The process of revising (or constructing) a policy at execution time---known as decision-time planning---has been key to achieving superhuman performance in perfect-information games like chess and Go. A recent line of work has extended decision-time planning to imperfect-information games, leading to superhuman performance in poker. However, these methods involve solving subgames whose sizes grow quickly in the amount of non-public information, making them unhelpful when the amount of non-public information is large. Motivated by this issue, we introduce an alternative framework for decision-time planning that is not based on solving subgames, but rather on update equivalence. In this update-equivalence framework, decision-time planning algorithms replicate the updates of last-iterate algorithms, which need not rely on public information. This facilitates scalability to games with large amounts of non-public information. Using this framework, we derive a provably sound search algorithm for fully cooperative games based on mirror descent and a search algorithm for adversarial games based on magnetic mirror descent. We validate the performance of these algorithms in cooperative and adversarial domains, notably in Hanabi, the standard benchmark for search in fully cooperative imperfect-information games. Here, our mirror descent approach exceeds or matches the performance of public information-based search while using two orders of magnitude less search time. This is the first instance of a non-public-information-based algorithm outperforming public-information-based approaches in a domain they have historically dominated.

In their seminal work, Nayyar et al. (2013) showed that imperfect information can be abstracted away from common-payoff games by having players publicly announce their policies as they play. This insight underpins sound solvers and decision-time planning algorithms for common-payoff games. Unfortunately, a naive application of the same insight to two-player zero-sum games fails because Nash equilibria of the game with public policy announcements may not correspond to Nash equilibria of the original game. As a consequence, existing sound decision-time planning algorithms require complicated additional mechanisms that have unappealing properties. The main contribution of this work is showing that certain regularized equilibria do not possess the aforementioned non-correspondence problem—thus, computing them can be treated as perfect-information problems. Because these regularized equilibria can be made arbitrarily close to Nash equilibria, our result opens the door to a new perspective to solving two-player zero-sum games and yields a simplified framework for decision-time planning in two-player zero-sum games, void of the unappealing properties that plague existing decision-time planning approaches.

We extend the classic regret minimization framework for approximating equilibria in normal-form games by greedily weighing iterates based on regrets observed at runtime. Theoretically, our method retains all previous convergence rate guarantees. Empirically, experiments on large randomly generated games and normal-form subgames of the AI benchmark Diplomacy show that greedy weights outperforms previous methods whenever sampling is used, sometimes by several orders of magnitude.

We consider the task of accurately modeling strong human policies in multi-agent decision-making problems, given examples of human behavior. Imitation learning is effective at predicting human actions but may not match the strength of expert humans (e.g., by sometimes committing blunders), while self-play learning and search techniques such as AlphaZero lead to strong performance but may produce policies that differ markedly from human behavior. In chess and Go, we show that regularized search algorithms that penalize KL divergence from an imitation-learned policy yield higher prediction accuracy of strong humans and better performance than imitation learning alone. We then introduce a novel regret minimization algorithm that is regularized based on the KL divergence from an imitation-learned policy, and show that using this algorithm for search in no-press Diplomacy yields a policy that matches the human prediction accuracy of imitation learning while being substantially stronger.

Stackelberg equilibrium is a solution concept in two-player games where the leader has commitment rights over the follower. In recent years, it has become a cornerstone of many security applications, including airport patrolling and wildlife poaching prevention. Even though many of these settings are sequential in nature, existing techniques pre-compute the entire solution ahead of time. In this paper, we present a theoretically sound and empirically effective way to apply search, which leverages extra online computation to improve a solution, to the computation of Stackelberg equilibria in general-sum games. Instead of the leader attempting to solve the full game upfront, an approximate "blueprint" solution is first computed offline and is then improved online for the particular subgames encountered in actual play. We prove that our search technique is guaranteed to perform no worse than the pre-computed blueprint strategy, and empirically demonstrate that it enables approximately solving significantly larger games compared to purely offline methods. We also show that our search operation may be cast as a smaller Stackelberg problem, making our method complementary to existing algorithms based on strategy generation.

The standard problem setting in Dec-POMDPs is self-play, where the goal is to find a set of policies that play optimally together. Policies learned through self-play may adopt arbitrary conventions and implicitly rely on multi-step reasoning based on fragile assumptions about other agents’ actions and thus fail when paired with humans or independently trained agents at test time. To address this, we present off-belief learning (OBL). At each timestep OBL agents follow a policy $\pi_1$ that is optimized assuming past actions were taken by a given, fixed policy ($\pi_0$), but assuming that future actions will be taken by $\pi_1$. When $\pi_0$ is uniform random, OBL converges to an optimal policy that does not rely on inferences based on other agents’ behavior (an optimal grounded policy). OBL can be iterated in a hierarchy, where the optimal policy from one level becomes the input to the next, thereby introducing multi-level cognitive reasoning in a controlled manner. Unlike existing approaches, which may converge to any equilibrium policy, OBL converges to a unique policy, making it suitable for zero-shot coordination (ZSC). OBL can be scaled to high-dimensional settings with a fictitious transition mechanism and shows strong performance in both a toy-setting and the benchmark human-AI & ZSC problem Hanabi.

Prior AI successes in complex games have largely focused on settings with at most hundreds of actions at each decision point. In contrast, Diplomacy is a game with more than 10^20 possible actions per turn. Previous attempts to address games with large branching factors, such as Diplomacy, StarCraft, and Dota, used human data to bootstrap the policy or used handcrafted reward shaping. In this paper, we describe an algorithm for action exploration and equilibrium approximation in games with combinatorial action spaces. This algorithm simultaneously performs value iteration while learning a policy proposal network. A double oracle step is used to explore additional actions to add to the policy proposals. At each state, the target state value and policy for the model training are computed via an equilibrium search procedure. Using this algorithm, we train an agent, DORA, completely from scratch for a popular two-player variant of Diplomacy and show that it achieves superhuman performance. Additionally, we extend our methods to full-scale no-press Diplomacy and for the first time train an agent from scratch with no human data. We present evidence that this agent plays a strategy that is incompatible with human-data bootstrapped agents. This presents the first strong evidence of multiple equilibria in Diplomacy and suggests that self play alone may be insufficient for achieving superhuman performance in Diplomacy.

Lookahead search has been a critical component of recent AI successes, such as in the games of chess, go, and poker. However, the search methods used in these games, and in many other settings, are tabular. Tabular search methods do not scale well with the size of the search space, and this problem is exacerbated by stochasticity and partial observability. In this work we replace tabular search with online model-based fine-tuning of a policy neural network via reinforcement learning, and show that this approach outperforms state-of-the-art search algorithms in benchmark settings. In particular, we use our search algorithm to achieve a new state-of-the-art result in self-play Hanabi, and show the generality of our algorithm by also showing that it outperforms tabular search in the Atari game Ms. Pacman.

Recent superhuman results in games have largely been achieved in a variety of zero-sum settings, such as Go and Poker, in which agents need to compete against others. However, just like humans, real-world AI systems have to coordinate and communicate with other agents in cooperative partially observable environments as well. These settings commonly require participants to both interpret the actions of others and to act in a way that is informative when being interpreted. Those abilities are typically summarized as theory of mind and are seen as crucial for social interactions. In this paper we propose two different search techniques that can be applied to improve an arbitrary agreed-upon policy in a cooperative partially observable game. The first one, single-agent search, effectively converts the problem into a single agent setting by making all but one of the agents play according to the agreed-upon policy. In contrast, in multi-agent search all agents carry out the same common-knowledge search procedure whenever doing so is computationally feasible, and fall back to playing according to the agreed-upon policy otherwise. We prove that these search procedures are theoretically guaranteed to at least maintain the original performance of the agreed-upon policy (up to a bounded approximation error). In the benchmark challenge problem of Hanabi, our search technique greatly improves the performance of every agent we tested and when applied to a policy trained using RL achieves a new state-of-the-art score of 24.61 / 25 in the game, compared to a previous-best of 24.08 / 25.

The combination of deep reinforcement learning and search at both training and test time is a powerful paradigm that has led to a number of successes in single-agent settings and perfect-information games, best exemplified by AlphaZero. However, prior algorithms of this form cannot cope with imperfect-information games. This paper presents ReBeL, a general framework for self-play reinforcement learning and search that provably converges to a Nash equilibrium in any two-player zero-sum game. In the simpler setting of perfect-information games, ReBeL reduces to an algorithm similar to AlphaZero. Results in two different imperfect-information games show ReBeL converges to an approximate Nash equilibrium. We also show ReBeL achieves superhuman performance in heads-up no-limit Texas hold'em poker, while using far less domain knowledge than any prior poker AI.

Counterfactual regret minimization (CFR) is a family of iterative algorithms that are the most popular and, in practice, fastest approach to approximately solving large imperfectinformation games. In this paper we introduce novel CFR variants that 1) discount regrets from earlier iterations in various ways (in some cases differently for positive and negative regrets), 2) reweight iterations in various ways to obtain the output strategies, 3) use a non-standard regret minimizer and/or 4) leverage “optimistic regret matching”. They lead to dramatically improved performance in many settings. For one, we introduce a variant that outperforms CFR+, the prior state-of-the-art algorithm, in every game tested, including large-scale realistic settings. CFR+ is a formidable benchmark: no other algorithm has been able to outperform it. Finally, we show that, unlike CFR+, many of the important new variants are compatible with modern imperfect-informationgame pruning techniques and one is also compatible with sampling in the game tree.

Counterfactual Regret Minimization (CFR) is the leading algorithm for solving large imperfect-information games. It converges to an equilibrium by iteratively traversing the game tree. In order to deal with extremely large games, abstraction is typically applied before running CFR. The abstracted game is solved with tabular CFR, and its solution is mapped back to the full game. This process can be problematic because aspects of abstraction are often manual and domain specific, abstraction algorithms may miss important strategic nuances of the game, and there is a chicken-and-egg problem because determining a good abstraction requires knowledge of the equilibrium of the game. This paper introduces Deep Counterfactual Regret Minimization, a form of CFR that obviates the need for abstraction by instead using deep neural networks to approximate the behavior of CFR in the full game. We show that Deep CFR is principled and achieves strong performance in large poker games. This is the first non-tabular variant of CFR to be successful in large games.

The CFR framework has been a powerful tool for solving large-scale extensive-form games in practice. However, the theoretical rate at which past CFR-based algorithms converge to the Nash equilibrium is on the order of $O(T^{-1/2})$, where $T$ is the number of iterations. In contrast, first-order methods can be used to achieve a $O(T^{-1})$ dependence on iterations, yet these methods have been less successful in practice. In this work we present the first CFR variant that breaks the square-root dependence on iterations. By combining and extending recent advances on predictive and stable regret minimizers for the matrix-game setting we show that it is possible to leverage “optimistic” regret minimizers to achieve a $O(T^{-3/4})$ convergence rate within CFR. This is achieved by introducing a new notion of stable-predictivity, and by setting the stability of each counterfactual regret minimizer relative to its location in the decision tree. Experiments show that this method is faster than the original CFR algorithm, although not as fast as newer variants, in spite of their worst-case $O(T^{-1/2})$ dependence on iterations.

A fundamental challenge in imperfect-information games is that states do not have well-defined values. As a result, depth-limited search algorithms used in single-agent settings and perfect-information games do not apply. This paper introduces a principled way to conduct depth-limited solving in imperfect-information games by allowing the opponent to choose among a number of strategies for the remainder of the game at the depth limit. Each one of these strategies results in a different set of values for leaf nodes. This forces an agent to be robust to the different strategies an opponent may employ. We demonstrate the effectiveness of this approach by building a master-level heads-up no-limit Texas hold'em poker AI that defeats two prior top agents using only a 4-core CPU and 16 GB of memory. Developing such a powerful agent would have previously required a supercomputer.

Regret minimization is widely used in determining strategies for imperfect-information games and in online learning. In large games, computing the regrets associated with a single iteration can be slow. For this reason, pruning — in which parts of the decision tree are not traversed in every iteration — has emerged as an essential method for speeding up iterations in large games. The ability to prune is a primary reason why the Counterfactual Regret Minimization (CFR) algorithm using regret matching has emerged as the most popular iterative algorithm for imperfect-information games, despite its relatively poor convergence bound. In this paper, we introduce dynamic thresholding, in which a threshold is set at every iteration such that any action in the decision tree with probability below the threshold is set to zero probability. This enables pruning for the first time in a wide range of algorithms. We prove that dynamic thresholding can be applied to Hedge while increasing its convergence bound by only a constant factor in terms of number of iterations. Experiments demonstrate a substantial improvement in performance for Hedge as well as the excessive gap technique.

Iterative algorithms such as Counterfactual Regret Minimization (CFR) are the most popular way to solve large zero-sum imperfect-information games. In this paper we introduce Best-Response Pruning (BRP), an improvement to iterative algorithms such as CFR that allows poorly-performing actions to be temporarily pruned. We prove that when using CFR in zero-sum games, adding BRP will asymptotically prune any action that is not part of a best response to some Nash equilibrium. This leads to provably faster convergence and lower space requirements. Experiments show that BRP results in a factor of 7 reduction in space, and the reduction factor increases with game size.

No-limit Texas Hold'em is the most popular variant of poker in the world. Heads-up no-limit Texas Hold'em is the main benchmark challenge for AI in imperfect-information games. We present Libratus, the first - and so far only - AI to defeat top human professionals in that game. Libratus's architecture features three main modules, each of which has new algorithms: pre-computing a solution to an abstraction of the game which provides a high-level blueprint for the strategy of the AI, a new nested subgame-solving algorithm which repeatedly calculates a more detailed strategy as play progresses, and a self-improving module which augments the pre-computed blueprint over time.

In imperfect-information games, the optimal strategy in a subgame may depend on the strategy in other, unreached subgames. Thus a subgame cannot be solved in isolation and must instead consider the strategy for the entire game as a whole, unlike perfect-information games. Nevertheless, it is possible to first approximate a solution for the whole game and then improve it in individual subgames. This is referred to as subgame solving. We introduce subgame-solving techniques that outperform prior methods both in theory and practice. We also show how to adapt them, and past subgame-solving techniques, to respond to opponent actions that are outside the original action abstraction; this significantly outperforms the prior state-of-the-art approach, action translation. Finally, we show that subgame solving can be repeated as the game progresses down the game tree, leading to far lower exploitability. These techniques were a key component of Libratus, the first AI to defeat top humans in heads-up no-limit Texas hold'em poker.

Counterfactual Regret Minimization (CFR) is a popular iterative algorithm for approximating Nash equilibria in imperfect-information multi-step two-player zero-sum games. We introduce the first general, principled method for warm starting CFR. Our approach requires only a strategy for each player, and accomplishes the warm start at the cost of a single traversal of the game tree. The method provably warm starts CFR to as many iterations as it would have taken to reach a strategy profile of the same quality as the input strategies, and does not alter the convergence bounds of the algorithms. Unlike prior approaches to warm starting, ours can be applied in all cases. Our method is agnostic to the origins of the input strategies. For example, they can be based on human domain knowledge, the observed strategy of a strong agent, the solution of a coarser abstraction, or the output of some algorithm that converges rapidly at first but slowly as it gets closer to an equilibrium. Experiments demonstrate that one can improve overall convergence in a game by first running CFR on a smaller, coarser abstraction of the game and then using the strategy in the abstract game to warm start CFR in the full game.

The leading approach for solving large imperfect-information games is automated abstraction followed by running an equilibrium-finding algorithm. We introduce a distributed version of the most commonly used equilibrium-finding algorithm, counterfactual regret minimization (CFR), which enables CFR to scale to dramatically larger abstractions and numbers of cores. The new algorithm begets constraints on the abstraction so as to make the pieces running on different computers disjoint. We introduce an algorithm for generating such abstractions while capitalizing on state-of-the-art abstraction ideas such as imperfect recall and the earth-mover's-distance similarity metric. Our techniques enabled an equilibrium computation of unprecedented size on a supercomputer with a high inter-blade memory latency. Prior approaches run slowly on this architecture. Our approach also leads to a significant improvement over using the prior best approach on a large shared-memory server with low memory latency. Finally, we introduce a family of post-processing techniques that outperform prior ones. We applied these techniques to generate an agent for two-player no-limit Texas Hold'em. It won the 2014 Annual Computer Poker Competition, beating each opponent with statistical significance.

Counterfactual Regret Minimization (CFR) is a leading algorithm for finding a Nash equilibrium in large zero-sum imperfect-information games. CFR is an iterative algorithm that repeatedly traverses the game tree, updating regrets at each information set.We introduce an improvement to CFR that prunes any path of play in the tree, and its descendants, that has negative regret. It revisits that sequence at the earliest subsequent CFR iteration where the regret could have become positive, had that path been explored on every iteration. The new algorithm maintains CFR's convergence guarantees while making iterations significantly faster---even if previously known pruning techniques are used in the comparison. This improvement carries over to CFR+, a recent variant of CFR. Experiments show an order of magnitude speed improvement, and the relative speed improvement increases with the size of the game.

Regret matching is a widely-used algorithm for learning how to act. We begin by proving that regrets on actions in one setting (game) can be transferred to warm start the regrets for solving a different setting with same structure but different payoffs that can be written as a function of parameters. We prove how this can be done by carefully discounting the prior regrets. This provides, to our knowledge, the first principled warm-starting method for no-regret learning. It also extends to warm-starting the widely-adopted counterfactual regret minimization (CFR) algorithm for large incomplete-information games; we show this experimentally as well. We then study optimizing a parameter vector for a player in a two-player zero-sum game (e.g., optimizing bet sizes to use in poker). We propose a custom gradient descent algorithm that provably finds a locally optimal parameter vector while leveraging our warm-start theory to significantly save regret-matching iterations at each step. It optimizes the parameter vector while simultaneously finding an equilibrium. We present experiments in no-limit Leduc Hold'em and no-limit Texas Hold'em to optimize bet sizing. This amounts to the first action abstraction algorithm (algorithm for selecting a small number of discrete actions to use from a continuum of actions---a key preprocessing step for solving large games using current equilibrium-finding algorithms) with convergence guarantees for extensive-form games.