3) Eliminating the Bias in Scoring at Oxshott:
During this year's Tilling Trophy Compatition, there were some suggestions that there might be a bias in our Scoring system at Oxshott Bridge Club favouring North/South Pairs. The Scorer requested help from Peter Wardle and he has provided an excellent anaysis with recommendations to improve our Scoring for the future.
Here is the initial request sent to Peter:-
From: Mike Sadler [mailto:sadler252@btopenworld.com] Sent: 17 March 2011 10:32 To: 'WARDLE' Cc: 'Eric Austin'; 'George Gardiner'; 'Susan Sadler' Subject: Eliminating the Bias in Scoring
Peter,
You seem to know quite a bit about the “ins” and “outs” of scoring for Bridge. I wonder if you could help by giving explanations to the following two Questions:-
1) Do we have a bias towards N/S in our scoring at Oxshott in the Tilling Trophy?
2) How many Rounds should be “Arrow-switched” to most fairly develop a single Leaderboard between N/S and E/W? Is there a fairer system for establishing one “Order”?
The background for these Questions arises out of your excellent work trying to explain the recent BGB Sims scoring. Well, last night we had Round 6 of our Tilling Trophy (the annual Pairs Championship competition) at the Oxshott Bridge Club. There were 14 + ½ Tables present and, as usual, we played 12 Rounds of two Boards with an “Arrow-switch” for the last two Rounds. Now the players playing East were boasting about receiving great hands while the players playing North were belly-aching about not having a hand to play all evening. When the results were computed, it turmed out that those Pairs playing North/South performed better than those playing East/West; indeed SIX out of the top SEVEN on the final Leaderboard were playing North/South. That has raised the thorny question AGAIN, which you addressed before, about the suspicion that there is a bias against East/West in the scoring – as we examined in the recent BGB Sims competition.
Now, I have looked through the results of the six Rounds that we have played this year of the Tilling Trophy to see whether there is a consistent unevenness in the results between N/S and E/W. Here is the result:-
Round Number of North/South
Pairs featuring in the Top Ten
6 7
7 6
8 7
9 7
10 6
11 3
As you can see, in five of the six Rounds, the Pairs playing North/South have featured in more than half of the Top Ten placings on the final Leaderboards. Indeed over the six Rounds they fill 60% of the Top Ten placings. How can this be explained?
Secondly, I came across a short article in one of the Bridge magazines recently that recommended that, in order to fairly establish a single ranking from an evening’s Bridge, it was best to “Arrow-switch” for one-quarter of the Rounds played. This article offered no justification or explanation as to why this was the best number. Do you have any information as to the fairest method of arriving at a single Leaderboard on such occasions? Is there a better way for us to determine our Pairs Champion for the Year at Oxshott?
I know that everyone found your answers on the BGB Sims questions to be very interesting. I hope that you will be happy to attempt an answer to these Questions, and allow me to use them also on the Website, so that everyone can learn.
Hope all goes well with you.
Kind regards,
Mike
The responses from Peter Wardle make interesting reading:-
Mike,
I have now completed a similar analysis for the 12 and 14 table Mitchell movements used at Oxshott. The analysis of the 12 table movement (with tables 1 and 1 sharing boards and with a relay) has taken a lot of time because, unlike the 13/14/15 table movement, it is not symmetrical and this means that the “fairest” number of arrow-switch rounds varies according which pair’s point of view is considered – I have done the calculation from each pairs point of view and then taken an average to arrive at a recommendation as to the “fairest” number of arrow-switch rounds.
The definition of fairness relates to the number of times that each of the other pairs have played the same hands as a particular pair. Ideally the number would be the same for each of the other pairs – but this is not possible because of the nature of the movements and the number of pairs playing etc.
The following is an extension of the table in my earlier email to include the 14 table movement.
|
Movement |
|
No. of arrow-switch rounds |
|
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
13 table straight Mitchell 12 x 2 board rounds |
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
|
Min |
0 |
0 |
2 |
4 |
6 |
2 |
0 |
|
|
STDV |
10.99 |
7.54 |
5.04 |
4.06 |
4.22 |
4.65 |
4.82 |
|
14 table Mitchell 12 x 2 board rounds |
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
Skip after 7 rounds. No relay No sharing |
Min |
0 |
0 |
2 |
2 |
4 |
0 |
0 |
|
|
STDV |
10.16 |
7.02 |
4.88 |
4.19 |
4.37 |
4.97 |
5.00 |
|
15 table straight Mitchell 12 x 2 board rounds |
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
|
Min |
0 |
0 |
2 |
2 |
2 |
0 |
0 |
|
|
STDV |
9.48 |
6.64 |
4.77 |
4.29 |
4.57 |
5.14 |
5.27 |
You will see that for 13, 14 and 15 tables the “fairest” result is achieved when 3 out of the 12 rounds are “arrow-switched”.
In the case of the 12 table (share and relay movement) the average standard deviation (“STDV”) over all 24 pairs is lowest when 4 out of the 12 rounds are “arrow—switched”
|
Movement |
|
No. of arrow-switch rounds |
|
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
12 table Mitchell 12 x 2 board rounds |
Average STDV |
11.99 |
8.27 |
5.46 |
4.07 |
3.96 |
4.40 |
4.69 |
|
Tables 1 &12 share boards. Relay between tables 6 & 7 |
Max STDV |
11.99 |
8.61 |
6.28 |
5.18 |
5.05 |
4.91 |
4.91 |
|
|
Min STDV |
11.99 |
8.20 |
4.91 |
3.19 |
3.40 |
4.14 |
4.38 |
Please let me know if you have questions about the above or would like a more detailed explanation of the calculations.
Regards,
Peter Wardle
From: WARDLE [mailto:wardle_1@ntlworld.com] Sent: 10 May 2011 20:29 To: 'Mike Sadler'; 'Eric Austin'; 'George Gardiner'; 'Susan Sadler' Subject: RE: Eliminating the Bias in Scoring
Mike,
As discussed I have now completed the analysis for a 15 table Mitchell movement (without relay or board sharing). Whilst doing this analysis I realised that I could use Standard Deviation (a statistical function which describes the degree of scatter about an average value) as a proxy for “fairness”. This is on the basis that if the movement is totally “fair” then every other pair in the room will have shared the same number of hands as a given pair. The following table summarises the impact of the number of arrow-switch rounds upon the standard deviation for the number of boards shared with each of the other pairs in the room – the lower the standard deviation the “Fairer” the result.
|
Movement |
|
No. of arrow-switch rounds |
|
|
|
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
13 table straight Mitchell 12 x 2 board rounds |
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
|
Min |
0 |
0 |
2 |
4 |
6 |
2 |
0 |
|
|
STDV |
10.99 |
7.54 |
5.04 |
4.06 |
4.22 |
4.65 |
4.82 |
|
|
|
|
|
|
|
|
|
|
|
15 table straight Mitchell 12 x 2 board rounds |
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
|
Min |
0 |
0 |
2 |
2 |
2 |
0 |
0 |
|
|
STDV |
9.48 |
6.64 |
4.77 |
4.29 |
4.57 |
5.14 |
5.27 |
You will see that in both cases the “fairest” result is obtained when 3 rounds out of the 12 are arrow-switched. In no case is a completely “fair” result obtained since there are always a spread in the number of hands shared with the different pairs in the room – once again, this is where the skill lies in sitting at a table which lies between two tables occupied by weak pairs!
If someone can provide me with the details of the share/relay for 12, 14 and 16 table Mitchell movements, I will do a similar analysis for these rounds.
Regards,
Peter
From: WARDLE [mailto:wardle_1@ntlworld.com] Sent: 04 May 2011 16:33 To: 'Mike Sadler'; 'Eric Austin'; 'George Gardiner'; 'Susan Sadler' Subject: RE: Eliminating the Bias in Scoring
Mike,
As I mentioned to you last time we saw each other, I was going to do some further analysis and have a further think about the issues raised by your interesting question, but that I wouldn’t be able to complete the analysis before I went on holiday. I’ve now been able to complete things (though it has taken a lot longer than I had hoped as I’ve been completing and submitting an OU End of Course Assessment) and this email is a summary of my analysis and thoughts.
In an ideal world, if one was trying to rank all the pairs in a club session, each pair should have played some of the hands that each of the other pairs in the room have played (ideally the number of hands in common should be the same for all the other competing pairs). This is clearly not achieved when playing a Mitchell movement without arrow-switching. The Howell movement (where both N/S and E/W move) is inherently better HOWEVER it is better suited to situations where there are only a small number of tables (the normal number of tables at Oxshott means that it is not really an option) AND it does tend to confuse people (more people moving and people sometimes get confused about where they should move to).
Thus if trying to get a single result list for a club event the normal approach is to adopt arrow-switching (for multi-club events the Ecats approach of combining the EW and NS results is the only viable option). We then come to the question you raise about the number of rounds which should be arrow-switched to get the “fairest” result.
Firstly whilst arrow-switching can help equalise things a bit by increasing the number of pairs that have played some of the same hands (board and direction) as a particular pair, it is not perfect since it doesn’t help even up the number of hands that one has played against each pair playing on the evening concerned. To demonstrate this I have looked at an example where there are 13 tables and 24 boards are played (12 times two boards) in a simple Mitchell movement. There are 26 pairs (#1-#26, of which #1-#13 play NS in round 1 and #14-#26 play EW in round 1) and I have looked at which pairs have played the same hands as pair #1. The analysis is easier to do than explain, so I apologise if the following explanation is difficult to follow – just ask if you need clarification.
|
Number of boards where pair play same hand as Pair #1 |
|
|
Number of arrow-switch rounds |
|
Pair |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
#2 |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
#3 |
22 |
18 |
16 |
16 |
16 |
16 |
16 |
|
#4 |
22 |
18 |
14 |
12 |
12 |
12 |
12 |
|
#5 |
22 |
18 |
14 |
10 |
8 |
8 |
8 |
|
#6 |
22 |
18 |
14 |
10 |
6 |
4 |
4 |
|
#7 |
22 |
18 |
14 |
10 |
6 |
2 |
0 |
|
#8 |
22 |
18 |
14 |
10 |
6 |
2 |
0 |
|
#9 |
22 |
18 |
14 |
10 |
6 |
4 |
4 |
|
#10 |
22 |
18 |
14 |
10 |
8 |
8 |
8 |
|
#11 |
22 |
18 |
14 |
12 |
12 |
12 |
12 |
|
#12 |
22 |
18 |
16 |
16 |
16 |
16 |
16 |
|
#13 |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
#14 |
0 |
2 |
6 |
6 |
10 |
10 |
12 |
|
#15 |
0 |
4 |
6 |
10 |
10 |
14 |
10 |
|
#16 |
0 |
4 |
8 |
10 |
14 |
12 |
12 |
|
#17 |
0 |
4 |
8 |
12 |
14 |
14 |
10 |
|
#18 |
0 |
4 |
8 |
12 |
14 |
12 |
12 |
|
#19 |
0 |
4 |
8 |
12 |
12 |
12 |
10 |
|
#20 |
0 |
4 |
8 |
10 |
10 |
10 |
10 |
|
#21 |
0 |
4 |
8 |
8 |
8 |
8 |
12 |
|
#22 |
0 |
4 |
6 |
6 |
6 |
10 |
10 |
|
#23 |
0 |
4 |
4 |
4 |
8 |
8 |
12 |
|
#24 |
0 |
2 |
2 |
6 |
6 |
10 |
10 |
|
#25 |
0 |
0 |
4 |
4 |
8 |
8 |
12 |
|
#26 |
0 |
4 |
4 |
8 |
8 |
12 |
12 |
|
Max |
22 |
20 |
20 |
20 |
20 |
20 |
20 |
|
Min |
0 |
0 |
2 |
4 |
6 |
8 |
0 |
What can be seen from the above table is that in this particular example arrow-switching never gives a truly “fair” result in that there are always differences in the number of boards on which one pair’s performance is directly measured against the other pairs. It is clear that a single round of arrow-switching is insufficient as there is one pair who have never played any of the same hands as pair #1. In the range of 2-5 arrow-switched rounds the “fairness” (as measured by the minimum number of boards on which the other pairs have played to same hand as pair #1) improves. However in this example, when the arrow-switch is made at the 50% point we get back to a situation where two pairs have not played any of the hands played by pair A!!!!
It is also clear from this analysis that where one sits is very important – ideally one wants to sit so that there are two weak pairs sitting in the same direction on the table on either side of the one where one is sitting (this is because these pairs will be the ones who play the maximum number of the same hands played by pair #1). This confirms what I have said in my earlier emails – the choice of table/direction having regard for where the strong pairs are going to be sitting can affect ones chances of getting a good result.
This analysis is specific to a 13 table simple Mitchell movement and the impact of arrow-switching is likely to vary according to the number of tables (particular where there are an even number of tables as there will either be a skip, a relay of board-sharing between tables). For competitions such as the Tilling trophy where you want a single winner and decide to achieve this by arrow-switching you should have a minimum of 2 rounds of arrow-switching and it is probably best to arrow-switch after (N*0.5-1) rounds where N is the number of rounds to be played in the course of the evening.
Regards,
Peter
P.S. Please note the correct to my earlier email – I missed out “NOT” in the penultimate sentence
From: WARDLE [mailto:wardle_1@ntlworld.com] Sent: 17 March 2011 13:09 To: 'Mike Sadler' Cc: 'Eric Austin'; 'George Gardiner'; 'Susan Sadler' Subject: RE: Eliminating the Bias in Scoring
Mike,
Many thanks for this interesting question. I can’t claim to be an expert but am interested in statistics and the mechanics of the scoring system and will attempt to put an answer together. I’ll need a bit of time to look at the data and reflect on things, but I do have an initial thought on a possible explanation.
In common with most (if not all) clubs, you use a “free seating” system (i.e. pairs decide where they are going to sit). I suspect that there is a preference to sit North/South if possible (avoids the disruption of moving). I know from my point of view that I prefer to sit N/S because my partner and I get alternating opportunities to open the bidding, whereas for E/W one of the pair doesn’t get a chance to open the bidding until at least half-way through the evening (if the pair are of different skill levels, then whether or not the weaker partner gets the opportunities to open the bidding can affect their results) .
If the best/keenest players get to the club earlier than the others and chose to sit N/S then there could be a North/South bias when an “arrow-switch” is not used. If this does play a role in the “N/S” bias then you could consider using a random allocated seating system so that pairs play roughly the same number of times N/S and E/W.
Regards,
Peter
|