april 2017

# The vacancy quality

Ulrik Dickow and Peter Smulders

The approach to the improvement of a pairs movement used in program balans is to minimize the spread in the "scores" for combinations of 2 pairs, by swapping ("arrow switching") the NS and EW pairs at one or more tables.

This 2-pair "score" or "amount of competition" is a measure of the way the score of one pair influences the result of the other pair, and depends for any board on whether the two pairs played the board in the same compass direction, in opposite directions, or as direct opponents. The spread in these scores may be expressed by the quality factor Qf. When all scores are equal Qf = 100 and the movement is said to be perfect. For details see http://www.pjms.nl/BALANS/balance.html

However we know already that Qf alone is not sufficient to characterize a movement. Various "perfect" 4-table Howells for instance, give rise to quite different results in the "2 strong pair" model and in the Bussemaker model. Apparently it is not enough to consider only 2-pair interactions.

A next step might be to also take 3-pair interactions into account. Rather than taking this neat but also very complicated road we pay attention to the optimal properties when one pair is absent. We calculate for each pair p the quality factor Qf1(p) when this pair is absent, and from these the average value for all possible choices of the absent pair, Qf1av. This Qf1av or "vacancy quality" is used to distinguish between movements with the same Qf.

It may be argued that the vacancy quality is closely related to the 3-pair interaction. In any case we found that optimal vacancy quality also improves the quality of the movement for the full number of pairs.

In the new setup of balans (version 7 and later) the following quantities are optimalised, and in that priority order

1. the quality factor Qf
2. a combination of Qf1av and Qf1max in the form of a weighting function
Fw = w1 * Qf1max + w2 * P * Qf1av ,   where P is the number of pairs.
3. the fourth moment d4 of the distribution of scores.

Remarks.

• Rather than the Qf's mentioned in 1) and 2) above the program internally works with the "sum of squares" for a more efficient calculation. The maximum of Qf coincides with the minimum of the sum of squares.
• For a contest with an odd number of pairs, one usually uses a movement intended for an even number, with one absent pair. You want a good balance, not only for the full movement, but also for the case of an absent pair. For this reason we introduce the weighting function, which enables the user to search for a compromise between best full movement and best movement with one absent pair.
• The d4 option may be turned off.

### Example

As an example, consider a movement for 8 pairs and 6 rounds, taken from F. Schiereck's Groot Schemaboek:
``` 8  4  6  6 0
1- 2 A  3- 4 B  5- 6 C  7- 8 D
3- 2 E  4- 6 F  7- 5 A  1- 8 C
3- 1 D  4- 7 E  2- 5 F  6- 8 B
1- 6 E  5- 4 D  2- 7 B  3- 8 F
6- 3 A  2- 4 C  7- 1 F  5- 8 E
7- 3 C  6- 2 D  1- 5 B  4- 8 A

#Qf=80   Qo=85.71
#Qf1av=66 Qf1max=66, for pair(s) 1 2 3 4 5 6 7 8
```
This movement already has the same Qf1 = 66 for any absent pair. With the new version of balans we quickly find a solution with Qf1 = 78.57 for any absent pair:
``` 8  4  6  6 0
1- 2 A  3- 4 B  5- 6 C  7- 8 D
2- 3 E  4- 6 F  7- 5 A  1- 8 C
3- 1 D  4- 7 E  5- 2 F  6- 8 B
6- 1 E  5- 4 D  2- 7 B  3- 8 F
6- 3 A  2- 4 C  1- 7 F  5- 8 E
7- 3 C  6- 2 D  1- 5 B  4- 8 A

#Qf=80   Qo=85.71
#Qf1av=78.57 Qf1max=78.57, for pair(s) 1 2 3 4 5 6 7 8
```
All in all, the quality factor Qf has not changed, but we found a considerable improvement for Qf1

For the above, the table 4 and round 1 were kept fixed in the run of balans. Releasing these restrictions does not lead to a further improvement.

There is no doubt that the resulting movement with Qf1=78.57 is better than the original with Qf1=66.00.

The improvement is demonstrated by the probability tables from the Bussemaker model

Before optimalisation

```Pair  1  2  3  4  5  6  7  8
Rank
1   79 19  2  .
2   19 58 17  4  2
3    2 20 53 17  8  .
4    .  3 25 54 16  2
5          2 16 54 25  3  .
6          .  8 17 53 20  2
7             2  4 17 58 19
8                .  2 19 79
Qd 61.0
```

After optimalisation

```Pair  1  2  3  4  5  6  7  8
Rank
1   83 15  2  .
2   16 63 16  4  2
3    1 21 55 16  9
4       2 26 56 14  2
5          2 14 56 26  2
6             9 16 55 21  1
7             2  4 16 63 16
8                .  2 15 83
Qd 64.1
```

Another striking example is the 3-table movement with 5 rounds and 2 board sets per round mentioned in the main text. One such movement, given in "Movements a Fair Approach", has Qf1av=66.67. Previous versions of "balans" found no improvement since the movement is already "perfect". But optimizing the vacancy quality leads directly to solutions with Qf1av=100. In this case even the probability table from the Bussemaker model becomes ideal:

```Pair  1  2  3  4  5  6
Rank
1  100
2     100
3        100
4           100
5              100
6                 100
Qd 100.0
```

These examples show that with the help of the vacancy quality it is possible to find improvements to pairs movements that can not be discovered by the conventional method of least squares of scores.