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In recent years, the growing adoption of autobidding has motivated the study of auction design with value-maximizing auto-bidders. It is known that under mild assumptions, uniform bid-scaling is an optimal bidding strategy in truthful auctions, e.g., Vickrey-Clarke-Groves auction (VCG), and the price of anarchy for VCG is $2$. However, for other auction formats like First-Price Auction (FPA) and Generalized Second-Price auction (GSP), uniform bid-scaling may not be an optimal bidding strategy, and bidders have incentives to deviate to adopt strategies with non-uniform bid-scaling. Moreover, FPA can achieve optimal welfare if restricted to uniform bid-scaling, while its price of anarchy becomes $2$ when non-uniform bid-scaling strategies are allowed. All these price of anarchy results have been focused on welfare approximation in the worst-case scenarios. To complement theoretical understandings, we empirically study how different auction formats (FPA, GSP, VCG) with different levels of non-uniform bid-scaling perform in an autobidding world with a synthetic dataset for auctions. Our empirical findings include: * For both uniform bid-scaling and non-uniform bid-scaling, FPA is better than GSP and GSP is better than VCG in terms of both welfare and profit; * A higher level of non-uniform bid-scaling leads to lower welfare performance in both FPA and GSP, while different levels of non-uniform bid-scaling have no effect in VCG. Our methodology of synthetic data generation may be of independent interest.

We study the price of anarchy of the generalized second-price auction where bidders are value maximizers (i.e., autobidders). We show that in general the price of anarchy can be as bad as $0$. For comparison, the price of anarchy of running VCG is $1/2$ in the autobidding world. We further show a fined-grained price of anarchy with respect to the discount factors (i.e., the ratios of click probabilities between lower slots and the highest slot in each auction) in the generalized second-price auction, which highlights the qualitative relation between the smoothness of the discount factors and the efficiency of the generalized second-price auction.

In digital online advertising, advertisers procure ad impressions simultaneously on multiple platforms, or so-called channels, such as Google Ads, Meta Ads Manager, etc., each of which consists of numerous ad auctions. We study how an advertiser maximizes total conversion (e.g. ad clicks) while satisfying aggregate return-on-investment (ROI) and budget constraints across all channels. In practice, an advertiser does not have control over, and thus cannot globally optimize, which individual ad auctions she participates in for each channel, and instead authorizes a channel to procure impressions on her behalf: the advertiser can only utilize two levers on each channel, namely setting a per-channel budget and per-channel target ROI. In this work, we first analyze the effectiveness of each of these levers for solving the advertiser's global multi-channel problem. We show that when an advertiser only optimizes over per-channel ROIs, her total conversion can be arbitrarily worse than what she could have obtained in the global problem. Further, we show that the advertiser can achieve the global optimal conversion when she only optimizes over per-channel budgets. In light of this finding, under a bandit feedback setting that mimics real-world scenarios where advertisers have limited information on ad auctions in each channels and how channels procure ads, we present an efficient learning algorithm that produces per-channel budgets whose resulting conversion approximates that of the global optimal problem. Finally, we argue that all our results hold for both single-item and multi-item auctions from which channels procure impressions on advertisers' behalf.

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.

Online advertising channels have commonly focused on maximizing total advertiser value (or welfare) to enhance long-run retention and channel healthiness. Previous literature has studied auction design by incorporating machine learning predictions on advertiser values (also known as machine-learned advice) through various forms to improve total welfare. Yet, such improvements could come at the cost of individual bidders' welfare and do not shed light on how particular advertiser bidding strategies impact welfare. Motivated by this, we present an analysis on an individual bidder's welfare loss in the autobidding world for auctions with and without machine-learned advice, and also uncover how advertiser strategies relate to such losses. In particular, we demonstrate how ad platforms can utilize ML advice to improve welfare guarantee on the aggregate and individual bidder level by setting ML advice as personalized reserve prices when the platform consists of autobidders who maximize value while respecting a return-on-ad spent (ROAS) constraint. Under parallel VCG auctions with such ML advice-based reserves, we present a worst-case welfare lower-bound guarantee for an individual autobidder, and show that the lower-bound guarantee is positively correlated with ML advice quality as well the scale of bids induced by the autobidder's bidding strategies. Further, we prove an impossibility result showing that no truthful, and possibly randomized mechanism with anonymous allocations can achieve universally better individual welfare guarantees than VCG, in presence of personalized reserves based on ML-advice of equal quality. Moreover, we extend our individual welfare guarantee results to generalized first price (GFP) and generalized second price (GSP) auctions.

We study the price of anarchy of the first-price auction in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders). We show that with autobidders only, the price of anarchy of the first-price auction is $1/2$, and with both kinds of bidders, the price of anarchy degrades to about $0.457$ (the precise number is given by an optimization). These results complement the recent result by Jin and Lu [2022] showing that the price of anarchy of the first-price auction with traditional bidders only is $1 - 1/e^2$. We further investigate a setting where the seller can utilize machine-learned advice to improve the efficiency of the auctions. There, we show that as the accuracy of the advice increases, the price of anarchy improves smoothly from about $0.457$ to $1$.

Quantum spins, referred to the spin operator preserved by full SU(2) symmetry in the absence of the magnetic anistropy, have been proposed to host exotic interactions with superconductivity4. However, spin orbit coupling and crystal field splitting normally cause a significant magnetic anisotropy for d/f-shell spins on surfaces6,9, breaking SU(2) symmetry and fabricating the spins with Ising properties10. Recently, magnetic nanographenes have been proven to host intrinsic quantum magnetism due to their negligible spin orbital coupling and crystal field splitting. Here, we fabricate three atomically precise nanographenes with the same magnetic ground state of spin S=1/2 on Pb(111) through engineering sublattice imbalance in graphene honeycomb lattice. Scanning tunneling spectroscopy reveals the coexistence of magnetic bound states and Kondo screening in such hybridized system. Through engineering the magnetic exchange strength between the unpaired spin in nanographenes and cooper pairs, quantum phase transition from the singlet to the doublet state has been observed, in consistent with quantum models of spins on superconductors. Our work demonstrates delocalized graphene magnetism host highly tunable magnetic bound states with cooper pairs, which can be further developed to study the Majorana bound states and other rich quantum physics of low-dimensional quantum spins on superconductors., Comment: 13 pages, 4figures

Topological defects in graphene materials introduce exotic properties which are absent in their defect-free counterparts with both fundamental importance and technological implications. Although individual topological defects have been widely studied, collective magnetic behaviors originating from well-organized multiple topological defects remain a great challenge. Here, we studied the collective magnetic properties originating from three pentagon topological defects in truxene-based molecular graphenoids by using scanning tunneling microscopy and non-contact atomic force microscopy. Unpaired $\pi$ electrons are introduced into the aromatic topology of truxene molecular graphenoids one by one by dissociating hydrogen atoms at the pentagon defects via atom manipulation. Scanning tunneling spectroscopy measurements together with density functional theory calculations suggest that the unpaired electrons are ferromagnetically coupled, forming a collective high-spin quartet state of S=3/2. Our work demonstrates that the collective spin ordering can be realized through engineering regular patterned topological defects in molecular graphenoids, providing a new platform for designer one-dimensional ferromagnetic spin chains and two-dimensional ferromagnetic networks., Comment: 25 pages, 12 figures, include Supplementary Information

We study bilateral trade between two strategic agents. The celebrated result of Myerson and Satterthwaite states that in general, no incentive-compatible, individually rational and weakly budget balanced mechanism can be efficient. I.e., no mechanism with these properties can guarantee a trade whenever buyer value exceeds seller cost. Given this, a natural question is whether there exists a mechanism with these properties that guarantees a constant fraction of the first-best gains-from-trade, namely a constant fraction of the gains-from-trade attainable whenever buyer's value weakly exceeds seller's cost. In this work, we positively resolve this long-standing open question on constant-factor approximation, mentioned in several previous works, using a simple mechanism.

In classic auction theory, reserve prices are known to be effective for improving revenue for the auctioneer against quasi-linear utility maximizing bidders. The introduction of reserve prices, however, usually do not help improve total welfare of the auctioneer and the bidders. In this paper, we focus on value maximizing bidders with return on spend constraints -- a paradigm that has drawn considerable attention recently as more advertisers adopt auto-bidding algorithms in advertising platforms -- and show that the introduction of reserve prices has a novel impact on the market. Namely, by choosing reserve prices appropriately the auctioneer can improve not only the total revenue but also the total welfare. Our results also demonstrate that reserve prices are robust to bidder types, i.e., reserve prices work well for different bidder types, such as value maximizers and utility maximizers, without using bidder type information. We generalize these results for a variety of auction mechanisms such as VCG, GSP, and first-price auctions. Moreover, we show how to combine these results with additive boosts to improve the welfare of the outcomes of the auction further. Finally, we complement our theoretical observations with an empirical study confirming the effectiveness of these ideas using data from online advertising auctions.

A patient seller aims to sell a good to an impatient buyer (i.e., one who discounts utility over time). The buyer will remain in the market for a period of time $T$, and her private value is drawn from a publicly known distribution. What is the revenue-optimal pricing-curve (sequence of (price, time) pairs) for the seller? Is randomization of help here? Is the revenue-optimal pricing curve computable in polynomial time? We answer these questions in this paper. We give an efficient algorithm for computing the revenue-optimal pricing curve. We show that pricing curves, that post a price at each point of time and let the buyer pick her utility maximizing time to buy, are revenue-optimal among a much broader class of sequential lottery mechanisms. I.e., mechanisms that allow the seller to post a menu of lotteries at each point of time cannot get any higher revenue than pricing curves. We also show that the even broader class of mechanisms that allow the menu of lotteries to be adaptively set, can earn strictly higher revenue than that of pricing curves, and the revenue gap can be as big as the support size of the buyer's value distribution., Comment: Accepted to ACM-SIAM Symposium on Discrete Algorithms (SODA) 2023