The Computational Complexity of Single-Player Imperfect-Recall Games

We study single-player extensive-form games with imperfect recall, such as the Sleeping Beauty problem or the Absentminded Driver game. For such games, two natural equilibrium concepts have been proposed as alternative solution concepts to ex-ante optimality. One equilibrium concept uses generalized double halving (GDH) as a belief system and evidential decision theory (EDT), and another one uses generalized thirding (GT) as a belief system and causal decision theory (CDT). Our findings relate those three solution concepts of a game to solution concepts of a polynomial maximization problem: global optima, optimal points with respect to subsets of variables and Karush-Kuhn-Tucker (KKT) points. Based on these correspondences, we are able to settle various complexity-theoretic questions on the computation of such strategies. For ex-ante optimality and (EDT,GDH)-equilibria, we obtain NP-hardness and inapproximability, and for (CDT,GT)-equilibria we obtain CLS-completeness results.Comment: Long version of the paper that got accepted to the Thirty-Second International Joint Conference on Artificial Intelligence (IJCAI-23). 10 pages and 2 figures in the main body. 17 pages and 4 figures in the appendi

Rate of Price Discovery in Iterative Combinatorial Auctions

We study a class of iterative combinatorial auctions which can be viewed as subgradient descent methods for the problem of pricing bundles to balance supply and demand. We provide concrete convergence rates for auctions in this class, bounding the number of auction rounds needed to reach clearing prices. Our analysis allows for a variety of pricing schemes, including item, bundle, and polynomial pricing, and the respective convergence rates confirm that more expressive pricing schemes come at the cost of slower convergence. We consider two models of bidder behavior. In the first model, bidders behave stochastically according to a random utility model, which includes standard best-response bidding as a special case. In the second model, bidders behave arbitrarily (even adversarially), and meaningful convergence relies on properly designed activity rules

A marriage between adversarial team games and 2-player games: enabling abstractions, no-regret learning, and subgame solving

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 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 sub game solving. This work shows that we can re cover from this weakness by bridging the gap be tween 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 repre sented as a single coordinator who only knows information common to the whole team and pre scribes to each member an action for any pos sible 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 he original extensive form when designing ab stractions. Furthermore, we prove payoff equiva lence of our representation, and we provide tech niques that, starting directly from the extensive form, generate dramatically more compact repre sentations without information loss. Finally, we experimentally evaluate our techniques when ap plied to a standard testbed, comparing their per formance with the current state of the art

Computing Optimal Equilibria and Mechanisms via Learning in Zero-Sum Extensive-Form Games

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