Contract bridge is a game characterized by incomplete information, posing an exciting challenge for artificial intelligence methods. This paper proposes the BridgeHand2Vec approach, which leverages a neural network to embed a bridge player's hand (consisting of 13 cards) into a vector space. The resulting representation reflects the strength of the hand in the game and enables interpretable distances to be determined between different hands. This representation is derived by training a neural network to estimate the number of tricks that a pair of players can take. In the remainder of this paper, we analyze the properties of the resulting vector space and provide examples of its application in reinforcement learning, and opening bid classification. Although this was not our main goal, the neural network used for the vectorization achieves SOTA results on the DDBP2 problem (estimating the number of tricks for two given hands).

Champion bridge player Sharon Osberg once wrote, "Playing bridge is like running a business. While it's little surprise chess fell to number-crunching supercomputers long ago, you'd expect humans to maintain a more unassailable advantage in bridge, a game of incomplete information, cooperation, and sly communication. Over millennia, our brains have evolved to read subtle facial queues and body language. We've assembled sprawling societies dependent on the competition and cooperation of millions. Surely such skills are beyond the reach of machines? In recent years, the most advanced AI has begun encroaching on some of our most proudly held territory; the ability to navigate an uncertain world where information is limited, the game is infinitely nuanced, and no one succeeds alone. Last week, French startup NukkAI took another step when its NooK bridge-playing AI outplayed eight bridge world champions in a competition held in Paris. The game was simplified, and NooK didn't exactly go ...

Skat is a fascinating combinatorial card game, show-casing many of the intrinsic challenges for modern AI systems such as cooperative and adversarial behaviors (among the players), randomness (in the deal), and partial knowledge (due to hidden cards). Given the larger number of tricks and higher degree of uncertainty, reinforcement learning is less effective compared to classical board games like Chess and Go. As within the game of Bridge, in Skat we have a bidding and trick-taking stage. Prior to the trick-taking and as part of the bidding process, one phase in the game is to select two skat cards, whose quality may influence subsequent playing performance drastically. This paper looks into different skat selection strategies. Besides predicting the probability of winning and other hand strength functions we propose hard expert-rules and a scoring functions based on refined skat evaluation features. Experiments emphasize the impact of the refined skat putting algorithm on the playing performance of the bots, especially for AI bidding and AI game selection.

This paper proposes \emph{knowledge-based paraonoia search} (KBPS) to find forced wins during trick-taking in the card game Skat; for some one of the most interesting card games for three players. It combines efficient partial information game-tree search with knowledge representation and reasoning. This worst-case analysis, initiated after a small number of tricks, leads to a prioritized choice of cards. We provide variants of KBPS for the declarer and the opponents, and an approximation to find a forced win against most worlds in the belief space. Replaying thousands of expert games, our evaluation indicates that the AIs with the new algorithms perform better than humans in their play, achieving an average score of over 1,000 points in the agreed standard for evaluating Skat tournaments, the extended Seeger system.

Assessing the skill level of players to predict the outcome and to rank the players in a longer series of games is of critical importance for tournament play. Besides weaknesses, like an observed continuous inflation, through a steadily increasing playing body, the ELO ranking system, named after its creator Arpad Elo, has proven to be a reliable method for calculating the relative skill levels of players in zero-sum games. The evaluation of player strength in trick-taking card games like Skat or Bridge, however, is not obvious. Firstly, these are incomplete information partially observable games with more than one player, where opponent strength should influence the scoring as it does in existing ELO systems. Secondly, they are game of both skill and chance, so that besides the playing strength the outcome of a game also depends on the deal. Last but not least, there are internationally established scoring systems, in which the players are used to be evaluated, and to which ELO should align. Based on a tournament scoring system, we propose a new ELO system for Skat to overcome these weaknesses.

Bridge is a trick-taking card game requiring the ability to evaluate probabilities since it is a game of incomplete information where each player only sees its cards. In order to choose a strategy, a player needs to gather information about the hidden cards in the other players' hand. We present a methodology allowing us to model a part of card playing in Bridge using Probabilistic Logic Programming.

{\alpha}{\mu} is an anytime heuristic search algorithm for incomplete information games that assumes perfect information for the opponents. {\alpha}{\mu} addresses the strategy fusion and non-locality problems encountered by Perfect Information Monte Carlo sampling. In this paper {\alpha}{\mu} is applied to the game of Bridge.

After impressive successes in deterministic and fully-observable board games to significantly outclass humans, game playing research shifts towards non-deterministic and imperfect information card games, where humans are still persistently better. In this paper we devise a player that challenges human supremacy in Skat. We provide a complete player for playing selected variants of the game, with effective solutions for bidding and Skat putting, extracting knowledge from several million games. For trick play we combine expert rules with engineered tree exploration for optimal open card play. For dealing with uncertainty especially in Ouvert games we search the belief space.

The game of bridge consists of two stages: bidding and playing. While playing is proved to be relatively easy for computer programs, bidding is very challenging. During the bidding stage, each player knowing only his/her own cards needs to exchange information with his/her partner and interfere with opponents at the same time. Existing methods for solving perfect-information games cannot be directly applied to bidding. Most bridge programs are based on human-designed rules, which, however, cannot cover all situations and are usually ambiguous and even conflicting with each other. In this paper, we, for the first time, propose a competitive bidding system based on deep learning techniques, which exhibits two novelties. First, we design a compact representation to encode the private and public information available to a player for bidding. Second, based on the analysis of the impact of other players' unknown cards on one's final rewards, we design two neural networks to deal with imperfect information, the first one inferring the cards of the partner and the second one taking the outputs of the first one as part of its input to select a bid. Experimental results show that our bidding system outperforms the top rule-based program.