We study the constrained variant of the multi-armed bandit (MAB) problem, in which the learner aims not only at minimizing the total loss incurred during the learning dynamic, but also at controlling the violation of multiple unknown constraints, under both full and bandit feedback. We consider a non-stationary environment that subsumes both stochastic and adversarial models and where, at each round, both losses and constraints are drawn from distributions that may change arbitrarily over time. In such a setting, it is provably not possible to guarantee both sublinear regret and sublinear violation. Accordingly, prior work has mainly focused either on settings with stochastic constraints or on relaxing the benchmark with fully adversarial constraints (e.g., via competitive ratios with respect to the optimum). We provide the first algorithms that achieve optimal rates of regret and positive constraint violation when the constraints are stochastic while the losses may vary arbitrarily, and that simultaneously yield guarantees that degrade smoothly with the degree of adversariality of the constraints. Specifically, under full feedback we propose an algorithm attaining $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ regret and $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ positive violation, where $C$ quantifies the amount of non-stationarity in the constraints. We then show how to extend these guarantees when only bandit feedback is available for the losses. Finally, when bandit feedback is available for the constraints, we design an algorithm achieving $\widetilde{\mathcal{O}}(\sqrt{T}+C)$ positive violation and $\widetilde{\mathcal{O}}(\sqrt{T}+C\sqrt{T})$ regret.

Bilateral trade is a central problem in algorithmic economics, and recent work has explored how to design trading mechanisms that use no-regret learning algorithms. However, no-regret learning is impossible when budget balance has to be enforced at each time step. Bernasconi et al. show how this impossibility result can be circumvented by relaxing the budget balance constraint to hold only globally over all time steps. In particular, they design an algorithm achieving regret of the order of \(\tilde{O}(T^{^3/_4})\) and provide a lower bound of \(\Omega(T^{^5/_7})\).

We study the computational complexity of the problem of computing local min-max equilibria of games with a nonconvex-nonconcave utility function f. From the work of Daskalakis, Skoulakis, and Zampetakis (STOC 2021), this problem was known to be hard in the restrictive case in which players are required to play strategies that are jointly constrained, leaving open the question of its complexity under more natural constraints. In this paper, we settle the question and show that the problem is PPAD-hard even under product constraints and, in particular, over the hypercube.

We study the sample complexity of learning a uniform approximation of an n-dimensional cumulative distribution function (CDF) within an error є > 0, when observations are restricted to a minimal one-bit feedback. This serves as a counterpart to the multivariate DKW inequality under “full feedback”, extending it to the setting of “bandit feedback”. Our main result shows a near-dimensional-invariance in the sample complexity: we get a uniform є-approximation with a sample complexity 1/є3log(1/є)O(n) over a arbitrary fine grid, where the dimensionality n only affects logarithmic terms. As direct corollaries, we provide tight sample complexity bounds and novel regret guarantees for learning fixed-price mechanisms in small markets, such as bilateral trade settings.

Stackelberg games (SGs) constitute the most fundamental and acclaimed models of strategic interactions involving some form of commitment. Moreover, they form the basis of more elaborate models of this kind, such as, e.g., Bayesian persuasion and principal-agent problems. Addressing learning tasks in SGs and related models is crucial to operationalize them in practice, where model parameters are usually unknown. In this paper, we revise the sample complexity of learning an optimal strategy to commit to in SGs. We provide a novel algorithm that (i) does not require any of the limiting assumptions made by state-of-the-art approaches and (ii) deals with a trade-off between sample complexity and termination probability arising when leader’s strategies representation has finite precision. Such a trade-off has been completely neglected by existing algorithms and, if not properly managed, it may result in them using exponentially-many samples. Our algorithm requires novel techniques, which also pave the way to addressing learning problems in other models with commitment ubiquitous in the real world.

Bilateral trade models the problem of facilitating trades between a seller and a buyer having private valuations for the item being sold. In the online version of the problem, the learner faces a new seller and buyer at each time step, and has to post a price for each of the two parties without any knowledge of their valuations. We consider a scenario where, at each time step, before posting prices the learner observes a context vector containing information about the features of the item for sale. The valuations of both the seller and the buyer follow an unknown linear function of the context. In this setting, the learner could leverage previous transactions in an attempt to estimate private valuations. We characterize the regret regimes of different settings, taking as a baseline the best context-dependent prices in hindsight. First, in the setting in which the learner has two-bit feedback and strong budget balance constraints, we propose an algorithm with $O(\log T)$ regret. Then, we study the same set-up with noisy valuations, providing a tight $\widetilde O(T^{2/3})$ regret upper bound. Finally, we show that loosening budget balance constraints allows the learner to operate under more restrictive feedback. Specifically, we show how to address the one-bit, global budget balance setting through a reduction from the two-bit, strong budget balance setup. This established a fundamental trade-off between the quality of the feedback and the strictness of the budget constraints.

We study online learning in constrained MDPs (CMDPs), focusing on the goal of attaining sublinear strong regret and strong cumulative constraint violation. Differently from their standard (weak) counterparts, these metrics do not allow negative terms to compensate positive ones, raising considerable additional challenges. Efroni et al. (2020) were the first to propose an algorithm with sublinear strong regret and strong violation, by exploiting linear programming. Thus, their algorithm is highly inefficient, leaving as an open problem achieving sublinear bounds by means of policy optimization methods, which are much more efficient in practice. Very recently, Muller et al. (2024) have partially addressed this problem by proposing a policy optimization method that allows to attain $\widetilde{\mathcal{O}}(T^{0.93})$ strong regret/violation. This still leaves open the question of whether optimal bounds are achievable by using an approach of this kind. We answer such a question affirmatively, by providing an efficient policy optimization algorithm with $\widetilde{\mathcal{O}}(\sqrt{T})$ strong regret/violation. Our algorithm implements a primal-dual scheme that employs a state-of-the-art policy optimization approach for adversarial (unconstrained) MDPs as primal algorithm, and a UCB-like update for dual variables.

Principal-agent problems model scenarios where a principal aims at incentivizing an agent to take costly, unobservable actions through the provision of payments. Such interactions are ubiquitous in several real-world applications, ranging from blockchain to the delegation of machine learning tasks. In this paper, we initiate the study of hidden-action principal-agent problems under approximate best responses, in which the agent may select any action that is not too much suboptimal given the principal's payment scheme (a.k.a. contract). Our main result is a polynomial-time algorithm to compute an optimal contract under approximate best responses. This is perhaps surprising, as computing an optimal commitment under approximate best responses is known to be computationally intractable in Stackelberg games. We also investigate the learnability of contracts under approximate best responses, by providing a no-regret learning algorithm for a natural application scenario where the principal does not know anything about the environment.

In the bandits with knapsacks framework (BwK) the learner has $m$ resource-consumption (i.e., packing) constraints. We focus on the generalization of BwK in which the learner has a set of general long-term constraints. The goal of the learner is to maximize their cumulative reward, while at the same time achieving small cumulative constraints violations. In this scenario, there exist simple instances where conventional methods for BwK fail to yield sublinear violations of constraints. We show that it is possible to circumvent this issue by requiring the primal and dual algorithm to be weakly adaptive. Indeed, even without any information on the Slater's parameter $\rho$ characterizing the problem, the interaction between weakly adaptive primal and dual regret minimizers leads to a ``self-bounding'' behavior of dual variables. In particular, their norm remains suitably upper bounded across the entire time horizon even without explicit projection steps. By exploiting this property, we provide best-of-both-worlds guarantees for stochastic and adversarial inputs. In the first case, we show that the algorithm guarantees sublinear regret. In the latter case, we establish a tight competitive ratio of $\rho/(1+\rho)$. In both settings, constraints violations are guaranteed to be sublinear in time. Finally, this results allow us to obtain new result for the problem of contextual bandits with linear constraints, providing the first no-$\alpha$-regret guarantees for adversarial contexts.

We study online learning in constrained Markov decision processes (CMDPs) with adversarial losses and stochastic hard constraints, under bandit feedback. We consider three scenarios. In the first one, we address general CMDPs, where we design an algorithm attaining sublinear regret and cumulative positive constraints violation. In the second scenario, under the mild assumption that a policy strictly satisfying the constraints exists and is known to the learner, we design an algorithm that achieves sublinear regret while ensuring that constraints are satisfied at every episode with high probability. In the last scenario, we only assume the existence of a strictly feasible policy, which is not known to the learner, and we design an algorithm attaining sublinear regret and constant cumulative positive constraints violation. Finally, we show that in the last two scenarios, a dependence on the Slater's parameter is unavoidable. To the best of our knowledge, our work is the first to study CMDPs involving both adversarial losses and hard constraints. Thus, our algorithms can deal with general non-stationary environments subject to requirements much stricter than those manageable with existing ones, enabling their adoption in a much wider range of applications.

We study online learning in constrained Markov decision processes (CMDPs) in which rewards and constraints may be either stochastic or adversarial. In such settings, stradi et al. (2024) proposed the first best-of-both-worlds algorithm able to seamlessly handle stochastic and adversarial constraints, achieving optimal regret and constraint violation bounds in both cases. This algorithm suffers from two major drawbacks. First, it only works under full feedback, which severely limits its applicability in practice. Moreover, it relies on optimizing over the space of occupancy measures, which requires solving convex optimization problems, an highly inefficient task. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with bandit feedback. Specifically, when the constraints are stochastic, the algorithm achieves $\widetilde{\mathcal{O}}(\sqrt{T})$ regret and constraint violation, while, when they are adversarial, it attains $\widetilde{\mathcal{O}}(\sqrt{T})$ constraint violation and a tight fraction of the optimal reward. Moreover, our algorithm is based on a policy optimization approach, which is much more efficient than occupancy-measure-based methods.

In online marketing, the advertisers aim to balance achieving high volumes and high profitability. The companies' business units address this tradeoff by maximizing the volumes while guaranteeing a minimum Return On Investment (ROI) level. Such a task can be naturally modeled as a combinatorial optimization problem subject to ROI and budget constraints that can be solved online. In this picture, the learner's uncertainty over the constraints' parameters plays a crucial role since the algorithms' exploration choices might lead to their violation during the entire learning process. Such violations represent a major obstacle to adopting online techniques in real-world applications. Thus, controlling the algorithms' exploration during learning is paramount to making humans trust online learning tools. This paper studies the nature of both optimization and learning problems. In particular, we show that the learning problem is inapproximable within any factor (unless P = NP) and provide a pseudo-polynomial-time algorithm to solve its discretized version. Subsequently, we prove that no online learning algorithm can violate the (ROI or budget) constraints a sublinear number of times during the learning process while guaranteeing a sublinear regret. We provide the GCB algorithm that guarantees sublinear regret at the cost of a linear number of constraint violations and GCBsafe that guarantees w.h.p.a constant upper bound on the number of constraint violations at the cost of a linear regret. Moreover, we designed GCBsafe(ψ, φ), which guarantees both sublinear regret and safety w.h.p. at the cost of accepting tolerances ψ and φ in the satisfaction of the ROI and budget constraints, respectively. Finally, we provide experimental results to compare the regret and constraint violations of GCB, GCBsafe, and GCBsafe(ψ, φ).

Online learning algorithms for designing optimal bilateral trade mechanisms have recently received significant attention. This paper addresses a key inefficiency in prior two-bit feedback models, which synchronously query both the buyer and the seller for their willingness to trade. This approach is inherently inefficient as it offers a trade to the seller even if the buyer rejects the offer. We propose an asynchronous mechanism that queries the seller only if the buyer has already accepted the offer. Consequently, the mechanism receives one bit of feedback from the buyer and a "censored" bit from the seller---a signal richer than the standard one-bit (trade/no-trade) feedback, but less informative than the two-bit model. Assuming independent valuations with bounded densities---the same distributional conditions underlying the two-bit results of Cesa-Bianchi et al. [2024a]---we design an algorithm that achieves $\tilde{O}(T^{2/3})$ regret against the best fixed price in hindsight. This matches the lower bound for the strictly richer two-bit model, showing that our mechanism elicits the minimal feedback necessary to attain optimal rates.

In Bayesian persuasion, an informed sender strategically discloses information to a receiver so as to persuade them to undertake desirable actions. Recently, Markov persuasion processes (MPPs) have been introduced to capture sequential scenarios where a sender faces a stream of myopic receivers in a Markovian environment. The MPPs studied so far in the literature suffer from issues that prevent them from being fully operational in practice, e.g., they assume that the sender knows receivers' rewards. We fix such issues by addressing MPPs where the sender has no knowledge about the environment. We design a learning algorithm for the sender, working with partial feedback. We prove that its regret with respect to an optimal information-disclosure policy grows sublinearly in the number of episodes, as it is the case for the loss in persuasiveness cumulated while learning. Moreover, we provide lower bounds for our setting matching the guarantees of our algorithm.

Correlated equilibria —and their generalization $\Phi$-equilibria— are a fundamental object of study in game theory, offering a more tractable alternative to Nash equilibria in multi-player settings. While computational aspects of equilibrium computation are well-understood in some settings, fundamental questions are still open in _generalized games_, that is, games in which the set of strategies allowed to each player depends on the other players' strategies. These classes of games model fundamental settings in economics and have been a cornerstone of economics research since the seminal paper of Arrow and Debreu [1954]. Recently, there has been growing interest, both in economics and in computer science, in studying correlated equilibria in generalized games. It is known that finding a social welfare maximizing correlated equilibrium in generalized games is NP-hard. However, the existence of efficient algorithms to find _any_ equilibrium remains an important open question. In this paper, we answer this question negatively, showing that this problem is PPAD-complete.

This paper initiates the study of data-dependent regret bounds in constrained MAB settings. These are bounds that depend on the sequence of losses that characterize the problem instance. Thus, in principle they can be much smaller than classical $\widetilde{\mathcal{O}}(\sqrt{T})$ regret bounds, while being equivalent to them in the worst case. Despite this, data-dependent regret bounds have been completely overlooked in constrained MABs. The goal of this paper is to answer the question: Can data-dependent regret bounds be derived in the presence of constraints? We provide an affirmative answer in constrained MABs with adversarial losses and stochastic constraints. Specifically, our main focus is on the most challenging and natural settings with hard constraints, where the learner must ensure that the constraints are always satisfied with high probability. We design an algorithm with a regret bound consisting of two data-dependent terms. The first one captures the difficulty of satisfying the constraints, while the second one encodes the complexity of learning independently of their presence. We also prove a lower bound showing that these two terms are not artifacts of our specific approach and analysis, but rather the fundamental components that inherently characterize the problem complexity. Finally, in designing our algorithm, we also derive some novel results in the related (and easier) soft constraints settings, which may be of independent interest.

We study online decision making problems under resource constraints, where both reward and cost functions are drawn from distributions that may change adversarially over time. We focus on two canonical settings: $(i)$ online resource allocation where rewards and costs are observed before action selection, and $(ii)$ online learning with resource constraints where they are observed after action selection, under full feedback or bandit feedback. It is well known that achieving sublinear regret in these settings is impossible when the rewards and cost distributions may change arbitrarily over time. To address this challenge, we analyze a framework in which the learner is guided by a spending plan—a sequence prescribing expected resource usage across rounds. We design general (primal-)dual methods that achieve sublinear regret with respect to baselines that follow the spending plan. Crucially, the performance of our algorithms improves when the spending plan ensures a well-balanced distribution of the budget across rounds. We additionally provide a robust variant of our methods to handle worst-case scenarios where the spending plan is highly imbalanced. To conclude, we study the regret of our algorithms when competing against benchmarks that deviate from the prescribed spending plan.

In constrained MDPs (CMDPs) with adversarial rewards and constraints, a known impossibility result prevents any algorithm from attaining sublinear regret and constraint violation, when competing against a best-in-hindsight policy that satisfies the constraints on average. In this paper, we show how to ease such a negative result, by considering settings that generalize both stochastic CMDPs and adversarial ones. We provide algorithms whose performances smoothly degrade as the level of environment adverseness increases. In this paper, we show that this negative result can be eased in CMDPs with non-stationary rewards and constraints, by providing algorithms whose performances smoothly degrade as non-stationarity increases. Specifically, they attain $\widetilde{\mathcal{O}} (\sqrt{T} + C)$ regret and positive constraint violation under bandit feedback, where $C$ measures the adverseness of rewards and constraints. This is $C = \Theta(T)$ in the worst case, coherently with the impossibility result for adversarial CMDPs. First, we design an algorithm with the desired guarantees when $C$ is known. Then, in the case $C$ is unknown, we obtain the same results by embedding multiple instances of such an algorithm in a general meta-procedure, which suitably selects them so as to balance the trade-off between regret and constraint violation.

We study a new class of contract design problems where a principal delegates the execution of multiple projects to a set of agents. The principal's expected reward from each project is a combinatorial function of the agents working on it. Each agent has limited capacity and can work on at most one project, and the agents are heterogeneous, with different costs and contributions for participating in different projects. The main challenge of the principal is to decide how to allocate the agents to projects when the number of projects grows in scale.

Many microeconomic models involve optimizing a piecewise linear function, such as in contract design, posted-price auctions, and first-price auctions. When key parameters are unknown and stochastic, the task becomes learning to optimize an unknown piecewise linear reward function. This is typically cast as an online learning problem, where the goal is to minimize regret relative to the best fixed decision in hindsight.

Information design and contract design are two important topics in microeconomics. In a principal-agent model, information design concerns the information asymmetry between two players, while contract design addresses the moral hazard challenge. Though they are typically studied as two disjoint problems, in many real-world scenarios, e.g., recruitment, information asymmetry and moral hazard could appear simultaneously.

The problem of contract design addresses the challenge of moral hazard in principle-agent setups. The agent exerts costly efforts that produce a random outcome with an associated reward for the principal. Moral hazard refers to the tension that the principal cannot observe the agent’s effort level hence needs to incentivize the agent only through rewarding the realized effort outcome, i.e., the contract. Bayesian contract design studies the principal’s design problem of an optimal contract when facing an unknown agent characterized by a private Bayesian type. In its most general form, the agent’s type is inherently “multi-parameter” and can arbitrarily affect both the agent’s productivity and effort costs. In contrast, a natural single-parameter setting of much recent interest simplifies the agent’s type to a single value that describes the agent’s cost per unit of effort, whereas agents’ efforts are assumed to be equally productive.

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme-called contract-in order to induce an agent to take a costly, unobservable action leading to favorable outcomes. We consider a generalization of the classical (single-round) version of the problem in which the principal interacts with the agent by committing to contracts over multiple rounds. The principal has no information about the agent, and they have to learn an optimal contract by only observing the outcome realized at each round. We focus on settings in which the size of the agent's action space is small. We design an algorithm that learns an approximately-optimal contract with high probability in a number of rounds polynomial in the size of the outcome space, when the number of actions is constant. Our algorithm solves an open problem by Zhu et al. [1]. Moreover, it can also be employed to provide a ((O) over tildeT(4/5)) regret bound in the related online learning setting in which the principal aims at maximizing their cumulative utility over rounds, considerably improving previously-known regret bounds.

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme---called contract---in order to induce an agent to take a costly, unobservable action leading to favorable outcomes. We consider a generalization of the classical (single-round) version of the problem in which the principal interacts with the agent by committing to contracts over multiple rounds. The principal has no information about the agent, and they have to learn an optimal contract by only observing the outcome realized at each round. We focus on settings in which the size of the agent's action space is small. We design an algorithm that learns an approximately-optimal contract with high probability in a number of rounds polynomial in the size of the outcome space, when the number of actions is constant. Our algorithm solves an open problem by Zhu et al. [2022]. Moreover, it can also be employed to provide a $\widetilde{\mathcal{O}}(T^{4/5})$ regret bound in the related online learning setting in which the principal aims at maximizing their cumulative utility, thus considerably improving previously-known regret bounds.

The bandits with knapsacks (BwK) framework models online decision-making problems in which an agent makes a sequence of decisions subject to resource consumption constraints. The traditional model assumes that each action consumes a non-negative amount of resources and the process ends when the initial budgets are fully depleted. We study a natural generalization of the BwK framework which allows non-monotonic resource utilization, i.e., resources can be replenished by a positive amount. We propose a best-of-both-worlds primal-dual template that can handle any online learning problem with replenishment for which a suitable primal regret minimizer exists. In particular, we provide the first positive results for the case of adversarial inputs by showing that our framework guarantees a constant competitive ratio $\alpha$ when $B=\Omega(T)$ or when the possible per-round replenishment is a positive constant. Moreover, under a stochastic input model, our algorithm yields an instance-independent $\tilde{\mathcal{O}}(T^{1/2})$ regret bound which complements existing instance-dependent bounds for the same setting. Finally, we provide applications of our framework to some economic problems of practical relevance.

We investigate the mechanism design problem faced by a principal who hires \emph{multiple} agents to gather and report costly information. Then, the principal exploits the information to make an informed decision. We model this problem as a game, where the principal announces a mechanism consisting in action recommendations and a payment function, a.k.a. scoring rule. Then, each agent chooses an effort level and receives partial information about an underlying state of nature based on the effort. Finally, the agents report the information (possibly non-truthfully), the principal takes a decision based on this information, and the agents are paid according to the scoring rule. While previous work focuses on single-agent problems, we consider multi-agents settings. This poses the challenge of coordinating the agents' efforts and aggregating correlated information. Indeed, we show that optimal mechanisms must correlate agents' efforts, which introduces externalities among the agents, and hence complex incentive compatibility constraints and equilibrium selection problems. First, we design a polynomial-time algorithm to find an optimal incentive compatible mechanism. Then, we study an online problem, where the principal repeatedly interacts with a group of unknown agents. We design a no-regret algorithm that provides $\widetilde{\mathcal{O}}(T^{2/3})$ regret with respect to an optimal mechanism, matching the state-of-the-art bound for single-agent settings.

We study online learning problems in which a decision maker has to make a sequence of costly decisions, with the goal of maximizing their expected reward while adhering to budget and return-on-investment (ROI) constraints. Existing primal-dual algorithms designed for constrained online learning problems under adversarial inputs rely on two fundamental assumptions. First, the decision maker must know beforehand the value of parameters related to the degree of strict feasibility of the problem (i.e. Slater parameters). Second, a strictly feasible solution to the offline optimization problem must exist at each round. Both requirements are unrealistic for practical applications such as bidding in online ad auctions. In this paper, we show how such assumptions can be circumvented by endowing standard primal-dual templates with weakly adaptive regret minimizers. This results in a “dual-balancing” framework which ensures that dual variables stay sufficiently small, even in the absence of knowledge about Slater’s parameter. We prove the first best-of-both-worlds no-regret guarantees which hold in absence of the two aforementioned assumptions, under stochastic and adversarial inputs. Finally, we show how to instantiate the framework to optimally bid in various mechanisms of practical relevance, such as first- and second-price auctions.

In this paper, we propose a novel generalization of rested and restless bandits where the evolution of the arms’ expected rewards is governed by a graph defined over the arms. An edge connecting a pair of arms $(i,j)$ represents the fact that a pull of arm $i$ triggers the evolution of arm $j$, and vice versa. Interestingly, rested and restless bandits are both special cases of our model for some suitable (degenerate) graphs. Still, the model can represent way more general and interesting scenarios. We first tackle the problem of computing the optimal policy when no specific structure is assumed on the graph, showing that it is NP-hard. Then, we focus on a specific structure forcing the graph to be composed of a set of fully connected subgraphs (i.e., cliques), and we prove that the optimal policy can be easily computed in closed form. Then, we move to the learning problem presenting regret minimization algorithms for deterministic and stochastic cases. Our regret bounds highlight the complexity of the learning problem by incorporating instance-dependent terms that encode specific properties of the underlying graph structure. Moreover, we illustrate how the knowledge of the underlying graph is not necessary for achieving the no-regret property.

We study online learning in episodic constrained Markov decision processes (CMDPs), where the learner aims at collecting as much reward as possible over the episodes, while satisfying some long-term constraints during the learning process. Rewards and constraints can be selected either stochastically or adversarially, and the transition function is not known to the learner. While online learning in classical (unconstrained) MDPs has received considerable attention over the last years, the setting of CMDPs is still largely unexplored. This is surprising, since in real-world applications, such as, e.g., autonomous driving, automated bidding, and recommender systems, there are usually additional constraints and specifications that an agent has to obey during the learning process. In this paper, we provide the first best-of-both-worlds algorithm for CMDPs with long-term constraints, in the flavor of Balseiro et al. (2023). Our algorithm is capable of handling settings in which rewards and constraints are selected either stochastically or adversarially, without requiring any knowledge of the underling process. Moreover, our algorithm matches state-of-the-art regret and constraint violation bounds for settings in which constraints are selected stochastically, while it is the first to provide guarantees in the case in which they are chosen adversarially.

We study online Bayesian persuasion problems in which an informed sender repeatedly faces a receiver with the goal of influencing their behavior through the provision of payoff-relevant information. Previous works assume that the sender has knowledge about either the prior distribution over states of nature or receiver's utilities, or both. We relax such unrealistic assumptions by considering settings in which the sender does not know anything about the prior and the receiver. We design an algorithm that achieves sublinear---in the number of rounds T---regret with respect to an optimal signaling scheme, and we also provide a collection of lower bounds showing that the guarantees of such an algorithm are tight. Our algorithm works by searching a suitable space of signaling schemes in order to learn receiver's best responses. To do this, we leverage a non-standard representation of signaling schemes that allows to cleverly overcome the challenge of not knowing anything about the prior over states of nature and receiver's utilities. Finally, our results also allow to derive lower/upper bounds on the sample complexity of learning signaling schemes in a related Bayesian persuasion PAC-learning problem.

We address a generalization of the bandit with knapsacks problem, where a learner aims to maximize rewards while satisfying an arbitrary set of long-term constraints. Our goal is to design best-of-both-worlds algorithms that perform optimally under both stochastic and adversarial constraints. Previous works address this problem via primal-dual methods, and require some stringent assumptions, namely the Slater's condition, and in adversarial settings, they either assume knowledge of a lower bound on the Slater's parameter, or impose strong requirements on the primal and dual regret minimizers such as requiring weak adaptivity. We propose an alternative and more natural approach based on optimistic estimations of the constraints. Surprisingly, we show that estimating the constraints with an UCB-like approach guarantees optimal performances. Our algorithm consists of two main components: (i) a regret minimizer working on moving strategy sets and (ii) an estimate of the feasible set as an optimistic weighted empirical mean of previous samples. The key challenge in this approach is designing adaptive weights that meet the different requirements for stochastic and adversarial constraints. Our algorithm is significantly simpler than previous approaches, and has a cleaner analysis. Moreover, ours is the first best-of-both-worlds algorithm providing bounds logarithmic in the number of constraints. Additionally, in stochastic settings, it provides $\widetilde O(\sqrt{T})$ regret without Slater's condition.

In this paper, we introduce a novel variation of multi-armed bandits called bandits with ranking feedback. Unlike traditional bandits, this variation provides feedback to the learner that allows them to rank the arms based on previous pulls, without quantifying numerically the difference in performance. This type of feedback is well-suited for scenarios where the arms' values cannot be precisely measured using metrics such as monetary scores, probabilities, or occurrences. Common examples include human preferences in matchmaking problems. Furthermore, its investigation answers the theoretical question on how numerical rewards are crucial in bandit settings. In particular, we study the problem of designing no-regret algorithms with ranking feedback both in the stochastic and adversarial settings. We show that, with stochastic rewards, differently from what happens with non-ranking feedback, no algorithm can suffer a logarithmic regret in the time horizon $T$ in the instance-dependent case. Furthermore, we provide two algorithms. The first, namely DREE, guarantees a superlogarithmic regret in $T$ in the instance-dependent case thus matching our lower bound, while the second, namely R-LPE, guarantees a regret of $\mathcal{\widetilde O}(\sqrt{T})$ in the instance-independent case. Remarkably, we show that no algorithm can have an optimal regret bound in both instance-dependent and instance-independent cases. Finally, we prove that no algorithm can achieve a sublinear regret when the rewards are adversarial.

We study the problem faced by a service provider that has to sell services to a user. In our model the service provider proposes various payment options (a menu) to the user which may be based, for example, on the quality of the service. Then, the user chooses one of these options and pays an amount to the service provider, contingent on the observed final outcome. Users are not able to observe directly the action performed by the service provide to reach the final outcome. This might incentivize misconduct. Therefore, we propose a model that enforces trust through economics incentives. The problem has two crucial features: i) the service provider is responsible for both formulating the contract and performing the action for which the user issues payments, and ii) the user is unaware of the true action carried out by the service provider, which is hidden. We study this delegation problem through the lens of contract design, with the overarching goal of enabling the computation of contracts that guarantee that the user can trust the service provider, even if their action is hidden.

We study hidden-action principal-agent problems with multiple agents, in which each agent has an arbitrary number of actions, and the joint action induces outcomes according to an arbitrary distribution. We study two classes of mechanisms: deterministic mechanisms, in which the agents play a Nash equilibrium of the game induced by the contract, and randomized mechanisms, inspired by single-agent randomized contracts and correlated equilibria. First, we compare the two classes, designing a polynomial-time algorithm to compute almost optimal contracts and showing that randomized mechanisms provide an arbitrarily larger principal's utility than deterministic ones.

Bilateral trade models the problem of intermediating between two rational agents — a seller and a buyer — both characterized by a private valuation for an item they want to trade. We study the online learning version of the problem, in which at each time step a new seller and buyer arrive and the learner has to set prices for them without any knowledge about their (adversarially generated) valuations.

Bayesian persuasion studies how an informed sender should influence beliefs of rational receivers that take decisions through Bayesian updating of a common prior. We focus on the online Bayesian persuasion framework, in which the sender repeatedly faces one or more receivers with unknown and adversarially selected types. First, we show how to obtain a tight $\tilde O(T^{1/2})$ regret bound in the case in which the sender faces a single receiver and has bandit feedback, improving over the best previously known bound of $\tilde O(T^{4/5})$. Then, we provide the first no-regret guarantees for the multi-receiver setting under bandit feedback. Finally, we show how to design no-regret algorithms with polynomial per-iteration running time by exploiting type reporting, thereby circumventing known complexity results on online Bayesian persuasion. We provide efficient algorithms guaranteeing a $O(T^{1/2})$ regret upper bound both in the single- and multi-receiver scenario when type reporting is allowed.

The computational study of equilibria involving constraints on players’ strategies has been largely neglected. However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored. In this paper, we introduce and computationally characterize constrained Phi-equilibria—a more general notion than constrained CEs—in normal-form games. We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players’ actions is fixed. Moreover, in the special case in which a player’s constraints do not depend on other players’ strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.

We study the problem of designing mechanisms for information acquisition scenarios. This setting models strategic interactions between a uniformed receiver and a set of informed senders. In our model the senders receive information about the underlying state of nature and communicate their observation (either truthfully or not) to the receiver, which, based on this information, selects an action. Our goal is to design mechanisms maximizing the receiver's utility while incentivizing the senders to report truthfully their information. First, we provide an algorithm that efficiently computes an optimal incentive compatible (IC) mechanism. Then, we focus on the online problem in which the receiver sequentially interacts in an unknown game, with the objective of minimizing the cumulative regret w.r.t. the optimal IC mechanism, and the cumulative violation of the incentive compatibility constraints. We investigate two different online scenarios, i.e., the full and bandit feedback settings. For the full feedback problem, we propose an algorithm that guarantees $\tilde{O}(\sqrt{T})$ regret and violation, while for the bandit feedback setting we present an algorithm that attains $\tilde{O}(T^{\alpha})$ regret and $\tilde{O}(T^{1-\alpha/2})$ violation for any $\alpha \in [1/2, 1]$. Finally, we complement our results providing a tight lower bound.

Bayesian persuasion studies the problem faced by an informed sender who strategically discloses information to influence the behavior of an uninformed receiver. Recently, a growing attention has been devoted to settings where the sender and the receiver interact sequentially, in which the receiver's decision-making problem is usually modeled as a Markov decision process (MDP). However, the literature focuses on computing optimal information-revelation policies (a.k.a. signaling schemes) under the restrictive assumption that the receiver acts myopically, selecting actions to maximize the one-step utility and disregarding future rewards. This is justified by the fact that, when the receiver is farsighted and thus considers future rewards, finding an optimal Markovian signaling scheme is NP-hard. In this paper, we show that Markovian signaling schemes do not constitute the "right" class of policies. Indeed, differently from most of the MDPs settings, we show that Markovian signaling schemes are not optimal, and general history-dependent signaling schemes should be considered. Moreover, we also show that history-dependent signaling schemes circumvent the negative complexity results affecting Markovian signaling schemes. Formally, we design an algorithm that computes an optimal and $\epsilon$-persuasive history-dependent signaling scheme in time polynomial in ${1}/{\epsilon}$ and in the instance size. The crucial challenge is that general history-dependent signaling schemes cannot be represented in polynomial space. Nevertheless, we introduce a convenient subclass of history-dependent signaling schemes, called promise-form, which are as powerful as general history-dependent ones and efficiently representable. Intuitively, promise-form signaling schemes compactly encode histories in the form of honest promises on future receiver's rewards.

We study hidden-action principal-agent problems with multiple agents. These are problems in which a principal commits to an outcome-dependent payment scheme (called contract) in order to incentivize some agents to take costly, unobservable actions that lead to favorable outcomes. Previous works study models where the principal observes a single outcome determined by the actions of all the agents. This considerably limits the contracting power of the principal, since payments can only depend on the joint result achieved by the agents.

We study hidden-action principal-agent problems in which a principal commits to an outcome-dependent payment scheme (called contract) so as to incentivize the agent to take a costly, unobservable action leading to favorable outcomes. In particular, we focus on Bayesian settings where the agent has private information. This is collectively encoded by the agent's type, which is unknown to the principal but randomly drawn according to a finitely-supported, commonly-known probability distribution. In our model, the agent's type determines both the probability distribution over outcomes and the cost associated with each agent's action. In Bayesian principal-agent problems, the principal may be better off by committing to a menu of contracts specifying a contract for each agent's type, rather than committing to a single contract. This induces a two-stage process that resembles interactions studied in classical mechanism design: after the principal has committed to a menu, the agent first reports a type to the principal, and, then, the latter puts in place the contract in the menu that corresponds to the reported type. Thus, the principal's computational problem boils down to designing a menu of contracts that incentivizes the agent to report their true type and maximizes expected utility. Previous works showed that, in Bayesian principal-agent problems, computing an optimal menu of contracts or an optimal (single) contract is APX-hard, which is in sharp contrast from what happens in non-Bayesian settings, where an optimal contract can be computed efficiently. Crucially, previous works focus on menus of deterministic contracts. Surprisingly, in this paper, we show that, if one instead considers menus of randomized contracts defined as probability distributions over payment vectors, then an “almost-optimal” menu can be computed in polynomial time. Indeed, the problem of computing a principal-optimal menu of randomized contracts may not admit a maximum, but only a supremum. Nevertheless, we show how to design a polynomial-time algorithm that guarantees the principal with an expected utility arbitrarily close to the supremum. Besides this main result, we also close several gaps in the computational complexity analysis of the problem of computing menus of deterministic contracts. In particular, we prove that the problem cannot be approximated up to within any multiplicative factor and it does not admit an additive FPTAS unless P = NP, even in basic instances with a constant number of actions and only four outcomes. This considerably extends previously-known negative results. Then, we show that our hardness result is tight, by providing an additive PTAS that works in instances with a constant number of outcomes. We complete our analysis by showing that an optimal menu of deterministic contracts can be computed in polynomial time when either there are only two outcomes or there is a constant number of types.

We study single-item single-unit Bayesian posted price auctions, where buyers arrive sequentially and their valuations for the item being sold depend on a random, unknown state of nature. The seller has complete knowledge of the actual state and can send signals to the buyers so as to disclose information about it. For instance, the state of nature may reflect the condition and/or some particular features of the item, which are known to the seller only. The problem faced by the seller is about how to partially disclose information about the state so as to maximize revenue. Unlike classical signaling problems, in this setting, the seller must also correlate the signals being sent to the buyers with some price proposals for them. This introduces additional challenges compared to standard settings. As a preliminary step, we show that, w.l.o.g., the seller can deterministically propose a price to each buyer on the basis of the signal being sent to that buyer, rather than selecting prices stochastically and arbitrarily correlating them with signals sent to all the buyers. Next, we consider two cases: the one where the seller can only send signals publicly visible to all buyers, and the case in which the seller can privately send a different signal to each buyer. As a first step, we prove that, in both settings, the problem of maximizing the seller's revenue does not admit an additive FPTAS unless P=NP, even for basic instances with a single buyer. As a result, in the rest of the paper, we focus on designing additive PTASs. In order to do so, we first introduce a unifying framework encompassing both public and private signaling, whose core result is a decomposition lemma that allows focusing on a finite set of possible buyers' posteriors. This forms the basis on which our additive PTASs are developed. In particular, in the public signaling setting, our PTAS employs some ad hoc techniques based on linear programming, while our PTAS for the private setting relies on the ellipsoid method to solve an exponentially-sized LP in polynomial time. In the latter case, we need a custom approximate separation oracle, which we implement with a dynamic programming approach.

Abstract We study algorithmic Bayesian persuasion problems in which the principal (a.k.a. the sender) has to persuade multiple agents (a.k.a. receivers) by using public communication channels. Specifically, our model follows the multi-receiver model with no inter-agent externalities introduced by Arieli and Babichenko (J Econ Theory 182:185–217, 2019). It is known that the problem of computing a sender-optimal public persuasive signaling scheme is not approximable even in simple settings. Therefore, prior works usually focus on determining restricted classes of the problem for which efficient approximation is possible. Typically, positive results in this space amounts to finding bi-criteria approximation algorithms yielding an almost optimal and almost persuasive solution in polynomial time. In this paper, we take a different perspective and study the persuasion problem in the general setting where the space of the states of nature, the action space of the receivers, and the utility function of the sender can be arbitrary. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling scheme in such settings. In particular, we show that, assuming the Exponential Time Hypothesis, solving this problem requires at least a quasi-polynomial number of steps even in instances with simple utility functions and binary action spaces such as an election with the k-voting rule. In doing so, we prove that a relaxed version of the Maximum Feasible Subsystem of Linear Inequalities problem requires at least quasi-polynomial time to be solved. Finally, we close the gap by providing a quasi-polynomial time bi-criteria approximation algorithm for arbitrary public persuasion problems that, under mild assumptions, yields a QPTAS.

Most economic reports suggest that almost half of the market value unlocked by artificial intelligence (AI) by the next decade (about 9 trillion USD per year) will be in marketing&sales. In particular, AI will allow the optimization of more and more intricate economic settings in which multiple different activities can be automated jointly. A relatively recent example is that one of ad auctions in which similar products or services are displayed together with their price, thus merging advertising and pricing in a unique website. This is the case, e.g., of Google Hotel Ads and TripAdvisor. More precisely, as in a classical ad auction, the ranking of the ads depends on the advertisers' bids, while, differently from classical ad auctions, the price is displayed together with the ad, so as to provide a direct comparison among the prices and thus dramatically affect the behavior of the users. This paper investigates how displaying prices and ads together conditions the properties of the main economic mechanisms such as VCG and GSP. Initially, we focus on the direct-revelation mechanism, showing that prices are chosen by the mechanisms once given the advertisers' reports. We also provide an efficient algorithm to compute the optimal allocation given the private information reported by the advertisers. Then, with both VCG and GSP payments, we show the inefficiency in terms of Price of Anarchy (PoA) and Stability (PoS) over the social welfare and mechanism's revenue when the advertisers choose the prices. The main results show that, with both VCG and GSP, PoS over the revenue may be unbounded even with two slots, while PoA over the social welfare may be as large as the number of slots. Finally, we show that, under some assumptions, simple modifications to VCG and GSP allow us to obtain a better PoS over the revenue.

We study single-item single-unit Bayesian posted price auctions, where buyers arrive sequentially and their valuations for the item being sold depend on a random, unknown state of nature. The seller has complete knowledge of the actual state and can send signals to the buyers so as to disclose information about it. For instance, the state of nature may reflect the condition and/or some particular features of the item, which are known to the seller only. The problem faced by the seller is about how to partially disclose information about the state so as to maximize revenue. Unlike classical signaling problems, in this setting, the seller must also correlate the signals being sent to the buyers with some price proposals for them. This introduces additional challenges compared to standard settings. We consider two cases: the one where the seller can only send signals publicly visible to all buyers, and the case in which the seller can privately send a different signal to each buyer. As a first step, we prove that, in both settings, the problem of maximizing the seller's revenue does not admit an FPTAS unless P=NP, even for basic instances with a single buyer. As a result, in the rest of the paper, we focus on designing PTASs. In order to do so, we first introduce a unifying framework encompassing both public and private signaling, whose core result is a decomposition lemma that allows focusing on a finite set of possible buyers' posteriors. This forms the basis on which our PTASs are developed. In particular, in the public signaling setting, our PTAS employs some ad hoc techniques based on linear programming, while our PTAS for the private setting relies on the ellipsoid method to solve an exponentially-sized LP in polynomial time. In the latter case, we need a custom approximate separation oracle, which we implement with a dynamic programming approach.

We study online learning problems in which a decision maker wants to maximize their expected reward without violating a finite set of $m$ resource constraints. By casting the learning process over a suitably defined space of strategy mixtures, we recover strong duality on a Lagrangian relaxation of the underlying optimization problem, even for general settings with non-convex reward and resource-consumption functions. Then, we provide the first best-of-both-worlds type framework for this setting, with no-regret guarantees both under stochastic and adversarial inputs. Our framework yields the same regret guarantees of prior work in the stochastic case. On the other hand, when budgets grow at least linearly in the time horizon, it allows us to provide a constant competitive ratio in the adversarial case, which improves over the $O(m \log T)$ competitive ratio of Immorlica et al. [FOCS’19]. Moreover, our framework allows the decision maker to handle non-convex reward and cost functions. We provide two game-theoretic applications of our framework to give further evidence of its flexibility.

We study decision making problems in which an agent sequentially interacts with a stochastic environment defined by means of a tree structure. The agent repeatedly faces the environment over time, and, after each round, it perceives a utility and a cost, which are both stochastic. The goal of the agent is to learn an optimal strategy in an online fashion, while, at the same time, keeping costs below a given safety threshold. Our model naturally fits many real-world scenarios, such as, e.g., opponent exploitation in games and web link selection. We study the hard-threshold problem of achieving sublinear regret while guaranteeing that the threshold constraint is satisfied at every iteration with high probability. First, we show that, in general, any algorithm with such a guarantee incurs in a linear regret. This motivates the introduction of a relaxed problem, namely the soft-threshold problem, in which we only require that the cumulative violation of the threshold constraint grows sublinearly, and, thus, we can provide an algorithm with sublinear regret. Next, we show how, in the hard-threshold problem, a sublinear regret algorithm can be designed under the additional assumption that there exists a known strategy strictly satisfying the threshold constraint. We also show that our regret bounds are tight. Finally, we cast the opponent exploitation problem to our model, and we experimentally evaluate our algorithms on a standard testbed of games.

We study signaling in Bayesian ad auctions, in which bidders' valuations depend on a random, unknown state of nature. The auction mechanism has complete knowledge of the actual state of nature, and it can send signals to bidders so as to disclose information about the state and increase revenue. For instance, a state may collectively encode some features of the user that are known to the mechanism only, since the latter has access to data sources unaccessible to the bidders. We study the problem of computing how the mechanism should send signals to bidders in order to maximize revenue. While this problem has already been addressed in the easier setting of second-price auctions, to the best of our knowledge, our work is the first to explore ad auctions with more than one slot. In this paper, we focus on public signaling and VCG mechanisms, under which bidders truthfully report their valuations. We start with a negative result, showing that, in general, the problem does not admit a PTAS unless P = NP, even when bidders' valuations are known to the mechanism. The rest of the paper is devoted to settings in which such negative result can be circumvented. First, we prove that, with known valuations, the problem can indeed be solved in polynomial time when either the number of states d or the number of slots m is fixed. Moreover, in the same setting, we provide an FPTAS for the case in which bidders are single minded, but d and m can be arbitrary. Then, we switch to the random valuations setting, in which these are randomly drawn according to some probability distribution. In this case, we show that the problem admits an FPTAS, a PTAS, and a QPTAS, when, respectively, d is fixed, m is fixed, and bidders' valuations are bounded away from zero.

The increasing competition in digital advertising induced a proliferation of media agencies playing the role of intermediaries between advertisers and platforms selling ad slots. When a group of competing advertisers is managed by a common agency, many forms of collusion, such as bid rigging, can be implemented by coordinating bidding strategies, dramatically increasing advertisers' value. We study the problem of finding bids and monetary transfers maximizing the utility of a group of colluders, under GSP and VCG mechanisms. First, we introduce an abstract bid optimization problem---called weighted utility problem (WUP)---, which is useful in proving our results. We show that the utilities of bidding strategies are related to the length of paths in a directed acyclic weighted graph, whose structure and weights depend on the mechanism under study. This allows us to solve WUP in polynomial time by finding a shortest path of the graph. Next, we switch to our original problem, focusing on two settings that differ for the incentives they allow for. Incentive constraints ensure that colluders do not leave the agency, and they can be enforced by implementing monetary transfers between the agency and the advertisers. In particular, we study the arbitrary transfers setting, where any kind of monetary transfer to and from the advertisers is allowed, and the more realistic limited liability setting, in which no advertiser can be paid by the agency. In the former, we cast the problem as a WUP instance and solve it by our graph-based algorithm, while, in the latter, we formulate it as a linear program with exponentially-many variables efficiently solvable by applying the ellipsoid algorithm to its dual. This requires to solve a suitable separation problem in polynomial time, which can be done by reducing it to the weighted utility problem a WUP instance.

We study a repeated information design problem faced by an informed sender who tries to influence the behavior of a self-interested receiver. We consider settings where the receiver faces a sequential decision making (SDM) problem. At each round, the sender observes the realizations of random events in the SDM problem. This begets the challenge of how to incrementally disclose such information to the receiver to persuade them to follow (desirable) action recommendations. We study the case in which the sender does not know random events probabilities, and, thus, they have to gradually learn them while persuading the receiver. Our goal is to design online learning algorithms that are no-regret for the sender, while at the same time being persuasive for the receiver. We start by providing a non-trivial polytopal approximation of the set of sender's persuasive information structures. This is crucial to design efficient learning algorithms. Next, we prove a negative result: no learning algorithm can be persuasive. Thus, we relax persuasiveness requirements by focusing on algorithms that guarantee that the receiver's regret in following recommendations grows sub-linearly. In the full-feedback setting---where the sender observes all random events realizations---, we provide an algorithm with $\tilde{O}(\sqrt{T})$ regret for both the sender and the receiver. Instead, in the bandit-feedback setting---where the sender only observes the realizations of random events actually occurring in the SDM problem---, we design an algorithm that, given an $\alpha \in [1/2, 1]$ as input, ensures $\tilde{O}({T^\alpha})$ and $\tilde{O}( T^{\max \{ \alpha, 1-\frac{\alpha}{2} \} })$ regrets for the sender and the receiver, respectively. This result is complemented by a lower bound showing that such a regrets trade-off is essentially tight.

We study online learning problems in which a decision maker has to take a sequence of decisions subject to $m$ long-term constraints. The goal of the decision maker is to maximize their total reward, while at the same time achieving small cumulative constraints violations across the $T$ rounds. We present the first best-of-both-world type algorithm for this general class of problems, with no-regret guarantees both in the case in which rewards and constraints are selected according to an unknown stochastic model, and in the case in which they are selected at each round by an adversary. Our algorithm is the first to provide guarantees in the adversarial setting with respect to the optimal fixed strategy that satisfies the long-term constraints. In particular, it guarantees a $\rho/(1+\rho)$ fraction of the optimal utility and sublinear regret, where $\rho$ is a feasibility parameter related to the existence of strictly feasible solutions. Our framework employs traditional regret minimizers as black-box components. Therefore, by instantiating it with an appropriate choice of regret minimizers it can handle both the full-feedback as well as the bandit-feedback setting. Moreover, it allows the decision maker to seamlessly handle scenarios with non-convex reward and constraints. We show how our framework may be applied in the context of budget-management mechanisms for repeated auctions in order to guarantee long-term constraints which are not packing (e.g., ROI constraints).

We study hidden-action principal-agent problems in which a principal commits to an outcome-dependent payment scheme (called contract) so as to incentivize the agent to take a costly, unobservable action leading to favorable outcomes. In particular, we focus on Bayesian settings where the agent has private information. This is collectively encoded by the agent's type, which is unknown to the principal, but randomly drawn according to a finitely-supported, commonly-known probability distribution. In our model, the agent's type determines both the probability distribution over outcomes and the cost associated with each agent's action. In Bayesian principal-agent problems, the principal may be better off by committing to a menu of contracts specifying a contract for each agent's type, rater than committing to a single contract. This induces a two-stage process that resembles interactions studied in classical mechanism design: after the principal has committed to a menu, the agent first reports a type to the principal, and, then, the latter puts in place the contract in the menu that corresponds to the reported type. Thus, the principal's computational problem boils down to designing a menu of contracts that incentivizes the agent to report their true type and maximizes expected utility.

We focus on the scenario in which an agent can exploit his information advantage to manipulate the outcome of an election. In particular, we study district-based elections with two candidates, in which the winner of the election is the candidate that wins in the majority of the districts. District-based elections are adopted worldwide (e.g., UK and USA) and are a natural extension of widely studied voting mechanisms (e.g., k-voting and plurality voting). We resort to the Bayesian persuasion framework, where the manipulator (sender) strategically discloses information to the voters (receivers) that update their beliefs rationally. We study both private signaling in which the sender can use a private communication channel per receiver and public signaling in which the sender can use a single communication channel for all the receivers. Furthermore, for the first time, we introduce semi-public signaling in which the sender can use a single communication channel per district. We show that there is a sharp distinction between private and (semi-)public signaling. In particular, optimal private signaling schemes can provide an arbitrarily better probability of victory than (semi-)public ones and can be computed efficiently, while optimal (semi-)public signaling schemes cannot be approximated to within any factor in polynomial time unless P=NP. However, we show that reasonable relaxations allow the design of multi-criteria PTASs for optimal (semi-)public signaling schemes. In doing so, we introduce a novel property, namely comparative stability, and we design a bi-criteria PTAS for public signaling in general Bayesian persuasion problems beyond elections when the sender's utility function is state-dependent.

Network congestion games are a well-understood model of multi-agent strategic interactions. Despite their ubiquitous applications, it is not clear whether it is possible to design information structures to ameliorate the overall experience of the network users. We focus on Bayesian games with atomic players, where network vagaries are modeled via a (random) state of nature which determines the costs incurred by the players. A third-party entity—the sender—can observe the realized state of the network and exploit this additional information to send a signal to each player. A natural question is the following: is it possible for an informed sender to reduce the overall social cost via the strategic provision of information to players who update their beliefs rationally? The paper focuses on the problem of computing optimal ex ante persuasive signaling schemes, showing that symmetry is a crucial property for its solution. Indeed, we show that an optimal ex ante persuasive signaling scheme can be computed in polynomial time when players are symmetric and have affine cost functions. Moreover, the problem becomes NP-hard when players are asymmetric, even in non-Bayesian settings.

Bayesian persuasion studies how an informed sender should partially disclose information to influence the behavior of a self-interested receiver. Classical models make the stringent assumption that the sender knows the receiver’s utility. This can be relaxed by considering an online learning framework in which the sender repeatedly faces a receiver of an unknown, adversarially selected type. We study, for the first time, an online Bayesian persuasion setting with multiple receivers. We focus on the case with no externalities and binary actions, as customary in offline models. Our goal is to design no-regret algorithms for the sender with polynomial per-iteration running time. First, we prove a negative result: for any 0 < $\alpha$ $\leq$ 1, there is no polynomial-time no-$\alpha$-regret algorithm when the sender’s utility function is supermodular or anonymous. Then, we focus on the setting of submodular sender’s utility functions and we show that, in this case, it is possible to design a polynomial-time no-(1-1/e)-regret algorithm. To do so, we introduce a general online gradient descent framework to handle online learning problems with a finite number of possible loss functions. This requires the existence of an approximate projection oracle. We show that, in our setting, there exists one such projection oracle which can be implemented in polynomial time.

We focus on the election manipulation problem through social influence, where a manipulator exploits a social network to make her most preferred candidate win an election. Influence is due to information in favor of and/or against one or multiple candidates, sent by seeds and spreading through the network according to the independent cascade model. We provide a comprehensive theoretical study of the election control problem, investigating two forms of manipulations: seeding to buy influencers given a social network and removing or adding edges in the social network given the set of the seeds and the information sent. In particular, we study a wide range of cases distinguishing in the number of candidates or the kind of information spread over the network. Our main result shows that the election manipulation problem is not affordable in the worst-case, even when one accepts to get an approximation of the optimal margin of victory, except for the case of seeding when the number of hard-to-manipulate voters is not too large, and the number of uncertain voters is not too small, where we say that a voter that does not vote for the manipulator's candidate is hard-to-manipulate if there is no way to make her vote for this candidate, and uncertain otherwise. We also provide some results showing the hardness of the problems in special cases. More precisely, in the case of seeding, we show that the manipulation is hard even if the graph is a line and that a large class of algorithms, including most of the approaches recently adopted for social-influence problems (e.g., greedy, degree centrality, PageRank, VoteRank), fails to compute a bounded approximation even on elementary networks, such as undirected graphs with every node having a degree at most two or directed trees. In the case of edge removal or addition, our hardness results also apply to election manipulation when the manipulator has an unlimited budget, being allowed to remove or add an arbitrary number of edges, and to the basic case of social influence maximization/minimization in the restricted case of finite budget. Interestingly, our hardness results for seeding and edge removal/addition still hold in a re-optimization variant, where the manipulator already knows an optimal solution to the problem and computes a new solution once a local modification occurs, e.g., the removal/addition of a single edge.

We focus on the following natural question: is it possible to influence the outcome of a voting process through the strategic provision of information to voters who update their beliefs rationally? We investigate whether it is computationally tractable to design a signaling scheme maximizing the probability with which the sender's preferred candidate is elected. We resort to the model recently introduced by Arieli and Babichenko (2019) (i.e., without inter-agent externalities), and focus on, as illustrative examples, k-voting rules and plurality voting. There is a sharp contrast between the case in which private signals are allowed and the more restrictive setting in which only public signals are allowed. In the former, we show that an optimal signaling scheme can be computed efficiently both under a k-voting rule and plurality voting. In establishing these results, we provide two contributions applicable to general settings beyond voting. Specifically, we extend a well-known result by Dughmi and Xu (2017) to more general settings and prove that, when the sender's utility function is anonymous, computing an optimal signaling scheme is fixed-parameter tractable in the number of receivers' actions. In the public signaling case, we show that the sender's optimal expected return cannot be approximated to within any factor under a k-voting rule. This negative result easily extends to plurality voting and problems where utility functions are anonymous.

We focus on the scenario in which messages pro and/or against one or multiple candidates are spread through a social network in order to affect the votes of the receivers. Several results are known in the literature when the manipulator can make seeding by buying influencers. In this paper, instead, we assume the set of influencers and their messages to be given, and we ask whether a manipulator (e.g., the platform) can alter the outcome of the election by adding or removing edges in the social network. We study a wide range of cases distinguishing for the number of candidates or for the kind of messages spread over the network. We provide a positive result, showing that, except for trivial cases, manipulation is not affordable, the optimization problem being hard even if the manipulator has an unlimited budget (i.e., he can add or remove as many edges as desired). Furthermore, we prove that our hardness results still hold in a reoptimization variant, where the manipulator already knows an optimal solution to the problem and needs to compute a new solution once a local modification occurs (e.g., in bandit scenarios where estimations related to random variables change over time).

In Bayesian persuasion, an informed sender has to design a signaling scheme that discloses the right amount of information so as to influence the behavior of a self-interested receiver. This kind of strategic interaction is ubiquitous in real economic scenarios. However, the original model by Kamenica and Gentzkow makes some stringent assumptions which limit its applicability in practice. One of the most limiting assumptions is arguably that, in order to compute an optimal signaling scheme, the sender is usually required to know the receiver's utility function. In this paper, we relax this assumption through an online learning framework in which the sender faces a receiver with unknown type. At each round, the receiver's type is chosen adversarially from a finite set of possible types. We are interested in no-regret algorithms prescribing a signaling scheme at each round of the repeated interaction with performances close to that of the best-in-hindsight signaling scheme. First, we prove a hardness result on the per-iteration running time required to achieve the no-regret property. Then, we provide algorithms for the full and partial information model which exhibit regret sublinear in the number of rounds and polynomial in the parameters of the game.

We study the problem of computing correlated strategies to commit to in games with multiple leaders and followers. To the best of our knowledge, this problem is widely unexplored so far, as the majority of the works in the literature focus on games with a single leader and one or more followers. The fundamental ingredient of our model is that a leader can decide whether to participate in the commitment or to defect from it by taking on the role of follower. This introduces a preliminary stage where, before the underlying game is played, the leaders make their decisions to reach an agreement on the correlated strategy to commit to. We distinguish three solution concepts on the basis of the constraints that they enforce on the agreement reached by the leaders. Then, we provide a comprehensive study of the properties of our solution concepts, in terms of existence, relation with other solution concepts, and computational complexity.

We study the problem of finding Stackelberg equilibria in games with a massive number of players. So far, the only known game instances in which the problem is solved in polynomial time are some particular congestion games. However, a complete characterization of hard and easy instances is still lacking. In this paper, we extend the state of the art along two main directions. First, we focus on games where players' actions are made of multiple resources, and we prove that the problem is NP-hard and not in Poly-APX unless P = NP, even in the basic case in which players are symmetric, their actions are made of only two resources, and the cost functions are monotonic. Second, we focus on games with singleton actions where the players are partitioned into classes, depending on which actions they have available. In this case, we provide a dynamic programming algorithm that finds an equilibrium in polynomial time, when the number of classes is fixed and the leader plays pure strategies. Moreover, we prove that, if we allow for leader's mixed strategies, then the problem becomes NP-hard even with only four classes and monotonic costs. Finally, for both settings, we provide mixed-integer linear programming formulations, and we experimentally evaluate their scalability on both random game instances and worst-case instances based on our hardness reductions.