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The problem of contract design addresses the challenge of moral hazard in principle-agent setups. The agent exerts costly efforts that produce a random outcome with an associated reward for the principal. Moral hazard refers to the tension that the principal cannot observe the agent’s effort level hence needs to incentivize the agent only through rewarding the realized effort outcome, i.e., the contract. Bayesian contract design studies the principal’s design problem of an optimal contract when facing an unknown agent characterized by a private Bayesian type. In its most general form, the agent’s type is inherently “multi-parameter” and can arbitrarily affect both the agent’s productivity and effort costs. In contrast, a natural single-parameter setting of much recent interest simplifies the agent’s type to a single value that describes the agent’s cost per unit of effort, whereas agents’ efforts are assumed to be equally
productive.
The main result of this paper is an almost approximation-preserving polynomial-time reduction from the most general multi-parameter Bayesian contract design (BCD) to single-parameter BCD. That is, for any multi-parameter BCD instance I^M, we construct a single-parameter instance I^S such that any β-approximate contract (resp. menu of contracts) of I^S can in turn be converted to a (β − ϵ)-approximate contract (resp. menu of contracts) of I^M. The reduction is in time polynomial in the input size and log(1/ϵ); moreover, when β = 1 (i.e., the given single-parameter solution is exactly optimal), the dependence on 1/ϵ can be removed, leading to a polynomial-time exact reduction. This efficient reduction is somewhat surprising because in the closely related problem of Bayesian mechanism design, a polynomial-time reduction from multi-parameter to single-parameter setting is believed to not exist. Our result demonstrates the intrinsic difficulty of addressing moral hazard in Bayesian contract design, regardless of being single-parameter or multi-parameter.
As byproducts, our reduction answers two open questions in recent literature of algorithmic contract design: (a) it implies that optimal contract design in single-parameter BCD is not in APX unless P=NP even when the agent’s type distribution is regular, answering the open question of [3] in the negative; (b) it implies that the principal’s (order-wise) tight utility gap between using a menu of contracts and a single contract is Θ(n) where n is the number of actions, answering the major open question of [27] for the single-parameter case.
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Auto-bidding and Auctions in Online Advertising: A Survey
Ashwinkumar Badanidiyuru Varadaraja
Christopher Liaw
Haihao (Sean) Lu
Andres Perlroth
Georgios Piliouras
Ariel Schvartzman
Kelly Spendlove
Hanrui Zhang
Mingfei Zhao
ACM SIGecom Exchanges, 22 (2024)
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In this survey, we summarize recent developments in research fueled by the growing adoption of automated bidding strategies in online advertising. We explore the challenges and opportunities that have arisen as markets embrace this autobidding and cover a range of topics in this area, including bidding algorithms, equilibrium analysis and efficiency of common auction formats, and optimal auction design.
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Efficiency of the Generalized Second-Price Auction for Value Maximizers
Hanrui Zhang
Proceedings of the ACM on Web Conference 2024, 46–56
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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.
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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.
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Non-uniform Bid-scaling and Equilibria for Different Auctions: An Empirical Study
Proceedings of the ACM on Web Conference 2024, 256–266
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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.
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The recent increasing adoption of autobidding has inspired the growing interest in analyzing the performance of classic mechanism with value-maximizing auto-bidders both theoretically and empirically. It is known that optimal welfare can be obtained in first-price auctions if auto-bidders are restricted to uniform bid-scaling and the price of anarchy is $2$ when non-uniform bid-scaling strategies are allowed.
In this paper, we provide a fine-grained price of anarchy analysis for non-uniform bid-scaling strategies in first-price auctions, demonstrating the reason why more powerful (individual) non-uniform bid-scaling strategies may lead to worse (aggregated) performance in social welfare. Our theoretical results match recent empirical findings that a higher level of non-uniform bid-scaling leads to lower welfare performance in first-price auctions.
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Bayesian Calibrated Click-Through Auctions
Junjie Chen
Minming Li
Haifeng Xu
51st International Colloquium on Automata, Languages, and Programming (ICALP 2024), Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik, 44:1-44:18
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We study information design in click-through auctions, in which the bidders/advertisers bid for winning an opportunity to show their ads but only pay for realized clicks. The payment may or may not happen, and its probability is called the click-through rate (CTR). This auction format is widely used in the industry of online advertising. Bidders have private values, whereas the seller has private information about each bidder's CTRs. We are interested in the seller's problem of partially revealing CTR information to maximize revenue. Information design in click-through auctions turns out to be intriguingly different from almost all previous studies in this space since any revealed information about CTRs will never affect bidders' bidding behaviors -- they will always bid their true value per click -- but only affect the auction's allocation and payment rule. In some sense, this makes information design effectively a constrained mechanism design problem.
Our first result is an FPTAS to compute an approximately optimal mechanism under a constant number of bidders. The design of this algorithm leverages Bayesian bidder values which help to "smooth" the seller's revenue function and lead to better tractability. The design of this FPTAS is complex and primarily algorithmic. Our second main result pursues the design of "simple" mechanisms that are approximately optimal yet more practical. We primarily focus on the two-bidder situation, which is already notoriously challenging as demonstrated in recent works. When bidders' CTR distribution is symmetric, we develop a simple prior-free signaling scheme, whose construction relies on a parameter termed optimal signal ratio. The constructed scheme provably obtains a good approximation as long as the maximum and minimum of bidders' value density functions do not differ much.
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Complex Dynamics in Autobidding Systems
Georgios Piliouras
Kelly Spendlove
Proceedings of the 25th ACM Conference on Economics and Computation (2024)
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It has become the default in markets such as ad auctions for participants to bid in an auction through automated bidding agents (autobidders) which adjust bids over time to satisfy return-over-spend constraints. Despite the prominence of such systems for the internet economy, their resulting dynamical behavior is still not well understood. Although one might hope that such relatively simple systems would typically converge to the equilibria of their underlying auctions, we provide a plethora of results that show the emergence of complex behavior, such as bi-stability, periodic orbits and quasi periodicity. We empirically observe how the market structure (expressed as motifs) qualitatively affects the behavior of the dynamics. We complement it with theoretical results showing that autobidding systems can simulate both linear dynamical systems as well logical boolean gates.
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Mechanism Design for Large Language Models
Paul Duetting
Haifeng Xu
Proceedings of the ACM on Web Conference 2024, Association for Computing Machinery, New York, NY, USA, 144–155
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We investigate auction mechanisms for AI-generated content, focusing on applications like ad creative generation. In our model, agents' preferences over stochastically generated content are encoded as large language models (LLMs). We propose an auction format that operates on a token-by-token basis, and allows LLM agents to influence content creation through single dimensional bids. We formulate two desirable incentive properties and prove their equivalence to a monotonicity condition on output aggregation. This equivalence enables a second-price rule design, even absent explicit agent valuation functions. Our design is supported by demonstrations on a publicly available LLM.
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Optimal Mechanisms for a Value Maximizer: The Futility of Screening Targets
Proceedings of the 25th ACM Conference on Economics and Computation (EC) (2024)
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Motivated by the increased adoption of autobidding algorithms in internet advertising markets, we study the design of optimal mechanisms for selling items to a value-maximizing buyer with a return-on-spend constraint. The buyer's values and target ratio in the return-on-spend constraint are private. We restrict attention to deterministic sequential screening mechanisms that can be implemented as a menu of prices paid for purchasing an item or not. The main result of this paper is to provide a characterization of an optimal mechanism. Surprisingly, we show that the optimal mechanism does not require target screening, i.e., offering a single pair of prices is optimal for the seller. The optimal mechanism is a subsidized posted price that provides a subsidy to the buyer to encourage participation and then charges a fixed unit price for each item sold. The seller's problem is a challenging non-linear mechanism design problem, and a key technical contribution of our work is to provide a novel approach to analyze non-linear pricing contracts.
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