Autobidding Auctions in the Presence of User Costs

We study autobidding ad auctions with user costs, where each bidder is value-maximizing subject to a return-over-investment (ROI) constraint, and the seller aims to maximize the social welfare taking into consideration the user's cost of viewing an ad. We show that in the worst case, the approximation ratio of social welfare by running the vanilla VCG auctions with user costs could as bad as 0. To improve the performance of VCG, We propose a new variant of VCG based on properly chosen cost multipliers, and prove that there exist auction-dependent and bidder-dependent cost multipliers that guarantee approximation ratios of 1/2 and 1/4 respectively in terms of the social welfare.


Introduction
As 2022, over 90% of online advertising transact through automation technologies (Mitchell, 2022).As one key application of automation in online advertising, auto-bidding significantly simplifies the tasks of advertisers by delegating the bidding to automated agents who submit realtime bids on behalf of the advertisers in each ad auction to optimize with pre-specified high-level objectives and constraints.One popular and representative example of auto-bidding strategies is to maximize the total advertiser value subject to a target return-on-investment (ROI) constraint declared by the advertiser.This type of auto-bidders is referred as value maximizers by a recent line of research (Balseiro et al., 2021a;Deng et al., 2021) as these auto-bidders aim to maximize the total advertiser value from ad auctions, such as the (weighted) number of clicks or conversions, while guaranteeing the ROI no less than the advertiser-specified threshold.
The differences in behavior models between the classic (quasi-linear) utility maximizers (who optimize for value minus payment) and value maximizers results in different conclusions for many basic problems in auction theory.As one demonstrating example, Vickrey-Clarke-Groves (VCG) auction is known to achieve the optimal social welfare when bidders are utility maximizers.However, Aggarwal et al. (2019) showed that in the worst case, the welfare of VCG auction can be as bad as 1/2 of the optimal social welfare when bidders are value maximizers.Similar distinctions have been discovered by recent works revisiting the effectiveness of various existing mechanisms in the presence of value maximizers (Balseiro et al., 2021a;Deng et al., 2022;Liaw et al., 2022;Mehta, 2022).
User Costs User costs to the auctioneer do not receive much attention in the literature on auction design for online advertising with utility maximizers, despite of the prevalence of the externalities on user experience of displaying ads to users (Abrams and Schwarz, 2007).One reason is that when bidders are utility maximizers, many commonly studied auctions and their properties directly generalize from models without user costs to models with user costs.For example, VCG auction by definition directly applies to the model with user costs and continues to achieve the optimal social welfare.Abrams and Schwarz (2007) generalize General-Second-Price (GSP) auction to accommodate the cost from user experience impact in online advertising by treating the original bids minus the costs as the effective bids from advertisers.They further prove that GSP preserves the property that there exists an equilibrium achieving the optimal social welfare after the generalization.
Challenges from User Cost in Autobidding.However, such nice generalization of concepts and results for user costs no longer works when the bidders are value maximizers.As we will show later in Section 4.2, VCG auction can have 0 approximation to the optimal social welfare in the worst case when bidders are value maximizers.One intuitive explanation of the worsened approximation guarantee of VCG auction in this case lies behind the so-called uniform-bidding strategy widely adopted by auto-bidding agents (Bateni et al., 2014;Feldman et al., 2007).Under uniform-bidding strategies, each bidder bids its value of each auction times the bid multiplier that is constant across all auctions for this bidder.From the perspective of value maximizers, the best response is always achievable by uniform-bidding with a properly optimized bid multiplier under VCG auction with/without user costs (Aggarwal et al., 2019;Deng et al., 2021).In contrast, from the perspective of the auctioneer, bidders participating in the same auction could have different bid multipliers, meaning that their base values are scaled up by different magnitudes.Thus, the ranking of their bids can be very different from the efficient ranking, leading to an inefficient allocation.The inefficiency becomes worse off in the presence of user costs, since the cost associated to each bidder is not scaled by the corresponding bid multiplier in VCG auction.For bidders with bid multipliers much larger than 1, the effect of their costs in ranking is then vanished, resulting in additional allocation inefficiency.
Based on the above observation, one naïve and minimal fix to VCG auction is to artificially scale user costs by a multiplier and hope that VCG auction can retain a good approximation guarantee with a properly chosen cost multiplier.Unfortunately, using one global cost multiplier is not enough as we will show later in Section 4.2, the approximation guarantee for VCG auction with a global cost multiplier is still 0 in the worst case.

Our Results
In this paper, we present how the constant approximation of social welfare can be retained with one step further on top of the above idea, namely, VCG auction with cost multipliers.In particular, we prove that there exists a set of cost multipliers corresponding to each auction (i.e., auction-dependent but bidder-independent cost multipliers) such that the welfare in any equilibrium among value maximizers under VCG auction with those cost multipliers is at least 1/2 of the optimal social welfare (Section 4.3).In addition, we further show that there exists a set of cost multipliers corresponding to each bidder (i.e., bidder-dependent but auction-independent cost multipliers) such that the welfare performance is at least 1/4 of the optimal social welfare (Section 4.4).
More recently, there's a line of research focusing on value maximizing bidders motivated by auto-bidding (e.g.target CPA bidders and target ROAS bidders) in online advertising.Aggarwal et al. (2019) characterizes the optimal bidding strategies in truthful auction for value maximizing bidders and shows that in the worst case, the welfare of VCG auction can be as bad as 1/2 of the optimal social welfare.Balseiro et al. (2021b); Deng et al. (2022Deng et al. ( , 2021) ) show how boosts and reserves can be used to improve the welfare efficiency in auctions with value maximizing bidders.Balseiro et al. (2021aBalseiro et al. ( , 2022a) ) characterize the revenue-optimal single-stage auctions with value maximizers with or without budget constraints under various information structure.In addition to the above, there are other related work studying slightly different bidding models, e.g.Babaioff et al. (2021); Goel et al. (2019Goel et al. ( , 2014)); Golrezaei et al. (2021) study utility maximizers with ROI constraints, and Fadaei and Bichler (2016); Wilkens et al. (2017Wilkens et al. ( , 2016) ) study value maximizing bidders with per-auction target constraints.
As far as we know, Abrams and Schwarz (2007) is the first paper to study auction design with user costs for utility maximizing bidders, and Li et al. (2013) extends the study to include positionspecific information.User costs are motivated by studies on "banner blindness" Benway and Lane (1998); Drèze and Hussherr (2003); hoan Cho and Cheon (2004) which show how user experience affects future user behavior with ads.

Preliminaries
Ad auctions with user costs.Following prior work on autobidding ad auctions (Aggarwal et al., 2019;Balseiro et al., 2021b;Deng et al., 2021;Mehta, 2022), we consider the following multiauction model.There are bidders [ ] = {1, 2, . . ., } and auctions [ ] = {1, 2, . . ., }.We generally use to index bidders, and to index auctions.In each auction , each bidder has a value , , as well as a user cost , .The user cost is suffered by the user who views the ad provided by bidder in auction when wins.As such, it does not affect the behavior of the bidder directly, but counts negatively towards the social welfare.We assume the value , is private, but the user cost , is public and can be directly observed by the auction mechanism.In each auction , each bidder submits a bid , , and all bidders' bids { ′ , } ′ and user costs { ′ , } ′ together determine each 's allocation , (i.e., , = 1 if wins in , and 0 otherwise) and payment , , in the way prescribed by the auction mechanism (we will discuss auction mechanisms momentarily).We omit the dependence of , and , on { ′ , } ′ when it is clear from the context.ROI-constrained value maximizers.We assume bidders are ROI-constrained value maximizers.That is, each bidder maximizes the total value receives in all auctions, subject to the constraint that the ratio between this total value and the total payment makes in all auctions is at least some target quantity.Following prior work (Balseiro et al., 2021b;Deng et al., 2022Deng et al., , 2021)), without loss of generality, we assume this target ratio is 1 for all bidders.Formally, each bidder solves the following optimization problem to decide how to bid in all auctions: We always assume there is a way of bidding such that the ROI constraint is satisfied (this is true for all quasilinear truthful auction mechanisms).
Quasilinear truthfulness and uniform bidding.In this paper, we focus on auction mechanisms that are quasilinear truthful.An auction mechanism is quasilinear truthful, if each bidder would be motivated to bid their true value if they were to maximize their quasilinear utility.In other words, for each and , fixing any , and { ′ , } ′ ≠ , the following holds for any possible bid : One example of quasilinear truthful mechanisms is the second-price auction: in each auction , the bidder with the highest bid wins, and the payment makes is the second highest bid.In addition, quasilinear truthful mechanisms enjoy the following nice property: without loss of generality, the optimal bidding strategy of an ROI-constrained value maximizer is always uniform bidding (Aggarwal et al., 2019).That is, for each bidder , there exists a bid multiplier ≥ 1, such that fixing other bidders' bids, the optimal (i.e., value-maximizing subject to the ROI constraint) bidding strategy of is to bid , = • , in each auction .Note that the bid multiplier is never smaller than 1, which in particular means no bidder should ever bid below their true value.In the rest of the paper, we only consider such bidding strategies, and sometimes use the corresponding bid multiplier to represent a bidding strategy.
Equilibria and the price of anarchy.To measure the efficiency of an auction mechanism, we consider the behavior of bidders in equilibrium under that mechanism.We say the bidding strategies { } form an equilibrium, if each bidder 's bidding strategy is a best response to all other bidders' strategies − = { ′ } ′ ≠ .Moreover, we use the price of anarchy (PoA) to quantify the efficiency of a mechanism.The PoA of a mechanism is the ratio between the worst-case welfare in equilibrium under that mechanism and the optimal welfare.Formally, fixing a mechanism, .
There are 2 things worth noting in the above definition: • If every bidder would contribute negatively to the social welfare if they were to win in a particular auction (i.e., max ( , − , ) < 0), then auction contributes 0 to the optimal welfare.
• In order for the PoA to be well defined, we always assume the optimal welfare is positive.

Challenges Posed by User Costs
To gain some intuition, we first briefly review existing approaches to autobidding auctions through quasilinear truthful mechanisms, particularly the second-price auction, and discuss why it fails in the presence of user costs.In a second-price auction, the bidder with the highest bid wins, and the payment of the winner is the second highest bid.It is known that the second-price auction has a PoA of 1/2 (Aggarwal et al., 2019) when (1) bidders are ROI-constrained value maximizers and (2) there is no user cost.Variants of the second-price auction has been analyzed in other cost-free settings, sometimes leading to improved efficiency guarantees (e.g., in settings with machine-learned advice (Balseiro et al., 2021b)).Nonetheless, in the core of these analyses lies the following conceptually simple argument: Theorem 1 ( (Aggarwal et al., 2019;Balseiro et al., 2021b;Deng et al., 2021)).Suppose there is no user cost.The second-price auction achieves a PoA of 1/2.
Proof.Suppose there is no user cost.For each auction , consider the "rightful winner" rw( ) who has the highest value in , i.e., rw( ) = argmax , (ties are broken arbitrarily).Since the secondprice auction is quasilinear truthful, in equilibrium, all bidders, including the rightful winner, must bid at least their own value, i.e., , ≥ , for all and .Therefore, in each auction , one of the following two cases must happen: (1) the rightful winner rw( ) wins, and collects value , from auction , or (2) the rightful winner rw( ) does not win, in which case the second highest bid is at least rw( ), = max , , and therefore the winner must pay at least this amount in auction .Summing over all auctions, the above reasoning implies that the sum of the following two quantities must be at least the optimal welfare: (1) the total value in all auctions collected by the respective rightful winners, and (2) the total payment made in all auctions by bidders other than the respective rightful winners.The first quantity clearly lower bounds the welfare in equilibrium.In fact, the second quantity is also a lower bound of the welfare: recall that the ROI constraints require that the total value each bidder receives should be at least the total payment that bidder makes, and summing over all bidders, one can see the second quantity also lower bounds the welfare in equilibrium.Now since the sum of the two quantities is at least the optimal welfare, at least one of the two must be no smaller than half of the optimal welfare, which means the welfare in equilibrium is no smaller than half of the optimal welfare.This gives a lower bound of 1/2 on the PoA of the second-price auction.
In words, the above argument lower bounds the total value collected in equilibrium and the total payment made in equilibrium, respectively, and use the larger one between the two as a lower bound of the welfare in equilibrium.Given this argument, it might appear that the most natural mechanism in the presence of user costs is the second price auction with "cost-adjusted bids".That is, bidders are sorted by the following adjusted bid: , − , (negative adjusted bids are discarded); the bidder with the highest adjusted bid wins, and pays the minimum bid to win (by Myerson's characterization (Myerson, 1981), this is the unique payment rule that guarantees quasilinear truthfulness). 1ow let us try to generalize the argument above to this new mechanism.For each auction , the rightful winner rw( ) is the one with the largest cost-adjusted value, i.e., rw( ) = argmax ( , − , ).Again there are two cases: (1) if the rightful winner rw( ) wins in auction , then the welfare collected by rw( ) is rw( ), − rw( ), ; (2) if the rightful winner does not win, then the second highest cost-adjusted bid must be at least max ( , − , ), so payment made by the winner (say * ) is at least * , + max ( , − , ).Now one might be tempted to apply the following (faulty) reasoning and argue the PoA is again at least 1/2: in case (1), the contribution of auction to the total value collected minus user cost incurred is rw( ), − rw( ), = max ( , − , ), which is the same as the contribution of to the optimal welfare.Similarly, in case (2), the contribution of to the total payment made minus user cost incurred is max ( , − , ), which is again the same as the contribution of to the optimal welfare.Now if we sum over all , then we know that the sum of (1) the total value minus user cost and (2) the total payment minus user cost is at least the optimal welfare, so the larger one is at least half of the optimal welfare, which gives a PoA lower bound of 1/2.
The problem of this reasoning is in the last step: when summing over , the user cost is only counted once, so what we actually get is: the sum of the total value and the total payment, minus the total user cost incurred, is at least the optimal welfare.This unfortunately does not give us any nontrivial bound on the welfare in equilibrium.
In fact, we will show in Section 4.2 that the second-price auction with cost-adjusted bids has a PoA of 0. Indeed, to handle user costs we need fundamentally new ideas and techniques.

Our Mechanisms
We present two quasilinear truthful mechanisms with provable efficiency guarantees in the presence of user costs.The first mechanism is anonymous, i.e., the outcome that any bidder receives is indepedent of that bidder's identity (or index).The second mechanism is auction-oblivious, i.e., it works in the same way across all auctions, and may treat different bidders in different ways.

Cost Multipliers
We first introduce the key idea behind both of our mechanisms, the use of cost multipliers.Instead of adjusting each bidder's bid by subtracting the user cost as is (as in the second-price auction with cost-adjusted bids discussed in the previous section), we apply a multiplier to the cost before performing the subtraction.We illustrate the power of this idea in the single-bidder setting, which also serves as a warm-up for our subsequent discussion.
Cost multipliers in the single-bidder setting.Suppose = 1.For each auction , let , , and be the only bidder's value, user cost, and bid in , respectively.Let be the set of auctions where the bidder's value is no smaller than the respective cost, i.e., = { | ≥ }.Then, the optimal welfare is simply ∈ ( − ).
Let ≥ 1 be the unique number such that ∈ = • ∈ .Consider the following mechanism: in each auction , the only bidder wins if the bid ≥ • , in which case the bidder pays the minimum bid to win, i.e., = • ; otherwise, no bidder wins and no payment happens.Observe that the bidder's optimal bidding strategy in response to this mechanism is to set their bid multiplier to exactly , such that = • in each .This is because when the bid multiplier is , the bidder wins in all auctions in and no other auctions, so the total value collected by the bidder is ∈ .On the other hand, in each ∈ , the payment made by the bidder is = • , so by the choice of , the total payment is ∈ • = ∈ , which is the same as the total value.In other words, when the bid multiplier is , the bidder's ROI constraint is binding.This means there is no way to win in more auctions without violating the ROI constraint, because the marginal ROI ratio of winning in any other auction is strictly smaller than 1/ ≤ 1.So, the only equilibrium is fully efficient, and the PoA of this mechanism in the single-bidder setting is 1.

Mechanisms with Global Cost Multipliers
However, it turns out that it is not enough to use a global cost multiplier.In particular, mechanisms with global cost multipliers can get arbitrarily worse PoA.
Theorem 2. For any ∈ (0, 1/3), there exists an instance in which any mechanism with a global cost multiplier gets PoA ≤ 3 .
Proof.Consider the instance with = ⌊1/ ⌋ and = 2 .Bidder only has non-zero values and costs in auction 2 − 1 and 2 .The values and costs of bidder are specified as the follow table: It is easy to check that the optimal welfare can be achieved by letting bidder only wins auction 2 , i.e.OPT = max 0, max ( , − , ) = .
Notice that ,2 ,2 = 1 + , For any mechanism with a global cost multiplier , if > 1 + , bidder will win nothing and contribute 0 to the welfare.And if ≤ 1 + +1 , bidder will win both auction 2 − 1 and auction 2 , and the overall welfare is + 1 Therefore, for each bidder ∈ [ ], its contribution to welfare is at most 2 if ∉ (1+ +1 , 1+ ], and at most 1 otherwise.Note that these intervals (1+ +1 , 1+ ] are disjoint, and thus the overall welfare of a mechanism with any global multiplier is at most Such an impossibility result implies that it is necessary to incorporate additional information into the design of cost multipliers to be effective.In Section 4.3 and Section 4.4, we show the existence of effective auction-dependent cost multipliers and bidder-dependent cost multipliers, respectively.The constructions of these cost multipliers in the existence proofs rely on knowing additional auction-specific and/or bidder-specific value information, which may not be accessible in a real system.One could consider applying -driven techniques to construct cost multipliers in practice, which is outside of the scope of this paper.
One may check that the above mechanism is in fact quasilinear truthful.There is one ambiguity in the mechanism: for an auction , if rw( ), = 0, then the mechanism sets = ∞, which may introduce undefined behavior.Although this corner case is unlikely to be a problem in practice, for consistency in theory, we use the following convention: if = ∞, then for any bidder , • , = ∞ if , > 0, and • , = 1 2 rw( ), if , = 0.One may check that with this convention, our analysis correctly handles infinite multipliers.
Analysis of the mechanism.The efficiency of the mechanism is captured by the following theorem.
Theorem 3. The mechanism with auction-dependent cost multipliers has a PoA of 1/2.
Proof.First observe that by Myerson's characterization (Myerson, 1981), the mechanism is quasilinear truthful, and so each bidder 's bid in each auction is at least their value, i.e., , ≥ , .Using this fact, we will lower bound the difference between the total payment made in all auctions and the total cost incurred in all auctions in equilibrium.That is, we lower bound the following quantity: Due to the ROI constraints, this is a lower bound of the welfare in equilibrium.Fix any equilibrium.We compare the contribution of each auction to the above difference against the contribution of the same auction to the optimal welfare.For each auction , consider the following 3 cases: • max ( , − , ) < 0 and no bidder wins.In this case, the payment made and the cost incurred in are both 0. On ther other hand, the contribution of to the optimal welfare is also 0. So the contribution of to the difference is (trivially) at least 1/2 of the contribution of to the optimal welfare.
• rw( ) wins in auction , i.e., 1 ( ) = rw( ).In this case, by the choice of , the payment that rw( ) makes is at least The cost incurred in is rw( ), , so the contribution of to the difference is at least 1 2 ( rw( ), − rw( ), ).On the other hand, the contribution of to the optimal welfare is rw( ), − rw( ), , and the ratio between the contribution to the difference and that to the optimal welfare is at least 1/2.
• A bidder other than the rightful winner wins.In this case, we know the second highest score is at least the score of the rightful winner rw( ).Since the rightful winner bids at least their value, and by the choice of , we have Then the payment of the winner is at least The cost incurred in is 1 ( ), , so the contribution of to the difference is at least In other words, the contribution of to the difference is at least 1/2 of the contribution of to the optimal welfare.
Summarizing the 3 cases, we see that the contribution of each auction to the difference between the total payment and the total cost is at least 1/2 of the contribution of the same auction to the optimal welfare.As a result, the difference between the total payment and the total cost is at least 1/2 of the optimal welfare.By the ROI constraints of the bidders, the former quantity is a lower bound of the welfare in equilibrium, which gives a lower bound of 1/2 on the PoA.

Bidder-Dependent Cost Multipliers
We now proceed to show the existence of bidder-dependent (and auction-independent) cost multipliers with a PoA of 1/4.The mechanism.For each bidder , let be the set of auctions where is the rightful winner (as defined in the previous subsection), i.e., = { | rw( ) = }.Note that by definition, for each and ∈ , , − , ≥ 0. For each bidder , let ≥ 0 be the unique number such that In each auction , first discard any bidder where , − (1+ ) • , < 0. If all bidders are discarded, then no bidder wins in auction .Otherwise, let each bidder 's score be the cost-adjusted bid , − , and sort all remaining bidders by their scores.The bidder among the remaining ones with the highest score, 1 ( ), wins, and pays the minimum bid to win.That is, if 1 ( ) is the only remaining bidder, then 1 ( ) pays 1 ( ), = (1 + 1 ( ) ) • 1 ( ), .Otherwise, let 2 ( ) be the bidder among the remaining ones with the second highest score. 1 ( ) pays 1 ( ), = max{(1 + 1 ( ) ) • 1 ( ), , 2 ( ), − 2 ( ), + 1 ( ), }.Again, one can check the above mechanism is in fact quasilinear truthful.We also note that this mechanism, unlike our anonymous mechanism, uses different quantities to prescreen and sort bidders.In particular, the quantity used for prescreening is no larger than the score used for sorting.This is necessary for the mechanism to provide the desired efficiency guarantee.
Analysis of the mechanism.The efficiency of the mechanism is captured by the following theorem.
Theorem 4. The mechanism with bidder-dependent cost multipliers has a PoA of 1/4.The proof of the above theorem is fairly involved.We first introduce some useful notions and properties.For each bidder , let 's core auctions be the subset of where 's value is at least (1 + ) times 's user cost, i.e., We will use the following fact regarding : This establishes the claim.
Below we consider each bidder separately, and show that the contribution of auctions in to + is at least half of the contribution of the same auctions to the optimal welfare, i.e., ∈ ( , − , ).First consider each ∈ agg .In this case, each ∈ has a winner.This is because the bid , of in satisfies , = • , ≥ (1 + ) • , ≥ (1 + ) • , , so at least one bidder (i.e., ) passes the prescreening stage of the mechanism.Also, observe that each ∈ agg contributes nothing to .So we only need to argue that the total contribution of all auctions in to is at least 1/2 of the contribution of the same auctions to the optimal welfare.In fact, if wins in auction , then the payment makes in is at least (1 + ) • , , so the contribution to is at least • , .Summing over ∈ and by the choice of , the total contribution to (and therefore to + ) is at least which is 1/2 of the contribution to the optimal welfare.Now consider each ∈ con .For such a bidder , only auctions in may contribute to + .Observe that each ∈ has a winner.This is becasue the bid , is at least , , which by the choice of satisfies , ≥ (1 + ) • , .So , ≥ (1 + ) • , , and at least one bidder (i.e., ) passes the prescreening stage of the mechanism.Below we argue that the contribution of each ∈ to + is at least the contribution of the same auction to the optimal welfare.If wins in , then contributes to , and the contribution is precisely , − , , which is the same as the contribution of to the optimal welfare.If does not win in , then the score of 2 ( ) is at least the score of , i.e., 2 ( ), − 2 ( ), ≥ , − , ≥ , − , .This means the payment made by the winner, 1 ( ) must satisfy 1 ( ), ≥ 2 ( ), − 2 ( ), + 1 ( ), ≥ , − , + 1 ( ), .
So the contribution of to is at least , − , , which is the contribution of to the optimal welfare.Summing over ∈ , we see that the total contribution is at least This gives a lower bound of 1/4 on the PoA of the mechanism.

Conclusion
In this paper, we study the impact of user costs in autobidding auctions.We show that running vanilla VCG may result in poor performance.To improve the performance of VCG, we propose new variants with cost multipliers and provably show that they can guarantee constant approximations in terms of social welfare.For future works, as our paper examines one possibility to improve VCG by cost multipliers, it is interesting to explore other variants of VCG that may improve the performance.It is also very interesting to consider bridging the gap between theory and practice by designing practical mechanisms inspired by our theoretical results.
contribution of auctions in to the optimal welfare.Now we can conclude the proof by observing , , • ( , − , ) ≥ max{ ,