A Mathematical Way to Choose a Partner

There is a special game coming up at my club and I would like to win it. I have 2 very good non-expert partners whom I could ask to play. One is very cautious (Player A) and the other is very aggressive (Player B). Which one is more likely to produce the 3 tops over average which are needed to stand a chance of coming away with the trophy?

I decided to do a simple mathematical analysis of our chances based solely on the expected distribution of tops and bottoms. Player A is a very good card player who never takes big risks. He seldom generates a bottom and not very often generates a top through overt action. On most of the boards we are at the mercy of the opposition who provide us with a scattering of average pluses and minuses that average out over the session. My task is to be a reliable partner who occasionally adds a dash of initiative.

Player B is like a stock speculator who gambles on emerging markets with unsubstantiated optimism. He gets in early and often, forcing the action with aggressive bidding based largely on distribution rather than HCP. Many of the opposing pairs who would have made a mistake if given the chance are forced to silent but effective defence against a hopeless contract. Player B has a tendency to aim always for the best possible result – too narrow a target.  Sometimes it works, and partners must get out of the way and accept some bottoms along with the more frequent tops.

One might think that a swinging player is more likely to produce a big score than a cautious player, on the grounds that swingers are masters of their own fate and may pile up many more tops when the conditions prove favorable.  Assume the upcoming session is 26 boards, top of 12. On similar sessions at the club Player A and I produce on average an estimated 3 tops and 1 bottom through our own overt actions, that bottom may be due to my over-reaction to his quiet approach. Let’s suppose that Player B produces 5 tops and 3 bottoms, my role being reduced largely to that of scorekeeper. Both methods produce 2 tops over average, a decent 58% score. The question is this: with which partner am I more likely to score more than 3 tops over average (61.5% or more)?

To obtain a rough estimate of our future chances we assume the tops are randomly distributed in time so as to conform to a Poisson distribution with a given average. The average is also the variance of the distribution, so 5 tops can produce a large number of tops at times. That’s encouraging to those who habitually play for tops, but we can’t overlook the bottoms. Psychologically we may think of a top as a result of good practice and a bottom as a result of bad luck, but the two belong to the same family for which not every member turns out to be a smashing success.

We assume the number of bottoms also conforms to a Poisson distribution, one independent of the distribution of tops. Thus the joint probability density function of tops and bottoms taken together is merely the product of their individual probabilities. This is not true in general as early results affect later actions, however, we assume that such is not the case, and that each board is played on its own merits in a consistent manner.

Poisson Probability Distributions

Player B                Player A

Events Tops (5)    Bottoms(3)        Tops(3)    Bottoms(1)

0        0.0067        0.0498            0.0498        0.3679

1        0.0337        0.1494            0.1494        0.3679

2        0.0842        0.2240            0.2240        0.1839

3        0.1404        0.2240            0.2240        0.0613

4        0.1755        0.1670            0.1670        0.0153

5        0.1755        0.1008            0.1008        0.0031

6        0.1462        0.0504            0.0504        0.0005

7        0.1004        0.0216            0.0216           —

8        0.0653        0.0081            0.0081           —

9        0.0363           —

10        0.0181           —

For those who love numbers as I do, the columns show how the number of tops and bottoms (events) are likely to be distributed over many sessions. The numbers in each column sum to 1, so their products also sum to 1, as all cases are covered. We wish to extract those cases for which the tops exceed the bottoms by 3 or more. The combinations are 3-0, 4-0, 4-1, 5-0, 5-1, 5-2, and so on.  The sum of the products in these cases gives the proportion of sessions in which the condition for a good score holds.

For Player B I estimate the proportion of good scores is just above 30%, whereas for Player A it is about 40%. Thus, swinging for tops lessens the chance of a good score when the associated number of bottoms is also high. Surprisingly, just playing a large number of hands for boring averages is a good policy if one can combine that with the ability to score well when the few opportunities arise. Failure to take advantage of erring opponents by avoiding penalty doubles is carrying caution to an unacceptable extreme. One needs to generate some tops in order to expect to win, but careful Player A appears to be the better choice.

And yet… the winners seem to score many tops, so it can’t be an entirely bad strategy to force the action during the auction. The secret is to avoid the bottoms while striving for tops. If we assume that the average number of tops created by overt action is 5, what is the average number of bottoms that is required to match the performance of Player A? It turns out that average has to be reduced to 2.2 per session.  Two bottoms are an acceptable number of disasters provided that one generates 5 tops through hyperactivity. Player B is a little too error prone and doesn’t always come up to his full potential.

Memo to Myself

When playing with cautious Player A, remind him not to always play down to the field. Encourage him to take advantage of clearly advantageous positions, such as bidding to a cold minor suit slam rather than stopping much too short in 3NT.

When playing with active Player B, don’t get upset by the occasional foolish result and needlessly add to the evitable number of bottoms by trying to recover the loss.

Remember: all raises are invitational and the minor suits are mere stepping stones.

Confession

I must now confess that the results given above are restricted by practical considerations based on my experiences with these players. I have not included cases where we might generate 6 or more boards above average (73%). Quite probably it is as much as, if not more than, my fault than theirs, but there it is. One shouldn’t be a slave to mathematical theory when practical experience over-rides the idealistic assumptions. However, suppose that Player B and I were in the expert class so that extremely good scores should not be ruled out, then we can add 10% to the total of Player B, giving him about a 40% chance of scoring over 61.5%. Those great sessions, although rare, add up. If Player A were promoted to expert class as well, he would have a 4% chance of scoring over 73%, thus keeping ahead of Player B in the ‘good score’ category, while falling behind in the ‘excellent score’ category. Thus, if the game were a world-wide pairs contest, I should choose Player B, as a big score is needed to place amongst all those unknowns who get over 70% playing in small clubs in remote locations.

These are the results for a one-session matchpoint event. One difficulty for swingers is that there may not be enough boards on which to swing tops to make up for the inevitable self-generated bottoms.  Often it is merely a matter of running out of boards. Eight tops and five bottoms constitute half the boards in play. There is a logical reason for playing a tight game in a short contest and the mathematics should reflect that.

In a 2-session (or longer) event there is more scope for recovery from early bottoms. The Poisson distributions for larger averages more closely resemble a normal (Gaussian) distribution symmetrical about the mean. The cautious player loses some of the advantage due to the skew of the Poisson distribution concentrated between zero and two bottoms. (Refer to the right-hand extreme of the table above for evidence of this effect.) As a result of the symmetry the key characteristic becomes the difference between the average number of tops and bottoms.

Swingers must combine patience with their aggression. A convenient way of achieving this is to play a system that differs from that of the majority and then to keep faithfully to that system. This removes the emotional element. One waits for the opportunities for tops that will inevitably arise because one is operating under different bidding conventions. At my club the 2/1 system is the almost exclusive choice. I prefer Precision, but suspect that Polish Club is best. The measure of success should be how many more tops than bottoms are produced systemically. Moving to Poland, I might switch to ACOL, but I very much doubt that would be a move in the right direction.

If one’s main concern is avoiding bottoms, one should adopt the communal bidding system, perhaps, for the sake of ego, adding a few trendy conventions that seldom arise. The worse the communal system the better it is for the experienced players who can make intelligent adjustments, (sometimes incorrectly referred to as ‘lies’), and who can take comfort in the belief that, come what may, they will never do worse than the dumbest pair in the me-too crowd.