arXiv subjects

Paul Duetting

Publications and source records attributed to Paul Duetting.

17 recordsLinked to original sources

arXiv2026

Combinatorial Contracts Through Demand Types

In the combinatorial action model of contract design, a principal delegates a complex project to an agent, incentivizing a subset of actions from a ground set of $n$ actions, via a linear contract. Computing the optimal contract is a challenging problem that generally hinges on two factors: (i) the number of "critical values" - values of the linear contract parameter at which the agent's best response changes from one set to another, and (ii) the complexity of the agent's best-response problem (demand query). Prior work has used this approach to devise polynomial-time algorithms for the optimal contract problem under specific reward functions: gross substitutes, supermodular, and ultra. We develop a unified geometric framework for algorithmic contract design by establishing a fundamental link to the theory of demand types from consumer theory. Under this geometric view, bounding the number of critical values reduces to counting the best-response regions which the "contract ray" pierces. Leveraging this connection, we introduce the class of All Substitutes and Complements (ASC) functions, and show that it admits at most $O(n^2)$ critical values, strictly generalizing and unifying all previously known classes admitting poly-many critical values. We conjecture that, under some mild assumptions, ASC is the maximal such class. Turning to the demand query aspect, we develop a new technique for efficiently computing a demand query using value queries, which works in general for "succinct" demand types. Combining these structural and algorithmic results, we obtain polynomial-time algorithms for new classes of reward functions that exhibit substitutes and complements simultaneously.

cs.GT↗

arXiv2026

Anonymous Contracts

We study a multi-agent contracting problem where agents exert costly effort to achieve individually observable binary outcomes. While the principal can theoretically extract the full social welfare using a discriminatory contract that tailors payments to individual costs, such contracts may be perceived as unfair. In this work, we introduce and analyze anonymous contracts, where payments depend solely on the total number of successes, ensuring identical treatment of agents. We first establish that every anonymous contract admits a pure Nash equilibrium. However, because general anonymous contracts can suffer from multiple equilibria with unbounded gaps in principal utility, we identify uniform anonymous contracts as a desirable subclass. We prove that uniform anonymous contracts guarantee a unique equilibrium, thereby providing robust performance guarantees. In terms of efficiency, we prove that under limited liability, anonymous contracts cannot generally approximate the social welfare better than a factor logarithmic in the spread of agent success probabilities. We show that uniform contracts are sufficient to match this theoretical limit. Finally, we demonstrate that removing limited liability significantly boosts performance: anonymous contracts generally achieve an $O(\log n)$ approximation to the social welfare and, surprisingly, can extract the full welfare whenever agents' success probabilities are distinct. This reveals a structural reversal: widely spread probabilities are the hardest case under limited liability, whereas identical probabilities become the hardest case when limited liability is removed.

cs.GT↗

arXiv2025

Information Design With Large Language Models

Information design is typically studied through the lens of Bayesian signaling, where signals shape beliefs purely based on their correlation with the true state of the world. However, behavioral economics and psychology emphasize that human decision-making is more complex and can depend on how information is framed. This paper formalizes a language-based notion of framing and bridges this to the popular Bayesian-persuasion model. We model framing as a possibly non-Bayesian, linguistic way to influence a receiver's prior belief, while a signaling/recommendation scheme can further refine this belief in the classic Bayesian way. A key challenge in systematically optimizing in this framework is the vast space of possible framings and the difficulty of predicting their effects on receivers. Based on growing evidence that Large Language Models (LLMs) can effectively serve as proxies for human behavior, we formulate a theoretical model based on access to a framing-to-belief mapping. This model then enables us to precisely characterize when solely optimizing framing or jointly optimizing framing and signaling is tractable. We substantiate our theoretical analysis with an empirical study that leverages LLMs to optimize over the natural-language framing space using an iterative prompt optimization method combined with analytical solvers for optimal signaling schemes.

cs.GT↗

arXiv2025

Succinct Ambiguous Contracts

Real-world contracts are often ambiguous. While recent work by D\"utting, Feldman, Peretz, and Samuelson (EC 2023, Econometrica 2024) demonstrates that ambiguous contracts can yield large gains for the principal, their optimal solutions often require deploying an impractically large menu of contracts. This paper investigates \emph{succinct} ambiguous contracts, which are restricted to consist of at most $k$ classic contracts. By letting $k$ range from $1$ to $n-1$, this yields an interpolation between classic contracts ($k=1$) and unrestricted ambiguous contracts ($k=n-1$). This perspective enables important structural and algorithmic results. First, we establish a fundamental separability property: for any number of actions $n$ and any succinctness level $k$, computing an optimal $k$-ambiguous contract reduces to finding optimal classic contracts over a suitable partition of the actions, up to an additive balancing shift acting as a base payment. Second, we show bounds on the principal's loss from using a $k$-ambiguous rather than an unrestricted ambiguous contract, which uncover a striking discontinuity in the principal's utility regarding contract size: lacking even a single contract option may cause the principal's utility to drop sharply by a multiplicative factor of $2$, a bound we prove to be tight. Finally, we characterize the tractability frontier of the optimal $k$-ambiguous contract problem. Our separability result yields a poly-time algorithm whenever the number of partitions of $n-1$ actions into $k$ sets is polynomial, recovering and extending known results for classic and unrestricted ambiguous contracts. We complement this with a tight hardness result, showing that the problem is \textsf{NP}-hard whenever the number of partitions is super-polynomial. Moreover, when $k \approx n/3$, the problem is even hard to approximate.

cs.GT↗

arXiv2025

The Pseudo-Dimension of Contracts

Algorithmic contract design studies scenarios where a principal incentivizes an agent to exert effort on her behalf. In this work, we focus on settings where the agent's type is drawn from an unknown distribution, and formalize an offline learning framework for learning near-optimal contracts from sample agent types. A central tool in our analysis is the notion of pseudo-dimension from statistical learning theory. Beyond its role in establishing upper bounds on the sample complexity, pseudo-dimension measures the intrinsic complexity of a class of contracts, offering a new perspective on the tradeoffs between simplicity and optimality in contract design. Our main results provide essentially optimal tradeoffs between pseudo-dimension and representation error (defined as the loss in principal's utility) with respect to linear and bounded contracts. Using these tradeoffs, we derive sample- and time-efficient learning algorithms, and demonstrate their near-optimality by providing almost matching lower bounds on the sample complexity. Conversely, for unbounded contracts, we prove an impossibility result showing that no learning algorithm exists. Finally, we extend our techniques in three important ways. First, we provide refined pseudo-dimension and sample complexity guarantees for the combinatorial actions model, revealing a novel connection between the number of critical values and sample complexity. Second, we extend our results to menus of contracts, showing that their pseudo-dimension scales linearly with the menu size. Third, we adapt our algorithms to the online learning setting, where we show that, a polynomial number of type samples suffice to learn near-optimal bounded contracts. Combined with prior work, this establishes a formal separation between expert advice and bandit feedback for this setting.

cs.GT↗

arXiv2024

Algorithmic Contract Theory: A Survey

A contract is an economic tool used by a principal to incentivize one or more agents to exert effort on her behalf, by defining payments based on observable performance measures. A key challenge addressed by contracts -- known in economics as moral hazard -- is that, absent a properly set up contract, agents might engage in actions that are not in the principal's best interest. Another common feature of contracts is limited liability, which means that payments can go only from the principal -- who has the deep pocket -- to the agents. With classic applications of contract theory moving online, growing in scale, and becoming more data-driven, tools from contract theory become increasingly important for incentive-aware algorithm design. At the same time, algorithm design offers a whole new toolbox for reasoning about contracts, ranging from additional tools for studying the tradeoff between simple and optimal contracts, through a language for discussing the computational complexity of contracts in combinatorial settings, to a formalism for analyzing data-driven contracts. This survey aims to provide a computer science-friendly introduction to the basic concepts of contract theory. We give an overview of the emerging field of "algorithmic contract theory" and highlight work that showcases the potential for interaction between the two areas. We also discuss avenues for future research.

cs.GT↗

arXiv2024

Multi-Agent Combinatorial Contracts

Combinatorial contracts are emerging as a key paradigm in algorithmic contract design, paralleling the role of combinatorial auctions in algorithmic mechanism design. In this paper we study natural combinatorial contract settings involving teams of agents, each capable of performing multiple actions. This scenario extends two fundamental special cases previously examined in the literature, namely the single-agent combinatorial action model of [Duetting et al., 2021] and the multi-agent binary-action model of [Babaioff et al., 2012, Duetting et al., 2023]. We study the algorithmic and computational aspects of these settings, highlighting the unique challenges posed by the absence of certain monotonicity properties essential for analyzing the previous special cases. To navigate these complexities, we introduce a broad set of novel tools that deepen our understanding of combinatorial contracts environments and yield good approximation guarantees. Our main result is a constant-factor approximation for submodular multi-agent multi-action problems with value and demand oracles access. This result is tight: we show that this problem admits no PTAS (even under binary actions). As a side product of our main result, we devise an FPTAS, with value and demand oracles, for single-agent combinatorial action scenarios with general reward functions, which is of independent interest. We also provide bounds on the gap between the optimal welfare and the principal's utility. We show that, for subadditive rewards, perhaps surprisingly, this gap scales only logarithmically (rather than linearly) in the size of the action space.

cs.GT↗

arXiv2023

Mechanism Design for Large Language Models

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.

cs.GT↗

arXiv2022

Multi-Agent Contracts

We study a natural combinatorial single-principal multi-agent contract design problem, in which a principal motivates a team of agents to exert effort toward a given task. At the heart of our model is a reward function, which maps the agent efforts to an expected reward of the principal. We seek to design computationally efficient algorithms for finding optimal (or near-optimal) linear contracts for reward functions that belong to the complement-free hierarchy. Our first main result gives constant-factor approximation algorithms for submodular and XOS reward functions, with value oracles for submodular reward functions and value and demand oracles for XOS reward functions. It relies on an unconventional use of ``prices'' and (approximate) demand queries for selecting the set of agents that the principal should contract with, and exploits a novel scaling property of XOS functions and their marginals, which may be of independent interest. As our second main result, we show that constant approximation is the best we can get for submodular reward functions, even with both value and demand oracles. For the larger class of subadditive reward functions, we establish an $\Omega(\sqrt{n})$ impossibility for settings with $n$ agents. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This rapid degradation presents a surprising departure from previous literature, e.g., on combinatorial auctions, where approximation guarantees tend to deteriorate more

cs.GT↗

arXiv2021

Combinatorial Contracts

We introduce a new model of combinatorial contracts in which a principal delegates the execution of a costly task to an agent. To complete the task, the agent can take any subset of a given set of unobservable actions, each of which has an associated cost. The cost of a set of actions is the sum of the costs of the individual actions, and the principal's reward as a function of the chosen actions satisfies some form of diminishing returns. The principal incentivizes the agents through a contract, based on the observed outcome. Our main results are for the case where the task delegated to the agent is a project, which can be successful or not. We show that if the success probability as a function of the set of actions is gross substitutes, then an optimal contract can be computed with polynomially many value queries, whereas if it is submodular, the optimal contract is NP-hard. All our results extend to linear contracts for higher-dimensional outcome spaces, which we show to be robustly optimal given first moment constraints. Our analysis uncovers a new property of gross substitutes functions, and reveals many interesting connections between combinatorial contracts and combinatorial auctions, where gross substitutes is known to be the frontier for efficient computation.

cs.GT↗

arXiv2021

Calibrated Click-Through Auctions: An Information Design Approach

We analyze the optimal information design in a click-through auction with fixed valuations per click, but stochastic click-through rates. While the auctioneer takes as given the auction rule of the click-through auction, namely the generalized second-price auction, the auctioneer can design the information flow regarding the click-through rates among the bidders. A natural requirement in this context is to ask for the information structure to be calibrated in the learning sense. With this constraint, the auction needs to rank the ads by a product of the bid and an unbiased estimator of the click-through rates, and the task of designing an optimal information structure is thus reduced to the task of designing an optimal unbiased estimator. We show that in a symmetric setting with uncertainty about the click-through rates, the optimal information structure attains both social efficiency and surplus extraction. The optimal information structure requires private (rather than public) signals to the bidders. It also requires correlated (rather than independent) signals, even when the underlying uncertainty regarding the click-through rates is independent. Beyond symmetric settings, we show that the optimal information structure requires partial information disclosure.

econ.TH↗

arXiv2020

The Complexity of Contracts

We initiate the study of computing (near-)optimal contracts in succinctly representable principal-agent settings. Here optimality means maximizing the principal's expected payoff over all incentive-compatible contracts---known in economics as "second-best" solutions. We also study a natural relaxation to approximately incentive-compatible contracts. We focus on principal-agent settings with succinctly described (and exponentially large) outcome spaces. We show that the computational complexity of computing a near-optimal contract depends fundamentally on the number of agent actions. For settings with a constant number of actions, we present a fully polynomial-time approximation scheme (FPTAS) for the separation oracle of the dual of the problem of minimizing the principal's payment to the agent, and use this subroutine to efficiently compute a delta-incentive-compatible (delta-IC) contract whose expected payoff matches or surpasses that of the optimal IC contract. With an arbitrary number of actions, we prove that the problem is hard to approximate within any constant c. This inapproximability result holds even for delta-IC contracts where delta is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear delta-IC contracts with constant delta are sufficient to achieve a constant-factor approximation of the "first-best" (full-welfare-extracting) solution, and that such a contract can be computed in polynomial time.

cs.DS↗

arXiv2013

Valuation Compressions in VCG-Based Combinatorial Auctions

The focus of classic mechanism design has been on truthful direct-revelation mechanisms. In the context of combinatorial auctions the truthful direct-revelation mechanism that maximizes social welfare is the VCG mechanism. For many valuation spaces computing the allocation and payments of the VCG mechanism, however, is a computationally hard problem. We thus study the performance of the VCG mechanism when bidders are forced to choose bids from a subspace of the valuation space for which the VCG outcome can be computed efficiently. We prove improved upper bounds on the welfare loss for restrictions to additive bids and upper and lower bounds for restrictions to non-additive bids. These bounds show that the welfare loss increases in expressiveness. All our bounds apply to equilibrium concepts that can be computed in polynomial time as well as to learning outcomes.

cs.GT↗

arXiv2013

Expressiveness and Robustness of First-Price Position Auctions

Since economic mechanisms are often applied to very different instances of the same problem, it is desirable to identify mechanisms that work well in a wide range of circumstances. We pursue this goal for a position auction setting and specifically seek mechanisms that guarantee good outcomes under both complete and incomplete information. A variant of the generalized first-price mechanism with multi-dimensional bids turns out to be the only standard mechanism able to achieve this goal, even when types are one-dimensional. The fact that expressiveness beyond the type space is both necessary and sufficient for this kind of robustness provides an interesting counterpoint to previous work on position auctions that has highlighted the benefits of simplicity. From a technical perspective our results are interesting because they establish equilibrium existence for a multi-dimensional bid space, where standard techniques break down. The structure of the equilibrium bids moreover provides an intuitive explanation for why first-price payments may be able to support equilibria in a wider range of circumstances than second-price payments.

cs.GT↗

arXiv2013

Polymatroid Prophet Inequalities

Consider a gambler and a prophet who observe a sequence of independent, non-negative numbers. The gambler sees the numbers one-by-one whereas the prophet sees the entire sequence at once. The goal of both is to decide on fractions of each number they want to keep so as to maximize the weighted fractional sum of the numbers chosen. The classic result of Krengel and Sucheston (1977-78) asserts that if both the gambler and the prophet can pick one number, then the gambler can do at least half as well as the prophet. Recently, Kleinberg and Weinberg (2012) have generalized this result to settings where the numbers that can be chosen are subject to a matroid constraint. In this note we go one step further and show that the bound carries over to settings where the fractions that can be chosen are subject to a polymatroid constraint. This bound is tight as it is already tight for the simple setting where the gambler and the prophet can pick only one number. An interesting application of our result is in mechanism design, where it leads to improved results for various problems.

cs.DS↗

arXiv2012

Auctions with Heterogeneous Items and Budget Limits

We study individual rational, Pareto optimal, and incentive compatible mechanisms for auctions with heterogeneous items and budget limits. For multi-dimensional valuations we show that there can be no deterministic mechanism with these properties for divisible items. We use this to show that there can also be no randomized mechanism that achieves this for either divisible or indivisible items. For single-dimensional valuations we show that there can be no deterministic mechanism with these properties for indivisible items, but that there is a randomized mechanism that achieves this for either divisible or indivisible items. The impossibility results hold for public budgets, while the mechanism allows private budgets, which is in both cases the harder variant to show. While all positive results are polynomial-time algorithms, all negative results hold independent of complexity considerations.

cs.GT↗

arXiv2012

Payment Rules through Discriminant-Based Classifiers

In mechanism design it is typical to impose incentive compatibility and then derive an optimal mechanism subject to this constraint. By replacing the incentive compatibility requirement with the goal of minimizing expected ex post regret, we are able to adapt statistical machine learning techniques to the design of payment rules. This computational approach to mechanism design is applicable to domains with multi-dimensional types and situations where computational efficiency is a concern. Specifically, given an outcome rule and access to a type distribution, we train a support vector machine with a special discriminant function structure such that it implicitly establishes a payment rule with desirable incentive properties. We discuss applications to a multi-minded combinatorial auction with a greedy winner-determination algorithm and to an assignment problem with egalitarian outcome rule. Experimental results demonstrate both that the construction produces payment rules with low ex post regret, and that penalizing classification errors is effective in preventing failures of ex post individual rationality.

cs.GT↗