This article sets out to show, both empirically and also with some fancy math, that arrow switching about 1/8 of the boards in a Mitchell allows one to declare an overall winner with a great degree of fairness, and that arrow switching more than this is incorrect.
I must acknowledge my debt to David Martin who ran a brilliant seminar on this topic in 1997 and got me really interested in the subject. I should also acknowledge the contributions of the late John Manning to the subject and his book of movements that every TD should carry.
As a start point we will consider the 8-round share-and-relay movement because it illustrates the effects of the switch very clearly. For simplicity we will have "1-board rounds". The matchpointing will be by the English method where 2 points are awarded for beating another pair and one for a tie. ACBLers should just divide by two.
We will designate that pair 1NS are world experts who always make 1NT and that at all other tables the hand is passed out. This is just to keep the numbers small and easy to follow. EW have added 10. The travellers are scored on first available line with Pair 1 above Pair 8 (the sharers) and Boards 1 & 8 are below. The heading MPs shows the scores without a switch and the Switch column shows the score where the last round is switched (denoted by *s. For Board 1 we can see that 13 vs 2 get 6 and 8 respectively).
|Board 1||Board 8|
Setting out the scores for Boards 1-8 on a recap sheet with no switch we get:
|etc||Check Total: 112x8 = 896|
This is as we expect in that Pair 1 takes 7.14% from Pairs 2-8 for their extra 50%
Now let us score it with the arrrow switch (denoted by the *), and something strange happens. Not only do pairs 2-8 get an extra 2 mps from the fact they get the benefit of the EW line when they switch, they also gain 2 mps from the board when Pair 1 switches and they don't. They gain 4 matchpoints.total.
Here is the recap sheet for the one round arrow switch:
|14-18||5x8 + 2x6 + 1x0||52||46.43|
|Check Total: 1x112 + 2x56 + 12x52 + 1x48 = 896|
Some Sponsoring Organisations tend to switch half the rounds. Let us see what happens if we arrow switch the last 4 rounds
|Check Total 112 + 60x2 + 56x2 + 52x6 + 48x5 = 896|
I promised you some fancy maths and here it comes:
While you are playing a board there are three other sorts of pair:
If you get a top then; A loses R-1 mps, all of B loses 1 mp and all of C gains 1 mp So the degree of competition (compared with average) between any two pairs (i,j) in one line but not near each other is expressed as (R- 1).Aij + Bij - Cij matchpoints.
For R rounds with Q rounds arrow switched the competition against each other pair in the same line is 0 + (R-2Q) -2Q matchpoints. There are three terms reflecting: direct opponents (A); the pairs in our line (B); the pairs in the other line (C). The 2Q happens because both we and they switch in our line and on different boards
Since one never plays any of the people in one's own line we can ignore (R-1).A, hence the zero term (A). The R-2Q refers to those playing this board in my line (B); and the -2Q refers to the opposite line's players that I'm not currently playing (C) So we can say that the competition with each other pair in our line is R-4Q
The total amount of competition is expressed as R(T-1) [the check total], so the average competition available is R(T-1)/(2T-1) mps where 2T-1 is the number of other pairs. In other words we want to fight for the same number of matchpoints with every other pair to a grand total equal to the check total.
So for fairness in a movement we can say R-4Q must be equal R(T-1)/(2T- 1). This expression says "The competition between any two pairs in our line" must be equal to "the average competition between any two pairs regardless of line"
We must arrow switch slightly more than 1/8 of the rounds for fairness. Anything else is WRONG!!!
In another article I'll be looking at multi-session, and multi-section events