilitymaximizers [ Borgs et al. , 2007 ] . Pai and Vohra [ 2014 ] characterize the revenue-optimal auction of utility-maximizing bidders with budget constraints . For applications in online advertising , Balseiro and Gur [ 2019 ] develop budget management strategies that are no-regret in a long run , which is extended in more complex settings [ Avadhanula et al. , 2021 , Balseiro et al. , 2020b , 2021b , Celli et al. , 2021 ] . Balseiro et al . [ 2017 , 2020a ] provide a thorough study to compare different commonly used budget management strategies in practice . Conitzer et al . [ 2017 , 2018 ] study pacing algorithms for budget constraints in both first price auctions and second price auctions . The techniques of reserve prices and additive boosts have been widely studied in the literature [ Amin et al. , 2013 , Deng et al. , 2021a , Golrezaei et al. , 2021b , Lavi et al. , 2003 , Paes Leme et al. , 2016 , Sandholm and Likhodedov , 2015 ] . Among these works , Deng et al . [ 2021b ] improve the welfare approximation ratio to ( c+ 1 ) / ( c+ 2 ) using boosted auctions with accurate signals about bidders ’ values in the environment with value maximizers only . Our results are more robust and more general from several aspects : We allow inaccurate signals that approximate bidders ’ values ; Our results hold for a mixture of behavior models in auctions with reserves and/or boosts ; We provide guarantees on revenue performance in addition to welfare performance . Some of our results rely on the assumption that the auto-bidders adopt uniform bidding strategies when the underlying auction is not incentive-compatible for utility-maximizing bidders . In practice , it is usually hard for bidders to adopt and optimize non-uniform bidding strategies , and moreover , uniform bidding has been shown to perform well against optimal non-uniform bidding strategies in ad auctions [ Balseiro and Gur , 2019 , Bateni et al. , 2014 , Deng et al. , 2020 , Feldman and Muthukrishnan , 2008 , Feldman et al. , 2007 ] .

Summary: This paper studies auction design for advertising slots when the agents are using auto-bidding optimization algorithms.  The bidders: The focus is on “value-maximizing agents”: this type of agent aims to maximize the total value he gets from his allocation, subject to the constraint that his value is at least his price (as opposed to a utility-maximizing agent, who aims to maximize his value minus his price; the authors also study interpolations between value maximizers and utility maximizers). The authors write that this is the prevalent adopted model for the behavior of auto-bidding agents. There are $m$ position auctions and each agent $i$ has a fixed (i.e., not Bayesian) value $v_{i,j}$ for each auction $j \in [m]$. Agent $i$’s value for the $k^{th}$ slot in auction $j$ is $v_{i,j}\cdot pos_{j,k}$, where $pos_{j,k}$ is a position normalizer that doesn’t depend on $i$ and is common knowledge.  The auctions: The paper studies VCG, GSP, and first-price auctions with reserves and boosts, the latter of which means that each bidder’s bid $b_{i,j}$ is increased by some boost $z_{i,j}$. The reserves and boosts are set to be equal to some noisy signals about the values $v_{i,j}$ which the authors write could be obtained, for example, using a machine learning model. These signals are assumed to be accurate up to some multiplicative error. Also, the bids are assumed to be undominated (which includes the support of Nash equilibrium bids as a subset).  The results: The authors prove approximation guarantees showing that for each of these three auctions, the revenue and welfare are approximately optimal, where the approximation factor depends on the accuracy of the noisy value signals. They also provide experiments on semi-synthetic data that validate these theoretical findings. From my understanding, the buyers’ values are based on real-world ad auction data, but the signals and bids are synthetically generated.