Block 2 · Auction Theory in the Autobidding World

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Time budget

~90 minutes. Read Aggarwal-Badanidiyuru-Mehta ^[1] first - it is short and everything else is a reaction to it. Then the value-vs-utility paper ^[2] and Liaw-Mehta-Perlroth ^[6] for the price-of-anarchy landscape. Skim the pacing-equilibrium papers ^[9][10] for the existence and uniqueness statements only. Read Golrezaei-Lobel-Paes Leme ^[20] against Balseiro et al. ^[4]. Keep the 2024 survey ^[15] open as a map and return to Balseiro-Gur ^[11] in Block 4.

The premise shifted under the theory §

Classical auction theory, including everything in Block 1, models a bidder as a utility maximizer: an agent with a private value $v$ per click who wants to maximize $v\cdot (\text{clicks})-(\text{payment})$. That is not what bids on Amazon Sponsored Products in 2026. A brand sets a daily budget and a target ACOS (advertising cost of sales, the inverse of ROAS) and hands the bidding to an algorithm - Amazon’s own dynamic bidding, a third-party tool, or an in-house system. The algorithm’s objective is to maximize conversions or sales subject to the budget and the target ratio. It does not have a per-click value in the classical sense. It has a constraint.

Google’s auction theorists noticed this around 2019, named the resulting agents autobidders, and asked the obvious question: what happens to the equilibrium and efficiency results when the bidders are constraint-satisfying optimizers rather than utility maximizers? The answer, developed over 2019-2024 and mapped in a 2024 survey with twenty-six authors ^[15], is that a surprising amount of the classical theory breaks - VCG stops being efficient, first-price and second-price trade places, randomization beats determinism - and that the new results are directly about the agents an Amazon advertiser is running or competing against.

1. Value maximizers and the uniform-scaling result §

The model §

Aggarwal, Badanidiyuru and Mehta ^[1] formalize autobidding with general affine constraints. A bidder participates in many auctions $j$, has value $v_{ij}$ for winning auction $j$, and wants to maximize total value ${\sum }_j{x}_{ij}{v}_{ij}$ subject to constraints of the form ${\sum }_j{x}_{ij}(a_{ij}-c_{ij}{p}_j)\le 0$. A budget is one such constraint; a target ROAS (spend $\le$ value / target) is another. The paper proves that in truthful auctions the optimal single-agent strategy has a strikingly simple form: bid $v_{ij}$ scaled by a uniform multiplier per constraint, $b_{ij}=\lambda \cdot v_{ij}$ for a budget-like constraint, and more generally a bid that is a fixed affine function of the value determined by the Lagrange multipliers of the binding constraints ^[1].

Read that once more, because it justifies a large fraction of practice. A budget-constrained bidder should not shade different auctions differently. It should bid its value everywhere, times one number, and tune the number until the budget clears. Every pacing controller in production is an algorithm for finding $\lambda$. Babaioff, Cole, Hartline, Immorlica and Lucier ^[18] supply the microfoundation from the other direction: drop a non-quasi-linear agent - one with an ROI constraint - into a truthful quasi-linear mechanism, and its best response is to scale every value by one constant. The uniform multiplier is not a modeling convenience; it is what the constrained agent would do anyway.

There is a companion result that reorganizes Block 1: Wilkens, Cavallo and Niazadeh ^[19] prove that for value maximizers, GSP - not VCG - is the truthful mechanism. The auction Amazon says it runs is incentive-compatible for the agents Amazon’s advertisers actually run.

The price of anarchy is one half §

The paper’s second result is the one that launched the literature: when every advertiser is such an autobidder, an equilibrium exists, and total advertiser value at equilibrium is at least half of the value a central planner could achieve ^[1]. This bound - a price of anarchy (PoA) of 2, or “1/2 efficiency” depending on convention - is tight even for VCG, which for utility maximizers is fully efficient. The intuition: two autobidders each with a ROAS constraint can both scale their bids up until the constraint binds, and end up in an allocation where the wrong one wins half the time because each is spending its slack on the other’s best auctions.

What this means for a Sponsored Products bidder with a daily budget and a target ACOS. Your optimal policy really is “value times a multiplier”; the hard part is the value ($p\text{CTR}\cdot p\text{CVR}\cdot \text{margin}$, Block 3) and the multiplier dynamics (Block 4), not the bid formula. And the market you are in can be leaving up to half its total value on the table through no fault of yours.

Section takeaway. For an agent maximizing value under budget and ROAS constraints, the optimal bid is value times a per-constraint multiplier, and a market of such agents can be only half-efficient even under VCG.

2. Value versus utility, and who the platform is designing for §

Balseiro, Deng, Mao, Mirrokni and Zuo ^[2] put the two bidder types side by side and ask what the revenue-optimal mechanism is for each. Their main results are about private information. When bidders are value maximizers with a target ratio, the platform can extract first-best revenue if either the values or the target ratios are private - but not when both are. For utility maximizers, first-best is never achievable and the paper characterizes the revenue-optimal mechanism instead ^[2]. The design problem is therefore not one problem. A platform that assumes its advertisers are utility maximizers when they are in fact running ROAS-constrained autobidders is solving the wrong optimization, and an autobidder that assumes the platform is treating it as a utility maximizer will be surprised by the reserves it meets.

Deng, Mao, Mirrokni and Zuo ^[3] then propose the first constructive fix for the PoA of 2: boosts. If the platform adds to each autobidder’s bid an amount proportional to the platform’s own signal about that bidder’s value, the equilibrium allocation improves, and the paper reports both theory and experiments showing that appropriately chosen boost weights raise welfare and revenue simultaneously ^[3]. Boosts are, mechanically, the platform putting its thumb on the ranking score - the same object as the quality weight $q$ in Block 1, now justified as an efficiency correction rather than a relevance correction.

Balseiro et al. ^[4] look at reserve prices from the same angle. In the autobidding world, appropriately chosen reserves improve both revenue and welfare, and - the robustness result - the improvement holds whether the bidders turn out to be value maximizers or utility maximizers, so the platform does not need to know which type it faces. The analysis covers VCG, GSP and first-price ^[4]. Put this next to Amazon’s soft reserves in Block 1: whatever the legal characterization, a reserve set from the platform’s predicted ROAS is exactly the object this theory says a welfare-maximizing platform facing autobidders should use.

Hold the reserve result against a field experiment that points the other way. Golrezaei, Lobel and Paes Leme ^[20] studied ROI-constrained buyers on Google’s AdX and found that when reserves rise, these buyers lower their bids - the constraint tightens, the multiplier falls - and that the revenue-optimal mechanism for such buyers has reduced reserves, or even subsidies. Balseiro et al. ^[4] model the same bidder type and conclude reserves help. The difference is what the reserve is set against: a reserve tuned to the platform’s value signal (Balseiro) versus a uniform reserve met by a binding-ROI buyer who responds by pacing down (Golrezaei). Both can be right, and an advertiser’s ACOS target is exactly the lever that decides which regime it is in. Deng, Golrezaei, Jaillet, Liang and Mirrokni ^[21] close the loop: using machine-learned value predictions as personalized reserves yields individual welfare guarantees for each autobidder, not just aggregate ones, with the guarantee improving as the prediction improves.

What this means for you. The platform is not a neutral clearing house. It has both an efficiency argument and a revenue argument for boosting some bids and flooring others based on its prediction of your value. Your bid competes with the platform’s model of your bid - and how you respond to its floors changes what floor is optimal for it to set.

Section takeaway. Which bidder type the platform designs for changes the optimal mechanism; boosts and reserves both restore efficiency lost to autobidding, and both amount to the platform reweighting your bid by its own prediction of your value.

3. Randomization beats determinism §

Here is the result most at odds with classical intuition. Mehta ^[5] asked whether any auction can beat VCG’s PoA of 2 for autobidders and showed that randomized allocation can. Liaw, Mehta and Perlroth ^[6] then charted the whole space in the prior-free setting. Three theorems: non-truthfulness buys nothing for deterministic mechanisms; every deterministic mechanism has PoA at least 2, even with two bidders; and first-price has PoA exactly 2. Then the constructive part: a randomized, non-truthful auction with PoA 1.8 for two bidders. And a limit: no prior-free auction can beat the PoA-2 bound as the number of advertisers grows ^[6]. The WWW 2023 version extends the characterization across the spectrum of auction formats and confirms that randomization’s advantage is a small-bidder-count phenomenon ^[7].

Why does randomness help? A deterministic auction gives the highest-scored autobidder the slot with certainty, so two autobidders with slack in their constraints escalate against each other on their contested auctions. A randomized allocation smooths the payoff, breaks the escalation, and lets both spend their slack more evenly across auctions. It is the auction-theory version of mixed strategies stabilizing a game. The Gumbel-noise mechanism from Amazon’s auction group in Block 1 is one concrete instantiation.

What this means for you. On queries with few serious competitors - the long tail again - you may be facing a stochastic allocation, and “I bid more and lost” is no longer a contradiction. A bidder that treats each auction outcome as a deterministic signal of the competitor’s bid will mislearn the landscape.

Section takeaway. For autobidders, no deterministic auction beats half-efficiency, randomized auctions can (to 1.8 for two bidders), and the advantage vanishes in thick markets.

4. First price, mixed populations, and machine-learned advice §

Google Ad Manager moved its display exchange to a unified first-price auction in 2019 ^[16], and the industry followed. That is a display-side event, not evidence about Amazon’s Sponsored Products rule, but the theory it prompted is general. Deng, Mao, Mirrokni, Zhang and Zuo ^[8] (NeurIPS 2024) analyze first-price auctions with a mixed population: some traditional utility maximizers, some autobidders. With autobidders alone the PoA is 1/2 (efficiency at least half). With both types the guarantee drops to about 0.457. Then the interesting part: if the seller supplies machine-learned advice - a signal about each bidder’s value - and bidders incorporate it, the efficiency guarantee improves smoothly from 0.457 toward 1 as the advice becomes accurate ^[8]. Accurate platform models are literally a welfare instrument. Add budgets and the picture darkens: Liaw, Mehta and Zhu ^[22] show that with budget constraints the first-price PoA is $n$ in general - as bad as the number of bidders - and 2 only when each bidder’s value is at most its budget, with randomized first-price again reaching 1.8 for two bidders.

Why did display markets move to first price at all? Paes Leme, Sivan and Teng ^[23] and Despotakis, Ravi and Sayedi ^[24] give the same answer from theory and from marketing science: inter-exchange competition. When several exchanges compete for the same publisher inventory, second-price exchanges are undercut by first-price ones, and the market converges on first price. Google’s 2019 announcement explicitly exempted Search ^[16]. Amazon sells its own inventory on its own results page, so the force that moved display does not act on Sponsored Products; whatever first-price behavior an Amazon bidder meets comes from reserves, not from format.

First-price also created the bid shading industry. In a second-price auction you bid your value; in a first-price auction you must decide how far below value to bid, which requires predicting the competitive threshold. Gligorijevic et al. ^[13] at Yahoo framed this as a supervised problem on non-censored logs (where the winning price is observed) and built a production shading model; Pan et al. ^[25] framed it as win-rate estimation plus surplus maximization; Zhou et al. ^[26] replaced the point estimate with a learned distribution over the minimum winning price and reported ROI gains of 2.4%, 2.4% and 8.6% on CPM, CPC and CPA campaigns. Wang, Yang, Deng and Kong ^[27] give the learning theory for a budgeted bidder in repeated first-price auctions. Qu and Kan ^[14] address the two sources of uncertainty a shader faces - the value of the opportunity and the distribution of competing bids - with a doubly distributionally robust formulation, which is the right response when the landscape shifts under the model. No fetchable Amazon page documents bid shading in Amazon DSP, so treat vendor claims about it as unverified. And the shading problem is not confined to first-price exchanges: under Amazon’s soft-reserve rule from Block 1, a bidder pays its own bid whenever it lands below the soft reserve, so an Amazon bidder is in a partially first-price auction whether or not it knows it.

The price-of-anarchy landscape in one table §

Result	Setting	Bidders	Efficiency guarantee	Source
GSP	Full information / Bayesian	Quasi-linear utility maximizers (incl. no-regret learners)	PoA $\le 1.282$ / $\le 2.927$	Caragiannis et al. [28]
VCG / any truthful auction	Truthful, deterministic	Value maximizers with affine constraints	$\ge 1/2$ (PoA 2), tight	Aggarwal et al. [1]
Any deterministic mechanism	Prior-free	Autobidders, even $n=2$	PoA $\ge 2$	Liaw et al. [6]
First-price	Prior-free, deterministic	Autobidders only	PoA exactly 2	Liaw et al. [6], Deng et al. [8]
First-price	Prior-free	Mixed utility + value maximizers	$\approx 0.457$	Deng et al. [8]
First-price + ML advice	Advice of accuracy $\to 1$	Mixed	$0.457\to 1$ smoothly	Deng et al. [8]
Randomized non-truthful	Prior-free, $n=2$	Autobidders	PoA 1.8	Liaw et al. [6][7]
First-price with budgets	Deterministic	Budgeted autobidders	PoA $=n$ in general; 2 if value $\le$ budget; randomized 1.8 for $n=2$	Liaw, Mehta, Zhu [22]
Throttling (vs pacing)	First or second price	Budgeted bidders	Liquid-welfare PoA 2, tight	Chen, Kroer, Kumar [29]
Any prior-free auction	$n\to \infty$	Autobidders	Cannot beat PoA 2	Liaw et al. [6]
VCG + boosts	Platform has value signal	Autobidders with ROAS + budget	Improves welfare and revenue	Deng et al. [3]
Gradient autobidders, any format	Repeated, bandit feedback	Budget + ROI constrained	Liquid welfare $\ge 1/2$ of optimum, no convergence needed	Lucier et al. [12]

Section takeaway. First-price is no worse than second-price for autobidders in the worst case, mixed populations make things worse, accurate platform-side prediction makes things better, and shading is a live problem for any bidder who can end up paying its own bid.

5. Budgets over time: pacing as an equilibrium object §

Everything above is static. Real budgets are daily, real auctions arrive over the day, and the multiplier $\lambda$ from Section 1 has to be found online. This is pacing, and the theory says pacing is not a delivery heuristic. It is an equilibrium concept.

Multiplicative pacing equilibria §

Conitzer, Kroer, Sodomka and Stier-Moses ^[9] define a multiplicative pacing equilibrium in second-price markets: each bidder has a multiplier ${\alpha }_i\in [0,1]$ scaling all its bids, and in equilibrium every bidder either exhausts its budget exactly or has ${\alpha }_i=1$ (is unconstrained). They prove existence, and then the uncomfortable part: markets can have multiple pacing equilibria with substantially different welfare, revenue and allocations; finding the welfare- or revenue-maximizing one is NP-hard; and bidders can have incentives to misreport bids or budgets ^[9]. The arXiv version dates from 2017; the Operations Research publication is 2022.

Pacing by multiplier is not the only way to spend a budget. Throttling - randomly sitting out a fraction of auctions - is the other, and Chen, Kroer and Kumar ^[29] analyze throttling equilibria: the liquid-welfare PoA is 2 in both first and second price, tight; the first-price throttling equilibrium is unique and reachable by tâtonnement, while computing the second-price one is PPAD-complete. Amazon’s own “not paced through the day, capped at 30x over the month” rule ^[30] is closer to throttling than to a multiplier, which is one reason external tools layer a pacer on top.

First-price pacing is better behaved §

Conitzer, Kroer, Panigrahi, Schrijvers, Sodomka, Stier-Moses and Wilkens ^[10] repeat the exercise for first-price markets and get a cleaner theory. The first-price pacing equilibrium is unique, monotone in the inputs, has desirable properties, and can be computed efficiently as the solution of an Eisenberg-Gale convex program - the same object that computes Fisher-market equilibria. On realistic instances, bidders have small regret relative to optimal ex-post strategies and little incentive to misreport, with budget-constrained bidders showing the smaller regret ^[10]. This is a second, independent reason the industry could move to first price without chaos: the pacing dynamics are stabler.

Two results from Balseiro and coauthors sit underneath both. Balseiro, Besbes and Weintraub ^[31] introduced the fluid mean-field equilibrium for repeated auctions with budgets in ad exchanges: budgets make bidders shade, and a platform that sets reserves ignoring budgets loses profit. Balseiro, Kroer and Kumar ^[32] then proved a revenue-equivalence theorem for the budgeted world: in pacing equilibrium, all standard auction formats raise the same revenue. The format debate of Section 4 is, in the long run and with budgets binding, a debate about dynamics rather than about the level of revenue.

Adaptive pacing from the bidder’s side §

Balseiro and Gur ^[11] model the individual budget-constrained advertiser facing a stream of auctions with uncertainty about the future, and give an adaptive pacing strategy: adjust the multiplier using the observed expenditure path, a dual-descent on the budget constraint. Against arbitrary competitor bids the strategy is asymptotically optimal - it achieves vanishing regret relative to the best fixed multiplier in hindsight. When every bidder uses it, the joint dynamics converge to a tractable steady state that constitutes an approximate Nash equilibrium ^[11]. This is the theoretical spine of every multiplicative pacing controller in production, including the ones catalogued in the ICML 2024 “field guide” mentioned in the map note. Feng, Padmanabhan and Wang ^[33] give the return-on-spend analogue: a primal-dual controller for an RoS-constrained advertiser with near-optimal regret that never violates the constraint - the algorithmic form of a target-ACOS bidder.

Regret and liquid welfare without convergence §

Lucier, Pattathil, Slivkins and Zhang ^[12] take the most realistic setup in the block: autobidders with both budget and ROI constraints, learning by gradient methods under bandit feedback (they see only their own outcomes), across first-price, second-price and everything between. Their algorithm satisfies the constraints and achieves vanishing individual regret, and - the headline - expected liquid welfare is at least half the optimum even if the dynamics never converge to equilibrium ^[12]. This is the guarantee an engineer actually wants, because production bidders never converge; competitors’ budgets, values and models change daily.

Liquid welfare, in one paragraph

Ordinary welfare sums the values of the winning bidders. With budgets, that is misleading: a bidder with a $10 budget who “wins” $1,000 of value cannot pay for it, so the value was never really available to the market. Liquid welfare caps each bidder’s contribution at its budget: ${\sum }_i\min ({\text{budget}}_i,{\text{value won}}_i)$. It is the natural efficiency measure when bidders are budget-constrained, it is what the pacing and autobidding papers bound, and it is why a “PoA 2” statement in this literature is a statement about liquid, not raw, welfare.

Deng, Golrezaei, Jaillet, Liang and Mirrokni ^[17] extend the constraint structure to multiple channels - think Sponsored Products plus Sponsored Brands plus DSP. Their result is a warning: per-channel ROI targets can make total conversions arbitrarily worse than a global target, while per-channel budget optimization can attain the global optimum in their model ^[17]. Constraints should be set where the objective lives.

What this means for you. Your daily budget makes you a participant in a pacing game whose equilibrium may not be unique (second price) or may be unique and computable (first price). A dual-descent pacer is near-optimal against anything. And if you run several Amazon ad types with separate ACOS targets, the theory says you are probably leaving conversions on the table.

Section takeaway. Pacing is an equilibrium problem: second-price markets can have many, first-price markets have one computable one, adaptive multiplier control is near-optimal for the individual, and gradient autobidders guarantee half of liquid welfare without ever converging.

6. Learning dynamics and the agency problem §

Everything above assumes bidders reach equilibrium. Feng, Guruganesh, Liaw, Mehta and Sethi ^[34] ask what mean-based no-regret learners - the algorithms an autobidder actually runs - converge to, and the answer splits by format: in second price and VCG they converge to truthful bidding; in first price they converge to the Bayes-Nash shading equilibrium. So learning agents do recover the textbook predictions, format by format. Block 6 covers the dynamics in depth, including the cases where they do not converge at all.

The uncomfortable corollary is that agents who learn together can learn to coordinate. Decarolis, Goldmanis and Penta ^[35] show that a common agency - one intermediary bidding for several advertisers, which is what every third-party Amazon tool is - can exploit GSP more than VCG by coordinating its clients’ bids. Decarolis and Rovigatti ^[36] measure it: on roughly 40 million Google auctions, growing agency concentration cut platform revenue by about 11%. Guan, Zhang, Feng and Lin ^[37] bring this to autobidders: stand-aside coordination, where RoS-constrained autobidders under one agent take turns, is provably profitable. An advertiser choosing a bidding tool is choosing whose portfolio its bids are coordinated with.

Section takeaway. No-regret learners recover the format-specific equilibria, and an agency bidding for many clients can coordinate them - profitably and, in the autobidding case, provably.

7. What the theory does not cover §

Three gaps, each picked up later in the day. The theory models competitors as either static or as identical learners; when they are heterogeneous learned agents the dynamics are an open problem, and the algorithmic-collusion evidence from Block 6 suggests the outcomes can be worse than any equilibrium. The theory takes values $v_{ij}$ as known; in practice they are model outputs with calibration error, and Block 3 is about how large that error is. And the theory treats the mechanism as fixed while the platform is optimizing it against you (Block 1, Section 4). Where the bidder is a learned agent and the mechanism is a learned agent, the right frame is a game between models, which is where Block 5 comes in.

Reading list for this block §

1. Aggarwal, Badanidiyuru, Mehta - Autobidding with Constraints, WINE 2019. 20 minutes. The model, the uniform-scaling theorem, the PoA-2 example.
2. Balseiro, Deng, Mao, Mirrokni, Zuo - The Landscape of Auto-bidding Auctions: Value versus Utility Maximization, EC 2021. 15 minutes. Introduction and the table of first-best results.
3. Liaw, Mehta, Perlroth - Efficiency of Non-Truthful Auctions in Auto-bidding: The Power of Randomization, WWW 2023. 15 minutes. Theorem statements; skip proofs.
4. Deng, Mao, Mirrokni, Zhang, Zuo - Efficiency of the First-Price Auction in the Autobidding World, 2022. 10 minutes. The 0.457 result and the ML-advice section.
5. Conitzer, Kroer, Sodomka, Stier-Moses - Multiplicative Pacing Equilibria in Auction Markets, OR 2022. 10 minutes. Definitions, existence, multiplicity example.
6. Lucier, Pattathil, Slivkins, Zhang - Autobidders with Budget and ROI Constraints, COLT 2024. 10 minutes. Setup and the liquid-welfare theorem.
7. Aggarwal et al. - Auto-bidding and Auctions in Online Advertising: A Survey, 2024. 10 minutes, as a map. Use its taxonomy to place every other paper.
8. Golrezaei, Lobel, Paes Leme - Auction Design for ROI-Constrained Buyers, WWW 2021. 10 minutes. The AdX field-experiment section, then the reduced-reserve result; hold it against reading 2 of Block 1’s reserve story.
9. Decarolis, Rovigatti - From Mad Men to Maths Men, AER 2021. 5 minutes. The 11% headline and the identification strategy.
10. Balseiro, Gur - Learning in Repeated Auctions with Budgets, Mgmt Sci 2019. Reference; re-read in Block 4.

Questions to carry forward §

• Amazon’s soft reserve is set from predicted ROAS. Is that a Deng-et-al. boost, a Balseiro-et-al. robust reserve, or something the theory has not modeled? What would distinguish them empirically?
• The uniform-scaling result assumes a truthful auction. Under quality-weighted GSP with reserves, is “value times one multiplier” still optimal, or does the denominator $q$ reintroduce per-auction shading?
• Randomization helps with two bidders and not with many. On which of your keywords are there two serious bidders, and does the win/loss pattern there look stochastic?
• If per-channel ROI targets are provably wasteful, why does every advertising dashboard expose them? What would a single-objective, multi-channel control surface look like?
• Golrezaei et al. found ROI-constrained buyers bid down when reserves rise. If Amazon’s soft reserve is set from predicted ROAS and your tool responds by lowering the multiplier, who wins the resulting dynamic - and does Amazon’s month-level budget cap change the answer?
• Every third-party Amazon bidding tool is a common agency. Should you expect it to coordinate you with its other clients, and what would that look like in your win rates?
• Lucier et al. guarantee half of liquid welfare without convergence. Can you measure how far a live market is from that bound, or is the bound only useful as a worst case?

References §

1. Aggarwal, Badanidiyuru, Mehta. Autobidding with Constraints. WINE 2019. springer
2. Balseiro, Deng, Mao, Mirrokni, Zuo. The Landscape of Auto-bidding Auctions: Value versus Utility Maximization. EC 2021. dl.acm.org
3. Deng, Mao, Mirrokni, Zuo. Towards Efficient Auctions in an Auto-bidding World. WWW 2021. arXiv:2103.13356, dl.acm.org
4. Balseiro, Deng, Mao, Mirrokni, Zuo. Robust Auction Design in the Auto-bidding World. NeurIPS 2021. arXiv:2111.02468, neurips.cc
5. Mehta. Auction Design in an Auto-bidding Setting: Randomization Improves Efficiency Beyond VCG. WWW 2022. arXiv:2204.10956, dl.acm.org
6. Liaw, Mehta, Perlroth. Efficiency of Non-Truthful Auctions under Auto-bidding. 2022. arXiv:2207.03630
7. Liaw, Mehta, Perlroth. Efficiency of Non-Truthful Auctions in Auto-bidding: The Power of Randomization. WWW 2023. dl.acm.org
8. Deng, Mao, Mirrokni, Zhang, Zuo. Efficiency of the First-Price Auction in the Autobidding World. NeurIPS 2024. arXiv:2208.10650
9. Conitzer, Kroer, Sodomka, Stier-Moses. Multiplicative Pacing Equilibria in Auction Markets. Operations Research 70(2), 2022 (arXiv 2017). arXiv:1706.07151, ideas.repec.org
10. Conitzer, Kroer, Panigrahi, Schrijvers, Sodomka, Stier-Moses, Wilkens. Pacing Equilibrium in First-Price Auction Markets. EC 2019; Management Science 2022. arXiv:1811.07166
11. Balseiro, Gur. Learning in Repeated Auctions with Budgets: Regret Minimization and Equilibrium. Management Science 65(9), 2019. doi.org/10.1287/mnsc.2018.3174
12. Lucier, Pattathil, Slivkins, Zhang. Autobidders with Budget and ROI Constraints: Efficiency, Regret, and Pacing Dynamics. COLT 2024. arXiv:2301.13306
13. Gligorijevic, Zhou, Shetty, Kitts, Pan, Pan, Flores. Bid Shading in The Brave New World of First-Price Auctions. CIKM 2020. arXiv:2009.01360
14. Qu, Kan. Double Distributionally Robust Bid Shading for First Price Auctions. 2024 (author list per arXiv record; verify before formal citation). arXiv:2410.14864
15. Aggarwal et al. (26 authors). Auto-bidding and Auctions in Online Advertising: A Survey. 2024. arXiv:2408.07685, dl.acm.org
16. Google. An Update on First Price Auctions for Google Ad Manager. May 2019. blog.google
17. Deng, Golrezaei, Jaillet, Liang, Mirrokni. Multi-channel Autobidding with Budget and ROI Constraints. 2023. arXiv:2302.01523
18. Babaioff, Cole, Hartline, Immorlica, Lucier. Non-quasi-linear Agents in Quasi-linear Mechanisms. ITCS 2021. arXiv:2012.02893
19. Wilkens, Cavallo, Niazadeh. GSP: The Cinderella of Mechanism Design. WWW 2017. doi.org/10.1145/3038912.3052687
20. Golrezaei, Lobel, Paes Leme. Auction Design for ROI-Constrained Buyers. WWW 2021. doi.org/10.1145/3442381.3449841
21. Deng, Golrezaei, Jaillet, Liang, Mirrokni. Individual Welfare Guarantees in the Autobidding World with Machine-Learned Advice. WWW 2024. arXiv:2209.04748
22. Liaw, Mehta & Zhu. Efficiency of Non-Truthful Auctions in Auto-bidding with Budget Constraints. WWW 2024. arXiv:2310.09271
23. Paes Leme, Sivan, Teng. Why Do Competitive Markets Converge to First-Price Auctions? WWW 2020. doi.org/10.1145/3366423.3380142
24. Despotakis, Ravi, Sayedi. First-Price Auctions in Online Display Advertising. Journal of Marketing Research 2021. doi.org/10.1177/00222437211030201
25. Pan et al. Bid Shading by Win-Rate Estimation and Surplus Maximization. AdKDD 2020. arXiv:2009.09259
26. Zhou et al. An Efficient Deep Distribution Network for Bid Shading in First-Price Auctions. KDD 2021. arXiv:2107.06650
27. Wang, Yang, Deng, Kong. Learning to Bid in Repeated First-Price Auctions with Budgets. ICML 2023. proceedings.mlr.press
28. Caragiannis, Kaklamanis, Kanellopoulos, Kyropoulou, Lucier, Paes Leme, Tardos. Bounding the Inefficiency of Outcomes in Generalized Second Price Auctions. Journal of Economic Theory 2015. doi.org/10.1016/j.jet.2014.04.010
29. Chen, Kroer, Kumar. Throttling Equilibria in Auction Markets. WINE 2021. arXiv:2107.10923
30. Amazon Ads. Best Practices for Your Sponsored Products Ads (daily budgets not paced; monthly cap). advertising.amazon.com
31. Balseiro, Besbes, Weintraub. Repeated Auctions with Budgets in Ad Exchanges: Approximations and Design. Management Science 2015. doi.org/10.1287/mnsc.2014.2022
32. Balseiro, Kroer, Kumar. Contextual Standard Auctions with Budgets: Revenue Equivalence and Efficiency Guarantees. EC 2022; Management Science 2023. arXiv:2102.10476
33. Feng, Padmanabhan, Wang. Online Bidding Algorithms for Return-on-Spend Constrained Advertisers. WWW 2023. arXiv:2208.13713
34. Feng, Guruganesh, Liaw, Mehta, Sethi. Convergence Analysis of No-Regret Bidding Algorithms in Repeated Auctions. AAAI 2021. arXiv:2009.06136
35. Decarolis, Goldmanis, Penta. Marketing Agencies and Collusive Bidding in Online Ad Auctions. Management Science 2020. doi.org/10.1287/mnsc.2019.3457
36. Decarolis, Rovigatti. From Mad Men to Maths Men: Concentration and Buyer Power in Online Advertising. American Economic Review 2021. doi.org/10.1257/aer.20190811
37. Guan, Zhang, Feng & Lin. On the Coordination of Value-Maximizing Bidders. ICML 2026 (per arXiv). arXiv:2511.04993

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