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.

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.

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.

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).

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.

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.

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.

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.

Policy gradient is one of the most famous algorithms in reinforcement learning. This paper studies the mean dynamics of the soft-max policy gradient algorithm and its properties in multi-agent settings by resorting to evolutionary game theory and dynamical system tools. Unlike most multi-agent reinforcement learning algorithms, whose mean dynamics are a slight variant of the replicator dynamics not affecting the properties of the original dynamics, the soft-max policy gradient dynamics presents a structure significantly different from that of the replicator. In particular, we show that the soft-max policy gradient dynamics in a given game are equivalent to the replicator dynamics in an auxiliary game obtained by a non-convex transformation of the payoffs of the original game. Such a structure gives the dynamics several non-standard properties. The first property we study concerns the convergence to the best response. In particular, while the continuous-time mean dynamics always converge to the best response, the crucial question concerns the convergence speed. Precisely, we show that the space of initializations can be split into two complementary sets such that the trajectories initialized from points of the first set (said good initialization region) directly move to the best response. In contrast, those initialized from points of the second set (said bad initialization region) move first to a series of sub-optimal strategies and then to the best response. Interestingly, in multi-agent adversarial machine learning environments, we show that an adversary can exploit this property to make any current strategy of the learning agent using the soft-max policy gradient fall inside a bad initialization region, thus slowing its learning process and exploiting that policy. When the soft-max policy gradient dynamics is studied in multi-population games, modeling the learning dynamics in self-play, we show that the dynamics preserve the volume of the set of initial points. This property proves that the dynamics cannot converge when the only equilibrium of the game is fully mixed, as the volume of the set of initial points would need to shrink. We also give empirical evidence that the volume expands over time, suggesting that the dynamics in games with fully-mixed equilibrium is chaotic.

In this paper, we investigate the optimal online configuration of episodic Markov decision processes when the space of the possible configurations is continuous. Specifically, we study the interaction between a learner (referred to as the configurator) and an agent with a fixed, unknown policy, when the learner aims to minimize her losses by choosing transition functions in online fashion. The losses may be unrelated to the agent's rewards. This problem applies to many real-world scenarios where the learner seeks to manipulate the Markov decision process to her advantage. We study both deterministic and stochastic settings, where the losses are either fixed or sampled from an unknown probability distribution. We design two algorithms whose peculiarity is to rely on occupancy measures to explore with optimism the continuous space of transition functions, achieving constant regret in deterministic settings and sublinear regret in stochastic settings, respectively. Moreover, we prove that the regret bound is tight with respect to any constant factor in deterministic settings. Finally, we compare the empiric performance of our algorithms with a baseline in synthetic experiments.

Autoregressive processes naturally arise in a large variety of real-world scenarios, including stock markets, sales forecasting, weather prediction, advertising, and pricing. When facing a sequential decision-making problem in such a context, the temporal dependence between consecutive observations should be properly accounted for guaranteeing convergence to the optimal policy. In this work, we propose a novel online learning setting, namely, Autoregressive Bandits (ARBs), in which the observed reward is governed by an autoregressive process of order $k$, whose parameters depend on the chosen action. We show that, under mild assumptions on the reward process, the optimal policy can be conveniently computed. Then, we devise a new optimistic regret minimization algorithm, namely, AutoRegressive Upper Confidence Bound (AR-UCB), that suffers sublinear regret of order $\tilde{O} ( \frac{(k+1)^{3/2}\sqrt{nT}}{(1-\Gamma)^2} )$, where $T$ is the optimization horizon, $n$ is the number of actions, and $\Gamma < 1$ is a stability index of the process. Finally, we empirically validate our algorithm, illustrating its advantages w.r.t. bandit baselines and its robustness to misspecification of key parameters.

Heavy-tailed distributions naturally arise in several settings, from finance to telecommunications. While regret minimization under subgaussian or bounded rewards has been widely studied, learning with heavy-tailed distributions only gained popularity over the last decade. In this paper, we consider the setting in which the reward distributions have finite absolute raw moments of maximum order $1+\epsilon$, uniformly bounded by a constant $u<+\infty$, for some $\epsilon \in (0,1]$. In this setting, we study the regret minimization problem when $\epsilon$ and $u$ are unknown to the learner and it has to adapt. First, we show that adaptation comes at a cost and derive two negative results proving that the same regret guarantees of the non-adaptive case cannot be achieved with no further assumptions. Then, we devise and analyze a fully data-driven trimmed mean estimator and propose a novel adaptive regret minimization algorithm, \texttt{AdaR-UCB}, that leverages such an estimator. Finally, we show that \texttt{AdaR-UCB} is the first algorithm that, under a known distributional assumption, enjoys regret guarantees nearly matching those of the non-adaptive heavy-tailed case.

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.

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.

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.

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 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.

In this paper, we study the class of games known as hidden-role games in which players are assigned privately to teams and are faced with the challenge of recognizing and cooperating with teammates. This model includes both popular recreational games such as the Mafia/Werewolf family and The Resistance (Avalon) and many real-world settings, such as distributed systems where nodes need to work together to accomplish a goal in the face of possible corruptions. There has been little to no formal mathematical grounding of such settings in the literature, and it was previously not even clear what the right solution concepts (notions of equilibria) should be. A suitable notion of equilibrium should take into account the communication channels available to the players (e.g., can they communicate? Can they communicate in private?). Defining such suitable notions turns out to be a nontrivial task with several surprising consequences. In this paper, we provide the first rigorous definition of equilibrium for hidden-role games, which overcomes serious limitations of other solution concepts not designed for hidden-role games. We then show that in certain cases, including the above recreational games, optimal equilibria can be computed efficiently. In most other cases, we show that computing an optimal equilibrium is at least NP-hard or coNP-hard. Lastly, we experimentally validate our approach by computing exact equilibria for complete 5- and 6-player Avalon instances whose size in terms of number of information sets is larger than 1056.

A mediator observes no-regret learners playing an extensive-form game repeatedly across T rounds. The mediator attempts to steer players toward some desirable predetermined equilibrium by giving (nonnegative) payments to players. We call this the steering problem. The steering problem captures problems several problems of interest, among them equilibrium selection and information design (persuasion). If the mediator's budget is unbounded, steering is trivial because the mediator can simply pay the players to play desirable actions. We study two bounds on the mediator's payments: a total budget and a per-round budget. If the mediator's total budget does not grow with T, we show that steering is impossible. However, we show that it is enough for the total budget to grow sublinearly with T, that is, for the average payment to vanish. When players' full strategies are observed at each round, we show that constant per-round budgets permit steering. In the more challenging setting where only trajectories through the game tree are observable, we show that steering is impossible with constant per-round budgets in general extensive-form games, but possible in normal-form games or if the per-round budget may itself depend on T. We also show how our results can be generalized to the case when the equilibrium is being computed online while steering is happening. We supplement our theoretical positive results with experiments highlighting the efficacy of steering in large games.

According to the main international reports, more pervasive industrial and business-process automation, thanks to machine learning and advanced analytic tools, will unlock more than 14 trillion USD worldwide annually by 2030. In the specific case of pricing problems, which constitute the class of problems we investigate in this paper, the estimated unlocked value will be about 0.5 trillion USD per year. In particular, this paper focuses on pricing in e-commerce when the objective function is profit maximization and only transaction data are available. This setting is one of the most common in real-world applications. Our work aims to find a pricing strategy that allows defining optimal prices at different volume thresholds to serve different classes of users. Furthermore, we face the major challenge, common in real-world settings, of dealing with limited data available. We design a two-phase online learning algorithm, namely PVD-B, capable of exploiting the data incrementally in an online fashion. The algorithm first estimates the demand curve and retrieves the optimal average price, and subsequently it offers discounts to differentiate the prices for each volume threshold. We ran a real-world 4-month-long A/B testing experiment in collaboration with an Italian e-commerce company, in which our algorithm PVD-B - corresponding to A configuration - has been compared with human pricing specialists - corresponding to B configuration. At the end of the experiment, our algorithm produced a total turnover of about 300 KEuros, outperforming the B configuration performance by about 55%. The Italian company we collaborated with decided to adopt our algorithm for more than 1,200 products since January 2022.

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.

We introduce a new approach for computing optimal equilibria via learning in games. It applies to extensive-form settings with any number of players, including mechanism design, information design, and solution concepts such as correlated, communication, and certification equilibria. We observe that optimal equilibria are minimax equilibrium strategies of a player in an extensive-form zero-sum game. This reformulation allows to apply techniques for learning in zero-sum games, yielding the first learning dynamics that converge to optimal equilibria, not only in empirical averages, but also in iterates. We demonstrate the practical scalability and flexibility of our approach by attaining state-of-the-art performance in benchmark tabular games, and by computing an optimal mechanism for a sequential auction design problem using deep reinforcement learning.

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.

Ex ante correlation is becoming the mainstream approach for sequential adversarial team games, where a team of players faces another team in a zero-sum game. It is known that team members’ asymmetric information makes both equilibrium computation \textsf{APX}-hard and team’s strategies not directly representable on the game tree. This latter issue prevents the adoption of successful tools for huge 2-player zero-sum games such as, e.g., abstractions, no-regret learning, and subgame solving. This work shows that we can recover from this weakness by bridging the gap between sequential adversarial team games and 2-player games. In particular, we propose a new, suitable game representation that we call team-public-information, in which a team is represented as a single coordinator who only knows information common to the whole team and prescribes to each member an action for any possible private state. The resulting representation is highly explainable, being a 2-player tree in which the team’s strategies are behavioral with a direct interpretation and more expressive than the original extensive form when designing abstractions. Furthermore, we prove payoff equivalence of our representation, and we provide techniques that, starting directly from the extensive form, generate dramatically more compact representations without information loss. Finally, we experimentally evaluate our techniques when applied to a standard testbed, comparing their performance with the current state of the art.

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.

There is a rising interest in industrial online applications where data becomes available sequentially. Inspired by the recommendation of playlists to users where their preferences can be collected during the listening of the entire playlist, we study a novel bandit setting, namely Multi-Armed Bandit with Temporally-Partitioned Rewards (TP-MAB), in which the stochastic reward associated with the pull of an arm is partitioned over a finite number of consecutive rounds following the pull. This setting, unexplored so far to the best of our knowledge, is a natural extension of delayed-feedback bandits to the case in which rewards may be dilated over a finite-time span after the pull instead of being fully disclosed in a single, potentially delayed round. We provide two algorithms to address TP-MAB problems, namely, TP-UCB-FR and TP-UCB-EW, which exploit the partial information disclosed by the reward collected over time. We show that our algorithms provide better asymptotical regret upper bounds than delayed-feedback bandit algorithms when a property characterizing a broad set of reward structures of practical interest, namely α-smoothness, holds. We also empirically evaluate their performance across a wide range of settings, both synthetically generated and from a real-world media recommendation problem.

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.

In several e-commerce scenarios, pricing long-tail products effectively is a central task for the companies, and there is broad agreement that Artificial Intelligence (AI) will play a prominent role in doing that in the next future. Nevertheless, dealing with long-tail products raises major open technical issues due to data scarcity which preclude the adoption of the mainstream approaches requiring usually a huge amount of data, such as, e.g., deep learning. In this paper, we provide a novel online learning algorithm for dynamic pricing that deals with non-stationary settings due to, e.g., the seasonality or adaptive competitors, and is very efficient in terms of the need for data thanks to assumptions such as, e.g., the monotonicity of the demand curve in the price that are customarily satisfied in long-tail markets. Furthermore, our dynamic pricing algorithm is paired with a clustering algorithm for the long-tail products which aggregates similar products such that the data of all the products of the same cluster are merged and used to choose their best price. We first evaluate our algorithms in an offline synthetic setting, comparing their performance with the state of the art and showing that our algorithms are more robust and data-efficient in long-tail settings. Subsequently, we evaluate our algorithms in an online setting with more than 8,000 products, including popular and long-tail, in an A/B test with humans for about two months. The increase of revenue thanks to our algorithms is about 18% for the popular products and about 90% for the long-tail products.

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).

In adversarial team games, a team of players sequentially faces a team of adversaries. These games are the simplest setting with multiple players where cooperation and competition coexist, and it is known that the information asymmetry among the team members makes equilibrium approximation computationally hard. Although much effort has been spent designing scalable algorithms, the problem of solving large game instances is open. In this paper, we extend the successful approach of solving huge two-player zero-sum games, where a blueprint strategy is computed offline by using an abstract version of the game and then it is refined online, that is, during a playthrough. In particular, to the best of our knowledge, our paper provides the first method for online strategy refinement via subgame solving in adversarial team games. Our method, based on the team belief DAG, generates a gadget game and then refine the blueprint strategy by using column-generation approaches in anytime fashion. If the blueprint is sparse, then our whole algorithm runs end-to-end in polynomial time given a best-response oracle; in particular, it avoids expanding the whole team belief DAG, which has exponential worst-case size. We apply our method to a standard test suite, and we empirically show the performance improvement of the strategies thanks to subgame solving.

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.

The existence of simple uncoupled no-regret learning dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and presence of private information in the game, correlation in extensive-form games possesses significantly different properties than in normal-form games, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device (a.k.a. mediator) must take into account the evolution of beliefs of each player as they make observations throughout the game. Due to that significant added complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics that converge to the set of EFCEs in n -player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play after T game repetitions is proven to be a \( O(1/\sqrt {T}) \) -approximate EFCE with high probability, and an EFCE almost surely in the limit.

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.

We study the problem of designing posted-price mechanisms in order to sell a single unit of a single item within a finite period of time. Motivated by real-world problems, such as, e.g., long-term rental of rooms and apartments, we assume that customers arrive online according to a Poisson process, and their valuations are drawn from an unknown distribution and discounted over time. We evaluate our mechanisms in terms of competitive ratio, measuring the worst-case ratio between their revenue and that of an optimal mechanism that knows the distribution of valuations. First, we focus on the identical valuation setting, where all the customers value the item for the same amount. In this setting, we provide a mechanism M_c that achieves the best possible competitive ratio, discussing its dependency on the parameters in the case of linear discount. Then, we switch to the random valuation setting. We show that, if we restrict the attention to distributions of valuations with a monotone hazard rate, then the competitive ratio of M_c is lower bounded by a strictly positive constant that does not depend on the distribution. Moreover, we provide another mechanism, called M_pc, which is defined by a piecewise constant pricing strategy and reaches performances comparable to those obtained with M_c. This mechanism is useful when the seller cannot change the posted price too often. Finally, we empirically evaluate the performances of our mechanisms in a number of experimental 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 problem of finding an optimal strategy for a team of players that faces an opponent in an imperfect-information zero-sum extensive-form game. Team members are not allowed to communicate during play but can coordinate before the game. In this setting, it is known that the best the team can do is sample a profile of potentially randomized strategies (one per player) from a joint (a.k.a. correlated) probability distribution at the beginning of the game. In this paper, we first provide new modeling results about computing such an optimal distribution by drawing a connection to a different literature on extensive-form correlation. Second, we provide an algorithm that allows one for capping the number of profiles employed in the solution. This begets an anytime algorithm by increasing the cap. We find that often a handful of well-chosen such profiles suffices to reach optimal utility for the team. This enables team members to reach coordination through a simple and understandable plan. Finally, inspired by this observation and leveraging theoretical concepts that we introduce, we develop an efficient column-generation algorithm for finding an optimal distribution for the team. We evaluate it on a suite of common benchmark games. It is three orders of magnitude faster than the prior state of the art on games that the latter can solve and it can also solve several games that were previously unsolvable.

The existence of uncoupled no-regret learning dynamics converging to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form games generalize normal-form games by modeling both sequential and simultaneous moves, as well as imperfect information. Because of the sequential nature and the presence of private information, correlation in extensive-form games possesses significantly different properties than in normal-form games. The extensive-form correlated equilibrium (EFCE) is the natural extensive-form counterpart to the classical notion of correlated equilibrium in normal-form games. Compared to the latter, the constraints that define the set of EFCEs are significantly more complex, as the correlation device ({\em a.k.a.} mediator) must take into account the evolution of beliefs of each player as they make observations throughout the game. Due to this additional complexity, the existence of uncoupled learning dynamics leading to an EFCE has remained a challenging open research question for a long time. In this article, we settle that question by giving the first uncoupled no-regret dynamics which provably converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. We show that each iterate can be computed in time polynomial in the size of the game tree, and that, when all players play repeatedly according to our learning dynamics, the empirical frequency of play after T game repetitions is guaranteed to be a O(T^-1/2)-approximate EFCE with high probability, and an EFCE almost surely in the limit.

Recently, game-playing agents based on AI techniques have demonstrated super-human performance in several sequential games, such as chess, Go, and poker. Surprisingly, the multi-agent learning techniques that allowed to reach these achievements do not take into account the actual behavior of the human player, potentially leading to an impressive gap in performances. In this paper, we address the problem of designing artificial agents that learn how to effectively exploit unknown human opponents while playing repeatedly against them in an online fashion. We study the case in which the agent's strategy during each repetition of the game is subject to constraints ensuring that the human's expected utility is within some lower and upper thresholds. Our framework encompasses several real-world problems, such as human engagement in repeated game playing and human education by means of serious games. As a first result, we formalize a set of linear inequalities encoding the conditions that the agent's strategy must satisfy at each iteration in order to do not violate the given bounds for the human's expected utility. Then, we use such formulation in an upper confidence bound algorithm, and we prove that the resulting procedure suffers from sublinear regret and guarantees that the constraints are satisfied with high probability at each iteration. Finally, we empirically evaluate the convergence of our algorithm on standard testbeds of sequential games.

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).

We study an information-structure design problem (a.k.a. a persuasion problem) with a single sender and multiple receivers with actions of a priori unknown types, independently drawn from action-specific marginal probability distributions. As in the standard Bayesian persuasion model, the sender has access to additional information regarding the action types, which she can exploit when committing to a (noisy) signaling scheme through which she sends a private signal to each receiver. The novelty of our model is in considering the much more expressive case in which the receivers interact in a sequential game with imperfect information, with utilities depending on the game outcome and the realized action types. After formalizing the notions of ex ante and ex interim persuasiveness (which differ by the time at which the receivers commit to following the sender's signaling scheme), we investigate the continuous optimization problem of computing a signaling scheme which maximizes the sender's expected revenue. We show that computing an optimal ex ante persuasive signaling scheme is NP-hard when there are three or more receivers. Instead, in contrast with previous hardness results for ex interim persuasion, we show that, for games with two receivers, an optimal ex ante persuasive signaling scheme can be computed in polynomial time thanks to the novel algorithm we propose, based on the ellipsoid method.

Security Games have been widely adopted to model scenarios in which one player, the Defender, has to decide how to deploy her resources to minimize the loss that can be caused by an attack performed by another player, the Attacker, aiming at maximizing such loss. In the present paper, we focus on scenarios in which the Defender has lexicographic-like preferences on the targets, being primarily interested in defending the integrity of a subset of the targets and, only secondarily, to reduce the amount of the other damaged targets. Our central motivation for studying this problem comes from the need to reduce the impact of malicious flows in networks, that can be either physical, like cities, or virtual, e.g., social networks. In this work, we introduce a new class of security games to model these scenarios, characterizing it and proving the NP-hardness of computing a leader-follower equilibrium, which is the most appropriate solution concept for this setting. To compute such an equilibrium, we then provide an exact exponential-time algorithm, capable of exploiting the topological properties of the network. Finally, we show that, with opportune optimizations, this algorithm can work efficiently even on network of 10000 nodes.

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.

The existence of simple, uncoupled no-regret dynamics that converge to correlated equilibria in normal-form games is a celebrated result in the theory of multi-agent systems. Specifically, it has been known for more than 20 years that when all players seek to minimize their internal regret in a repeated normal-form game, the empirical frequency of play converges to a normal-form correlated equilibrium. Extensive-form (that is, tree-form) games generalize normal-form games by modeling both sequential and simultaneous moves, as well as private information. Because of the sequential nature and presence of partial information in the game, extensive-form correlation has significantly different properties than the normal-form counterpart, many of which are still open research directions. Extensive-form correlated equilibrium (EFCE) has been proposed as the natural extensive-form counterpart to normal-form correlated equilibrium. However, it was currently unknown whether EFCE emerges as the result of uncoupled agent dynamics. In this paper, we give the first uncoupled no-regret dynamics that converge to the set of EFCEs in n-player general-sum extensive-form games with perfect recall. First, we introduce a notion of trigger regret in extensive-form games, which extends that of internal regret in normal-form games. When each player has low trigger regret, the empirical frequency of play is a close to an EFCE. Then, we give an efficient no-trigger-regret algorithm. Our algorithm decomposes trigger regret into local subproblems at each decision point for the player, and constructs a global strategy of the player from the local solutions at each decision point.

Abstract The search problem of computing aStackelberg(orleader-follower)equilibrium(also referred to as anoptimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although theoptimisticandpessimisticversions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomial-time algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers’ game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader’s commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless $$\textsf {P}=\textsf {NP}$$ P=NP . Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies—a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier—under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as apessimistic bilevel programming problem, we show that, with two followers, the problem is -hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless $$\textsf {P}=\textsf {NP}$$ P=NP . This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave quadratic function over an unbounded nonconvex feasible region defined by linear and quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-the-art methods, we propose an exactad hocalgorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader’s strategy where such value is attained, also an $$\alpha $$ α -approximate strategy where $$\alpha > 0$$ α>0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computational experiments on a well-established testbed of game instances.

Multi-Armed Bandit (MAB) techniques have been successfully applied to many classes of sequential decision problems in the past decades. However, non-stationary settings -- very common in real-world applications -- received little attention so far, and theoretical guarantees on the regret are known only for some frequentist algorithms. In this paper, we propose an algorithm, namely Sliding-Window Thompson Sampling (SW-TS), for nonstationary stochastic MAB settings. Our algorithm is based on Thompson Sampling and exploits a sliding-window approach to tackle, in a unified fashion, two different forms of non-stationarity studied separately so far: abruptly changing and smoothly changing. In the former, the reward distributions are constant during sequences of rounds, and their change may be arbitrary and happen at unknown rounds, while, in the latter, the reward distributions smoothly evolve over rounds according to unknown dynamics. Under mild assumptions, we provide regret upper bounds on the dynamic pseudo-regret of SW-TS for the abruptly changing environment, for the smoothly changing one, and for the setting in which both the non-stationarity forms are present. Furthermore, we empirically show that SW-TS dramatically outperforms state-of-the-art algorithms even when the forms of non-stationarity are taken separately, as previously studied in the literature.

Equilibrium refinements are important in extensive-form (i.e., tree-form) games, where they amend weaknesses of the Nash equilibrium concept by requiring sequential rationality and other beneficial properties. One of the most attractive refinement concepts is quasi-perfect equilibrium. While quasiperfection has been studied in extensive-form games, it is poorly understood in Stackelberg settings—that is, settings where a leader can commit to a strategy—which are important for modeling, for example, security games. In this paper, we introduce the axiomatic definition of quasi-perfect Stackelberg equilibrium. We develop a broad class of game perturbation schemes that lead to them in the limit. Our class of perturbation schemes strictly generalizes prior perturbation schemes introduced for the computation of (non-Stackelberg) quasi-perfect equilibria. Based on our perturbation schemes, we develop a branch-and-bound algorithm for computing a quasi-perfect Stackelberg equilibrium. It leverages a perturbed variant of the linear program for computing a Stackelberg extensive-form correlated equilibrium. Experiments show that our algorithm can be used to find an approximate quasi-perfect Stackelberg equilibrium in games with thousands of nodes.

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.

In the context of multi-player, general-sum games, there is a growing interest in solution concepts involving some form of communication among players, since they can lead to socially better outcomes with respect to Nash equilibria and may be reached through learning dynamics in a decentralized fashion. In this paper, we focus on coarse correlated equilibria (CCEs) in sequential games. First, we complete the picture on the complexity of finding social-welfare-maximizing CCEs by proving that the problem is not in Poly-APX, unless P = NP, in games with three or more players (including chance). Then, we provide simple arguments showing that CFR---working with behavioral strategies---may not converge to a CCE in multi-player, general-sum sequential games. In order to amend this issue, we devise two variants of CFR that provably converge to a CCE. The first one (CFR-S) is a simple stochastic adaptation of CFR which employs sampling to build a correlated strategy, whereas the second variant (called CFR-Jr) enhances CFR with a more involved reconstruction procedure to recover correlated strategies from behavioral ones. Experiments on a rich testbed of multi-player, general-sum sequential games show that both CFR-S and CFR-Jr are dramatically faster than the state-of-the-art algorithms to compute CCEs, with CFR-Jr being also a good heuristic to find socially-optimal CCEs.

In 2017, Internet ad spending reached 209 billion USD worldwide, while, e.g., TV ads brought in 178 billion USD. An Internet advertising campaign includes up to thousands of sub-campaigns on multiple channels, e.g., search, social, display, whose parameters (bid and daily budget) need to be optimized every day, subject to a (cumulative) budget constraint. Such a process is often unaffordable for humans and its automation is crucial. As also shown by marketing funnel models, the sub-campaigns are usually interdependent, e.g., display ads induce awareness, increasing the number of impressions-and, thus, also the number of conversions-of search ads. This interdependence is widely exploited by humans in the optimization process, whereas, to the best of our knowledge, no algorithm takes it into account. In this paper, we provide the first model capturing the sub-campaigns interdependence. We also provide the IDIL algorithm, which, employing Granger Causality and Gaussian Processes, learns from past data, and returns an optimal stationary bid/daily budget allocation. We prove theoretical guarantees on the loss of IDIL w.r.t. the clairvoyant solution, and we show empirical evidence of its superiority in both realistic and real-world settings when compared with existing approaches.

We provide, to the best of our knowledge, the first computational study of extensive-form adversarial team games. These games are sequential, zero-sum games in which a team of players, sharing the same utility function, faces an adversary. We define three different scenarios according to the communication capabilities of the team. In the first, the teammates can communicate and correlate their actions both before and during the play. In the second, they can only communicate before the play. In the third, no communication is possible at all. We define the most suitable solution concepts, and we study the inefficiency caused by partial or null communication, showing that the inefficiency can be arbitrarily large in the size of the game tree. Furthermore, we study the computational complexity of the equilibrium-finding problem in the three scenarios mentioned above, and we provide, for each of the three scenarios, an exact algorithm. Finally, we empirically evaluate the scalability of the algorithms in random games and the inefficiency caused by partial or null communication.

Leadership games provide a powerful paradigm to model many real-world settings. Most literature focuses on games with a single follower who acts optimistically, breaking ties in favour of the leader. Unfortunately, for real-world applications, this is unlikely. In this paper, we look for efficiently solvable games with multiple followers who play either optimistically or pessimistically, i.e., breaking ties in favour or against the leader. We study the computational complexity of finding or approximating an optimistic or pessimistic leader-follower equilibrium in specific classes of succinct games—polymatrix like—which are equivalent to 2-player Bayesian games with uncertainty over the follower, with interdependent or independent types. Furthermore, we provide an exact algorithm to find a pessimistic equilibrium for those game classes. Finally, we show that in general polymatrix games the computation is harder even when players are forced to play pure strategies.

Pay-per-click advertising includes various formats (e.g., search, contextual, and social) with a total investment of more than 140 billion USD per year. An advertising campaign is composed of some subcampaigns-each with a different ad-and a cumulative daily budget. The allocation of the ads is ruled exploiting auction mechanisms. In this paper, we propose, for the first time to the best of our knowledge, an algorithm for the online joint bid/budget optimization of pay-per-click multi-channel advertising campaigns. We formulate the optimization problem as a combinatorial bandit problem, in which we use Gaussian Processes to estimate stochastic functions, Bayesian bandit techniques to address the exploration/exploitation problem, and a dynamic programming technique to solve a variation of the Multiple-Choice Knapsack problem. We experimentally evaluate our algorithm both in simulation-using a synthetic setting generated from real data from Yahoo!-and in a real-world application over an advertising period of two months.

We initiate the study of equilibrium refinements based on trembling-hand perfection in extensive-form games with commitment strategies, that is, where one player commits to a strategy first. We show that the standard strong (and weak) Stackelberg equilibria are not suitable for trembling-hand perfection, because the limit of a sequence of such strong (weak) Stackelberg commitment strategies of a perturbed game may not be a strong (weak) Stackelberg equilibrium itself. However, we show that the universal set of all Stackelberg equilibria (i.e., those that are optimal for at least some follower response function) is natural for trembling- hand perfection: it does not suffer from the problem above. We also prove that determining the existence of a Stackelberg equilibrium--refined or not--that gives the leader expected value at least v is NP-hard. This significantly extends prior complexity results that were specific to strong Stackelberg equilibrium.

We study Stackelberg games where the underlying structure is a congestion game. We recall that, while leadership in 2-player games has been widely investigated, only few results are known when the number of players is three or more. The intractability of finding a Stackelberg equilibrium (SE) in normal-form and polymatrix games is among them. In this paper, we focus on congestion games in which each player can choose a single resource (a.k.a. singleton congestion games) and a player acts as leader. We show that, without further assumptions, finding an SE when the followers break ties in favor of the leader is not in Poly-APX, unless P = NP. Instead, under the assumption that every player has access to the same resources and that the cost functions are monotonic, we show that an SE can be computed efficiently when the followers break ties either in favor or against the leader.

In the last two decades, online advertising has become the most effective way to sponsor a product or an event. The success of this advertising format is mainly due to the capability of the Internet channels to reach a broad audience and to target different groups of users with specific sponsored announces. This is of paramount importance for media agencies, companies whose primary goal is to design ad campaigns that target only those users who are interested in the sponsored product, thus avoiding unnecessary costs due to the display of ads to uninterested users. In the present work, we develop an automatic method to find the best user targets (a.k.a. contexts) that a media agency can use in a given Internet advertising campaign. More specifically, we formulate the problem of target optimization as a Learning from Logged Bandit Feedback (LLBF) problem, and we propose the TargOpt algorithm, which uses a tree expansion of the target space to learn the partition that efficiently maximizes the campaign revenue. Furthermore, since the problem of finding the optimal target is intrinsically exponential in the number of the features, we propose a tree-search method, called A-TargOpt, and two heuristics to drive the tree expansion, aiming at providing an anytime solution. Finally, we present empirical evidence, on both synthetically generated and real-world data, that our algorithms provide a practical solution to find effective targets for Internet advertising.

Nash equilibrium strategies have the known weakness that they do not prescribe rational play in situations that are reached with zero probability according to the strategies themselves, for example, if players have made mistakes. Trembling-hand refinements---such as extensive-form perfect equilibria and quasi-perfect equilibria---remedy this problem in sound ways. Despite their appeal, they have not received attention in practice since no known algorithm for computing them scales beyond toy instances. In this paper, we design an exact polynomial-time algorithm for finding trembling-hand equilibria in zero-sum extensive-form games. It is several orders of magnitude faster than the best prior ones, numerically stable, and quickly solves game instances with tens of thousands of nodes in the game tree. It enables, for the first time, the use of trembling-hand refinements in practice.

Recent milestones in equilibrium computation, such as the success of Libratus, show that it is possible to compute strong solutions to two-player zero-sum games in theory and practice. This is not the case for games with more than two players, which remain one of the main open challenges in computational game theory. This paper focuses on zero-sum games where a team of players faces an opponent, as is the case, for example, in Bridge, collusion in poker, and many non-recreational applications such as war, where the colluders do not have time or means of communicating during battle, collusion in bidding, where communication during the auction is illegal, and coordinated swindling in public. The possibility for the team members to communicate before game play—that is, coordinate their strategies ex ante—makes the use of behavioral strategies unsatisfactory. The reasons for this are closely related to the fact that the team can be represented as a single player with imperfect recall. We propose a new game representation, the realization form, that generalizes the sequence form but can also be applied to imperfect-recall games. Then, we use it to derive an auxiliary game that is equivalent to the original one. It provides a sound way to map the problem of finding an optimal ex-ante-correlated strategy for the team to the well-understood Nash equilibrium-finding problem in a (larger) two-player zero-sum perfect-recall game. By reasoning over the auxiliary game, we devise an anytime algorithm, fictitious team-play, that is guaranteed to converge to an optimal coordinated strategy for the team against an optimal opponent, and that is dramatically faster than the prior state-of-the-art algorithm for this problem.

Sponsored Search Auctions (SSAs) arguably represent the problem at the intersection of computer science and economics with the deepest applications in real life. Within the realm of SSAs, the study of the effects that showing one ad has on the other ads, a.k.a. externalities in economics, is of utmost importance and has so far attracted the attention of much research. However, even the basic question of modeling the problem has so far escaped a definitive answer. The popular cascade model is arguably too idealized to really describe the phenomenon yet it allows a good comprehension of the problem. Other models, instead, describe the setting more adequately but are too complex to permit a satisfactory theoretical analysis. In this work, we attempt to get the best of both approaches: firstly, we define a number of general mathematical formulations for the problem in the attempt to have a rich description of externalities in SSAs and, secondly, prove a host of results drawing a nearly complete picture about the computational complexity of the problem. We complement these approximability results with some considerations about mechanism design in our context.

The Team-maxmin equilibrium prescribes the optimal strategies for a team of rational players sharing the same goal and without the capability of correlating their strategies in strategic games against an adversary. This solution concept can capture situations in which an agent controls multiple resources - corresponding to the team members - that cannot communicate. It is known that such equilibrium always exists and it is unique (except degenerate cases) and these properties make it a credible solution concept to be used in real-world applications, especially in security scenarios. Nevertheless, to the best of our knowledge, the Team-maxmin equilibrium is almost completely unexplored in the literature. In this paper, we investigate bounds of (in)efficiency of the Team-maxmin equilibrium w.r.t. the Nash equilibria and w.r.t. the Maxmin equilibrium when the team members can play correlated strategies. Furthermore, we study a number of algorithms to find and/or approximate an equilibrium, discussing their theoretical guarantees and evaluating their performance by using a standard testbed of game instances.

We study the problem of computing an Extensive-Form Perfect Equilibrium (EFPE) in 2-player games. This equilibrium concept refines the Nash equilibrium requiring resilience with respect to a specific vanishing perturbation, representing mistakes of the players at each decision node. The scientific challenge is intrinsic to the EFPE definition: it requires a perturbation over the agent form, but the agent form is computationally inefficient due to the presence of highly nonlinear constraints. We show that the sequence form can be exploited in a non-trivial way and that, for general-sum games, finding an EFPE is equivalent to solving a suitably perturbed linear complementarity problem. We prove that Lemke's algorithm can be applied, showing that computing an EFPE is PPAD-complete. In the notable case of zero-sum games, the problem is in FP and can be solved by linear programming. Our algorithms also allow one to find a Nash equilibrium when players cannot perfectly control their moves, being subject to a given execution uncertainty, as is the case in most realistic physical settings.

We study, to the best of our knowledge, the first Bayesian algorithm for unimodal Multi-Armed Bandit (MAB) problems with graph structure. In this setting, each arm corresponds to a node of a graph and each edge provides a relationship, unknown to the learner, between two nodes in terms of expected reward. Furthermore, for any node of the graph there is a path leading to the unique node providing the maximum expected reward, along which the expected reward is monotonically increasing. Previous results on this setting describe the behavior of frequentist MAB algorithms. In our paper, we design a Thompson Sampling-based algorithm whose asymptotic pseudo-regret matches the lower bound for the considered setting. We show that -as it happens in a wide number of scenarios- Bayesian MAB algorithms dramatically outperform frequentist ones. In particular, we provide a thorough experimental evaluation of the performance of our and state-of-the-art algorithms as the properties of the graph vary.

The problem of computing the strategy to commit to has been widely investigated in the scientific literature for the case where a single-follower is present. In the multi-follower setting though, results are only sporadic. In this paper, we address the multi-follower case for normal-form games, assuming that, after observing the leader’s commitment, the followers play pure strategies and reach a Nash equilibrium. We focus on the pessimistic case where, among many equilibria, one minimizing the leader’s utility is chosen (the opposite case is computationally trivial). We show that the problem is NP-hard even with only two followers, and propose an exact exponential-time algorithm which, for any number of followers, either finds an equilibrium when the game admits a finite one or, if not, an α-approximation of the supremum of the leader’ utility, for any α > 0.

Logged data is one of the most widespread form of recorded information, since it can be acquired by almost any system and stored at a little cost. Customarily, the interaction logs between the system and a user (or environment) present the structure of a sequential decision process: given a context, the system performs an action and the user provides a feedback about it. This structure is common to a wide range of real-world micro-economic applications, e.g., e-commerce websites and advertisement campaigns. The problem of learning a policy from such logged interactions to take more profitable decisions in the future is known as the Learning from Logged Bandit Feedback (LLBF) problem. In this paper, we propose RADT, an algorithm specifically shaped for the LLBF setting and based on a risk-averse learning method which exploits the joint use of regression trees and statistical confidence bounds. Differently from existing techniques developed for this setting, RADT generates policies aiming to maximize a lower bound on the expected reward and provides a clear characterization of those features in the context that influence the process the most. Finally, we provide a wide experimental campaign over both synthetic and real-world datasets showing empirical evidence that RADT outperforms both state-of-the-art machine learning classification and regression techniques and existing methods addressing the LLBF setting.

We study the problem of computing an equilibrium in leader-follower games with a single leader and multiple followers where, after the leader’s commitment to a mixed strategy, the followers play simultaneously in a noncooperative way, reaching a Nash equilibrium. We tackle the problem from a bilevel programming perspective. Since, given the leader’s strategy, the followers’ subgame may admit multiple Nash equilibria, we consider the cases where the followers play either the best (optimistic) or the worst (pessimistic) Nash equilibrium in terms of the leader’s utility. For the optimistic case, we propose three formulations which cast the problem into a single level mixed-integer nonconvex program. For the pessimistic case, which, as we show, may admit a supremum but not a maximum, we develop an ad hoc branch-and-bound algorithm. Computational results are reported and illustrated.

Sponsored Search Auctions (SSAs) are one of the most successful applications of microeconomic mechanisms, with a revenue of about $72 billion in the US alone in 2016. However, the problem of designing the best economic mechanism for sponsored search auctions is far from being solved, and, given the amount at stake, it is no surprise that it has received growing attention over the past few years. The most common auction mechanism for SSAs is the Generalized Second Price (GSP). However, the GSP is known not to be truthful: the agents participating in the auction might have an incentive to report false values, generating economic inefficiency and suboptimal revenues in turn. Superior, efficient truthful mechanisms, such as the Vickrey-Clarke-Groves (VCG) auction, are well known in the literature. However, while the VCG auction is currently adopted for the strictly related scenario of contextual advertising, e.g., by Google and Facebook, companies are reluctant to extend it to SSAs, fearing prohibitive switching costs. Other than truthfulness, two issues are of paramount importance in designing effective SSAs. First, the choice of the user model; not only does an accurate user model better target ads to users, it also is a critical factor in reducing the inefficiency of the mechanism. Often an antagonist to this, the second issue is the running time of the mechanism, given the performance pressure these mechanisms undertake in real-world applications. In our work, we argue in favor of adopting the VCG mechanism based on the cascade model with ad/position externalities (APDC-VCG). Our study includes both the derivation of inefficiency bounds and the design and the experimental evaluation of exact and approximate algorithms.

Motivated by a number of security applications, among which border patrolling, we study, to the best of our knowledge, the first Security Game model in which patrolling strategies need to be combined with responses to signals raised by an alarm system, which is spatially uncertain (i.e., it is uncertain over the exact location the attack is ongoing) and is affected by false negatives (i.e., the missed detection rate of an attack may be positive). Ours is an infinite-horizon patrolling scenario on a graph, where a single patroller moves. We study the properties of the game model in terms of computational issues and form of the optimal strategies and we provide an approach to solve it. Finally, we provide an experimental analysis of our techniques.

The design of the best economic mechanism for Sponsored Search Auctions (SSAs) is a central task in computational mechanism design/game theory. Two open questions concern (i) the adoption of user models more accurate than the currently used one and (ii) the choice between Generalized Second Price auction (GSP) and Vickrey–Clark–Groves mechanism (VCG). In this paper, we provide some contributions to answer these questions. We study Price of Anarchy (PoA) and Price of Stability (PoS) over social welfare and auctioneer’s revenue of GSP w.r.t. the VCG when the users follow the famous cascade model. Furthermore, we provide exact, randomized, and approximate algorithms, showing that in real–world settings (Yahoo! Webscope A3 dataset, 10 available slots) optimal allocations can be found in less than 1s with up to 1,000 ads, and can be approximated in less than 20ms even with more than 1,000 ads with an average accuracy greater than 99%.

Evolutionary game theory provides the principal tools to model the dynamics of multi-agent learning algorithms. While there is a long-standing literature on evolutionary game theory in strategic-form games, in the case of extensive-form games few results are known and the exponential size of the representations currently adopted makes the evolutionary analysis of such games unaffordable. In this paper, we focus on dynamics for the sequence form of extensive-form games, providing three dynamics: one realization equivalent to the normal-form logit dynamic, one realization equivalent to the agent-form replicator dynamic, and one realization equivalent to the agent-form logit dynamic. All the considered dynamics require polynomial time and space, providing an exponential compression w.r.t. the dynamics currently known and providing thus tools that can be effectively employed in practice. Moreover, we use our tools to compare the agent-form and normal-form dynamics and to provide new "hybrid" dynamics.

Sponsored Search Auctions (SSAs) arguably represent the problem at the intersection of computer science and economics with the deepest applications in real life. Within the realm of SSAs, the study of the effects that showing one ad has on the other ads, a.k.a. externalities in economics, is of utmost importance and has so far attracted the attention of much research. However, even the basic question of modeling the problem has so far escaped a definitive answer. The popular cascade model is arguably too idealized to really describe the phenomenon yet it allows a good comprehension of the problem. Other models, instead, describe the setting more adequately but are too complex to permit a satisfactory theoretical analysis. In this work, we attempt to get the best of both approaches: firstly, we define a number of general mathematical formulations for the problem in the attempt to have a rich description of externalities in SSAs and, secondly, prove a host of results drawing a nearly complete picture about the computational complexity of the problem. We complement these approximability results with some considerations about mechanism design in our context.

Multi–Armed Bandits (MABs) have been widely considered in the last decade to model settings in which an agent wants to learn the action providing the highest expected reward among a fixed set of available actions during the operational life of a system. Classical techniques provide solutions that minimize the regret due to learning in settings where selecting an arm has no cost. Though, in many real world applications the learner has to pay some cost for pulling each arm and the learning process is constrained by a fixed budget B. This problem is addressed in the literature as the Budgeted MAB (BMAB). In this paper, for the first time, we study the problem of Budgeted Continuous–Armed Bandit (BCAB), where the set of the possible actions consists in a continuous set (e.g., a range of prices) and the learner suffers from a random reward and cost at each round. We provide a novel algorithm, named B–Zoom, which suffers a regret of, where d is the Zooming dimension of the problem. Finally, we provide an empirical analysis showing that, despite a lower average performance, the proposed approach is more robust to adverse settings as compared to existing algorithms designed for BMAB.

Most of the literature on delay tolerant networks (DTNs) focuses on optimal routing policies exploiting a priori knowledge about nodes mobility traces. For the case in which no a priori knowledge is available (very common in practice), apart from basic epidemic routing, the main approaches focus on controlling two-hop routing policies. However, these latter results commonly employ fluid approximation techniques, which, in principle, do not provide any theoretical bound over the approximation ratio. In our work, we focus on the case without a priori mobility knowledge and we provide approximation algorithms with theoretical guarantees that can be applied to cases where the number of hops allowed in the routing process is arbitrary. Our approach is rather flexible allowing us to address heterogeneous mobility patterns and transmission technologies, to consider explicitly the signaling and transmission costs, and to include also nodes discarding packets after a local timeout. We then provide a comprehensive performance evaluation of our algorithms, showing that two-hop routing provides the best tradeoff between delay and energy and that, in this case, they find solutions very close to the optimal ones with a low overhead. Finally, we compare our methods against some state-of-the-art approaches by means of a DTN simulation environment in realistic settings.

Mobile geo-location advertising, where mobile ads are targeted based on a user’s location, has been identified as a key growth factor for the mobile market. As with online advertising, a crucial ingredient for their success is the development of effective economic mechanisms. An important difference is that mobile ads are shown sequentially over time and information about the user can be learned based on their movements. Furthermore, ads need to be shown selectively to prevent ad fatigue. To this end, we introduce, for the first time, a user model and suitable economic mechanisms which take these factors into account. Specifically, we design two truthful mechanisms which produce an advertisement plan based on the user’s movements. One mechanism is allocatively efficient, but requires exponential compute time in the worst case. The other requires polynomial time, but is not allocatively efficient. Finally, we experimentally evaluate the trade off between compute time and efficiency of our mechanisms.

Multi-agent learning is a challenging open task in artificial intelligence. It is known an interesting connection between multi-agent learning algorithms and evolutionary game theory, showing that the learning dynamics of some algorithms can be modeled as replicator dynamics with a mutation term. Inspired by the recent sequence-form replicator dynamics, we develop a new version of the Q-learning algorithm working on the sequence form of an extensive-form game allowing thus an exponential reduction of the dynamics length w.r.t. those of the normal form. The dynamics of the proposed algorithm can be modeled by using the sequence-form replicator dynamics with a mutation term. We show that, although sequence-form and normal-form replicator dynamics are realization equivalent, the Q-learning algorithm applied to the two forms have non-realization equivalent dynamics. Originally from the previous works on evolutionary game theory models form multi-agent learning, we produce an experimental evaluation to show the accuracy of the model.

Most applications of wireless sensor networks (WSNs) rely on data about the positions of sensor nodes, which are not necessarily known beforehand. Several localization approaches have been proposed but most of them omit to consider that WSNs could be deployed in adversarial settings, where hostile nodes under the control of an attacker coexist with faithful ones. Verifiable multilateration (VM) was proposed to cope with this problem by leveraging on a set of trusted landmark nodes that act as verifiers. Although VM is able to recognize reliable localization measures, it allows for regions of undecided positions that can amount to the 40 percent of the monitored area. We studied the properties of VM as a noncooperative two-player game where the first player employs a number of verifiers to do VM computations and the second player controls a malicious node. The verifiers aim at securely localizing malicious nodes, while malicious nodes strive to masquerade as unknown and to pretend false positions. Thanks to game theory, the potentialities of VM are analyzed with the aim of improving the defender's strategy. We found that the best placement for verifiers is an equilateral triangle with edge equal to the power range $(R)$, and maximum deception in the undecided region is approximately $(0.27R)$. Moreover, we characterizedâin terms of the probability of choosing an unknown node to examine furtherâthe strategies of the players.

Evolutionary game theory combines game theory and dynamical systems and is customarily adopted to describe evolutionary dynamics in multi-agent systems. In particular, it has been proven to be a successful tool to describe multi-agent learning dynamics. To the best of our knowledge, we provide in this paper the first replicator dynamics applicable to the sequence form of an extensive-form game, allowing an exponential reduction of time and space w.r.t. the currently adopted replicator dynamics for normal form. Furthermore, our replicator dynamics is realization equivalent to the standard replicator dynamics for normal form. We prove our results for both discrete-time and continuous-time cases. Finally, we extend standard tools to study the stability of a strategy profile to our replicator dynamics.

Strong Nash equilibrium (SNE) is an appealing solution concept when rational agents can form coalitions. A strategy profile is an SNE if no coalition of agents can benefit by deviating. We present the first general-purpose algorithms for SNE finding in games with more than two agents. An SNE must simultaneously be a Nash equilibrium (NE) and the optimal solution of multiple non-convex optimization problems. This makes even the derivation of necessary and sufficient mathematical equilibrium constraints difficult. We show that forcing an SNE to be resilient only to pure-strategy deviations by coalitions, unlike for NEs, is only a necessary condition here. Second, we show that the application of Karush-Kuhn-Tucker conditions leads to another set of necessary conditions that are not sufficient. Third, we show that forcing the Pareto efficiency of an SNE for each coalition with respect to coalition correlated strategies is sufficient but not necessary. We then develop a tree search algorithm for SNE finding. At each node, it calls an oracle to suggest a candidate SNE and then verifies the candidate. We show that our new necessary conditions can be leveraged to make the oracle more powerful. Experiments validate the overall approach and show that the new conditions significantly reduce search tree size compared to using NE conditions alone.

Wireless access networks are often characterized by the interaction of different end users, communication technologies, and network operators. This paper analyzes the dynamics among these "actorsâ by focusing on the processes of wireless network selection, where end users may choose among multiple available access networks to get connectivity, and resource allocation, where network operators may set their radio resources to provide connectivity. The interaction among end users is modeled as a noncooperative congestion game, where players (end users) selfishly select the access network that minimizes their perceived selection cost. A method based on mathematical programming is proposed to find Nash equilibria and characterize their optimality under three cost functions, which are representative of different technological scenarios. System level simulations are then used to evaluate the actual throughput and fairness of the equilibrium points. The interaction among end users and network operators is then assessed through a two-stage multileader/multifollower game, where network operators (leaders) play in the first stage by properly setting the radio resources to maximize their users, and end users (followers) play in the second stage the aforementioned network selection game. The existence of exact and approximated subgame perfect Nash equilibria of the two-stage game is thoroughly assessed and numerical results are provided on the "qualityâ of such equilibria.

Sponsored search auctions constitute one of the most successful applications of microeconomic mechanisms. In mechanism design, auctions are usually designed to incentivize advertisers to bid their truthful valuations and, at the same time, to assure both the advertisers and the auctioneer a non--negative utility. Nonetheless, in sponsored search auctions, the click-through-rates (CTRs) of the advertisers are often unknown to the auctioneer and thus standard incentive compatible mechanisms cannot be directly applied and must be paired with an effective learning algorithm for the estimation of the CTRs. This introduces the critical problem of designing a learning mechanism able to estimate the CTRs as the same time as implementing a truthful mechanism with a revenue loss as small as possible compared to an optimal mechanism designed with the true CTRs. Previous works showed that in single-slot auctions the problem can be solved using a suitable exploration-exploitation mechanism able to achieve a per-step regret of order O(T-1/3) (where T is the number of times the auction is repeated). In this paper we extend these results to the general case of contextual multi-slot auctions with position- and ad-dependent externalities. In particular, we prove novel upper-bounds on the revenue loss w.r.t. to a VCG auction and we report numerical simulations investigating their accuracy in predicting the dependency of the regret on the number of rounds T, the number of slots K, and the number of advertisements n.

Recently, there has been a significant interest in studying security games to provide tools for addressing resource allocation problems in security applications. Patrolling security games (PSGs) constitute a special class of security games wherein the resources are mobile. One of the most relevant open problems in security games is the design of scalable algorithms to tackle realistic scenarios. While the literature mainly focuses on heuristics and decomposition techniques (e.g., double oracle), in this paper we provide, to the best of our knowledge, the first study on the use of abstractions in security games (specifically for PSGs) to design scalable algorithms. We define some classes of abstractions and we provide parametric algorithms to automatically generate abstractions. We show that abstractions allow one to relax the constraint of patrolling strategies' Markovianity (customary in PSGs) and to solve large game instances. We additionally pose the problem to search for the optimal abstraction and we develop an anytime algorithm to find it.

Recently there is an increase in smaller, domain-specific search engines that scour the deep web finding information that general-purpose engines are unable to discover. These search engines play a crucial role in the new generation of search paradigms where federated search engines (FSEs) integrate search results from heterogeneous sources. In this paper we pose, for the first time, the problem to design a revenue mechanism that ensures profits both to individual search engines and FSEs as a mechanism design problem. To this end, we extend the sponsored search auction models and we discuss possibility and impossibility results on the implementation of an incentive compatible mechanism. Specifically, we develop an execution-contingent VCG (where payments depend on the observed click behavior) that satisfies both individual rationality and weak budget balance in expectation.

Equilibrium computation in games is currently considered one of the most challenging issues in AI. In this paper, we provide, to the best of our knowledge, the first algorithm to compute a Selten's extensive-form perfect equilibrium (EFPE) with two--player games. EFPE refines the Nash equilibrium requiring the equilibrium to be robust to slight perturbations of both players' behavioral strategies. Our result puts the computation of an EFPE into the PPAD class, leaving open the question whether or not the problem is hard. Finally, we experimentally evaluate the computational time spent to find an EFPE and some relaxations of EFPE.

Use of game theoretical models to derive randomized mobile robot patrolling strategies has recently received a growing attention. We focus on the problem of patrolling environments with arbitrary topologies using multiple robots. We address two important issues cur rently open in the literature. We determine the smallest number of robots needed to patrol a given environment and we compute the optimal patrolling strategies along several coordination dimensions. Finally, we experimentally evaluate the proposed techniques.

Game theoretical approaches have been recently used to develop patrolling strategies for mobile robots. The idea is that the patroller and the intruder play a game, whose outcome depends on the combination of their actions. From the analysis of this game, an optimal strategy for the patrolling robot can be derived. Although game theoretical approaches are promising, their applicability in real settings is still an open problem. In this paper, we experimentally evaluate the practical applicability of the most general game theoretical approach for patrolling strategies, called BGA model. Experiments are conducted by using USARSim, with the goal of studying the behavior of the optimal patrolling strategy returned by the BGA model both in situations that violate its idealized assumptions and in comparison with other patrolling strategies that can be developed with much less computational effort.

In this paper we provide a mathematical ILP model for the Regeneration Placement Problem (RPP) which minimizes the total number of regeneration nodes allocated in a translucent optical network ensuring that all the node pairs can always reach one another via two link-disjoint lightpaths under physical-impairment constraints. Since RPP is NP-complete, large-site design problem can not be solved relying upon exact approaches. We then propose a game-theoretic approach to model RPP as a non-cooperative game and solve it applying the best response dynamic concept. Finally, we evaluate the performance of the proposed approach in terms of closeness of the obtained results to these provided by ILP: a MILP formulation is given in order to study the quality of the Nash equilibria by comparison to Price-of-Anarchy and Price-of-Stability bounds.

Cognitive radio networks aim at enhancing spectrum utilization by allowing cognitive devices to opportunistically access vast portions of the spectrum. To reach such ambitious goal, cognitive terminals must be geared with enhanced spectrum management capabilities including the detection of unused spectrum holes (spectrum sensing), the characterization of available bands (spectrum decision), the coordination with other cognitive devices in the access phase (spectrum sharing), and the capability to handover towards other spectrum holes when licensed users kick in or if a better spectrum opportunity becomes available (spectrum mobility). In this paper, a game theoretic framework is proposed to evaluate spectrum management functionalities in cognitive radio networks. The spectrum selection process is cast as a non-cooperative game among secondary users who can opportunistically select the "best" spectrum opportunity, under the tight constraint not to harm primary licensed users. Different quality measures for the spectrum opportunities are considered and evaluated in the game framework, including the spectrum bandwidth, and the spectrum opportunity holding time. The cost of spectrum mobility is also accounted in the analytical framework. Numerical results are reported to assess the quality of the game equilibria.

Using autonomous mobile robots to patrol environments for detecting intruders is a topic of increasing relevance for its possible applications. A large part of strategies for mobile patrolling robots proposed so far adopt some kind of random movements. Although these strategies are unpredictable for an intruder, they are not always efficient in getting the patroller a large expected utility. In this paper we propose an approach that considers a model of the adversary in a game theoretic framework to find optimally-efficient patrolling strategies. We show that our approach extends those proposed in literature and we experimentally analyze some of its features.

In artificial intelligence literature there is a rising interest in studying strategic interaction situations. In these situations a number of rational agents act strategically, being in competition, and their analysis is carried out by employing game theoretical tools. One of the most challenging strategic interaction situation is the strategic patrolling: a guard patrols a number of houses in the attempt to catch a rob, which, in its turn, chooses a house to rob in the attempt to be not catched by the guard. Our contribution in this paper is twofold. Firstly, we provide a critique concerning the models presented in literature and we propose a model that is game theoretical satisfactory. Secondly, by exploit the game theoretical analysis to design a solving algorithm more efficient than state-of-the-art's ones.

Simulating and controlling physiological phenomena are notoriously complex tasks to tackle and require accurate models of the phenomena of interest. Currently, most physiological processes are described by a set of partial models capturing specific aspects of the phenomena, and usually their composition does not produce effective comprehensive models. A current open issue is thus the development of techniques able to effectively describe a phenomenon starting from partial models. This is particularly relevant for heart rate regulation modeling where a large number of heterogeneous partial models exists. In this paper we make the original proposal of adopting a multiagent paradigm, called anthropic agency, to provide a powerful and flexible tool for combining partial models of heart rate regulation for adaptive cardiac pacing applications. The partial models are embedded in autonomous computational entities, called agents, that cooperatively negotiate in order to smooth their conflicts on the values of the variables forming the global model the multiagent system provides. We experimentally evaluate our approach and we analyze its properties.

The development of ambient intelligence (AmI) applications that effectively adapt to the needs of the users and environments requires, among other things, the presence of planning mechanisms for goal-oriented behavior. Planning is intended as the ability of an AmI system to build a course of actions that, when carried out by the devices in the environment, achieve a given goal. The problem of planning in AmI has not yet been adequately explored in literature. We propose a planning system for AmI applications, based on the hierarchical task network (HTN) approach and called distributed hierarchical task network (D- HTN), able to find courses of actions to address given goals. The plans produced by D-HTN are flexibly tailored to exploit the capabilities of the devices currently available in the environment in the best way. We discuss both the architecture and the implementation of D-HTN. Moreover, we present some of the experimental results that validated the proposed planner in a realistic application scenario in which an AmI system monitors and answers the needs of a diabetic patient.