Many, arbitrary and complicated that little can be said. There is
One type, however, that allows a precise statement and is at the
Same time sufficiently general to be of interest. (It is also funda-
Mental in the theory of regulation.)
From all possible tables let us eliminate those that make R’s
Game too easy to be of interest. Ex. 11/4/3 showed that if a column
Contains repetitions, R’s play need not be discriminating; that is,
R need not change his move with each change of D’s move. Let
Us consider, then, only those tables in which no column contains
A repeated outcome. When this is so R must select his move on full
Knowledge of D’s move; i.e. any change of D’s move must require
A change on R’s part. (Nothing is assumed here about how the out-
Comes in one column are related to those in another, so these rela-
Tions are unrestricted.) Such a Table is 11/5/1. Now, some target
Being given, let R specify what his move will be for each move by
D. What is essential is that, win or lose, he must specify one and
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↓
1
γ
2
α
3
β
… 9
… α
This transformation uniquely specifies a set of outcomes—
Those that will actually occur if D, over a sequence of plays,
includes every possible move at least once. For 1 and γ give the
Outcome k, and so on, leading to the transformation:
↓
(1, γ ) (2, α ) (3, β ) … (9, α )
kkk…l
It can now be stated that the variety in this set of outcomes cannot
Be less than
D ’s variety
----------------------------
R ’s variety
I.e., in this case, 9/3.
It is easily proved. Suppose R marks one element in each row
And concentrates simply on keeping the variety of the marked ele-
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A N I N T R O D UC T I O N T O C Y B E R NE T I C S
REQ U ISI TE V A RI ETY
Ments as small as possible (ignoring for the moment any idea of a
Target). He marks an element in the first row. In the second row he
Must change to a new column if he is not to increase the variety
By adding a new, different, element; for in the initially selected
Column the elements are all different, by hypothesis. To keep the
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Variety down to one element he must change to a new column at
Each row. (This is the best he can do; it may be that change from
Column to column is not sufficient to keep the variety down to one
Element, but this is irrelevant, for we are interested only in what
Is the least possible variety, assuming that everything falls as
Favourably as possible). So if R has n moves available (three in the
Example), at the n-th row all the columns are used, so one of the
Columns must be used again for the next row, and a new outcome
Must be allowed into the set of outcomes. Thus in Table 11/5/1,
Selection of the k’s in the first three rows will enable the variety
To be kept to one element, but at the fourth row a second element
Must be allowed into the set of outcomes.
In general: If no two elements in the same column are equal,
And if a set of outcomes is selected by R, one from each row, and
If the table has r rows and c columns, then the variety in the
Selected set of outcomes cannot be fewer than r/c.
THE LA W O F REQ U I SI TE VA RIE TY
We can now look at this game (still with the restriction that
No element may be repeated in a column) from a slightly different
Point of view. If R, S move is unvarying, so that he produces the
Same move, whatever D, S move, then the variety in the outcomes
Will be as large as the variety in D’S moves. D now is, as it were,
Exerting full control over the outcomes.
If next R uses, or has available, two moves, then the variety of
The outcomes can be reduced to a half (but not lower). If R has
Three moves, it can be reduced to a third (but not lower); and so
On. Thus if the variety in the outcomes is to be reduced to some
Assigned number, or assigned fraction of D, S variety, R, S variety
Must be increased to at least the appropriate minimum. Only vari-
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