# Mitchell, Arrow Switches and Balance

Our club, normally playing 8-11 tables, 24-27 boards, will need a single winner for an event shortly and I've been investigating the effect of the number of arrow switches on the movement balance.

The conventional wisdom is that approximately 1 in 8 rounds should be arrow switched to give a balanced single-winner ranking from Mitchell / Bowman two-winner movements.

Using what I consider to be an accurate algorithm to calculate the balance I find that the optimal number of arrow switches is 1 for 5 or fewer rounds/tables, 2 for 6 to 9 rounds and 3 for 10 to 13 rounds. I didn't look at more than 13 rounds.

The algorithm averages all [nPairs * (nPairs-1)] the imbalance counts [range 0..nRounds] and then flips and scales so 0 is 100% and nRounds is 0%.

Has anyone else concluded this or can say why this measure is wrong

## Comments

I have always thought it was 1 arrowswitch per number of boards not number of rounds. So for example I recently played a 17 table 34 board Mitchell with the last 2 rounds arrow switched.

https://www.ebu.co.uk/documents/media/bridge-movements-the-maths.pdf

It is based on boards - one club I play at has played a Mitchell of 4 rounds of 6 boards - to get 24 boards in a 4-table movement - a full Howell is only 21 or 28. Ideally half the boards of the last round should be arrowswitched. This can be done by splitting the event into two sessions, arrowswitching the last round of the second session or by creating a pseudo 8-round event with players only moving after rounds 2,4 and 6. EBUScore allows this relatively easily. (There is a 24-board movement provided - Baron Barclay which I have used elsewhere for the same reason, but I don't know how balanced it is)

I get a bit lost in the formulas and really appreciate David Stevenson's simpler explanation why we should switch much less than a quarter of the boards:

https://www.blakjak.org/lwz_ste3.htm

This would switch about one third to one quarter of the boards - too frequent.

Should use a Howell movement for 3, 4 or 5 tables - if not then you should switch a portion of the final round, preferably staggered at each table.