For common notions of correlated equilibrium in extensive-form games, computing an optimal ( e.g., welfare-maximizing) equilibrium is NP-hard. Other equilibrium notions-- communication [11] and certification [12] equilibria--augment the game with a mediator that has the power to both send and receive messages to and from the players--and, in particular, to remember the messages. In this paper, we investigate both notions in extensive-form games from a computational lens. We show that optimal equilibria in both notions can be computed in polynomial time, the latter under a natural additional assumption known in the literature. Our proof works by constructing a mediator-augmented game of polynomial size that explicitly represents the mediator's decisions and actions.

The extensive-form game has been studied considerably in recent years. It can represent games with multiple decision points and incomplete information, and hence it is helpful in formulating games with uncertain inputs, such as poker. We consider an extended-form game with two players and zero-sum, i.e., the sum of their payoffs is always zero. In such games, the problem of finding the optimal strategy can be formulated as a bilinear saddle-point problem. This formulation grows huge depending on the size of the game, since it has variables representing the strategies at all decision points for each player. To solve such large-scale bilinear saddle-point problems, the excessive gap technique (EGT), a smoothing method, has been studied. This method generates a sequence of approximate solutions whose error is guaranteed to converge at $\mathcal{O}(1/k)$, where $k$ is the number of iterations. However, it has the disadvantage of having poor theoretical bounds on the error related to the game size. This makes it inapplicable to large games. Our goal is to improve the smoothing method for solving extensive-form games so that it can be applied to large-scale games. To this end, we make two contributions in this work. First, we slightly modify the strongly convex function used in the smoothing method in order to improve the theoretical bounds related to the game size. Second, we propose a heuristic called centering trick, which allows the smoothing method to be combined with other methods and consequently accelerates the convergence in practice. As a result, we combine EGT with CFR+, a state-of-the-art method for extensive-form games, to achieve good performance in games where conventional smoothing methods do not perform well. The proposed smoothing method is shown to have the potential to solve large games in practice.

Standard models of multi-agent modal logic do not capture the fact that information is often ambiguous, and may be interpreted in different ways by different agents. We propose a framework that can model this, and consider different semantics that capture different assumptions about the agents' beliefs regarding whether or not there is ambiguity. We consider the impact of ambiguity on a seminal result in economics: Aumann's result saying that agents with a common prior cannot agree to disagree. This result is known not to hold if agents do not have a common prior; we show that it also does not hold in the presence of ambiguity. We then consider the tradeoff between assuming a common interpretation (i.e., no ambiguity) and a common prior (i.e., shared initial beliefs).

The Game Description Language is a high-level, rule-based formalisms for communicating the rules of arbitrary  games to general game-playing systems, whose challenging task is to learn to play previously unknown games without human intervention. Originally designed for deterministic games with complete information about the game state, the language was recently extended to include randomness and imperfect information. However, determining the extent to which this enhancement allows to describe truly arbitrary games was left as an open problem. We provide a positive answer to this question by relating the extended Game Description Language to the universal, mathematical concept of extensive-form games, proving that indeed just any such game can be described faithfully.