Learning Without Human Expertise: A Case Study of the Double Dummy Bridge Problem

Abstract

Artificial neural networks, trained only on sample deals, without presentation of any human knowledge or even rules of the game, are used to estimate the number of tricks to be taken by one pair of bridge players in the so-called double dummy bridge problem (DDBP). Four representations of a deal in the input layer were tested leading to significant differences in achieved results. In order to test networks' abilities to extract knowledge from sample deals, experiments with additional inputs representing estimators of hand's strength used by humans were also performed. The superior network trained solely on sample deals outperformed all other architectures, including those using explicit human knowledge of the game of bridge. Considering the suit contracts, this network, in a sample of 100,000 testing deals, output a perfect answer in 53.11% of the cases and only in 3.52% of them was mistaken by more than one trick. The respective figures for notrump contracts were equal to 37.80% and 16.36%. The above results were compared with the ones obtained by 24 professional human bridge players--members of The Polish Bridge Union--on test sets of sizes between 27 and 864 deals per player (depending on player's time availability). In case of suit contracts, the perfect answer was obtained in 53.06% of the testing deals for ten upper-classified players and in 48.66% of them, for the remaining 14 participants of the experiment. For the notrump contracts, the respective figures were equal to 73.68% and 60.78%. Except for checking the ability of neural networks in solving the DDBP, the other goal of this research was to analyze connection weights in trained networks in a quest for weights' patterns that are explainable by experienced human bridge players. Quite surprisingly, several such patterns were discovered (e.g., preference for groups of honors, drawing special attention to Aces, favoring cards from a trump suit, gradual importance of cards in one suit--from two to the Ace, etc.). Both the numerical figures and weight patterns are stable and repeatable in a sample of neural architectures (differing only by randomly chosen initial weights). In summary, the piece of research described in this paper provides a detailed comparison between various data representations of the DDBP solved by neural networks. On a more general note, this approach can be extended to a certain class of binary classification problems.

A work point count system coupled with back-propagation for solving double dummy bridge problem

“…The play a card of the suit led. A trick it or if no trumps were played by leads to the next stage and the announced during the bidding bridge, special focus in game representation is on sharing potentials of their hands [13].…”

“…The play proceeds clockwise and each player must, trick consists of four cards and is won by the maximum played by the maximum card of the suit led. The champion of aim of the declarer is to take at least the number bidding phase when the opponents try to prevent from bridge, special focus in game representation is on the fact that players cooperate in pairs, thus sharing potentials of their hands [13].…”

“…In the training phase exactly one out of 14 output values is set to a non-zero value, usually 1.0 and in the testing phase, the output neuron with the highest value defined the final prediction. Although the second method seems to be more suitable, in most of the experiments the networks using 14 output neurons achieved worse results than the corresponding networks having only one output neuron [13]. 13 Hence, the graphical demonstration of the methodology is furnished in Fig.…”

“…Although the second method seems to be more suitable, in most of the experiments the networks using 14 output neurons achieved worse results than the corresponding networks having only one output neuron [13]. 13 Hence, the graphical demonstration of the methodology is furnished in Fig. 4 …”

“…This is used by human players because modeling the strongest possible opponents provides a lower bound on the pay off that can be expected when the opponents are less informed. Of the two methods known as Work Point Count Method (WPCM) [8] [13] and Distributional Point Method (DPM) [9][13] to evaluate the hand strength during the game, is an exclusive and a popular system used to bid a final contract in bridge game. The knowledge of the game of bridge acquired by an individual player over a period of time is also considered and it is supplied to the neural network architecture in the form of input neurons by multiplying the card values by the appropriate numbers defined in WPCM.…”