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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 $(2+\epsilon)$-competitive online truthful mechanism for submodular/XOS valuations and any constant $\epsilon>0$. 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., Comment: Preliminary version in FOCS 2024

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. In the stochastic setting, we design an efficient learning algorithm achieving a regret bound of $O(T^{3/4})$. Our approach is based on an adaptive discretization scheme of the space of mechanisms, as any non-adaptive discretization fails to achieve sublinear regret. In the adversarial setting, we exploit the non-Lipschitzness of the problem to prove a strong negative result, namely that no learning algorithm can achieve more than half of the revenue of the best fixed mechanism in hindsight. We then consider the $\sigma$-smooth adversary; we construct an efficient learning algorithm that achieves a regret bound of $O(T^{2/3})$ and builds on a succinct encoding of exponentially many experts. Finally, we prove that no learning algorithm can achieve less than $\Omega(\sqrt T)$ regret in both the stochastic and the smooth setting, thus narrowing the range where the minimax regret rates for these two problems lie., Comment: Paper accepted at ACM EC 2024

The increasing deployment of AI is shaping the future landscape of the internet, which is set to become an integrated ecosystem of AI agents. Orchestrating the interaction among AI agents necessitates decentralized, self-sustaining mechanisms that harmonize the tension between individual interests and social welfare. In this paper we tackle this challenge by synergizing reinforcement learning with principal-agent theory from economics. Taken separately, the former allows unrealistic freedom of intervention, while the latter struggles to scale in sequential settings. Combining them achieves the best of both worlds. We propose a framework where a principal guides an agent in a Markov Decision Process (MDP) using a series of contracts, which specify payments by the principal based on observable outcomes of the agent's actions. We present and analyze a meta-algorithm that iteratively optimizes the policies of the principal and agent, showing its equivalence to a contraction operator on the principal's Q-function, and its convergence to subgame-perfect equilibrium. We then scale our algorithm with deep Q-learning and analyze its convergence in the presence of approximation error, both theoretically and through experiments with randomly generated binary game-trees. Extending our framework to multiple agents, we apply our methodology to the combinatorial Coin Game. Addressing this multi-agent sequential social dilemma is a promising first step toward scaling our approach to more complex, real-world instances.

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. The existence of such embedding enables a reduction from the version of the matroid secretary problem where the matroid is unknown to the version where the matroid is known in advance. We show that online matroid embeddings exist for binary (and hence graphic) and laminar matroids. We also show a negative result showing that no online matroid embedding exists for the class of all matroids. Finally, we define the notion of an approximate matroid embedding, generalizing the notion of {\alpha}-partition property, and provide an upper bound on the approximability of binary matroids by a partition matroid, matching the lower bound of Dughmi et al., Comment: 25 pages, 4 figures

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). We provide algorithms in this setting 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., Comment: To appear at ICML 24

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.

Algorithmic contract design is a new frontier in the intersection of economics and computation, with combinatorial contracts being a core problem in this domain. A central model within combinatorial contracts explores a setting where a principal delegates the execution of a task, which can either succeed or fail, to an agent. The agent can choose any subset among a given set of costly actions, where every subset is associated with a success probability. The principal incentivizes the agent through a contract that specifies the payment upon success of the task. A natural setting of interest is one with submodular success probabilities. It is known that finding the optimal contract for the principal is $\mathsf{NP}$-hard, but the hardness result is derived from the hardness of demand queries. A major open problem is whether the hardness arises solely from the hardness of demand queries, or if the complexity lies within the optimal contract problem itself. In other words: does the problem retain its hardness, even when provided access to a demand oracle? We resolve this question in the affirmative, showing that any algorithm that computes the optimal contract for submodular success probabilities requires an exponential number of demand queries, thus settling the query complexity problem.

We study the classic single-choice prophet inequality problem through a resource augmentation lens. Our goal is to bound the $(1-\varepsilon)$-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 $(1-\varepsilon)$-approximation to the expected offline optimum on a single copy. We show that block threshold algorithms, which set one threshold per copy, are optimal and give a tight bound of $k = \Theta(\log \log 1/\varepsilon)$. This shows that block threshold algorithms approach the offline optimum doubly-exponentially fast. For single threshold algorithms, we give a tight bound of $k = \Theta(\log 1/\varepsilon)$, establishing an exponential gap between block threshold algorithms and single threshold algorithms. Our model and results pave the way for exploring resource-augmented prophet inequalities in combinatorial settings. In line with this, we present preliminary findings for bipartite matching with one-sided vertex arrivals, as well as in XOS combinatorial auctions. Our results have a natural competition complexity interpretation in mechanism design and pricing applications., Comment: 33 pages

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., Comment: WWW'24 Best Paper

We study the combinatorial contracting problem of D\"utting et al. [FOCS '21], 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 $\alpha$ 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 $\alpha$, and that one can iterate through all of them efficiently in order to find the optimal contract. In this work we study to which extent GS rewards and additive costs constitute a tractability frontier for combinatorial contracts. We present an algorithm that for any rewards and costs, enumerates all critical values, with poly-many demand queries (in the number of critical values). This implies the tractability of the optimal contract for any setting with poly-many critical values and efficient demand oracle. A direct corollary is a poly-time algorithm for the optimal contract in settings with supermodular rewards and submodular costs. We also study a natural class of matching-based instances with XOS rewards and additive costs. While the demand problem for this setting is tractable, we show that it admits an exponential number of critical values. On the positive side, we present (pseudo-) polynomial-time algorithms for two natural special cases of this setting. Our work unveils a profound connection to sensitivity analysis, and designates matching-based instances as a crucial focal point for gaining a deeper understanding of combinatorial contract settings., Comment: 22 pages, 3 figures

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.

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., Comment: Accepted at ICML 2023

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. We show that for i.i.d. prices a single-threshold online algorithm achieves at least $1/2$ of the expected profit of the optimal offline algorithm and we prove that this is optimal. For non-i.i.d. prices in random order, where prices are no longer independent, we give a single-threshold online algorithm that achieves at least a $1/16$ fraction of the expected profit of the optimal offline algorithm. We also show that for this setting no online algorithm can yield a better than $1/3$ approximation, and thus establish a formal separation from the i.i.d. case. On the other hand, we present a threshold-based online algorithm for this setting that yields a $1/2-o(1)$ approximation. For non-i.i.d. prices no approximation is possible. We use the results for these base cases to solve a variety of more complex settings. For instance, we show a $1/2-o(1)$ approximation for settings where prices are affiliated and the online algorithm has only access to a single sample. We also extend our upper and lower bounds for the single item case to $k$ items, and thus in particular show that it is impossible to achieve $1-o(1)$ approximations. For the budgeted version, where fractions of an item can be bought, and gains can be reinvested, we show a constant-factor approximation to the optimal offline algorithm's growth rate. In a setting with $k$ item types and price streams, we achieve a $\Omega(1/k)$ approximation for the unit-capacity case, which is optimal.

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 \emph{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 $(1-\e)$-approximation in $O(n^{\text{poly} \log n\cdot f(\e)})$ time through careful discretization and non-trivial bundling processes. Using the toolbox developed for the QPTAS, coupled with a novel \emph{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 explore the deliberate infusion of ambiguity into the design of contracts. We show that when the agent is ambiguity-averse and hence chooses an action that maximizes their minimum utility, the principal can strictly gain from using an ambiguous contract, and this gain can be arbitrarily high. We characterize the structure of optimal ambiguous contracts, showing that ambiguity drives optimal contracts towards simplicity. We also provide a characterization of ambiguity-proof classes of contracts, where the principal cannot gain by infusing ambiguity. Finally, we show that when the agent can engage in mixed actions, the advantages of ambiguous contracts disappear.

We provide a justification for the prevalence of linear (commission-based) contracts in practice under the Bayesian framework. We consider a hidden-action principal-agent model, in which actions require different amounts of effort, and the agent's cost per-unit-of-effort is private. We show that linear contracts are near-optimal whenever there is sufficient uncertainty in the principal-agent setting.

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 and demand oracles, respectively. 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. Our second main result is an $\Omega(\sqrt{n})$ impossibility for settings with $n$ agents and subadditive reward functions, even with demand oracle access. A striking feature of this impossibility is that it applies to subadditive functions that are constant-factor close to submodular. This presents a surprising departure from previous literature, e.g., on combinatorial auctions.

Maximizing a submodular function is a fundamental task in machine learning and 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 $(4.597+O(\varepsilon))$-approximation algorithm with summary size $O( \frac{k+d}{\varepsilon^2}\log \frac{k}{\varepsilon})$ that is improved 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 $(9.435 + O(\varepsilon))$-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., Comment: Preliminary versions of this work appeared as arXiv:2201.13128 and in ICML'22. The main difference with respect to these versions consists in extending our results to non-monotone submodular functions

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. In this work we adopt a metric that was introduced in the context of bitcoin fee design markets: the percentage change in payment that can be achieved by being strategic. We study the behavior of this metric for single-unit and $k$-unit auction environments with $n$ i.i.d. buyers, and seek conditions under which the percentage change tends to zero as $n$ grows large. To the best of our knowledge, ours is the first rigorous study of the extent to which large multi-unit first price auctions are susceptible to manipulation. We provide an almost complete picture of the conditions under which they are truthful in the large, and exhibit some surprising boundaries., Comment: Extended Abstract at TheWebConference 2022

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 $O(k + \frac{d \log k}{\varepsilon^2})$. In the streaming setting we provide a $(5.582+O(\varepsilon))$-approximation algorithm with summary size and memory $O(k + \frac{d \log k}{\varepsilon^2})$. We complement our theoretical results with an in-depth experimental analysis showing the effectiveness of our algorithms on real-world datasets.

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 Grossman and Hart [1983] and the (procurement version) of single-dimensional mechanism design \`a la Myerson [1981]. 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 a linear programming (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 is 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., Comment: An extended abstract appeared in the Proceedings of the 22nd ACM Conference on Economics and Computation

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., Comment: Merges and extends arXiv:2103.13089 [cs.GT] and arXiv:2104.02050 [cs.DS]

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.

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.

We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $8$ that also uses just a single sample from each distribution. Finally, we show how to turn our $16$-approximate single-sample prophet inequality into a truthful single-sample mechanism for online bipartite matching with vertex arrivals.

The secretary problem is probably the purest model of decision making under uncertainty. In this paper we ask which advice can we 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, to more modern versions of advice in the form of samples, to an ML-inspired model where a classifier gives us noisy signal about whether or not the current secretary is the best on the market. Our main technique is a factor revealing LP that captures all of the problems above. 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 a new noisy binary advice model.

A prophet inequality states, for some $\alpha\in[0,1]$, 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 $\alpha$ 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 $\beta n$ samples for some $\beta\geq 0$. We first give improved lower bounds on $\alpha$ for a wide range of values of $\beta$; specifically, $\alpha\geq(1+\beta)/e$ when $\beta\leq 1/(e-1)$, which is tight, and $\alpha\geq 0.648$ when $\beta=1$, 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., Comment: Accepted to ITCS 2021

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.

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 {\em 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.

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., Comment: An extended abstract appeared in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, 2020 (SODA'20)

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: Given a sequence of random variables $X_1,\dots,X_n$ drawn independently from a distribution $F$, the goal is to choose a stopping time $\tau$ so as to maximize $\alpha$ such that for all distributions $F$ we have $\mathbb{E}[X_\tau] \geq \alpha \cdot \mathbb{E}[\max_tX_t]$. What makes this problem challenging is that the decision whether $\tau=t$ may only depend on the values of the random variables $X_1,\dots,X_t$ and on the distribution $F$. For quite some time the best known bound for the problem was $\alpha\geq1-1/e\approx0.632$ [Hill and Kertz, 1982]. Only recently this bound was improved by Abolhassani et al. [2017], and a tight bound of $\alpha\approx0.745$ was obtained by Correa et al. [2017]. The case where $F$ is unknown, such that the decision whether $\tau=t$ may depend only on the values of the first $t$ random variables but not on $F$, is equally well motivated (e.g., [Azar et al., 2014]) but has received much less attention. A straightforward guarantee for this case of $\alpha\geq1/e\approx0.368$ can be derived from the solution to the secretary problem. Our main result is that this bound is tight. Motivated by this impossibility result we investigate the case where the stopping time may additionally depend on a limited number of samples from~$F$. An extension of our main result shows that even with $o(n)$ samples $\alpha\leq 1/e$, so that the interesting case is the one with $\Omega(n)$ samples. Here we show that $n$ samples allow for a significant improvement over the secretary problem, while $O(n^2)$ samples are equivalent to knowledge of the distribution: specifically, with $n$ samples $\alpha\geq1-1/e\approx0.632$ and $\alpha\leq\ln(2)\approx0.693$, and with $O(n^2)$ samples $\alpha\geq0.745-\epsilon$ for any $\epsilon>0$.

We consider the classic principal-agent model of contract theory, in which a principal designs an outcome-dependent compensation scheme to incentivize an agent to take a costly and unobservable action. When all of the model parameters---including the full distribution over principal rewards resulting from each agent action---are known to the designer, an optimal contract can in principle be computed by linear programming. In addition to their demanding informational requirements, such optimal contracts are often complex and unintuitive, and do not resemble contracts used in practice. This paper examines contract theory through the theoretical computer science lens, with the goal of developing novel theory to explain and justify the prevalence of relatively simple contracts, such as linear (pure commission) contracts. First, we consider the case where the principal knows only the first moment of each action's reward distribution, and we prove that linear contracts are guaranteed to be worst-case optimal, ranging over all reward distributions consistent with the given moments. Second, we study linear contracts from a worst-case approximation perspective, and prove several tight parameterized approximation bounds., Comment: 30 pages, 4 figures, Latex; minor revision

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., Comment: An extended abstract appeared in ICML'19, along with a short Research Highlight in the Communications of the ACM

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), multi-dimensional 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., Comment: Extended abstract appeared in the Proceedings of the 58th IEEE Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA

We consider the problem of maximizing the expected revenue from selling $k$ homogeneous goods to $n$ unit-demand buyers who arrive sequentially with independent and identically distributed valuations. In this setting the optimal posted prices are dynamic in the sense that they depend on the remaining numbers of goods and buyers. We investigate how much revenue is lost when a single static price is used instead for all buyers and goods, and prove upper bounds on the ratio between the maximum revenue from dynamic prices and that from static prices. These bounds are tight for all values of $k$ and $n$ and vary depending on a regularity property of the underlying distribution. For general distributions we obtain a ratio of $2-k/n$, for regular distributions a ratio that increases in $n$ and is bounded from above by $1/(1-k^k/(e^{k}k!))$, which is roughly $1/(1-1/(\sqrt{2\pi k}))$. The lower bounds hold for the revenue gap between dynamic and static prices. The upper bounds are obtained via an ex-ante relaxation of the revenue maximization problem, as a consequence the tight bounds of $2-k/n$ in the general case and of $1/(1-1/(\sqrt{2\pi k}))$ in the regular case apply also to the potentially larger revenue gap between the optimal incentive-compatible mechanism and the optimal static price. Our results imply that for regular distributions the benefit of dynamic prices vanishes while for non-regular distributions dynamic prices may achieve up to twice the revenue of static prices.

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 strategize 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 $\Omega(1/\log m)$ 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 an $o(\log \log m/\log m)$ fraction of the optimal social welfare., Comment: Extended abstract in Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain

In the standard single-dimensional model of position auctions, bidders agree on the relative values of the positions and each of them submits a single bid that is interpreted in terms of these values. Motivated by current practice in sponsored search we consider a situation where the auctioneer uses estimates of the relative values, which may be imprecise, and show that under both complete and incomplete information a non-truthful mechanism is able to support an efficient outcome in equilibrium for a wider range of these estimates than the VCG mechanism. We thus exhibit a property of the VCG mechanism that may help explain the surprising rarity with which it is used even in settings with unit demand, a relative lack of robustness to misspecification of the bidding language. 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 result for incomplete information, which is driven by a surprising connection between equilibrium bids in the VCG mechanism and the generalized first-price mechanism., Comment: Abstract in Proceedings of the 17th ACM Conference on Economics and Computation

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 [SODA 2010] shows that combining a greedy algorithm that is an $\alpha$-approximation algorithm with a pay-your-bid payment rule yields a mechanism whose Price of Anarchy is $O(\alpha)$. 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. We demonstrate the far-reaching consequences of our result by showing its implications for sparse packing integer programs, such as multi-unit auctions and generalized matching, for the maximum traveling salesman problem, for combinatorial auctions, and for single source unsplittable flow problems. In all these problems our approach leads to novel simple, near-optimal mechanisms whose Price of Anarchy either matches or beats the performance guarantees of known mechanisms., Comment: Extended abstract appeared in Proc. of 16th ACM Conference on Economics and Computation (EC'15)

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.

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.

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.

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.

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 mechanism 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.

Matching markets play a prominent role in economic theory. A prime example of such a market is the sponsored search market. Here, as in other markets of that kind, market equilibria correspond to feasible, envy free, and bidder optimal outcomes. For settings without budgets such an outcome always exists and can be computed in polynomial-time by the so-called Hungarian Method. Moreover, every mechanism that computes such an outcome is incentive compatible. We show that the Hungarian Method can be modified so that it finds a feasible, envy free, and bidder optimal outcome for settings with budgets. We also show that in settings with budgets no mechanism that computes such an outcome can be incentive compatible for all inputs. For inputs in general position, however, the presented mechanism---as any other mechanism that computes such an outcome for settings with budgets---is incentive compatible.

If you recommend a product to me and I buy it, how much should you be paid by the seller? And if your sole interest 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 these questions is a key challenge in all forms of marketing that rely on and explore social ties; ranging from personal recommendations to viral marketing. In the first part of this paper, we show that there can be no pricing mechanism that is "truthful" with respect to the seller, and we use solution concepts from coalitional game theory, namely the Core, the Shapley Value, and the Nash Bargaining Solution, to derive provably "fair" prices for settings with one or multiple recommenders. We then investigate pricing mechanisms for the setting where recommenders have different "purchase arguments". Here we show that it might be beneficial for the recommenders to withhold some of their arguments, unless anonymity-proof solution concepts, such as the anonymity-proof Shapley value, are used. In the second part of this paper, we analyze the setting where the recommendee loses trust in the recommender for each irrelevant recommendation. Here we prove that even if the recommendee regains her initial trust on each successful recommendation, the expected total profit the recommender can make over an infinite period is bounded. This can only be overcome when the recommendee also incrementally regains trust during periods without any recommendation. Here, we see an interesting connection to "banner blindness", suggesting that showing fewer ads can lead to a higher long-term profit.