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Ashby W.R. (1958) Requisite variety and its implications for the control of complex systems, Cybernetica 1:2, p. 83-99. (available at http://pcp.vub.ac.be/Books/AshbyReqVar.pdf, republished on the web by F. Heylighen—Principia Cybernetica Project)
Requisite variety and its implications for the control of
complex systems
by W. Ross ASHBY, Director of Research, Barnwood House (Gloucester) Recent work on the fundamental processes of regulation in biology (Ashby, 1956) has shown the importance of a certain quantitative relation called the law of requisite variety. After this relation had been found, we appreciated that it was related to a theorem in a world far removed from the biological—that of Shannon on the quantity of noise or error that could be removed through a correction-channel (Shannon and Weaver, 1949; theorem 10). In this paper I propose to show the relationship between the two theorems, and to indicate something of their implications for regulation, in the cybernetic sense, when the system to be regulated is extremely complex. Since the law of requisite variety uses concepts more primitive than those used by entropy, I will start by giving an account of that law. I Variety. Given a set of elements, its variety is the number of elements that can be distinguished. Thus the set { g b c g g c } has a variety of 3 letters. (If two observers differ in the distinctions they can make, then they will differ in their estimates of the variety. Thus if the set is { b c a a C a B a } its variety in shapes is 5, but its variety in letters is 3. We shall not, however, have to treat this complication). For many purposes the variety may more conveniently be measured by the logarithm of this number. If the logarithm is taken to base 2, the unit is the bit. The context will make clear whether the number or its logarithm is being used as measure. Regulation and the pay-off matrix. Regulation achieves a “goal” against a set of disturbances. The disturbances may be actively hostile, as are those coming from an enemy, or merely irregular, as are those coming from the weather. The relations may be shown in the most general way by the
formalism that is already well known in the theory of games (Neumann and Morgenstern, 1947). A set D of disturbances di can be met by a set R of responses rj. The outcomes provide a table or matrix R r 1 r 2 r 3 … d 1 z 11 z 12 z 13 … d 2 z 21 z 22 z 23 … D d 3 z 31 z 32 z 33 … d 4 z 41 z 42 z 43 … … … … … … in which each cell shows an element zij from the set Z of possible outcomes. It is not implied that the elements must be numbers (though the possibility is not excluded). The form is thus general enough to include the case in which the events di and rj are themselves vectors, and have a complex internal structure. Thus the disturbances D might be all the attacks that can be made by a hostile army, and the responses R all the counter-measures that might be taken. What is required at this stage is that the sets are sufficiently well defined so that the facts determine a single-valued mapping of the product set D × R into the set Z of possible outcomes. (I use here the concepts as defined by Bourbaki, 1951). The “outcomes” so far are simple events, without any implication of desirability. In any real regulation, for the benefit of some defined person or organism or organisation, the facts usually determine a further mapping of the set Z of outcomes into a set E of values. E may be as simple as the 2-element set {good, bad}, and is commonly an ordered set, representing the preferences of the organism. Some subset of E is then defined as the “goal”. The set of values, with perhaps a scale of preference, is often obvious in human affairs; but in the biological world, and in the logic of the subject, it must have explicit mention. Thus if the outcome is “gets into deep water”, the valuation is uncertain until we know whether the organism is a cat or a fish. In the living organisms, the scale of values is usually related to their “essential variables”—those fundamental variables that must be kept within certain “physiological” limits if the organism is to survive. Other organisations also often have their essential variables: in an economic system, a firm’s profits is of this nature, for only if this variable keeps positive can the firm survive. Given the goal—the “good” or “acceptable” elements in E —the inverse mapping of this subset will define, over Z , the subset of “acceptable outcomes”. Their occurrence in the body of the table or matrix will thus mark a subset of the product set D × R. Thus is defined a binary relation S between D and R in which “the elements di and rj have the relation S ” is equivalent to “ri, as response to di , gives an acceptable outcome”.
which each column has all its elements different. (Nothing is assumed here about the relation between the contents of one column and those of another). What this implies is that if the set D had a certain variety, the outcomes in any one column will have the same variety. In this case, if R is inactive in responding to D (i. e. if R adheres to one value rj for all values of D ), then the variety in the outcomes will be as large as that in D. Thus in this case, and if R stays constant, D can be said to be exerting full control over the outcomes. R, however, aims at confining the actual outcomes to some subset of the possible outcomes Z. It is necessary, therefore, that R acts so as to lessen the variety in the outcomes. If R does so act, then there is a quantitative relation between the variety in D , the variety in R , and the smallest variety that can be achieved in the set of actual outcomes; namely, the latter cannot be less than the quotient of the number of rows divided by the number of columns (Ashby, 1956; S.11/5). If the varieties are measured logarithmically, this means that if the varieties of D , R , and actual outcomes are respectively Vd , Vr , and Vo then the minimal value of Vo is Vd – Vr. If now Vd is given, Vo ’s minimum can be lessened only by a corresponding increase in Vr. This is the law of requisite variety. What it means is that restriction of the outcomes to the subset that is valued as Good demands a certain variety in R. We can see the relation from another point of view. R , by depending on D for its value, can be regarded as a channel of communication between D and the outcomes (though R , by acting as a regulator, is using its variety subtractively from that of D ). The law of requisite variety says that R’s capacity as a regulator cannot exceed its capacity as a channel for variety. The functional dependencies can be represented as in Fig. 1. (This diagram is necessary for comparison with Figs. 2 and 3). FIG. 1 The value at D threatens to transmit, via the table T to the outcomes Z , the full variety that occurs at D. For regulation, another channel goes through R , which takes a value so paired to that of D that T gives values at Z with reduced variety. Nature of the limitation. The statement that some limit cannot be exceeded may seem rash, for Nature is full of surprises. What, then, would we say if a case were demonstrated in which objective measurements shows that the limit was being exceeded? Here we would be facing the case in which appropriate effects were occurring without the occurrence of the corresponding causes. We would face the case of the examination candidate who gives the appropriate answers before he has been given the corresponding questions! When such things have happened in the past we have always looked for, and found, a channel
of communication which has accounted for the phenomenon, and which has shown that the normal laws of cause and effect do apply. We may leave the future to deal similarly with such cases if they arise. Meanwhile, few doubt that we may proceed on the assumption that genuine overstepping of the limitation does not occur. Examples in biology. In the biological world, examples that approximate to this form are innumerable, though few correspond with mathematical precision. This inexactness of correspondence does not matter in our present context, for we shall not be concerned with questions involving high accuracy, but only with the existence of this particular limitation. An approximate example occurs when a organism is subject to attacks by bacteria (of species di ) so that, if the organism is to survive, it must produce the appropriate anti- toxin rj. If the bacterial species are all different, and if each species demands a different anti-toxin, then clearly the organism, for survival, must have at least as many anti-toxins in its repertoire of responses as there are bacterial species. Again, if a fencer faces an opponent who has various modes of attack available, the fencer must be provided with at least an equal number of modes of defence if the outcome is to have the single value: attack parried. Analysis of Sommerhoff. Sommerhoff (1950) has conducted an analysis in these matters that bears closely on the present topic. He did not develop the quantitative relation between the varieties, but he described the basic phenomenon of regulation in biological systems with a penetrating insight and with a wealth of examples. He recognises that the concept of “regulation” demands variety in the disturbances D. His “coenetic variable” is whatever is responsible for the values of D. He also considers the environmental conditions that the organism must take into account (but as, in his words, these are “epistemically dependent” on the values of the coenetic variable, our symbol D can represent both, since his two do not vary independently.) His work shows, irrefutably in my opinion, how the concepts of co-ordination, integration, and regulation are properly represented in abstract form by a relation between the coenetic variable and the response, such that the outcome of the two is the achievement of some “focal condition” (referred to as “goal” here). From our point of view, what is important is the recognition that without the regulatory response the values at the focal condition would be more widely scattered. Sommerhoff’s diagram (Fig. 2) is clearly similar. (I have modified it slightly, so as to make it uniform with Figs. 1 and 3).
that many important communications are non-ergodic, their occurrence being especially frequent in the biological world. Thus we frequently study a complex biological system by isolating it, giving it a stimulus, and then observing the complex trajectory that results. Thus the entomologist takes an ant-colony, places a piece of meat nearby, and then observes what happens over the next twenty-four hours, without disturbing it further. Or the social psychologist observes how a gang of juvenile criminals forms, becomes active, and then breaks up. In such cases even a single trajectory can provide abundant information by the comparison of part with part, but the only ergodic portion of the trajectory is that which occurs ultimately, when the whole has arrived at some equilibrium, in which nothing further of interest is happening. Thus the ergodic part is degenerate. It is to be hoped that the extension of the basic concepts of Shannon and Wiener to the non-ergodic case will be as fruitful in biology as the ergodic case has been in commercial communication. It seems likely that the more primitive concept of “variety” will have to be used, instead of probability; for in the biological cases, systems are seldom isolated long enough, or completely enough, for the relative frequencies to have a stationary limit. Among the ergodic cases there is one, however, that is obviously related to the law of requisite variety. It is as follows. Let D , R , and E be three variables, such that we may properly observe or calculate certain entropies over them. Our first assumption is that if R is constant, all the entropy at D will be transmitted to, and appear at, E. This is equivalent to Hr(E) = Hr(D) (1) By writing H ( D , R ) in two forms we have H(D) + Hd(R) = H(R) + Hr(D) Use of (1) gives H(D) + Hd(R) = H(R) + Hr(E) = H(R,E) ≤ H(R) + H(E) i.e. H(E) ≥ H(D) + Hd(R) – H(R) (2) The entropy of E thus has a certain minimum—the expression on the right of (2). If H ( E ) is the entropy of the actual outcomes, then, for regulation, it may have to be reduced to a certain value. Equation (2) shows what can reduce it; it can be reduced: (i) by making Hd ( R ) = 0, i.e. by making R a determinate function of D , (ii) by making H ( R ) larger. If Hd ( R ) = 0, and H ( R ) the only variable on the right of (2), then a decrease in H ( E ) demands at least an equal increase in H ( R ). This conclusion is clearly similar to that of the law of requisite variety.
A simple generalisation has been given (Ashby, 1956) in which, when R remains constant, only a certain fraction of D ’s variety or entropy shows in the outcomes or in H ( E ). The result is still that each decrease in H ( E ) demands at least an equal increase in H(R). With this purely algebraic result we can now see exactly how these ideas join on to Shannon’s. His theorem 10 uses a diagram which can be modified to Figure 3 (to match the two preceding Figures). FIG. 3 Our “disturbance D ”, which threatens to get through to the outcome, clearly corresponds to the noise; and his theorem says that the amount of noise that can be prevented from appearing in the outcomes is limited to the entropy that can be transmitted through the correction channel. The message of zero entropy. What of the “message”? In regulation, the “message” to be transmitted is a constant, i.e. has zero entropy. Since this matter is fundamental, let us consider some examples. The ordinary thermostat is set at, say, 70º F. “Noise”, in the form of various disturbances, providing heat or cold, threatens to drive the output from this value. If the thermostat is completely efficient, this variation will be completely removed, and an observer who watches the temperature will see continuously only the value that the initial controller has set. The “message” is here the constant value 70. Similarly, the homeostatic mechanism that keeps our bodies, in health, about 98 º F is set at birth to maintain this value. The control comes from the gene-pattern and has zero entropy, for the selected value is unchanging. The same argument applies similarly to all the regulations that occur in other systems, such as the sociological and economic. Thus an attempt to stabilise the selling price of wheat is an attempt to transmit, to the farmers, a “message” of zero entropy; for this is what the farmer would receive if he were to ask daily “what is the price of wheat today”? The stabilisation, so far as it is successful, frees the message from the effects of those factors that might drive the price from the selected value. Thus, all acts of regulation can be related to the concepts of communication theory by our noticing that the “goal” is a message of zero entropy, and that the “disturbances” correspond to noise.
or does not solve; such is the manager whose business prospers or fails; such is the economist who can or cannot control an inflationary spiral. Not only are these practical activities covered by the theorem and so subject to limitation, but also subject to it are those activities by which Man shows his “intelligence”. “Intelligence” today is defined by the method used for its measurement; if the tests used are examined they will be found to be all of the type: from a set of possibilities, indicate one of the appropriate few. Thus all measure intelligence by the power of appropriate selection (of the right answers from the wrong). The tests thus use the same operation as is used in the theorem on requisite variety, and must therefore be subject to the same limitation. ( D , of course, is here the set of possible questions, and R is the set of all possible answers). Thus what we understand as a man’s “intelligence” is subject to the fundamental limitation: it cannot exceed his capacity as a transducer. (To be exact, “capacity” must here be defined on a per-second or a per-question basis, according to the type of test.) The team as regulator. It should be noticed that the limitation on “the capacity of Man” is grossly ambiguous, according to whether we refer to a single person, to a team, or to the whole of organised society. Obviously, that one man has a limited capacity does not impose a limitation on a team of n men, if n may be increased without limit. Thus the limitation that holds over a team of n men may be much higher, possibly n times as high, as that holding over the individual man. To make use of the higher limitation, however, the team must be efficiently organised; and until recently our understanding of organisation has been pitifully small. Consider, for instance, the repeated attempts that used to be made (especially in the last century) in which some large Chess Club played the World Champion. Usually the Club had no better way of using its combined intellectual resources than either to take a simple majority vote on what move to make next (which gave a game both planless and mediocre), or to follow the recommendation of the Club’s best player (which left all members but one practically useless). Both these methods are grossly inefficient. Today we know a good deal more about organisation, and the higher degrees of efficiency should soon become readily accessible. But I do not want to consider this question now. I want to emphasise the limitation. Let us therefore consider the would-be regulator, of some capacity that cannot be increased, facing a system of great complexity. Such is the psychologist, facing a mentally sick person who is a complexly interacting mass of hopes, fears, memories, loves, hates, endocrines, and so on. Such is the sociologist, facing a society of mixed races, religions, trades, traditions, and so on. I want to ask: given his limitation, and the complexity of the system to be regulated, what scientific strategies should he use? In such a case, the scientist should beware of accepting the classical methods without scrutiny. The classical methods have come to us chiefly from physics and chemistry, and these branches of science, far from being all-embracing, are actually much specialised and by no means typical. They have two peculiarities. The first is that their systems are composed of parts that show an extreme degree of homogeneity: contrast the similarity between atoms of carbon with the dissimilarity between persons. The second is that the systems studied by the physicist and chemist have nothing like the
richness of internal interaction that have the systems studied by the sociologist and psychologist. Or take the case of the scientist who would study the brain. Here again is a system of high complexity, with much heterogeneity in the parts, and great richness of connexion and internal interaction. Here too the quantities of information involved may well go beyond the capacity of the scientist as a transducer. Both of these qualities of the complex system—heterogeneity in the parts, and richness of interaction between them—have the same implication: the quantities of information that flow, either from system to observer or from part to part, are much larger than those that flow when the scientist is physicist or chemist. And it is because the quantities are large that the limitation is likely to become dominant in the selection of the appropriate scientific strategy. As I have said, we must beware of taking our strategies slavishly from physics and chemistry. They gained their triumphs chiefly against systems whose parts are homogeneous and interacting only slightly. Because their systems were so specialised, they have developed specialised strategies. We who face the complex system must beware of accepting their strategies as universally valid. It is instructive to notice that their strategies have already broken down in one case, which is worth a moment’s attention. Until about 19 2 5, the rule “vary only one factor at a time” was regarded as the very touchstone of the scientific method. Then R. A. Fisher, experimenting with the yields of crops from agricultural soils, realised that the system he faced was so dynamic, so alive, that any alteration of one variable would lead to changes in an uncountable number of other variables long before the crop was harvested and the experiment finished. So he proposed formally to vary whole sets of variables simultaneously—not without peril to his scientific reputation. At first his method was ridiculed, but he insisted that his method was the truly scientific and appropriate one. Today we realise that the rule “vary only one factor at a time” is appropriate only to certain special types of system, not valid universally. Thus we have already taken one step in breaking away from the classical methods. Another strategy that deserves scrutiny is that of collecting facts “in case they should come in useful some time”—the collecting of truth “for truth’s sake”. This method may be efficient in the systems of physics and chemistry, in which the truth is often invariant with time; but it may be quite inappropriate in the systems of sociology and economics, whose surrounding conditions are usually undergoing secular changes, so that the parameters to the system are undergoing changes—which is equivalent to saying that the systems are undergoing secular changes. Thus, it may be worthwhile finding the density of pure hafnium, for if the value is wanted years later it will not be changed. But of what use today, to a sociologist studying juvenile delinquency, would a survey be that was conducted, however carefully, a century ago? It might be relevant and helpful; but we could know whether it was relevant or not only after a comparison of it with the facts of today; and when we know these, there would be no need for the old knowledge. Thus the rule “collect truth for truth’s sake” may be justified when the truth is unchanging; but when the system is not completely isolated from its surroundings, and is undergoing secular changes, the collection of truth is futile, for it will not keep.
What I suggest is that measurement of the quantity of information, even if it can be done only approximately, will tell the investigator where a complex system falls in relation to his limitation. If it is well below the limit, the classic methods may be appropriate; but should it be above the limit, then if his work is to be realistic and successful, he must alter his strategy to one more like that of operational research. My emphasis on the investigator’s limitation may seem merely depressing. That is not at all my intention. The law of requisite variety, and Shannon’s theorem 10, in setting a limit to what can be done, may mark this era as the law of conservation of energy marked its era a century ago. When the law of conservation of energy was first pronounced, it seemed at first to be merely negative, merely an obstruction; it seemed to say only that certain things, such as getting perpetual motion, could not be done. Nevertheless, the recognition of that limitation was of the greatest value to engineers and physicists, and it has not yet exhausted its usefulness. I suggest that recognition of the limitation implied by the law of requisite variety may, in time, also prove useful, by ensuring that our scientific strategies for the complex system shall be, not slavish and inappropriate copies of the strategies used in physics and chemistry, but new strategies, genuinely adapted to the special peculiarities of the complex system. REFERENCES. ASHBY, W. Ross, Design for a brain. 2nd. imp. Chapman & Hall, London, 1954. ASHBY, W. Ross, An introduction to cybernetics. Chapman & Hall, London, 1956. BOURBAKI, N., Théorie des ensembles. Fascicule de resultats. A.S.E.I. N°1141. Hermann et Cie, Paris, 1951. NEUMANN, J. (von) and MORGENSTERN, O., Theory of games and economic behaviour. Princeton, 1947. SHANNON, C. E. and WEAVER, W., The mathematical theory of communication. University of Illinois Press, Urbana, 1949, SOMMERHOFF, G., Analytical biology. Oxford, University Press, London, 1950. CYBERNETICA (Namur) Vol I — N° 2 — 1958.
