Andrea Celli

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

We investigate the computational complexity of min-max optimization under coupled constraints. The work of Daskalakis, Skoulakis, and Zampetakis [DSZ21] was the first to study min-max optimization through the lens of computational complexity, showing that min-max problems with nonconvex-nonconcave objectives are PPAD-hard under coupled constraints. By carefully exploiting the coupled constraints rather than the structure of the objective function, we are able to significantly simplify and strengthen the proof of the hardness result. More precisely, the first contribution of this paper is a fundamentally new proof of their main result, which improves it in multiple directions: it holds for degree-$2$ polynomials which are quadratic-linear, it improves the dependence on the parameters of the problem (also yielding constant inapproximability for gradient descent-ascent in $\ell_\infty$-norm), and it is much simpler than previous approaches. Second, we show that with general constraints (i.e., the min player and max player have different constraints), even convex-concave (bilinear) min-max optimization becomes PPAD-hard. Along the way, we also provide PPAD-membership of a general problem related to quasi-variational inequalities, which has applications beyond our problem.

Riccardo Poiani, Martino Bernasconi, Andrea Celli

We study principal-agent problems in which a principal commits to an outcome-dependent payment scheme (i.e., a contract) in order to induce an agent to take a costly action leading to a favorable outcome. We consider the online extension of the classical (one-shot) principal-agent problem, in which the principal repeatedly interacts with agents by proposing contracts over multiple rounds. The principal has no information about the agents and, crucially, does not observe their actions. As a result, the principal must learn an optimal contract using only the realized outcomes observed at each round. We focus on the setting with binary actions and single-dimensional agent types, where the agent's private type represents their cost per unit-of-effort. For adversarial-type sequences, we provide tight $Θ(T^{2/3})$ regret guarantees. Remarkably, this rate is completely independent of the number of outcomes $m$. The upper bound is based on two key components: 1) a reduction to a one-dimensional threshold optimization problem and 2) a non-uniform discretization to handle the non-Lipschitz nature of the problem. Moreover, in the case of a single (fixed) hidden type, we show that it is possible to improve the rates and provide a tight $\widetildeΘ(\sqrt{T})$ regret bound. Our algorithm is based on an explore-then-commit strategy where we first approximately learn the hidden type via a stochastic binary search, and then we commit to a ``robustified'' near-optimal contract.

Brian Zhang, Gabriele Farina, Andrea Celli et al.

We study the problem of finding optimal correlated equilibria of various sorts in extensive-form games: normal-form coarse correlated equilibrium (NFCCE), extensive-form coarse correlated equilibrium (EFCCE), and extensive-form correlated equilibrium (EFCE). We make two primary contributions. First, we introduce a new algorithm for computing optimal equilibria in all three notions. Its runtime depends exponentially only on a parameter related to the information structure of the game. We also prove a fundamental complexity gap: while our size bounds for NFCCE are similar to those achieved in the case of team games by Zhang et al., this is impossible to achieve for the other two concepts under standard complexity assumptions. Second, we propose a two-sided column generation approach for use when the runtime or memory usage of the previous algorithm is prohibitive. Our algorithm improves upon the one-sided approach of Farina et al. by means of a new decomposition of correlated strategies which allows players to re-optimize their sequence-form strategies with respect to correlation plans which were previously added to the support. Experiments show that our techniques outperform the prior state of the art for computing optimal general-sum correlated equilibria.

Brian Hu Zhang, Gabriele Farina, Ioannis Anagnostides et al.

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.

Matteo Castiglioni, Andrea Celli, Alberto Marchesi et al.

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 violation 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 $ρ/(1+ρ)$ fraction of the optimal reward and sublinear regret, where $ρ$ 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 the full-feedback as well as the bandit-feedback setting. Moreover, it allows the decision maker to seamlessly handle scenarios with non-convex rewards and constraints. We show how our framework can be applied in the context of budget-management mechanisms for repeated auctions in order to guarantee long-term constraints that are not packing (e.g., ROI constraints).

Matteo Castiglioni, Andrea Celli, Christian Kroer

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

Vashist Avadhanula, Andrea Celli, Riccardo Colini-Baldeschi et al.

We study fully dynamic online selection problems in an adversarial/stochastic setting that includes Bayesian online selection, prophet inequalities, posted price mechanisms, and stochastic probing problems subject to combinatorial constraints. In the classical ``incremental'' version of the problem, selected elements remain active until the end of the input sequence. On the other hand, in the fully dynamic version of the problem, elements stay active for a limited time interval, and then leave. This models, for example, the online matching of tasks to workers with task/worker-dependent working times, and sequential posted pricing of perishable goods. A successful approach to online selection problems in the adversarial setting is given by the notion of Online Contention Resolution Scheme (OCRS), that uses a priori information to formulate a linear relaxation of the underlying optimization problem, whose optimal fractional solution is rounded online for any adversarial order of the input sequence. Our main contribution is providing a general method for constructing an OCRS for fully dynamic online selection problems. Then, we show how to employ such OCRS to construct no-regret algorithms in a partial information model with semi-bandit feedback and adversarial inputs.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

Bilateral trade models the problem of intermediating between two rational agents -- a seller and a buyer -- both characterized by a private valuation for an item they want to trade. We study the online learning version of the problem, in which at each time step a new seller and buyer arrive and the learner has to set prices for them without any knowledge about their (adversarially generated) valuations. In this setting, known impossibility results rule out the existence of no-regret algorithms when budget balanced has to be enforced at each time step. In this paper, we introduce the notion of \emph{global budget balance}, which only requires the learner to fulfill budget balance over the entire time horizon. Under this natural relaxation, we provide the first no-regret algorithms for adversarial bilateral trade under various feedback models. First, we show that in the full-feedback model, the learner can guarantee $\tilde O(\sqrt{T})$ regret against the best fixed prices in hindsight, and that this bound is optimal up to poly-logarithmic terms. Second, we provide a learning algorithm guaranteeing a $\tilde O(T^{3/4})$ regret upper bound with one-bit feedback, which we complement with a $Ω(T^{5/7})$ lower bound that holds even in the two-bit feedback model. Finally, we introduce and analyze an alternative benchmark that is provably stronger than the best fixed prices in hindsight and is inspired by the literature on bandits with knapsacks.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

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

Matteo Russo, Andrea Celli, Riccardo Colini Baldeschi et al.

In online learning, a decision maker repeatedly selects one of a set of actions, with the goal of minimizing the overall loss incurred. Following the recent line of research on algorithms endowed with additional predictive features, we revisit this problem by allowing the decision maker to acquire additional information on the actions to be selected. In particular, we study the power of \emph{best-action queries}, which reveal beforehand the identity of the best action at a given time step. In practice, predictive features may be expensive, so we allow the decision maker to issue at most $k$ such queries. We establish tight bounds on the performance any algorithm can achieve when given access to $k$ best-action queries for different types of feedback models. In particular, we prove that in the full feedback model, $k$ queries are enough to achieve an optimal regret of $Θ\left(\min\left\{\sqrt T, \frac Tk\right\}\right)$. This finding highlights the significant multiplicative advantage in the regret rate achievable with even a modest (sublinear) number $k \in Ω(\sqrt{T})$ of queries. Additionally, we study the challenging setting in which the only available feedback is obtained during the time steps corresponding to the $k$ best-action queries. There, we provide a tight regret rate of $Θ\left(\min\left\{\frac{T}{\sqrt k},\frac{T^2}{k^2}\right\}\right)$, which improves over the standard $Θ\left(\frac{T}{\sqrt k}\right)$ regret rate for label efficient prediction for $k \in Ω(T^{2/3})$.

Emanuele Coccia, Martino Bernasconi, Andrea Celli

We study the problem of contextual online bilateral trade. At each round, the learner faces a seller-buyer pair and must propose a trade price without observing their private valuations for the item being sold. The goal of the learner is to post prices to facilitate trades between the two parties. Before posting a price, the learner observes a $d$-dimensional context vector that influences the agent's valuations. Prior work in the contextual setting has focused on linear models. In this work, we tackle a general nonparametric setting in which the buyer's and seller's valuations behave according to arbitrary Lipschitz functions of the context. We design an algorithm that leverages contextual information through a hierarchical tree construction and guarantees regret $\widetilde{O}(T^{{(d-1)}/d})$. Remarkably, our algorithm operates under two stringent features of the setting: (1) one-bit feedback, where the learner only observes whether a trade occurred or not, and (2) strong budget balance, where the learner cannot subsidize or profit from the market participants. We further provide a matching lower bound in the full-feedback setting, demonstrating the tightness of our regret bound.

Annalisa Barbara, Riccardo Poiani, Martino Bernasconi et al.

We study a setting in which two players play a (possibly approximate) Nash equilibrium of a bimatrix game, while a learner observes only their actions and has no knowledge of the equilibrium or the underlying game. A natural question is whether the learner can rationalize the observed behavior by inferring the players' payoff functions. Rather than producing a single payoff estimate, inverse game theory aims to identify the entire set of payoffs consistent with observed behavior, enabling downstream use in, e.g., counterfactual analysis and mechanism design across applications like auctions, pricing, and security games. We focus on the problem of estimating the set of feasible payoffs with high probability and up to precision $ε$ on the Hausdorff metric. We provide the first minimax-optimal rates for both exact and approximate equilibrium play, in zero-sum as well as general-sum games. Our results provide learning-theoretic foundations for set-valued payoff inference in multi-agent environments.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli et al.

We study the query complexity of min-max optimization of a nonconvex-nonconcave function $f$ over $[0,1]^d \times [0,1]^d$. We show that, given oracle access to $f$ and to its gradient $\nabla f$, any algorithm that finds an $\varepsilon$-approximate stationary point must make a number of queries that is exponential in $1/\varepsilon$ or $d$.

Martino Bernasconi, Matteo Castiglioni, Andrea Celli

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

Maria-Florina Balcan, Martino Bernasconi, Matteo Castiglioni et al.

We study the problem of online learning in Stackelberg games with side information between a leader and a sequence of followers. In every round the leader observes contextual information and commits to a mixed strategy, after which the follower best-responds. We provide learning algorithms for the leader which achieve $O(T^{1/2})$ regret under bandit feedback, an improvement from the previously best-known rates of $O(T^{2/3})$. Our algorithms rely on a reduction to linear contextual bandits in the utility space: In each round, a linear contextual bandit algorithm recommends a utility vector, which our algorithm inverts to determine the leader's mixed strategy. We extend our algorithms to the setting in which the leader's utility function is unknown, and also apply it to the problems of bidding in second-price auctions with side information and online Bayesian persuasion with public and private states. Finally, we observe that our algorithms empirically outperform previous results on numerical simulations.

Martino Bernasconi, Andrea Celli, Riccardo Colini-Baldeschi et al.

In the random-order model for online learning, the sequence of losses is chosen upfront by an adversary and presented to the learner after a random permutation. Any random-order input is \emph{asymptotically} equivalent to a stochastic i.i.d. one, but, for finite times, it may exhibit significant {\em non-stationarity}, which can hinder the performance of stochastic learning algorithms. While algorithms for adversarial inputs naturally maintain their regret guarantees in random order, simple no-regret algorithms exist for the stochastic model that fail against random-order instances. In this paper, we propose a general template to adapt stochastic learning algorithms to the random-order model without substantially affecting their regret guarantees. This allows us to recover improved regret bounds for prediction with delays, online learning with constraints, and bandits with switching costs. Finally, we investigate online classification and prove that, in random order, learnability is characterized by the VC dimension rather than the Littlestone dimension, thus providing a further separation from the general adversarial model.

Riccardo Poiani, Martino Bernasconi, Andrea Celli

In pure exploration problems, a statistician sequentially collects information to answer a question about some stochastic and unknown environment. The probability of returning a wrong answer should not exceed a maximum risk parameter $δ$ and good algorithms make as few queries to the environment as possible. The Track-and-Stop algorithm is a pioneering method to solve these problems. Specifically, it is well-known that it enjoys asymptotic optimality sample complexity guarantees for $δ\to 0$ whenever the map from the environment to its correct answers is single-valued (e.g., best-arm identification with a unique optimal arm). The Sticky Track-and-Stop algorithm extends these results to settings where, for each environment, there might exist multiple correct answers (e.g., $ε$-optimal arm identification). Although both methods are optimal in the asymptotic regime, their non-asymptotic guarantees remain unknown. In this work, we fill this gap and provide non-asymptotic guarantees for both algorithms.

Riccardo Poiani, Martino Bernasconi, Andrea Celli

We study pure exploration problems where the set of correct answers is possibly infinite, e.g., the regression of any continuous function of the means of the bandit. We derive an instance-dependent lower bound for these problems. By analyzing it, we discuss why existing methods (i.e., Sticky Track-and-Stop) for finite answer problems fail at being asymptotically optimal in this more general setting. Finally, we present a framework, Sticky-Sequence Track-and-Stop, which generalizes both Track-and-Stop and Sticky Track-and-Stop, and that enjoys asymptotic optimality. Due to its generality, our analysis also highlights special cases where existing methods enjoy optimality.

Andrea Celli, Matteo Castiglioni, Christian Kroer

We study online learning problems in which a decision maker wants to maximize their expected reward without violating a finite set of $m$ resource constraints. By casting the learning process over a suitably defined space of strategy mixtures, we recover strong duality on a Lagrangian relaxation of the underlying optimization problem, even for general settings with non-convex reward and resource-consumption functions. Then, we provide the first best-of-many-worlds type framework for this setting, with no-regret guarantees under stochastic, adversarial, and non-stationary inputs. Our framework yields the same regret guarantees of prior work in the stochastic case. On the other hand, when budgets grow at least linearly in the time horizon, it allows us to provide a constant competitive ratio in the adversarial case, which improves over the best known upper bound bound of $O(\log m \log T)$. Moreover, our framework allows the decision maker to handle non-convex reward and cost functions. We provide two game-theoretic applications of our framework to give further evidence of its flexibility. In doing so, we show that it can be employed to implement budget-pacing mechanisms in repeated first-price auctions.

Matteo Castiglioni, Alberto Marchesi, Andrea Celli et al.

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 $0 < α\leq 1$, there is no polynomial-time no-$α$-regret algorithm when the sender's utility function is supermodular or anonymous. Then, we focus on the case of submodular sender's utility functions and we show that, in this case, it is possible to design a polynomial-time no-$(1 - \frac{1}{e})$-regret algorithm. To do so, we introduce a general online gradient descent scheme 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.

Gabriele Farina, Andrea Celli, Alberto Marchesi et al.

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 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 its counterpart 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 must keep 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 is proven to be a O(T^-0.5)-approximate EFCE with high probability after T game repetitions, and an EFCE almost surely in the limit.

Federico Cacciamani, Andrea Celli, Marco Ciccone et al.

Many real-world scenarios involve teams of agents that have to coordinate their actions to reach a shared goal. We focus on the setting in which a team of agents faces an opponent in a zero-sum, imperfect-information game. Team members can coordinate their strategies before the beginning of the game, but are unable to communicate during the playing phase of the game. This is the case, for example, in Bridge, collusion in poker, and collusion in bidding. In this setting, model-free RL methods are oftentimes unable to capture coordination because agents' policies are executed in a decentralized fashion. Our first contribution is a game-theoretic centralized training regimen to effectively perform trajectory sampling so as to foster team coordination. When team members can observe each other actions, we show that this approach provably yields equilibrium strategies. Then, we introduce a signaling-based framework to represent team coordinated strategies given a buffer of past experiences. Each team member's policy is parametrized as a neural network whose output is conditioned on a suitable exogenous signal, drawn from a learned probability distribution. By combining these two elements, we empirically show convergence to coordinated equilibria in cases where previous state-of-the-art multi-agent RL algorithms did not.

Gabriele Farina, Andrea Celli, Nicola Gatti et al.

We focus on the problem of finding an optimal strategy for a team of two 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 that 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 computes such an optimal distribution by only using profiles where only one of the team members gets to randomize in each profile. We can also cap the number of such profiles we allow 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 relatively 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.

Andrea Celli, Alberto Marchesi, Gabriele Farina et al.

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 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.

Matteo Castiglioni, Andrea Celli, Alberto Marchesi et al.

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.

Matteo Castiglioni, Andrea Celli, Nicola Gatti

Persuasion studies how an informed principal may influence the behavior of agents by the strategic provision of payoff-relevant information. We focus on the fundamental multi-receiver model by Arieli and Babichenko (2019), in which there are no inter-agent externalities. Unlike prior works on this problem, we study the public persuasion problem in the general setting with: (i) arbitrary state spaces; (ii) arbitrary action spaces; (iii) arbitrary sender's utility functions. We fully characterize the computational complexity of computing a bi-criteria approximation of an optimal public signaling scheme. In particular, we show, in a voting setting of independent interest, that solving this problem requires at least a quasi-polynomial number of steps even in settings with a binary action space, assuming the Exponential Time Hypothesis. 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, in specific settings, yields a QPTAS.

Andrea Celli, Marco Ciccone, Raffaele Bongo et al.

Many real-world applications involve teams of agents that have to coordinate their actions to reach a common goal against potential adversaries. 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 collusion in bidding. The possibility for the team members to communicate before gameplay---that is, coordinate their strategies ex ante---makes the use of behavioral strategies unsatisfactory. We introduce Soft Team Actor-Critic (STAC) as a solution to the team's coordination problem that does not require any prior domain knowledge. STAC allows team members to effectively exploit ex ante communication via exogenous signals that are shared among the team. STAC reaches near-optimal coordinated strategies both in perfectly observable and partially observable games, where previous deep RL algorithms fail to reach optimal coordinated behaviors.

Matteo Castiglioni, Andrea Celli, Nicola Gatti

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 focus on the model recently introduced by Arieli and Babichenko (2019) (i.e., without inter-agent externalities), and consider, as explanatory examples, $k$-voting rule 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 general (i.e., applicable to settings beyond voting) contributions. 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 w.r.t. 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.

Andrea Celli, Stefano Coniglio, Nicola Gatti

We study an information-structure design problem (a.k.a. persuasion) with a single sender and multiple receivers with actions of a priori unknown types, independently drawn from action-specific marginal 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 case where 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 in 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. 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 a novel algorithm based on the ellipsoid method which we propose.

Andrea Celli, Stefano Coniglio, Nicola Gatti

We investigate the computation of equilibria in extensive-form games where ex ante correlation is possible, focusing on correlated equilibria requiring the least amount of communication between the players and the mediator. Motivated by the hardness results on the computation of normal-form correlated equilibria, we introduce the notion of normal-form coarse correlated equilibrium, extending the definition of coarse correlated equilibrium to sequential games. We show that, in two-player games without chance moves, an optimal (e.g., social welfare maximizing) normal-form coarse correlated equilibrium can be computed in polynomial time, and that in general multi-player games (including two-player games with Chance), the problem is NP-hard. For the former case, we provide a polynomial-time algorithm based on the ellipsoid method and also propose a more practical one, which can be efficiently applied to problems of considerable size. Then, we discuss how our algorithm can be extended to games with Chance and games with more than two players.

Andrea Celli, Nicola Gatti

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.

Nicola Basilico, Andrea Celli, Giuseppe De Nittis et al.

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 (unless degeneracy) 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.