Movement
I run a weekly game for one of our clubs on BBO, 20 boards, 2 boards a round, as desired by the club.
One day there were 9 tables and, knowing that the players would not favour reducing the number of boards to 18, I changed it to 7 rounds of 3 boards each.
It became a question of which of the two alternatives is mathematically preferable, 7 rounds where two pairs don't meet, or 10 rounds where each pair plays one other pair twice.
Any views?
Comments
The second is better (I've been told by a doctor or mathematics). If you follow the Manning method of calculating amounts of competition, in the first method there are two pairs against whom there is no competition whereas in the second you just have different amounts of competition with some pairs.
However, I'll be happy to have someone explain it better (or even disagree!)
If you're using a Virtual Club account, a BBO Howell would allow 10 rounds without playing a pair twice, although I don't know how fair the BBO Howells are compared to standard EBU movements.
Prompted by this thread, I've been analysing the "BBO Howell" movement from a balance point of view (Manning method).
A BBO "Howell" works like this:
The boards use a barometer movement, as usual for BBO. I've verified that this is the movement by manually comparing it to the first three rounds, and the last round, of an actual 6-table BBO club game I played on September 9. (Note that I'm not sure what pair and table numbering BBO uses internally because it doesn't show up in the hand records that I checked; so it's possible that I'm using a different numbering to BBO's, but the movement should be equivalent up to a renumbering of the tables and pairs.)
The BBO "Howell" movement is, unfortunately, horribly unbalanced. (Truncating it doesn't meaningfully affect its balance properties; the balance is just as bad either way.) Here's what the Manning method has to say about it (using the example of 9 tables, as in the OP):
Competition between each pair (full movement): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 22 18 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 22 8 2 22 22 18 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 8 3 18 22 22 18 14 10 6 2 -2 -6 -6 -2 2 6 10 14 8 4 14 18 22 22 18 14 10 6 2 -2 -6 -6 -2 2 6 10 8 5 10 14 18 22 22 18 14 10 6 2 -2 -6 -6 -2 2 6 8 6 6 10 14 18 22 22 18 14 10 6 2 -2 -6 -6 -2 2 8 7 2 6 10 14 18 22 22 18 14 10 6 2 -2 -6 -6 -2 8 8 -2 2 6 10 14 18 22 22 18 14 10 6 2 -2 -6 -6 8 9 -6 -2 2 6 10 14 18 22 22 18 14 10 6 2 -2 -6 8 10 -6 -6 -2 2 6 10 14 18 22 22 18 14 10 6 2 -2 8 11 -2 -6 -6 -2 2 6 10 14 18 22 22 18 14 10 6 2 8 12 2 -2 -6 -6 -2 2 6 10 14 18 22 22 18 14 10 6 8 13 6 2 -2 -6 -6 -2 2 6 10 14 18 22 22 18 14 10 8 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 22 22 18 14 8 15 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 22 22 18 8 16 18 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 22 22 8 17 22 18 14 10 6 2 -2 -6 -6 -2 2 6 10 14 18 22 8 18 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 Competition between each pair (truncated to 10 rounds): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 15 13 2 9 -2 5 -6 1 -8 3 -4 7 0 11 4 15 15 2 15 8 15 4 11 0 7 -4 1-10 1 -6 5 -2 9 11 15 3 13 8 8 15 4 11 0 7 -6 1-10 1 -6 5 7 9 13 4 2 15 8 8 15 4 11 0 5 -6 1-10 1 3 5 7 11 5 9 4 15 8 8 15 4 11 -2 5 -6 1 -1 1 3 -4 9 6 -2 11 4 15 8 8 15 4 9 -2 5 3 1 -1 -8 3 7 7 5 0 11 4 15 8 8 15 2 9 7 5 3 -8 -1 -8 5 8 -6 7 0 11 4 15 8 8 13 11 9 7 -4 3 -8 -1 3 9 1 -4 7 0 11 4 15 8 15 13 11 0 7 -4 3 -8 1 10 -8 1 -6 5 -2 9 2 13 15 17 6 13 2 9 -2 5 1 11 3-10 1 -6 5 -2 9 11 13 17 17 6 13 2 9 -2 -6 12 -4 1-10 1 -6 5 7 9 11 6 17 17 6 13 2 9 -4 13 7 -6 1-10 1 3 5 7 0 13 6 17 17 6 13 2 -2 14 0 5 -6 1 -1 1 3 -4 7 2 13 6 17 17 6 13 0 15 11 -2 5 3 1 -1 -8 3 -4 9 2 13 6 17 17 6 2 16 4 9 7 5 3 -8 -1 -8 3 -2 9 2 13 6 17 17 4 17 15 11 9 7 -4 3 -8 -1 -8 5 -2 9 2 13 6 17 6 18 15 15 13 11 9 7 5 3 1 1 -6 -4 -2 0 2 4 6The reason for the lack of balance is that pairs who start at adjacent tables, and sitting in the same direction, will continue to be sitting in the same direction on almost every round (thus being highly competitive, like in a Mitchell); whereas pairs who are "opposite each other" in the movement will be sitting in opposite directions on almost every round (thus being allies; like in a Mitchell, they're aiding each others' scores on almost every round they don't play each other, but unlike in a Mitchell, they play each other once out of 18 rounds rather than once out of 9 rounds, so their direct results against each other fail to cancel out the support they give each other on the other rounds).
I hadn't realised until today that a round robin could be quite so unbalanced! For what it's worth, the BBO "Howell" seems to be comparable (maybe equivalent?) to a regular Howell in terms of the nature of pair movement (each pair is following the next-numbered pair around), but boards get played in different directions to in a real Howell, and that has a major impact – it's comparable in some ways to arrow-switching too many rounds of a Mitchell, but worse because of the way in pattern in which the effective arrow-switches occur.
(This post is too long; continued in a second post…)
As a comparison, here's the balance table for a 9-table Barometer Mitchell where a round 10 is added using the same pairings as round 1:
Competition between each pair (extended to 10 rounds): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 10 10 10 10 10 10 10 10 8 -1 -1 -1 -1 -1 -1 -1 -1 2 10 10 10 10 10 10 10 10 -1 8 -1 -1 -1 -1 -1 -1 -1 3 10 10 10 10 10 10 10 10 -1 -1 8 -1 -1 -1 -1 -1 -1 4 10 10 10 10 10 10 10 10 -1 -1 -1 8 -1 -1 -1 -1 -1 5 10 10 10 10 10 10 10 10 -1 -1 -1 -1 8 -1 -1 -1 -1 6 10 10 10 10 10 10 10 10 -1 -1 -1 -1 -1 8 -1 -1 -1 7 10 10 10 10 10 10 10 10 -1 -1 -1 -1 -1 -1 8 -1 -1 8 10 10 10 10 10 10 10 10 -1 -1 -1 -1 -1 -1 -1 8 -1 9 10 10 10 10 10 10 10 10 -1 -1 -1 -1 -1 -1 -1 -1 8 10 8 -1 -1 -1 -1 -1 -1 -1 -1 10 10 10 10 10 10 10 10 11 -1 8 -1 -1 -1 -1 -1 -1 -1 10 10 10 10 10 10 10 10 12 -1 -1 8 -1 -1 -1 -1 -1 -1 10 10 10 10 10 10 10 10 13 -1 -1 -1 8 -1 -1 -1 -1 -1 10 10 10 10 10 10 10 10 14 -1 -1 -1 -1 8 -1 -1 -1 -1 10 10 10 10 10 10 10 10 15 -1 -1 -1 -1 -1 8 -1 -1 -1 10 10 10 10 10 10 10 10 16 -1 -1 -1 -1 -1 -1 8 -1 -1 10 10 10 10 10 10 10 10 17 -1 -1 -1 -1 -1 -1 -1 8 -1 10 10 10 10 10 10 10 10 18 -1 -1 -1 -1 -1 -1 -1 -1 8 10 10 10 10 10 10 10 10and here's the balance table for a 9-table Barometer Mitchell where rounds 8 and 9 are missing:
Competition between each pair (truncated to 7 rounds): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 7 7 7 7 7 7 7 7 2 -7 -7 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 2 2 -7 -7 2 2 2 2 2 3 7 7 7 7 7 7 7 7 2 2 2 -7 -7 2 2 2 2 4 7 7 7 7 7 7 7 7 2 2 2 2 -7 -7 2 2 2 5 7 7 7 7 7 7 7 7 2 2 2 2 2 -7 -7 2 2 6 7 7 7 7 7 7 7 7 2 2 2 2 2 2 -7 -7 2 7 7 7 7 7 7 7 7 7 2 2 2 2 2 2 2 -7 -7 8 7 7 7 7 7 7 7 7 -7 2 2 2 2 2 2 2 -7 9 7 7 7 7 7 7 7 7 -7 -7 2 2 2 2 2 2 2 10 2 2 2 2 2 2 2 -7 -7 7 7 7 7 7 7 7 7 11 -7 2 2 2 2 2 2 2 -7 7 7 7 7 7 7 7 7 12 -7 -7 2 2 2 2 2 2 2 7 7 7 7 7 7 7 7 13 2 -7 -7 2 2 2 2 2 2 7 7 7 7 7 7 7 7 14 2 2 -7 -7 2 2 2 2 2 7 7 7 7 7 7 7 7 15 2 2 2 -7 -7 2 2 2 2 7 7 7 7 7 7 7 7 16 2 2 2 2 -7 -7 2 2 2 7 7 7 7 7 7 7 7 17 2 2 2 2 2 -7 -7 2 2 7 7 7 7 7 7 7 7 18 2 2 2 2 2 2 -7 -7 2 7 7 7 7 7 7 7 7Interestingly, omitting rounds from a Mitchell when it isn't a barometer movement produces a somewhat different competition table; the different sequence of board movements mean that different boards end up unplayed. For a 9-table non-barometer Mitchell with rounds 8 and 9 missing, the competition levels show a substantial amount of competition between the N/S and E/W fields, looking somewhere intermediate between a Mitchell and Howell:
Competition between each pair (truncated to 7 rounds): 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 1 6 5 5 5 5 5 5 6 3 -6 -6 3 4 4 4 4 4 2 6 6 5 5 5 5 5 5 4 3 -6 -6 3 4 4 4 4 3 5 6 6 5 5 5 5 5 4 4 3 -6 -6 3 4 4 4 4 5 5 6 6 5 5 5 5 4 4 4 3 -6 -6 3 4 4 5 5 5 5 6 6 5 5 5 4 4 4 4 3 -6 -6 3 4 6 5 5 5 5 6 6 5 5 4 4 4 4 4 3 -6 -6 3 7 5 5 5 5 5 6 6 5 3 4 4 4 4 4 3 -6 -6 8 5 5 5 5 5 5 6 6 -6 3 4 4 4 4 4 3 -6 9 6 5 5 5 5 5 5 6 -6 -6 3 4 4 4 4 4 3 10 3 4 4 4 4 4 3 -6 -6 5 6 5 5 5 5 6 5 11 -6 3 4 4 4 4 4 3 -6 5 5 6 5 5 5 5 6 12 -6 -6 3 4 4 4 4 4 3 6 5 5 6 5 5 5 5 13 3 -6 -6 3 4 4 4 4 4 5 6 5 5 6 5 5 5 14 4 3 -6 -6 3 4 4 4 4 5 5 6 5 5 6 5 5 15 4 4 3 -6 -6 3 4 4 4 5 5 5 6 5 5 6 5 16 4 4 4 3 -6 -6 3 4 4 5 5 5 5 6 5 5 6 17 4 4 4 4 3 -6 -6 3 4 6 5 5 5 5 6 5 5 18 4 4 4 4 4 3 -6 -6 3 5 6 5 5 5 5 6 5I guess the main upshot from all this is that BBO need to work on their movements! I'm not sure whether there's an algorithm for generating "generic" Barometer Howells which work with any number of tables and are balanced, but there has to be something better than how the BBO "Howell" works at present – even naively truncating a (non-barometer) Mitchell and combining everyone into a single field (something you're never supposed to do with a Mitchell) produces a movement that's substantially more balanced than the full version of the BBO movement.
I doubt very much that BBO are able to improve their movements very easily otherwise they would have introduced arrow-switched Mitchells a long time ago. It seems to me that BBO's Howell movements are not nearly so bad when there are small numbers of tables say up to 7 (this is when they are most used) and the full Howell is played. All of these would benefit from at least one table being arrow-switched each round but I would be interested in the precise figures if ais523 can give them. Once we are at 8 tables then the Mitchell even if incomplete will be better I think. However half tables introduce a different consideration. Clubs often do not want stranger or robot substitute pairs and equally want a 2-board sit-out rather than a 3-board sit-out which means we play the Howell despite its unbalanced nature.
I am told that the RealBridge movement does not suffer from these defects and that even a truncated Howell is well-balanced. I would be interested if this can be verified easily.
I do know that RealBridge took care in constructing their movements and checked with Ian McKinnon about them.
I'm still working on how to arrow-switch the BBO Howells. Meanwhile, I thought I'd take a look at arrow-switching Mitchells in a barometer movement.
An arrow-switched Mitchell turns out to be very easy to implement under a barometer movement. Imagine an "arrow-switching marker" that can move between tables, and any table that's holding the marker plays arrow-switched. You then move the marker the same way that you'd move the boards in a regular Mitchell (i.e. it moves in the opposite direction from the players). For odd table counts, you can just do this directly (as in a regular Mitchell). For even table counts, you need to take the same precautions to avoid players being arrow-switched twice as you would need to take in a non-barometer Mitchell to prevent players playing the same boards twice, e.g. share-and-relay the arrow-switch marker (so that, e.g., the first round has no arrow-switches, and the first round of the second half arrow-switches two adjacent tables). If you place arrow-switch markers on N tables in this pattern in a barometer movement, you'll get identical results to arrow-switching the corresponding N rounds of a non-barometer Mitchell movement.
The optimal number of tables for a singly-arrow-switched movement is 7-8. Here's what it looks like with 7 tables, starting the marker on table 7 (so that the pattern is easier to spot):
Movement (NS v EW: boardset): 1v 8: 1 2v 9: 1 3v10: 1 4v11: 1 5v12: 1 6v13: 1 14v 7: 1 1v14: 2 2v 8: 2 3v 9: 2 4v10: 2 5v11: 2 12v 6: 2 7v13: 2 1v13: 3 2v14: 3 3v 8: 3 4v 9: 3 10v 5: 3 6v11: 3 7v12: 3 1v12: 4 2v13: 4 3v14: 4 8v 4: 4 5v 9: 4 6v10: 4 7v11: 4 1v11: 5 2v12: 5 13v 3: 5 4v14: 5 5v 8: 5 6v 9: 5 7v10: 5 1v10: 6 11v 2: 6 3v12: 6 4v13: 6 5v14: 6 6v 8: 6 7v 9: 6 9v 1: 7 2v10: 7 3v11: 7 4v12: 7 5v13: 7 6v14: 7 7v 8: 7 Competition between each pair: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 1 3 3 3 3 3 3 4 0 4 4 4 4 4 2 3 3 3 3 3 3 4 4 4 0 4 4 4 3 3 3 3 3 3 3 4 4 4 4 4 0 4 4 3 3 3 3 3 3 0 4 4 4 4 4 4 5 3 3 3 3 3 3 4 4 0 4 4 4 4 6 3 3 3 3 3 3 4 4 4 4 0 4 4 7 3 3 3 3 3 3 4 4 4 4 4 4 0 8 4 4 4 0 4 4 4 3 3 3 3 3 3 9 0 4 4 4 4 4 4 3 3 3 3 3 3 10 4 4 4 4 0 4 4 3 3 3 3 3 3 11 4 0 4 4 4 4 4 3 3 3 3 3 3 12 4 4 4 4 4 0 4 3 3 3 3 3 3 13 4 4 0 4 4 4 4 3 3 3 3 3 3 14 4 4 4 4 4 4 0 3 3 3 3 3 3and here's 8 tables, with a share-and-relay for the arrow-switch marker (the marker is on the relay on round 1 – thus no tables are arrow-switched – and being shared between tables 4 and 5 on round 5):
Movement (NS v EW: boardset): 1v 9: 1 2v10: 1 3v11: 1 4v12: 1 5v13: 1 6v14: 1 7v15: 1 8v16: 1 1v16: 2 2v 9: 2 3v10: 2 4v11: 2 5v12: 2 6v13: 2 7v14: 2 15v 8: 2 1v15: 3 2v16: 3 3v 9: 3 4v10: 3 5v11: 3 6v12: 3 13v 7: 3 8v14: 3 1v14: 4 2v15: 4 3v16: 4 4v 9: 4 5v10: 4 11v 6: 4 7v12: 4 8v13: 4 1v13: 5 2v14: 5 3v15: 5 16v 4: 5 9v 5: 5 6v10: 5 7v11: 5 8v12: 5 1v12: 6 2v13: 6 14v 3: 6 4v15: 6 5v16: 6 6v 9: 6 7v10: 6 8v11: 6 1v11: 7 12v 2: 7 3v13: 7 4v14: 7 5v15: 7 6v16: 7 7v 9: 7 8v10: 7 10v 1: 8 2v11: 8 3v12: 8 4v13: 8 5v14: 8 6v15: 8 7v16: 8 8v 9: 8 Competition between each pair: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 2 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 3 4 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 4 8 4 4 4 0 4 4 4 4 4 4 0 5 4 4 4 8 4 4 4 0 4 4 4 4 4 4 0 6 4 4 4 4 4 4 4 4 4 0 4 4 4 4 4 7 4 4 4 4 4 4 4 4 4 4 4 0 4 4 4 8 4 4 4 4 4 4 4 4 4 4 4 4 4 0 4 9 4 4 4 0 0 4 4 4 4 4 4 4 4 4 8 10 0 4 4 4 4 4 4 4 4 4 4 4 4 4 4 11 4 4 4 4 4 0 4 4 4 4 4 4 4 4 4 12 4 0 4 4 4 4 4 4 4 4 4 4 4 4 4 13 4 4 4 4 4 4 0 4 4 4 4 4 4 4 4 14 4 4 0 4 4 4 4 4 4 4 4 4 4 4 4 15 4 4 4 4 4 4 4 0 4 4 4 4 4 4 4 16 4 4 4 0 0 4 4 4 8 4 4 4 4 4 4I'm surprised that nothing like this has been implemented in BBO yet – it's fairly easy to do. (The balance gets worse the further you move away from 7-8 tables, especially if you need to arrow-switch multiple tables, but it does that for non-barometer arrow-switched Mitchells too; the situation with this sort of "barometer switching", and with non-barometer arrow switching, is identical.)
EDIT: I just realised that there's a trivial way to create fair barometer movements: if you have a fair non-barometer movement in which every pair plays every board, you can simply rearrange it by playing all the board 1s in round 1, all the board 2s in round 2, and so on, assigning table numbers arbitrarily. This is guaranteed to have the same balance properties as the original movement, because every pair is playing the same boards against the same opponents in the same direction. This should make it fairly easy to develop fair Barometer Howells simply by rearranging a (non-barometer) perfect Howell.
So on the subject of "could arrow-switching the BBO Howell make it fair": I haven't come up with systematic results yet, but it seems like this is indeed possible in the cases I checked.
Sometimes you can do it just by arrow-switching particular tables. However, there are some cases (such as the case of 7 tables) where there isn't any constant pattern of arrow-switching that produces a perfectly balanced movement. When playing 2-board rounds, though (as you typically are in a Howell), it's possible to get a perfectly balanced movement (in terms of amounts of competition) even with 7 tables, via arrow-switching the two boards of a round differently. (Specifically, arrow-switching tables 2 and 4 on the first board of each 2-board round, and table 5 on the second board of each 2-board round, produces perfectly balanced amounts of competition between each pair.)
I don't think this is a complete solution to the problem of making a fair barometer Howell, though, because it has a different problem: some pairs will be vulnerable more often than others (and probably some players will be dealing more often than others, although I haven't checked this). So there's probably a need for a second form of balancing, too: deciding which arrow-switching regime to use for which board in the round, individually for each round (i.e. deciding whether to swap tables 2 and 4 before or after swapping table 5 differently for each round – this doesn't affect the levels of competition but does affect dealer and vulnerability), and that's a problem I haven't solved yet. I also haven't found a general pattern for which tables you have to arrow-switch to make the movements work (although the number of cases to check is small enough that a computer can brute-force it pretty quickly).
On another note, the anomalies in the arrow-switched Mitchell I posted earlier bother me a little: the issue is that with a share-and-relay, there's one round on which two tables arrow-switch simultaneously, and this creates an anomaly in the competition chart (because the two tables who arrow-switch don't have fair levels of competition with each other). When the number of tables is a multiple of 4, this can be solved using a criss-cross Mitchell movement as the basis of the arrow-switching, rather than a regular Mitchell; you use a barometer "movement" for the boards, a criss-cross movement for the players, and move the arrow-switching marker criss-cross style in the opposite direction of the players. Odd table counts have a perfect Mitchell anyway, so that leaves only table counts that are twice an odd number with no known arrow-switching anomalies.
It strikes me that the trick of arrow-switching different boards in a round differently may potentially be useful in Mitchells in addition to Howells: for table counts around 12 or so, one arrow-switch is not enough to balance, and two is too many, so one-and-a-half arrow-switches is probably the correct number; I'm not sure whether there's a way to do this without any anomalies, though. One big advantage of designing movements for online play is that they can be more or less arbitrarily complex without any risk of a misboard or a player going to the wrong table – no matter how theoretically perfect they are, this sort of movement would be unwise to use in a club!
The hardest problem when it comes to designing a full set of movements for online play will be trying to find movements where the number of tables doesn't neatly correspond to the number of rounds; truncating movements tends to unbalance them, but table and round counts often don't match in practice. I'm not sure how solvable that will be, but it seems like an interesting topic to look into.