Federico Cacciamani

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.

Luca Carminati, Federico Cacciamani, Marco Ciccone et al.

\emph{Ex ante} correlation is becoming the mainstream approach for \emph{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, \emph{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 \emph{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 \emph{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.

Luca Carminati, Brian Hu Zhang, Federico Cacciamani et al.

Equilibrium finding in two-player zero-sum games with perfect recall is a well-studied topic that has led to many breakthroughs in computational game theory. This paper aims to generalize such techniques to (timeable) two-player zero-sum games with imperfect recall, or equivalently to two-team zero-sum games. In this setting, the problem of computing a mixed-strategy Nash equilibrium (or, equivalently, a team maxmin equilibrium with correlation) is known to be NP-hard. We connect the imperfect-recall setting with its perfect-recall counterpart through a novel construction we call the belief game. This is a perfect-recall game equivalent to a given (timeable) two-player zero-sum game with imperfect recall. The belief game may be exponentially larger than the original game but can be solved using any standard method. We then show that the strategy spaces of the two players in the belief game can be directly represented as a DAG, leading to a possibly exponential speedup. We call this the team belief DAG (TB-DAG). The TB-DAG simultaneously enjoys essentially optimal parameterized complexity bounds and the advantages of efficient regret minimization techniques. Along the way, we show $Δ_2^P$-completeness and $Σ_2^P$-completeness of finding Nash equilibria in both mixed and behavioral strategies for the class of games we consider. Experimentally, we show that the TB-DAG, when paired with existing learning techniques, yields state-of-the-art performance on a wide variety of benchmark team games.

Luca Carminati, Federico Cacciamani, Marco Ciccone et al.

The peculiarity of adversarial team games resides in the asymmetric information available to the team members during the play, which makes the equilibrium computation problem hard even with zero-sum payoffs. The algorithms available in the literature work with implicit representations of the strategy space and mainly resort to Linear Programming and column generation techniques to enlarge incrementally the strategy space. Such representations prevent the adoption of standard tools such as abstraction generation, game solving, and subgame solving, which demonstrated to be crucial when solving huge, real-world two-player zero-sum games. Differently from these works, we answer the question of whether there is any suitable game representation enabling the adoption of those tools. In particular, our algorithms convert a sequential team game with adversaries to a classical two-player zero-sum game. In this converted game, the team is transformed into a single coordinator player who only knows information common to the whole team and prescribes to the players an action for any possible private state. Interestingly, we show that our game is more expressive than the original extensive-form game as any state/action abstraction of the extensive-form game can be captured by our representation, while the reverse does not hold. Due to the NP-hard nature of the problem, the resulting Public Team game may be exponentially larger than the original one. To limit this explosion, we provide three algorithms, each returning an information-lossless abstraction that dramatically reduces the size of the tree. These abstractions can be produced without generating the original game tree. Finally, we show the effectiveness of the proposed approach by presenting experimental results on Kuhn and Leduc Poker games, obtained by applying state-of-art algorithms for two-player zero-sum games on the converted games

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.