We introduce the notion of an online matroid embedding, which is an algorithm for mapping an unknown matroid that is revealed in an online fashion to a larger-but-known matroid. We establish the existence of such an embedding for binary matroids, and use it to relate variants of the binary matroid secretary problem to each other, showing that seemingly simpler problems are in fact equivalent to seemingly harder ones (up to constant-factors). Specifically, we show this to be the case for the version of the matroid secretary problem in which the matroid is not known in advance, and where it is known in advance. We also show that the version with known matroid structure, is equivalent to the problem where weights are not fully adversarial but drawn from a known pairwise-independent distribution.

In the combinatorial-action contract model (Dütting et al., FOCS’21) a principal delegates the execution of a complex project to an agent, who can choose any subset from a given set of actions. Each set of actions incurs a cost to the agent, given by a set function \(c\), and induces an expected reward to the principal, given by a set function \(f\). To incentivize the agent, the principal designs a contract that specifies the payment upon success, with the optimal contract being the one that maximizes the principal’s utility.

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 gracefully.

A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to relax welfare efficiency and the use of approximation mechanisms. Such mechanisms in general make extensive use of the Bayesian priors. In this work, we investigate a question of increasing theoretical and practical importance: how much prior information is required to design mechanisms with near-optimal approximations?

Bilateral trade models the task of intermediating between two strategic agents, a seller and a buyer, willing to trade a good for which they hold private valuations. We study this problem from the perspective of a broker, in a regret minimization framework. At each time step, a new seller and buyer arrive, and the broker has to propose a mechanism that is incentivecompatible and individually rational, with the goal of maximizing profit. We propose a learning algorithm that guarantees a nearly tight regret in the stochastic setting when seller and buyer valuations are drawn i.i.d. from a fixed and possibly correlated unknown distribution. We further show that it is impossible to achieve sublinear regret in the non-stationary scenario where valuations are generated upfront by an adversary. Our ambitious benchmark for these results is the best incentive-compatible and individually rational mechanism. This separates us from previous works on efficiency maximization in bilateral trade, where the benchmark is a single number: the best fixed price in hindsight. A particular challenge we face is that uniform convergence for all mechanisms’ profits is impossible. We overcome this difficulty via a careful chaining analysis that proves convergence for a provably near-optimal mechanism at (essentially) optimal rate. We further showcase the broader applicability of our techniques by providing nearly optimal results for the joint ads problem.

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.

We study mechanism design when agents have private preferences and private information about a common payoff-relevant state. We show that standard message-driven mechanisms cannot implement socially efficient allocations when agents have multidimensional types, even under favorable conditions.

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.

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: the single-agent combinatorial action model of [18], and the multi-agent binary- action model of [4, 19].

In this work, we study online submodular maximization and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make, at most, a constant number of updates per step. We show a tight information-theoretic bound of 2/3 for general monotone submodular functions and an improved (also tight) bound of 3/4 for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a 0.51-approximation. Combined with an information-theoretic hardness of 1/2 for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

We study the deletion robust version of submodular maximization under matroid constraints. The goal is to extract a small-size summary of the data set that contains a high-value independent set even after an adversary deletes some elements. We present constant-factor approximation algorithms, whose space complexity depends on the rank $k$ of the matroid, the number $d$ of deleted elements, and the input precision $\varepsilon$. In the centralized setting we present a $(4.494+O(\varepsilon))$-approximation algorithm with summary size $O( \frac{k+d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ that improves to a $(3.582+O(\varepsilon))$-approximation with $O(k + \frac{d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ summary size when the objective is monotone. In the streaming setting we provide a $(9.294 + O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d}{\varepsilon^2}\log \frac{k}{\varepsilon})$; the approximation factor is then improved to $(5.582+O(\varepsilon))$ in the monotone case.

Abstract. 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.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this significant problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an \({\tilde{O}(\frac{{k^{2}}}{{\varepsilon}})}\) amortized update time (in the number of insertions and deletions) and yields a \({(4+O(\varepsilon))}\) -approximate solution with respect to the dynamic optimum, where \(k\) is the rank of the matroid.

In online combinatorial allocations/auctions,$n$bidders sequentially arrive, each with a combinatorial valuation (such as submodular/XOS) over subsets of$m$indivisible items. The aim is to immediately allocate a subset of the remaining items to maximize the total welfare, defined as the sum of bidder valuations. A long line of work has studied this problem when the bidder valuations come from known independent distributions. In particular, for submodular/XOS valuations, we know 2-competitive algorithms/mechanisms that set a fixed price for each item and the arriving bidders take their favorite subset of the remaining items given these prices. However, these algorithms traditionally presume the availability of the underlying distributions as part of the input to the algorithm. Contrary to this assumption, practical scenarios often require the learning of distributions, a task complicated by limited sample availability. This paper investigates the feasibility of achieving$O$(1) -competitive algorithms under the realistic constraint of having access to only a limited number of samples from the underlying bidder distributions. Our first main contribution shows that a mere single sample from each bidder distribution is sufficient to yield an$O$(1)-competitive algorithm for submodular/XOS valuations. This result leverages a novel extension of the secretary-style analysis, employing the sample to have the algorithm compete against itself. Although online, this first approach does not provide an online truthful mechanism. Our second main contribution shows that a polynomial number of samples suffices to yield a (2 + ∊) -competitive online truthful mechanism for submodular/XOS valuations and any constant ∊ > 0. This result is based on a generalization of the median-based algorithm for the single-item prophet inequality problem to combinatorial settings with multiple items.

Maximizing monotone submodular functions under cardinality constraints is a classic optimization task with several applications in data mining and machine learning. In this paper, we study this problem in a dynamic environment with consistency constraints: elements arrive in a streaming fashion, and the goal is maintaining a constant approximation to the optimal solution while having a stable solution (i.e., the number of changes between two consecutive solutions is bounded). In this setting, we provide algorithms with different trade-offs between consistency and approximation quality. We also complement our theoretical results with an experimental analysis showing the effectiveness of our algorithms in real-world instances.

Motivated by online retail, we consider the problem of selling one item (e.g., an ad slot) to two non-excludable buyers (say, a merchant and a brand). This problem captures, for example, situations where a merchant and a brand cooperatively bid in an auction to advertise a product, and both benefit from the ad being shown. A mechanism collects bids from the two and decides whether to allocate and which payments the two parties should make. This gives rise to intricate incentive compatibility constraints, e.g., on how to split payments between the two parties. We approach the problem of finding a revenue-maximizing incentive-compatible mechanism from an online learning perspective; this poses significant technical challenges. First, the action space (the class of all possible mechanisms) is huge; second, the function that maps mechanisms to revenue is highly irregular, ruling out standard discretization-based approaches.

We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the (1 - ε)-competition complexity of different types of online algorithms. This metric asks for the smallest k such that the expected value of the online algorithm on k copies of the original instance, is at least a (1 - ε)-approximation to the expected offline optimum on a single copy.

We study the combinatorial contracting problem of Dütting et al. [13], in which a principal seeks to incentivize an agent to take a set of costly actions. In their model, there is a binary outcome (the agent can succeed or fail), and the success probability and the costs depend on the set of actions taken. The optimal contract is linear, paying the agent an α fraction of the reward. For gross substitutes (GS) rewards and additive costs, they give a poly-time algorithm for finding the optimal contract. They use the properties of GS functions to argue that there are poly-many “critical values” of α, and that one can iterate through all of them efficiently in order to find the optimal contract.

We investigate auction mechanisms to support the emerging format of AI-generated content. We in particular study how to aggregate several LLMs in an incentive compatible manner. In this problem, the preferences of each agent over stochastically generated contents are described/encoded as an LLM. A key motivation is to design an auction format for AI-generated ad creatives to combine inputs from different advertisers. We argue that this problem, while generally falling under the umbrella of mechanism design, has several unique features. We propose a general formalism---the token auction model---for studying this problem. A key feature of this model is that it acts on a token-by-token basis and lets LLM agents influence generated contents through single dimensional bids.

Designing an incentive compatible auction that maximizes expected revenue is an intricate task. The single-item case was resolved in a seminal piece of work by Myerson in 1981, but more than 40 years later, a full analytical understanding of the optimal design still remains elusive for settings with two or more items. In this work, we initiate the exploration of the use of tools from deep learning for the automated design of optimal auctions. We model an auction as a multi-layer neural network, frame optimal auction design as a constrained learning problem, and show how it can be solved using standard machine learning pipelines. In addition to providing generalization bounds, we present extensive experimental results, recovering essentially all known solutions that come from the theoretical analysis of optimal auction design problems and obtaining novel mechanisms for settings in which the optimal mechanism is unknown.

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here $n$ agents with subadditive valuation functions compete for the assignment of $m$ items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of $O(\log m)$. We make major progress on this question by providing an $O(\log \log m)$ prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an $O(\log \log m)$ approximation to the optimal revenue for subadditive valuations under an item-independence assumption.

Maximizing monotone submodular functions under a matroid constraint is a classic algorithmic problem with multiple applications in data mining and machine learning. We study this classic problem in the fully dynamic setting, where elements can be both inserted and deleted in real-time. Our main result is a randomized algorithm that maintains an efficient data structure with an $\tilde{O}(k^2)$ amortized update time (in the number of additions and deletions) and yields a $4$-approximate solution, where $k$ is the rank of the matroid.

Inspired by applications in pricing and contract design, we study the maximization of one-sided Lipschitz functions, which only provide the (weaker) guarantee that they do not grow too quickly in one direction. We show that it is possible to learn a maximizer for such a function while incurring $O(\log \log T)$ total regret (with a universal constant independent of the number of discontinuities / complexity of the function). This regret bound is asymptotically optimal in $T$ due to a lower bound of Kleinberg and Leighton. By applying this algorithm, we show that one can sell digital goods to multiple buyers and learn the optimal linear contract in the principal-agent setting while incurring at most $O(\log \log T)$ regret.

Contract design involves a principal who establishes contractual agreements about payments for outcomes that arise from the actions of an agent. In this paper, we initiate the study of deep learning for the automated design of optimal contracts. We introduce a novel representation: the Discontinuous ReLU (DeLU) network, which models the principal's utility as a discontinuous piecewise affine function of the design of a contract where each piece corresponds to the agent taking a particular action. DeLU networks implicitly learn closed-form expressions for the incentive compatibility constraints of the agent and the utility maximization objective of the principal, and support parallel inference on each piece through linear programming or interior-point methods that solve for optimal contracts. We provide empirical results that demonstrate success in approximating the principal's utility function with a small number of training samples and scaling to find approximately optimal contracts on problems with a large number of actions and outcomes.

We study a generalization of both the classic single-dimensional mechanism design problem, and the hidden-action principal-agent problem of contract theory [c.f., Alon et al. 2021]. In this setting, the principal seeks to incentivize an agent with a private Bayesian type to take a costly action. The goal is to design an incentive compatible menu of contracts which maximizes the expected revenue.

In this work we initiate the study of buy-and-sell prophet inequalities. We start by considering what is arguably the most fundamental setting. In this setting the online algorithm observes a sequence of prices one after the other. At each time step, the online algorithm can decide to buy and pay the current price if it does not hold the item already; or it can decide to sell and collect the current price as a reward if it holds the item.

In this paper we introduce a model of ambiguous contracts, capturing many real-life scenarios where agents engage in contractual relations that leave some degree of uncertainty. In this paper we introduce a model of ambiguous contracts, capturing many real-life scenarios where agents engage in contractual relations that leave some degree of uncertainty. Our starting point is the celebrated hidden-action model and the classic notion of a contract, where the principal commits to an outcome-contingent payment scheme for incentivizing an agent to take a costly action.

We study the prophet secretary problem, a well-studied variant of the classic prophet inequality, where values are drawn from independent known distributions but arrive in uniformly random order. Upon seeing a value at each step, the decision-maker has to either select it and stop or irrevocably discard it. Traditionally, the chosen benchmark is the expected reward of the prophet, who knows all the values in advance and can always select the maximum one. In this work, we study the prophet secretary problem against a less pessimistic but equally well-motivated benchmark; the online optimal. Here, the main goal is to find polynomial-time algorithms that guarantee near-optimal expected reward. As a warm-up, we present a quasi-polynomial time approximation scheme (QPTAS) achieving a (1 − ε)-approximation in O(npoly log n· f(ε)) time through careful discretization and non-trivial bundling processes. Using the toolbox developed for the QPTAS, coupled with a novel frontloading technique that enables us to reduce the number of decisions we need to make, we are able to remove the dependence on n in the exponent and obtain a polynomial time approximation scheme (PTAS) for this problem.

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.

Maximizing a monotone submodular function is a fundamental task in machine learning. In this paper we study the deletion robust version of the problem under the classic matroids constraint. Here the goal is to extract a small size summary of the dataset that contains a high value independent set even after an adversary deleted some elements. We present constant-factor approximation algorithms, whose space complexity depends on the rank $k$ of the matroid and the number $d$ of deleted elements. In the centralized setting we present a $(3.582+O(\varepsilon))$-approximation algorithm with summary size $O(k + \frac{d}{\eps^2}\log \frac{k}{\eps})$. In the streaming setting we provide a $(5.582+O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d}{\eps^2}\log \frac{k}{\eps})$. We complement our theoretical results with an in-depth experimental analysis showing the effectiveness of our algorithms on real-world datasets.

We study the competition complexity of dynamic pricing relative to the optimal auction in the fundamental single-item setting. In prophet inequality terminology, we compare the expected reward Am(F) achievable by the optimal online policy on m i.i.d. random variables drawn from F to the expected maximum Mn(F) of n i.i.d. draws from the same distribution. We ask how big does m have to be to ensure that (1+ε) Am(F) ≥ Mn(F) for all F.

We study single-sample prophet inequalities (SSPIs), i.e., prophet inequalities where only a single sample from each prior distribution is available. Besides a direct, and optimal, SSPI for the basic single choice problem [Rubinstein et al., 2020], most existing SSPI results were obtained via an elegant, but inherently lossy reduction to order-oblivious secretary (OOS) policies [Azar et al., 2014]. Motivated by this discrepancy, we develop an intuitive and versatile greedy-based technique that yields SSPIs directly rather than through the reduction to OOSs. Our results can be seen as generalizing and unifying a number of existing results in the area of prophet and secretary problems. Our algorithms significantly improve on the competitive guarantees for a number of interesting scenarios (including general matching with edge arrivals, bipartite matching with vertex arrivals, and certain matroids), and capture new settings (such as budget additive combinatorial auctions). Complementing our algorithmic results, we also consider mechanism design variants. Finally, we analyze the power and limitations of different SSPI approaches by providing a partial converse to the reduction from SSPI to OOS given by Azar et al.

We analyze the optimal information design in a click-through auction with stochastic click-through rates and known valuations per click. The auctioneer takes as given the auction rule of the click-through auction, namely the generalized second-price auction. Yet, the auctioneer can design the information flow regarding the click-through rates among the bidders. We require that 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 value and a calibrated prediction of the click-through rates. The task of designing an optimal information structure is thus reduced to the task of designing an optimal calibrated prediction.

First-price auctions have many desirable properties, including uniquely possessing some, like credibility. However, first-price auctions are also inherently non-truthful, and non-truthfulness may result in instability and inefficiencies. Given these pros and cons, we seek to quantify the extent to which first-price auctions are susceptible to manipulation.

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.

There is growing awareness and concern about fairness in machine learning and algorithm design. This is particularly true in online selection problems where decisions are often biased, for example, when assessing credit risks or hiring staff. We address the issues of fairness and bias in online selection by introducing multi-color versions of the classic secretary and prophet problem. Interestingly, existing algorithms for these problems are either very unfair or very inefficient, so we develop optimal fair algorithms for these new problems and provide tight bounds on their competitiveness. We validate our theoretical findings on real-world data.

A prophet inequality states, for some $α\in[0,1]$, that the expected value achievable by a gambler who sequentially observes random variables $X_1,\dots,X_n$ and selects one of them is at least an $α$ fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional $βn$ samples for some $β\geq 0$. We first give improved lower bounds on $α$ for a wide range of values of $β$; specifically, $α\geq(1+β)/e$ when $β\leq 1/(e-1)$, which is tight, and $α\geq 0.648$ when $β=1$, which improves on a bound of around $0.635$ due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of $1/e$ for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications.

Economic theory distinguishes between principal-agent settings in which the agent has a private type and settings in which the agent takes a hidden action. Many practical problems, however, involve aspects of both. For example, brand X may seek to hire an influencer Y to create sponsored content to be posted on social media platform Z. This problem has a hidden action component (the brand may not be able or willing to observe the amount of effort exerted by the influencer), but also a private type component (influencers may have different costs per unit-of-effort). This "effort" and "cost per unit-of-effort" perspective naturally leads to a principal-agent problem with hidden action and single-dimensional private type, which generalizes both the classic principal-agent hidden action model of contract theory a la Grossmann and Hart [1986] and the (procurement version) of single-dimensional mechanism design a la Myerson [1983]. A natural goal in this model is to design an incentive-compatible contract, which consist of an allocation rule that maps types to actions, and a payment rule that maps types to payments for the stochastic outcomes of the chosen action. Our main contribution is an LP-duality based characterization of implementable allocation rules for this model, which applies to both discrete and continuous types. This characterization shares important features of Myerson's celebrated characterization result, but also departs from it in significant ways. We present several applications, including a polynomial-time algorithm for finding the optimal contract with a constant number of actions. This in sharp contrast to recent work on hidden action problems with multi-dimensional private information, which has shown that the problem of computing an optimal contract for constant numbers of actions is APX-hard.

The secretary problem is probably the purest model of decision making under uncertainty. In this paper, we ask which advice we can give the algorithm to improve its success probability. We propose a general model that unifies a broad range of problems: from the classic secretary problem with no advice to the variant where the quality of a secretary is drawn from a known distribution and the algorithm learns each candidate’s quality on arrival, more modern versions of advice in the form of samples, and a machine-learning inspired model where a classifier gives us a noisy signal about whether the current secretary is the best on the market. Our main technique is a factor-revealing linear program (LP) that captures all of these problems. We use this LP formulation to gain structural insight into the optimal policy. Using tools from linear programming, we present a tight analysis of optimal algorithms for secretaries with samples, optimal algorithms when secretaries’ qualities are drawn from a known distribution, and optimal algorithms for a new noisy binary advice model.

A celebrated impossibility result by Myerson and Satterthwaite (1983) shows that any truthful mechanism for two-sided markets that maximizes social welfare must run a deficit, resulting in a necessity to relax welfare efficiency and the use of approximation mechanisms. Such mechanisms in general make extensive use of the Bayesian priors. In this work, we investigate a question of increasing theoretical and practical importance: how much prior information is required to design mechanisms with near-optimal approximations? Our first contribution is a more general impossibility result stating that no meaningful approximation is possible without any prior information, expanding the famous impossibility result of Myerson and Satterthwaite. Our second contribution is that one single sample (one number per item), arguably a minimum-possible amount of prior information, from each seller distribution is sufficient for a large class of two-sided markets. We prove matching upper and lower bounds on the best approximation that can be obtained with one single sample for subadditive buyers and additive sellers, regardless of computational considerations. Our third contribution is the design of computationally efficient blackbox reductions that turn any one-sided mechanism into a two-sided mechanism with a small loss in the approximation, while using only one single sample from each seller. On the way, our blackbox-type mechanisms deliver several interesting positive results in their own right, often beating even the state of the art that uses full prior information.

Prophet inequalities compare the expected performance of an online algorithm for a stochastic optimization problem to the expected optimal solution in hindsight. They are a major alternative to classic worst-case competitive analysis, of particular importance in the design and analysis of simple (posted-price) incentive compatible mechanisms with provable approximation guarantees. A central open problem in this area concerns subadditive combinatorial auctions. Here n agents with subadditive valuation functions compete for the assignment of m items. The goal is to find an allocation of the items that maximizes the total value of the assignment. The question is whether there exists a prophet inequality for this problem that significantly beats the best known approximation factor of O(log m). We make major progress on this question by providing an O(log log m) prophet inequality. Our proof goes through a novel primal-dual approach. It is also constructive, resulting in an online policy that takes the form of static and anonymous item prices that can be computed in polynomial time given appropriate query access to the valuations. As an application of our approach, we construct a simple and incentive compatible mechanism based on posted prices that achieves an O(log log m) approximation to the optimal revenue for subadditive valuations under an item-independence assumption.

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 δ-incentive-compatible (δ-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 δ-IC contracts where δ is a sufficiently rapidly-decaying function of c. On the positive side, we show that simple linear δ-IC contracts with constant δ 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.

Designing an incentive compatible auction that maximizes expected revenue is an intricate task. The single-item case was resolved in a seminal piece of work by Myerson in 1981. Even after 30-40 years of intense research the problem remains unsolved for seemingly simple multi-bidder, multi-item settings. In this work, we initiate the exploration of the use of tools from deep learning for the automated design of optimal auctions. We model an auction as a multi-layer neural network, frame optimal auction design as a constrained learning problem, and show how it can be solved using standard pipelines. We prove generalization bounds and present extensive experiments, recovering essentially all known analytical solutions for multi-item settings, and obtaining novel mechanisms for settings in which the optimal mechanism is unknown.

We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing price-based online approximation algorithms, a natural extension of threshold algorithms for settings beyond binary selection. Our analysis takes the form of an extension theorem: we derive sufficient conditions on prices when all weights are known in advance, then prove that the resulting approximation guarantees extend directly to stochastic settings. Our framework unifies and simplifies much of the existing literature on prophet inequalities and posted price mechanisms and is used to derive new and improved results for combinatorial markets (with and without complements), multidimensional matroids, and sparse packing problems. Finally, we highlight a surprising connection between the smoothness framework for bounding the price of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be recast as posted price mechanisms with comparable performance guarantees.

In a combinatorial auction with item bidding, agents participate in multiple single-item second-price auctions at once. As some items might be substitutes, agents need to strate- gize in order to maximize their utilities. A number of results indicate that high welfare can be achieved this way, giving bounds on the welfare at equilibrium. Recently, however, criticism has been raised that equilibria are hard to compute and therefore unlikely to be attained. In this paper, we take a different perspective. We study simple best-response dynamics. That is, agents are activated one after the other and each activated agent updates his strategy myopically to a best response against the other agents’ current strategies. Often these dynamics may take exponentially long before they converge or they may not converge at all. However, as we show, convergence is not even necessary for good welfare guarantees. Given that agents’ bid updates are aggressive enough but not too aggressive, the game will remain in states of good welfare after each agent has updated his bid at least once. In more detail, we show that if agents have fractionally subadditive valuations, natural dynamics reach and remain in a state that provides a 1/3 approximation to the optimal welfare after each agent has updated his bid at least once. For subadditive valuations, we can guarantee an Ω(1/log m) approximation in case of m items that applies after each agent has updated his bid at least once and at any point after that. The latter bound is complemented by a negative result, showing that no kind of best-response dynamics can guarantee more than a an o(log log m/ log m) fraction of the optimal social welfare.

We exhibit a property of the VCG mechanism that can help explain the surprising rarity with which it is used even in settings with unit demand: a relative lack of robustness to inaccuracies in the choice of its parameters. For a standard position auction environment in which the auctioneer may not know the precise relative values of the positions, we show that under both complete and incomplete information a non-truthful mechanism supports the truthful outcome of the VCG mechanism for a wider range of these values than the VCG mechanism itself. The result for complete information concerns the generalized second-price mechanism and lends additional theoretical support to the use of this mechanism in practice. Particularly interesting from a technical perspective is the case of incomplete information, where a surprising combinatorial equivalence helps us to avoid confrontation with an unwieldy differential equation.

The algorithmic requirements for dominant strategy incentive compatibility, or truthfulness, are well understood. Is there a similar characterization of algorithms that when combined with a suitable payment rule yield near-optimal welfare in all equilibria? We address this question by providing a tight characterization of a (possibly randomized) mechanism's Price of Anarchy provable via smoothness, for single-parameter settings. The characterization assigns a unique value to each allocation algorithm; this value provides an upper and a matching lower bound on the Price of Anarchy of a derived mechanism provable via smoothness. The characterization also applies to the sequential or simultaneous composition of single-parameter mechanisms. Importantly, the factor that we identify is typically not in one-to-one correspondence to the approximation guarantee of the algorithm. Rather, it is usually the product of the approximation guarantee and the degree to which the mechanism is loser independent.

Many algorithms, that are originally designed without explicitly considering incentive properties, are later combined with simple pricing rules and used as mechanisms. The resulting mechanisms are often natural and simple to understand. But how good are these algorithms as mechanisms? Truthful reporting of valuations is typically not a dominant strategy (certainly not with a pay-your-bid, first-price rule, but it is likely not a good strategy even with a critical value, or second-price style rule either). Our goal is to show that a wide class of approximation algorithms yields this way mechanisms with low Price of Anarchy. The seminal result of Lucier and Borodin [2010] shows that combining a greedy algorithm that is an α-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is O(α). In this paper we significantly extend the class of algorithms for which such a result is available by showing that this close connection between approximation ratio on the one hand and Price of Anarchy on the other also holds for the design principle of relaxation and rounding provided that the relaxation is smooth and the rounding is oblivious.

It is desirable for an economic mechanism that its properties hold in a robust way across multiple equilibria and under varying assumptions regarding the information available to the participants. In this paper we focus on the design of position auctions and seek mechanisms that guarantee high revenue in every efficient equilibrium under both complete and incomplete information. Our main result identifies a generalized first-price auction with multi-dimensional bids as the only standard design capable of achieving this goal, even though valuations are one-dimensional. The fact that expressiveness beyond the valuation space is necessary for robustness provides an interesting counterpoint to previous work, which has highlighted the benefits of simple bid spaces. From a technical perspective, our results are interesting because they establish equilibrium existence for a multi-dimensional bid space, where standard techniques for establishing equilibrium existence break down.

Deferred-acceptance auctions are auctions for binary single-parameter mechanism design problems whose allocation rule can be implemented using an adaptive reverse greedy algorithm. Milgrom and Segal [2014] recently introduced these auctions and proved that they satisfy a remarkable list of incentive guarantees: in addition to being dominant-strategy incentive-compatible, they are weakly group-strategyproof, can be implemented by ascending-clock auctions, and admit outcome-equivalent full-information pay-as-bid versions. Neither forward greedy mechanisms nor the VCG mechanism generally possess any of these additional incentive properties. The goal of this paper is to initiate the study of deferred-acceptance auctions from an approximation standpoint. We study these auctions through the lens of two canonical welfare-maximization problems, in knapsack auctions and in combinatorial auctions with single-minded bidders.

The existence of a unique equilibrium is the classic tool for ensuring predictiveness of game theory. Typical uniqueness results, however, are for Nash and Bayes-Nash equilibria and do not guarantee that natural game playing dynamic converges to this equilibrium. In fact, there are well known examples in which the equilibrium is unique, yet natural learning behavior does not converge to it. Motivated by this, we strive for stronger uniqueness results. We do not only require that there is a unique equilibrium, but also that this equilibrium must be learnable. We adopt correlated equilibrium as our solution concept, as simple and natural learning algorithms guarantee that the empirical distribution of play converges to the space of correlated equilibria. Our main result is to show uniqueness of correlated equilibria in a large class of single-parameter mechanisms with matroid structure. We also show that our uniqueness result extends to problems with polymatroid structure under some conditions. Our model includes a number of special cases interesting on their own right, such as procurement auctions and Bertrand competitions. An interesting feature of our model is that we do not need to assume that the players have quasi-linear utilities, and hence can incorporate models with risk averse players and certain forms of externalities.

Designing double auctions is a complex problem, especially when there are restrictions on the sets of buyers and sellers that may trade with one another. The goal of this paper is to develop ``black-box reductions'' from double-auction design to the exhaustively-studied problem of designing single-sided mechanisms.

Suppose your sole interest in recommending a product to me is to maximize the amount paid to you by the seller for a sequence of recommendations. How should you recommend optimally if I become more inclined to ignore you with each irrelevant recommendation you make? Finding an answer to this question is a key challenge in all forms of marketing that rely on and explore social ties; ranging from personal recommendations to viral marketing.

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.

A fundamental result in mechanism design theory, the so-called revelation principle, asserts that for many questions concerning the existence of mechanisms with a given outcome one can restrict attention to truthful direct-revelation mechanisms. In practice, however, many mechanisms use a restricted message space. This motivates the study of the tradeoffs involved in choosing simplified mechanisms, which can sometimes bring benefits in precluding bad or promoting good equilibria, and other times impose costs on welfare and revenue. We study the simplicity-expressiveness tradeoff in two representative settings, sponsored search auctions and combinatorial auctions, each being a canonical example for complete information and incomplete information analysis, respectively. We observe that the amount of information available to the agents plays an important role for the tradeoff between simplicity and expressiveness.

Auctions are widely used on the Web. Applications range from internet advertising to platforms such as eBay. In most of these applications the auctions in use are single/multi-item auctions with unit demand. The main drawback of standard mechanisms for this type of auctions, such as VCG and GSP, is the limited expressiveness that they offer to the bidders. The General Auction Mechanism (GAM) of [1] is taking a first step towards addressing the problem of limited expressiveness by computing a bidder optimal, envy free outcome for linear utility functions with identical slopes and a single discontinuity per bidder-item pair. We show that in many practical situations this does not suffice to adequately model the preferences of the bidders, and we overcome this problem by presenting the first mechanism for piece-wise linear utility functions with non-identical slopes and multiple discontinuities. Our mechanism runs in polynomial time. Like GAM it is incentive compatible for inputs that fulfill a certain non-degeneracy requirement, but our requirement is more general than the requirement of GAM. For discontinuous utility functions that are non-degenerate as well as for continuous utility functions the outcome of our mechanism is a competitive equilibrium. We also show how our mechanism can be used to compute approximately bidder optimal, envy free outcomes for a general class of continuous utility functions via piece-wise linear approximation. Finally, we prove hardness results for even more expressive settings.

Two-sided matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market where $n$ advertisers compete for the assignment of one of $k$ sponsored search results, also known as ``slots'', for certain keywords they are interested in. Here, as in other markets of that kind, market equilibria correspond to stable matchings. In this paper, we show how to modify Kuhn's Hungarian Method (Kuhn, 1955) so that it finds an optimal stable matching between advertisers and advertising slots in settings with generalized linear utilities, per-bidder-item reserve prices, and per-bidder-item maximum prices. The only algorithm for this problem presented so far (Aggarwal et al., 2009) requires the market to be in ``general position''. We do not make this assumption.

Suppose you buy a new laptop and, simply because you like it so much, you recommend it to friends, encouraging them to purchase it as well. What would be an adequate price for the vendor of the laptop to pay for your recommendation?