Bridge Play and Statistics

In our previous blog we discussed a virtual experiment involving 360 school children playing a 4-door Monty Hall game. We extended of the application of the underlying principles to the selection from a club suit consisting of 3 of the 4 cards ♣Q-8-6-2.

The box on the left below indicates the possible combinations when the LHO holds 3 card and the RHO, 1. The first column represents the card held by the RHO. Each line represents 90 samples, 360 in all, so the initial probability that the ♣Q is held by the LHO is ¼ . The samples are presented to 360 bridge players who are asked to imagine themselves defending a 3NT contract and to choose a card other than the ♣Q. Thus, from line 1 the 90 players have a choice of 3 cards, whereas for the other lines the other 270 children can choose from 2 cards only. In an ideal experiment the players would choose the spot cards equally at random, so as to minimize the information transmitted to a declarer. The expected numbers of ballots are in the middle box and the resultant conditional probabilities are shown in the box on the right. All very nice and regular.

Initial Conditions

Trial Results

Probabilities P (X | Y)

Q	8	6	2		—	30	30	30		—	1/4	1/4	1/4
8	Q	6	2		—	0	45	45		—	0	3/8	3/8
6	8	Q	2		—	45	0	45		—	3/8	0	3/8
2	8	6	Q		—	45	45	0		—	3/8	3/8	0

Totals

120

120

120

Chosen

♣8

♣6

♣2

 

With regard to human behavior it is inappropriate expect perfection. Inevitably one encounters natural variability. There is always an oddball in the crowd (maybe it’s me!) To the extent that a statistical study can be thought of as being perfect, it is with regard to the conditions under which the study was conducted rather than to the results obtained. Even the conditions of an experiment may be questioned. Why 3NT? one may ask. What does the rest of the hand look like? Let’s not get sidetracked. The relevant question here is: why assume equally probable choices?

The Maximum Entropy Principle

In the 19th century applications of statistics were condemned by those who prefer to think in terms of causes and effects. Pierre-Simon Laplace (1749-1827) caused a stir when he stated publicly to Napoleon that he had no need for divine intervention in his explanation of celestial mechanics. This amused Bonaparte but subsequently angered theologians and Newtonian scientists who maintained that some unseen hand was required to keep everything eternally rotating. With regard to statistical inference, Laplace maintained that as the sun had risen regularly for 500 years, he was willing to give odds of 1,826,214 to 1 that it would do so on the following morning as well. Some have taken this jest seriously, and have continued to argue about causality and such.

More relevant to bridge (where the hidden hand is an integral part of the game) is the Laplacian concept that all possible unknown conditions are equally probable. Metaphysicians have argued that if nothing is known about conditions, they could just as easily be assumed to possess any probability distribution one might wish to assign. Modern information theory has given us this explanation. Maximum ignorance concerning a set of conditions is a state of maximum entropy in which all probabilities are equal. If some knowledge is made available concerning these conditions, their probabilities must reflect this new knowledge, and so are equal no longer.

So we come to the play of the cards at bridge. By the time the opening lead is made, much information has been conveyed through the bidding that will affect the various probabilities. A declarer should adjust the probabilities in accordance with the information he has received on this particular hand as well as with his general knowledge of how the game is played. Returning to our survey concerning cards led from ♣8-6-2, the results may indicate that the choices are equally probable, but this is true only in a statistical sense. Some players will lead the ♣2 (low from odd), others the ♣8 (top of nothing) or ♣6 (MUD). Each lead is informative as there is a deterministic rule behind it, if declarer takes the time to look at the back of the convention card. The information is degraded to the extent to which players will deviate from the stated rules. The choices would be maximally uninformative if the opening lead were chosen at random every time. In summary, one does not play against everyone at once, but against one pair at a time.

The situation is different when a defender is required to follow to a lead by a declarer. The defender may choose to play low cards at random in order to reduce the information conveyed. The statistics of these plays are relevant in the analysis of card play in a way that a survey of opening leads is not. For opening leads, the statistics of opening leads tell us how many prefer to play MUD, the least informative of the 3 possibilities. As for following to a declarer’s lead, the statistics show us how random are the choices from insignificant cards. There is an essential difference.

The Testing of a Hypothesis

When one wishes to devise a statistical test, one must first have in mind what one is attempting to discover. The conditions of a test should be tailored with a particular question in mind. Assume we have devised a scenario discussed above where the LHO holds 3 of the 4 club cards and it doesn’t matter in any practical sense which insignificant card he chooses. Does the probability that the ♣Q is held by the RHO remain unchanged regardless of which low card appears on the first round from the LHO? In other words, in this situation do players choose from their low cards equally at random (our null hypothesis) or is there a bias? We collect the results from the 360 bridge players in the manner indicated previously and perform a test of the results to see to what degree the null hypothesis can be said to be confirmed. Here is a set of results we might obtain.

Initial Conditions

Trial Results

Probabilities P (X | Y)

Q	8	6	2		—	20	34	36		—	.19	.27	.28
8	Q	6	2		—	0	41	49		—	0	.33	.39
6	8	Q	2		—	48	0	42		—	.45	0	.33
2	8	6	Q		—	39	51	0		—	.36	.40	0

Totals

107

126

127

Chosen

♣8

♣6

♣2

 

There are methods one can use to discover to what degree one may say the results are obtainable when taken from a uniform distribution of choices. In total we have 360 choices represented of which 107 were of the ♣8, 126 of the ♣6, and 127 of the ♣2. The expected number was 120 for each. The null hypothesis that each card is equally likely to be played can be accepted at the 25% level, meaning that such variation from the norm would be generated by a random sampling of 360 trials more than 25% of the time.

Of course, the numbers represent not just one experiment but 4. The results in the first line give rise to suspicions that the ♣8 is less likely to be played from a combination of ♣8-6-2 than either the ♣6 or the ♣2. The variation evident in this mix would occur from a random sampling of equal distributions less than 10% of the time. One might conclude that more experiments are required for this combination in particular.

Alternatively, one might change the assumptions for this line. The hypothesis one is testing should not be formulated from the data themselves, for in that case one would always get a good fit, but must be proposed before the experiment is performed. Let’s assume the expected numbers are 15, 30, and 45, respectively, our guess expressed in the previous blog. The goodness-of-fit for this hypothesis is very good as variations greater than this would occur in more than 50% of samples of the same size. The biased choice model is more acceptable than the unbiased choice model.

Our Bridge Experiences Our encounters at the bridge table, successful or otherwise, represent but a very small sample of experiences from the great experiment which is Life. If it is difficult to draw conclusions from a controlled experiment, how much harder it is to do so from the chaotic conditions we encounter at the local bridge club. Our results, good or bad, are subject to a natural, random variability. Some impatiently attempt to ‘time the market’ by taking huge risks, thus increasing the variability, while others, akin to bond holders, ride out the storms with stable, but uninspired, adherence to standard textbook advice. Most tend to ‘go with the field’, which involves guessing the actions of the majority of the surrounding players. This acts to widen the statistical base, and has the advantage of minimizing variability at the cost of not attempting to maximize gain. Rarely does an expert play for averages; the late Paul Soloway said he never did so. Ideally one should prefer to employ methods based on the sound principles of probability theory, tempered by experience, without the egotistical expectation of always being right.