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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 conse- quences. 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 recent paper by Farina & Pipis (2023) established the existence of uncoupled no-linear-swap regret dynamics with polynomial-time iterations in extensive-form games. The equilibrium points reached by these dynamics, known as linear correlated equilibria, are currently the tightest known relaxation of correlated equilibrium that can be learned in polynomial time in any finite extensive-form game. However, their properties remain vastly unexplored, and their computation is onerous. In this paper, we provide several contributions shedding light on the fundamental nature of linear-swap regret. First, we show a connection between linear deviations and a generalization of communication deviations in which the player can make queries to a “mediator” who replies with action recommendations, and, critically, the player is not constrained to match the timing of the game as would be the case for communication deviations. We coin this latter set the untimed communication (UTC) deviations. We show that the UTC deviations coincide precisely with the linear deviations, and therefore that any player minimizing UTC regret also minimizes linear-swap regret. We then leverage this connection to develop state-of-the-art no-regret algorithms for computing linear correlated equilibria, both in theory and in practice. In theory, our algorithms achieve polynomially better per-iteration runtimes; in practice, our algorithms represent the state of the art by several orders of magnitude. 

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. 

Policy gradient methods enjoy strong practical performance in numerous tasks in reinforcement learning. Their theoretical understanding in multiagent settings, however, remains limited, especially beyond two-player competitive and potential Markov games. In this paper, we develop a new framework to characterize optimistic policy gradient methods in multi-player Markov games with a single controller. Specifically, under the further assumption that the game exhibits an equilibrium collapse, in that the marginals of coarse correlated equilibria (CCE) induce Nash equilibria (NE), we show convergence to stationary ϵ-NE in O(1/ϵ2) iterations, where O(⋅) suppresses polynomial factors in the natural parameters of the game. Such an equilibrium collapse is well-known to manifest itself in two-player zero-sum Markov games, but also occurs even in a class of multi-player Markov games with separable interactions, as established by recent work. As a result, we bypass known complexity barriers for computing stationary NE when either of our assumptions fails. Our approach relies on a natural generalization of the classical Minty property that we introduce, which we anticipate to have further applications beyond Markov games. 

Most of the literature on learning in games has focused on the restrictive setting where the underlying repeated game does not change over time. Much less is known about the convergence of no-regret learning algorithms in dynamic multiagent settings. In this paper, we characterize the convergence of optimistic gradient descent (OGD) in time-varying games. Our framework yields sharp convergence bounds for the equilibrium gap of OGD in zero-sum games parameterized on natural variation measures of the sequence of games, subsuming known results for static games. Furthermore, we establish improved second-order variation bounds under strong convexity-concavity, as long as each game is repeated multiple times. Our results also extend to time-varying general-sum multi-player games via a bilinear formulation of correlated equilibria, which has novel implications for meta-learning and for obtaining refined variation-dependent regret bounds, addressing questions left open in prior papers. Finally, we leverage our framework to also provide new insights on dynamic regret guarantees in static games. 1 

Recent algorithms have achieved superhuman performance at a number of twoplayer zero-sum games such as poker and go. However, many real-world situations are multi-player games. Zero-sum two-team games, such as bridge and football, involve two teams where each member of the team shares the same reward with every other member of that team, and each team has the negative of the reward of the other team. A popular solution concept in this setting, called TMECor, assumes that teams can jointly correlate their strategies before play, but are not able to communicate during play. This setting is harder than two-player zerosum games because each player on a team has different information and must use their public actions to signal to other members of the team. Prior works either have game-theoretic guarantees but only work in very small games, or are able to scale to large games but do not have game-theoretic guarantees. In this paper we introduce two algorithms: Team-PSRO, an extension of PSRO from twoplayer games to team games, and Team-PSRO Mix-and-Match which improves upon Team PSRO by better using population policies. In Team-PSRO, in every iteration both teams learn a joint best response to the opponent’s meta-strategy via reinforcement learning. As the reinforcement learning joint best response approaches the optimal best response, Team-PSRO is guaranteed to converge to a TMECor. In experiments on Kuhn poker and Liar’s Dice, we show that a tabular version of Team-PSRO converges to TMECor, and a version of Team PSRO using deep cooperative reinforcement learning beats self-play reinforcement learning in the large game of Google Research Football.