Zero-sum Extensive-form Games Research Articles

Asymmetric Action Abstractions for Planning in Real-Time Strategy Games

Action abstractions restrict the number of legal actions available for real-time planning in zero-sum extensive-form games, thus allowing algorithms to focus their search on a set of promising actions. Even though unabstracted game trees can lead to optimal policies, due to real-time constraints and the tree size, they are not a practical choice. In this context, we introduce an action abstraction scheme which we call asymmetric action abstraction. Asymmetric abstractions allow search algorithms to “pay more attention” to some aspects of the game by unevenly dividing the algorithm’s search effort amongst different aspects of the game. We also introduce four algorithms that search in asymmetrically abstracted game trees to evaluate the effectiveness of our abstraction schemes. Two of our algorithms are adaptations of algorithms developed for searching in action-abstracted spaces, Portfolio Greedy Search and Stratified Strategy Selection, and the other two are adaptations of an algorithm developed for searching in unabstracted spaces, NaïveMCTS. An extensive set of experiments in a real-time strategy game shows that search algorithms using asymmetric abstractions are able to outperform all other search algorithms tested.

Team Correlated Equilibria in Zero-Sum Extensive-Form Games via Tree Decompositions

Despite the many recent practical and theoretical breakthroughs in computational game theory, equilibrium finding in extensive-form team games remains a significant challenge. While NP-hard in the worst case, there are provably efficient algorithms for certain families of team game. In particular, if the game has common external information, also known as A-loss recall---informally, actions played by non-team members (i.e., the opposing team or nature) are either unknown to the entire team, or common knowledge within the team---then polynomial-time algorithms exist. In this paper, we devise a completely new algorithm for solving team games. It uses a tree decomposition of the constraint system representing each team's strategy to reduce the number and degree of constraints required for correctness (tightness of the mathematical program). Our approach has the bags of the tree decomposition correspond to team-public states---that is, minimal sets of nodes (that is, states of the team) such that, upon reaching the set, it is common knowledge among the players on the team that the set has been reached. Our algorithm reduces the problem of solving team games to a linear program with at most O(NW^(w+1)) nonzero entries in the constraint matrix, where N is the size of the game tree, w is a parameter that depends on the amount of uncommon external information, and W is the treewidth of the tree decomposition. In public-action games, our program size is bounded by the tighter 2^(O(nt))N for teams of n players with t types each. Our algorithm is based on a new way to write a custom, concise tree decomposition, and its fast run time does not assume that the decomposition has small treewidth. Since our algorithm describes the polytope of correlated strategies directly, we get equilibrium finding in correlated strategies for free---instead of, say, having to run a double oracle algorithm. We show via experiments on a standard suite of games that our algorithm achieves state-of-the-art performance on all benchmark game classes except one. We also present, to our knowledge, the first experiments for this setting where both teams have more than one member.

Computing Ex Ante Coordinated Team-Maxmin Equilibria in Zero-Sum Multiplayer Extensive-Form Games

Computational game theory has many applications in the modern world in both adversarial situations and the optimization of social good. While there exist many algorithms for computing solutions in two-player interactions, finding optimal strategies in multiplayer interactions efficiently remains an open challenge. This paper focuses on computing the multiplayer Team-Maxmin Equilibrium with Coordination device (TMECor) in zero-sum extensive-form games. TMECor models scenarios when a team of players coordinates ex ante against an adversary. Such situations can be found in card games (e.g., in Bridge and Poker), when a team works together to beat a target player but communication is prohibited; and also in real world, e.g., in forest-protection operations, when coordinated groups have limited contact during interdicting illegal loggers. The existing algorithms struggle to find a TMECor efficiently because of their high computational costs. To compute a TMECor in larger games, we make the following key contributions: (1) we propose a hybrid-form strategy representation for the team, which preserves the set of equilibria; (2) we introduce a column-generation algorithm with a guaranteed finite-time convergence in the infinite strategy space based on a novel best-response oracle; (3) we develop an associated-representation technique for the exact representation of the multilinear terms in the best-response oracle; and (4) we experimentally show that our algorithm is several orders of magnitude faster than prior state-of-the-art algorithms in large games.

Large Scale Learning of Agent Rationality in Two-Player Zero-Sum Games

With the recent advances in solving large, zero-sum extensive form games, there is a growing interest in the inverse problem of inferring underlying game parameters given only access to agent actions. Although a recent work provides a powerful differentiable end-to-end learning frameworks which embed a game solver within a deep-learning framework, allowing unknown game parameters to be learned via backpropagation, this framework faces significant limitations when applied to boundedly rational human agents and large scale problems, leading to poor practicality. In this paper, we address these limitations and propose a framework that is applicable for more practical settings. First, seeking to learn the rationality of human agents in complex two-player zero-sum games, we draw upon well-known ideas in decision theory to obtain a concise and interpretable agent behavior model, and derive solvers and gradients for end-to-end learning. Second, to scale up to large, real-world scenarios, we propose an efficient first-order primal-dual method which exploits the structure of extensive-form games, yielding significantly faster computation for both game solving and gradient computation. When tested on randomly generated games, we report speedups of orders of magnitude over previous approaches. We also demonstrate the effectiveness of our model on both real-world one-player settings and synthetic data.

Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games

Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework enables a new proof of CFR even in the known setting, which is derived from a perspective of decomposing polytope regret, thereby leading to an arguably simpler interpretation of the algorithm. Our generalization to convex compact sets and convex losses allows us to develop new algorithms for several problems: regularized sequential decision making, regularized Nash equilibria in zero-sum extensive-form games, and computing approximate extensive-form perfect equilibria. Our generalization also leads to the first regret-minimization algorithm for computing reduced-normal-form quantal response equilibria based on minimizing local regrets. Experiments show that our framework leads to algorithms that scale at a rate comparable to the fastest variants of counterfactual regret minimization for computing Nash equilibrium, and therefore our approach leads to the first algorithm for computing quantal response equilibria in extremely large games. Our algorithms for (quadratically) regularized equilibrium finding are orders of magnitude faster than the fastest algorithms for Nash equilibrium finding; this suggests regret-minimization algorithms based on decreasing regularization for Nash equilibrium finding as future work. Finally we show that our framework enables a new kind of scalable opponent exploitation approach.

Faster algorithms for extensive-form game solving via improved smoothing functions

Sparse iterative methods, in particular first-order methods, are known to be among the most effective in solving large-scale two-player zero-sum extensive-form games. The convergence rates of these methods depend heavily on the properties of the distance-generating function that they are based on. We investigate both the theoretical and practical performance improvement of first-order methods (FOMs) for solving extensive-form games through better design of the dilated entropy function—a class of distance-generating functions related to the domains associated with the extensive-form games. By introducing a new weighting scheme for the dilated entropy function, we develop the first distance-generating function for the strategy spaces of sequential games that has only a logarithmic dependence on the branching factor of the player. This result improves the overall convergence rate of several FOMs working with dilated entropy function by a factor of $$\Omega (b^dd)$$, where b is the branching factor of the player, and d is the depth of the game tree. Thus far, counterfactual regret minimization methods have been faster in practice, and more popular, than FOMs despite their theoretically inferior convergence rates. Using our new weighting scheme and a practical parameter tuning procedure we show that, for the first time, the excessive gap technique, a classical FOM, can be made faster than the counterfactual regret minimization algorithm in practice for large games, and that the aggressive stepsize scheme of CFR+ is the only reason that the algorithm is faster in practice.

An Exact Double-Oracle Algorithm for Zero-Sum Extensive-Form Games with Imperfect Information

Developing scalable solution algorithms is one of the central problems in computational game theory. We present an iterative algorithm for computing an exact Nash equilibrium for two-player zero-sum extensive-form games with imperfect information. Our approach combines two key elements: (1) the compact sequence-form representation of extensive-form games and (2) the algorithmic framework of double-oracle methods. The main idea of our algorithm is to restrict the game by allowing the players to play only selected sequences of available actions. After solving the restricted game, new sequences are added by finding best responses to the current solution using fast algorithms. We experimentally evaluate our algorithm on a set of games inspired by patrolling scenarios, board, and card games. The results show significant runtime improvements in games admitting an equilibrium with small support, and substantial improvement in memory use even on games with large support. The improvement in memory use is particularly important because it allows our algorithm to solve much larger game instances than existing linear programming methods. Our main contributions include (1) a generic sequence-form double-oracle algorithm for solving zero-sum extensive-form games; (2) fast methods for maintaining a valid restricted game model when adding new sequences; (3) a search algorithm and pruning methods for computing best-response sequences; (4) theoretical guarantees about the convergence of the algorithm to a Nash equilibrium; (5) experimental analysis of our algorithm on several games, including an approximate version of the algorithm.

ALTERNATIVE SELECTION FUNCTIONS FOR INFORMATION SET MONTE CARLO TREE SEARCH

We evaluate the performance of various selection methods for the Monte Carlo Tree Search algorithm in two-player zero-sum extensive-form games with imperfect information. We compare the standard Upper Confident Bounds applied to Trees (UCT) along with the less common Exponential Weights for Exploration and Exploitation (Exp3) and novel Regret matching (RM) selection in two distinct imperfect information games: Imperfect Information Goofspiel and Phantom Tic-Tac-Toe. We show that UCT after initial fast convergence towards a Nash equilibrium computes increasingly worse strategies after some point in time. This is not the case with Exp3 and RM, which also show superior performance in head-to-head matches.

An exact double-oracle algorithm for zero-sum extensive-form games with imperfect information

Developing scalable solution algorithms is one of the central problems in computational game theory. We present an iterative algorithm for computing an exact Nash equilibrium for two-player zero-su...