[MickyB]

I've seen very little on this topic - in general, people just seem to run the same movements as they would with only one section.

I suspect producing a genuine balanced movement would be nigh on impossible. If you just use two mitchells, you want to arrow-switch one-eighth of the boards to balance within the section, but to arrow-switch one-quarter of boards to balance within the other section.

It also occurred to me that you could use any amount of arrow-switching, then

-calculate the average score for each line of players
-calculate the amount of competition between each line

I think this would give you simultaneous equations which could be solved to produce an adjustment for each line, thus simulating a balanced movement.

Thoughts?

[Fluffy]

I also think that more on this area has to be done, for example a poor movement I ahve played twice:

In a multi session tournament there is a half section where you play a howel against your own line. But there are no arrow-switches.

This means that for normal sessions you score against your line since they have the same cards as you. This gives you almost the same comparison points than playing against a pair ourself. However then you go to the howell and have to play against them face to face, so you are comparing against them too much.

[campboy]

There is something you can do to recover the balance of an unbalanced movement by manipulating the scores afterwards, but not much. The rank of the competition matrix is the limiting factor; there are some linear equations which the scores must satisfy as a result of the movement, and attempting to compensate for the imbalance can't remove those constraints in a meaningful way. (Simplest case: if you run an unswitched Mitchell with no half table, the average NS score will be the same as the average EW score, and there is no way to tell which field was stronger.) But the competition matrix is bijective on its image so you can take some sort of pseudo-inverse which I guess is the same as what you are suggesting. I wrote down some details for this about 10 years ago but don't have a copy. I can't imagine any sort of score-adjustment to compensate for the movement would be popular with the players though.

The question of how you should arrow-switch for multiple sections is also something I've thought about (gordontd asked about it). I take the view that you should just define some sensible metric of the overall balance of a movement, and then see what arrow-switching schedule does best by that. Here's what I said then; I assume I'd come to the same conclusions again if I hadn't been able to find this

Quote

For a three-session event with 9-table Mitchell + 5-table Howell, arrow-switching two rounds of each Mitchell is best (by a long way). If you can't switch in the last session (barometer) then I would still just switch two rounds in both the other sessions.

The rough (i.e. wrong, but in ways which cancel out on careful analysis) explanation for this is as follows. Normally when you play a Mitchell you are only competing with other pairs sitting in the same direction as you. Arrow-switching about 1/8 of the rounds transfers about half of the competition to the other side, so you then compete in roughly equal amounts against NS pairs and EW pairs. In this event, however, you already get all the competition you need against your own line from the Howell. So you ideally want to transfer all the competition in the Mitchell to the other side. This means about twice as many switches, so 1/4 of the number of tables in the Mitchell. This argument also applies to a two-session event consisting of a Mitchell followed by two simultaneous Howells.

(The movement in question has three lines of 9 pairs and 1 stationary pair. In each session two lines play a Mitchell and the other plus the stationary pair play a Howell.)

[MickyB]

Thanks for the responses so far.

Say you are running two sections, both mitchells. Normally, the NS of section 1 spend most of their time on the same side as the EW of section 2, which feels somewhat bizarre. If you arrow-switch any boards in the conventional manner, this reduces this problem, but it exacerbates another one - that the people you compete with most are the ones you've actually played against.

I think the easiest way to balance the movement is -

One section runs as normal, with no arrow-switches
The other section arrow-switches half the boards, but by board number, not round - eg they arrow-switch boards 14-26

For two 13-table, 26-board mitchells, this would produce a total competition of -

-> 0 between any pair in section 1 and any pair in section 2 [13x1 - 13x1]
-> 26 with each of the other pairs in their section [26x1 for the same line, 2x25 - 24x1 for the other line]

How would this work in practice?

Either you make sure the stationary pairs carry out the arrow-switch, they would only have to switch directions at most twice per session. Obviously you could arrow-switch 15-26 instead of 14-26 to make life a little easier.

Alternatively, it must be possible somehow to arrow-switch the boards at source. This would probably require custom labels for dealer+vul.

[MickyB]

For comparison -

Two mitchells without any arrow-switching achieves competition of -

-> 26 with each pair in the same section, or in the same direction
-> -26 with each pair in the opposite section and opposite direction

Two mitchells with two rounds of arrow-switch achieves competition of -

-> 18 with most pairs in the same direction, regardless of section [22x1-4x1]
-> 34 with most pairs in the same section but opposite direction [50x2 - 20x1 + 4x1]
-> -18 with most pairs in the opposite direction in the other section [-22x1 + 4x1]

But in all three cases, arrow-switching the same boards as another pair will affect the competition between the two pairs.