Set-Based Retrograde Analysis: Precomputing the Solution to 24-card Bridge Double Dummy Deals

Retrograde analysis is used in game-playing programs to solve states at the end of a game, working backwards toward the start of the game. The algorithm iterates through and computes the perfect-play value for as many states as resources allow. We introduce setrograde analysis which achieves the same results by operating on sets of states that have the same game value. The algorithm is demonstrated by computing exact solutions for Bridge double dummy card-play. For deals with 24 cards remaining to be played ($10^{27}$ states, which can be reduced to $10^{15}$ states using preexisting techniques), we strongly solve all deals. The setrograde algorithm performs a factor of $10^3$ fewer search operations than a standard retrograde algorithm, producing a database with a factor of $10^4$ fewer entries. For applicable domains, this allows retrograde searching to reach unprecedented search depths.

Transformer Based Planning in the Observation Space with Applications to Trick Taking Card Games

Traditional search algorithms have issues when applied to games of imperfect information where the number of possible underlying states and trajectories are very large. This challenge is particularly evident in trick-taking card games. While state sampling techniques such as Perfect Information Monte Carlo (PIMC) search has shown success in these contexts, they still have major limitations. We present Generative Observation Monte Carlo Tree Search (GO-MCTS), which utilizes MCTS on observation sequences generated by a game specific model. This method performs the search within the observation space and advances the search using a model that depends solely on the agent's observations. Additionally, we demonstrate that transformers are well-suited as the generative model in this context, and we demonstrate a process for iteratively training the transformer via population-based self-play. The efficacy of GO-MCTS is demonstrated in various games of imperfect information, such as Hearts, Skat, and "The Crew: The Quest for Planet Nine," with promising results.

Clique Analysis and Bypassing in Continuous-Time Conflict-Based Search

While the study of unit-cost Multi-Agent Pathfinding (MAPF) problems has been popular, many real-world problems require continuous time and costs due to various movement models. In this context, this paper studies symmetry-breaking enhancements for Continuous-Time Conflict-Based Search (CCBS), a solver for continuous-time MAPF. Resolving conflict symmetries in MAPF can require an exponential amount of work. We adapt known enhancements from unit-cost domains for CCBS: bypassing, which resolves cost symmetries and biclique constraints which resolve spatial conflict symmetries. We formulate a novel combination of biclique constraints with disjoint splitting for spatial conflict symmetries. Finally, we show empirically that these enhancements yield a statistically significant performance improvement versus previous state of the art, solving problems for up to 10% or 20% more agents in the same amount of time on dense graphs.

History Filtering in Imperfect Information Games: Algorithms and Complexity

Historically applied exclusively to perfect information games, depth-limited search with value functions has been key to recent advances in AI for imperfect information games. Most prominent approaches with strong theoretical guarantees require subgame decomposition - a process in which a subgame is computed from public information and player beliefs. However, subgame decomposition can itself require non-trivial computations, and its tractability depends on the existence of efficient algorithms for either full enumeration or generation of the histories that form the root of the subgame. Despite this, no formal analysis of the tractability of such computations has been established in prior work, and application domains have often consisted of games, such as poker, for which enumeration is trivial on modern hardware. Applying these ideas to more complex domains requires understanding their cost. In this work, we introduce and analyze the computational aspects and tractability of filtering histories for subgame decomposition. We show that constructing a single history from the root of the subgame is generally intractable, and then provide a necessary and sufficient condition for efficient enumeration. We also introduce a novel Markov Chain Monte Carlo-based generation algorithm for trick-taking card games - a domain where enumeration is often prohibitively expensive. Our experiments demonstrate its improved scalability in the trick-taking card game Oh Hell. These contributions clarify when and how depth-limited search via subgame decomposition can be an effective tool for sequential decision-making in imperfect information settings.

Iterative Budgeted Exponential Search

We tackle two long-standing problems related to re-expansions in heuristic search algorithms. For graph search, A* can require $\Omega(2^{n})$ expansions, where $n$ is the number of states within the final $f$ bound. Existing algorithms that address this problem like B and B' improve this bound to $\Omega(n^2)$. For tree search, IDA* can also require $\Omega(n^2)$ expansions. We describe a new algorithmic framework that iteratively controls an expansion budget and solution cost limit, giving rise to new graph and tree search algorithms for which the number of expansions is $O(n \log C)$, where $C$ is the optimal solution cost. Our experiments show that the new algorithms are robust in scenarios where existing algorithms fail. In the case of tree search, our new algorithms have no overhead over IDA* in scenarios to which IDA* is well suited and can therefore be recommended as a general replacement for IDA*.

Policy Based Inference in Trick-Taking Card Games

Trick-taking card games feature a large amount of private information that slowly gets revealed through a long sequence of actions. This makes the number of histories exponentially large in the action sequence length, as well as creating extremely large information sets. As a result, these games become too large to solve. To deal with these issues many algorithms employ inference, the estimation of the probability of states within an information set. In this paper, we demonstrate a Policy Based Inference (PI) algorithm that uses player modelling to infer the probability we are in a given state. We perform experiments in the German trick-taking card game Skat, in which we show that this method vastly improves the inference as compared to previous work, and increases the performance of the state-of-the-art Skat AI system Kermit when it is employed into its determinized search algorithm.

Front-to-End Bidirectional Heuristic Search with Near-Optimal Node Expansions

It is well-known that any admissible unidirectional heuristic search algorithm must expand all states whose $f$-value is smaller than the optimal solution cost when using a consistent heuristic. Such states are called "surely expanded" (s.e.). A recent study characterized s.e. pairs of states for bidirectional search with consistent heuristics: if a pair of states is s.e. then at least one of the two states must be expanded. This paper derives a lower bound, VC, on the minimum number of expansions required to cover all s.e. pairs, and present a new admissible front-to-end bidirectional heuristic search algorithm, Near-Optimal Bidirectional Search (NBS), that is guaranteed to do no more than 2VC expansions. We further prove that no admissible front-to-end algorithm has a worst case better than 2VC. Experimental results show that NBS competes with or outperforms existing bidirectional search algorithms, and often outperforms A* as well.

Monte Carlo Tree Search with Heuristic Evaluations using Implicit Minimax Backups

Monte Carlo Tree Search (MCTS) has improved the performance of game engines in domains such as Go, Hex, and general game playing. MCTS has been shown to outperform classic alpha-beta search in games where good heuristic evaluations are difficult to obtain. In recent years, combining ideas from traditional minimax search in MCTS has been shown to be advantageous in some domains, such as Lines of Action, Amazons, and Breakthrough. In this paper, we propose a new way to use heuristic evaluations to guide the MCTS search by storing the two sources of information, estimated win rates and heuristic evaluations, separately. Rather than using the heuristic evaluations to replace the playouts, our technique backs them up implicitly during the MCTS simulations. These minimax values are then used to guide future simulations. We show that using implicit minimax backups leads to stronger play performance in Kalah, Breakthrough, and Lines of Action.