Self-regulation and requisite variety

[W. R. Ashby](, _Introduction to Cybernetics_, chapter 11, Wiley, 1956, pp. 202- 18. [[on](] Snippets of text (from the beginning, maybe the middle, and the end) appear below, to give a sense of the content for the chapter. For more depth, the original source is cited, above. ---

1.1. In the previous chapter [not included here] we considered regulation from the biological point of view, taking it as something sufficiently well understood. In this chapter we shall examine the process of regulation itself, with the aim of finding out exactly what is involved and implied. In particular we shall develop ways of measuring the amount or degree of regulation achieved, and we shall show that this amount has an upper limit.

1.2. The subject of regulation is very wide in its applications, covering as it does most of the activities in physiology, sociology, ecology, economics, and much of the activities in almost every branch of science and life. Further, the types of regulator that exist are almost bewildering in their variety. One way of treating the subject would be to deal seriatim with the various types; and Chapter 12 [not included here] will, in fact, indicate them. In this chapter, however, we shall be attempting to get at the core of the subject - to find what is common to all. What is common to all regulators, however, is not, at first sight, much like any particular form. We will therefore start anew in the next section, making no explicit reference to what has gone before. Only after the new subject has been sufficiently developed will we begin to consider any relation it may have to regulation.

1.3. _Play and outcome_. Let us therefore forget all about regulation and simply suppose that we are watching two players, R and D, who are engaged in a game. We shall follow the fortunes of R, who is attempting to score an α. The rules are as follows. They have before them Table 1, which can be seen by both. [p. 105] D must play first, by selecting a number, and thus a particular row. R, knowing this number, then selects a Greek letter, and thus a particular column. The italic letter specified by the intersection of the row and column is the outcome. If it is an α, R wins if not, R loses. [p. 106] [....]

## The Law of Requisite Variety 1.6. 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 Z)'s variety, R's variety _must_ be increased to at least the appropriate minimum. _Only variety in R's moves can force down the variety in the outcomes._ [p. 110]

1.7. If the varieties are measured logarithmically (as is almost always convenient), and if the same conditions hold, then the theorem takes a very simple form. [....] This is the Law of Requisite Variety. To put it more picturesquely : _only variety in R can force down the variety due to D; only variety can destroy variety._ [p. 110] [....]

1.11. _Regulation again_. We can now take up again the subject of regulation, ignored since the beginning of this chapter, for the Law of Requisite Variety enables us to apply a measure to regulation. Let us go back and reconsider what is meant, essentially, by 'regulation'. There is first a set of disturbances D, that start in the world outside the organism, often far from it, and that threaten, if the regulator R does nothing, to drive the essential variables E outside their proper range of values. The values of E correspond to the 'outcomes' of the previous sections. Of all these E-values only a few (η) are compatible with the organism's life, or are unobjectionable, so that the regulator R, to be successful, must take its value in a way so related to that of D that the outcome is, if possible, always within the acceptable set η, i.e. within physiological limits. [p. 113] [....]

We can now select a portion of the diagram, and focus attention on R as a transmitter (Figure 2).

digraph Ashby { D[shape=box] R[shape=box] T[shape=box] rankdir = LR D -> R R -> T }

The Law of Requisite Variety says that _R's capacity as a regulator cannot exceed R's capacity as a channel of communication_. In the form just given, the Law of Requisite Variety can be shown m exact relation to Shannon's Theorem 10, which says that if noise appears in a message, the amount of noise that can be removed by a correction channel is limited to the amount of information that can be carried by that channel. [p. 115] [....]

1.12. The diagram of immediate effects given in the previous section is clearly related to the formulation for 'directive correlation' given by Sommerhoff (Figure 3. [p. 116] [....]

## Control 1.14. The formulations given in this chapter have already suggested that regulation and control are intimately related. Thus, Table 1 enables R not only to achieve a as outcome in spite of all D's variations; but equally to achieve b or c at will. [p. 118] [....]

## Some Variations 1.16. In 1.4 the essential facts implied by regulation were shown as a simple rectangular table, as if it were a game between two players D and R. The reader may feel that this formulation is much too simple and that there are well-known regulations that it is insufficient to represent. The formulation, however, is really much more general than it seems, and in the remaining sections of this chapter we shall examine various complications that prove, on closer examination, to be really included in the basic formulation of 1.4. [p. 122]

1.17. _Compound disturbance_. The basic formulation of 1.4 included only one source of disturbance D, .... ... nothing in this chapter excludes the possibility that D may be a vector, with any number of components. A vectorial D is thus able to represent all such compound disturbances within the basic formulation.

1.18. _Noise_. A related case occurs when T is 'noisy' - when T has an extra input that is affected by some disturbance that interferes with it. [....] [p. 122] The suggestion is that if we start again from the beginning, and re-define D and T then some new transformation of D may be able to restore regulation. The new transformation will, of course, have to be more complex than the old, for D will have more components.

1.19. _Initial states_. A related case occurs when T is some machine that shows its behavior by a trajectory, with the out- come E depending on the properties of T's trajectory. [....] ... D, as a vector, can be re-defined to include T's initial state. Then the variety brought to E by the variety in T's initial state is allotted its proper place in the formulation.

1.20. _Compound target_. It may happen that the acceptable states η at E may have more than one condition. [p. 122] ... by allowing E to become a vector, the basic formulation of 1 .4 can be made to include all cases in which the target is complex, or conditional, or qualified. [pp. 121-122]

1.21. _Internal complexities_. As a last example, showing how comprehensive the basic formulation really is, consider the case in which the major problem seems to be not so much a regulation as an interaction between several regulations. Thus a signalman may have to handle several trains coming to his section simultaneously. To handle any one by itself would be straightforward, but here the problem is the control of them as a complex whole pattern. [....] It will be seen therefore that the basic formulation is capable, in principle, of including cases of any degree of internal complexity. [p. 124]