Variety, Part 1

The cybernetic concept of variety is enjoying some increase in usage. And that’s both in frequency and in a number of different contexts. Even typing “Ross Ashby” in Google Trends confirms that impression.RossyAshby_as_seen_by_GoogleTrends In the last two years, the interest seems stable, while in the previous six – it was non-existent, save for the lonely peak in May 2010. Google Trends is not a source of data to draw serious conclusions from, yet it confirms the impression coming from tweets, blogs, articles, and books. On the one hand, that’s good news. I still find the usage insignificant compared to what I believe it should be. Nevertheless, little attention is better than none. On the other hand, it attracts some interpretations, leading to a misapprehension of the concept. That’s why I hope it’s worth exchanging more ideas about variety, and those having more variety themselves would either enjoy wider adoption or those using them – more benefits, or both.

The concept of variety as a measure of complexity had been preceded and inspired by the information entropy of Claude Shannon, also known as the “amount of surprise” in a message. That, although stimulated by the development of communication technologies in the first half of the twentieth century, had its roots in statistical mechanics and Boltzmann’s definition of entropy. Boltzmann, unlike classical mechanics and thermodynamics, defined entropy as the number of possible microstates corresponding to the macro-state of a system.

Variety is defined as the number of possible states in a system. It is also applied to a set of elements. The number of different members determines the variety of a set. It can be applied to the members themselves, which can be in different states, and then the set of possible transitions has a certain variety. This is the first important property of variety. It’s recursive. I’ll come back to this later. Now, to clarify what is meant by “state”:

By a state of a system is meant any well-defined condition or property that can be recognised if it occurs again.

Ross Ashby

Variety can sometimes be easy to count. For example, after opening the game in chess with a pawn on D4, the queen has a variety of three: not to move or move to one of the two possible squares. If only the temporary variety gain is counted, then choosing D2 as the next move would give a variety of 9, and D3 would give 16. That’s not enough to tell if the move is good or bad, especially keeping in mind that some of that gained variety is not effective. However, in case of uncertainty, in games and elsewhere, moving to a place that both increases our future options and decreases those of the opponent seems good advice.

Variety can be expressed as a number, as it was done in the chess example, but in many cases, it’s more convenient to use the logarithm of that number (in case that sounds like a distant memory from school years, nowadays there are easy ways to refresh it in minutes). The common practice, maybe because of the first areas of application, is to use binary logarithms. When that is the case, variety can be expressed in bits. It is indeed more convenient to say the variety of a four-letter code using the English alphabet is 18.8 bits instead of 456 976. There is an extra bonus. When the logarithmic expression is used, varieties of elements are combined by adding instead of multiplying.

Variety is sometimes referred to and counted as permutations. That might be fine in certain cases but as a rule it is not. To use the example with the 4-letter code, it has 358 800 permutations (26 factorial divided by 22 factorial), while the variety is 456 976 (26 to the power of 4).

Variety is relative. It depends on the observer. That’s obvious even from the word “recognised” in the definition of state. If, for example, there is a clock with two hands that are exactly the same or at least to the extent that an observer can’t make the difference, then, from the point of view of the observer, the clock will have a much lower variety than a regular one. The observer will not be able to distinguish, for example, 12:30 and 6:03 as they will be seen as the same state of the clock.

Clock with indistiguishable hands

This can be seen as another dependency. That of the capacity of the channel or the variety of the transducer. For example, it is estimated that regular humans can distinguish up to 10 million colours, while tetrachromats – at least ten times more. The variety of the transducer and the capacity of the channel should always be taken into account.

When working with variety, it is useful to study the relevant constraints. If we throw a stone from the surface of Earth, certain constraints, including those we call “gravity” and the “resistance of the air”, would allow a much smaller range of possible states than if those constraints were not present. Ross Ashby made the following observation: “every law of nature is a constraint”, “science looks for laws; it is therefore much concerned with looking for constraints”.

There is this popular way of defining a system as something which is more than the sum of its parts. Let’s see this statement through the lens of varieties and constraints. If we have two elements, A and B, and each can be in two possible states on their own but when linked to each other A can bring B to another, third state, and B can bring A to another state as well. In this case, the system AB has certainly more variety than the sum of A and B unbound. But if, when linking A and B they inhibit each other, allowing one state instead of two, then it is clearly the opposite. That motivates rephrasing the popular statement to “a system might have different variety than the combined variety of its parts”.

If that example with A and B is too abstract, imagine a canoe sprint kayak with two paddlers working in sync and then compare it with a similar setting, with one of the paddlers rowing while the other holds her paddle in the water.

Yet, “is more than the sum of” can be retained but then another modification is needed. Here’s one suggested by Heinz von Foerster:

The measure of the sum of the parts is greater than the sum of the measures of the parts. One is the measure of the sum; the other is the sum of the measures. Take, for example, the measurement function “to square,” which makes this immediately apparent. I have two parts, one is a, the other b. Now I have the measure of the sum of the parts. What does that look like? a + b as the sum of the parts squared, (a + b)2 gives us a2 + 2ab + b2. Now I need the sum of the measures of the parts, and with this I have the measure of a (= a2) and the measure of b (= b2): a2 + b2. Now I claim that the measure of the sums of the parts is greater than the sum of the measures of the parts and state that: a2 + b2 + 2ab is greater than a2 + b2. So the measure of the sum is greater than the sum of the measures. Why? a and b squared already have a relation together

Heinz von Foerster. The Beginning of Heaven and Earth Has No Name (Meaning Systems) (p. 18)

And now about the law of requisite variety. It’s stated as “variety can destroy variety” by Ashby and as “only variety can absorb variety” by Beer, and has other formulations such as “The larger the variety of actions available to control system, the larger the variety of perturbations it is able to compensate”. Basically, when the variety of the regulator is lower than the variety of the disturbance, that gives high variety of the outcome. A regulator can only achieve the desired outcome variety if its own variety is the same or higher than that of the disturbance. The recursive nature mentioned earlier can now be easily seen if we look at the regulator as a channel between the disturbance and the outcome or if we account for the variety of the channels at the level of recursion with which we started.

To really understand the significance of this law, it should be seen how it exerts itself in various situations, which we wouldn’t normally describe with words such as “regulator”, “perturbations” and “variety”.

In the chess example, the power of each piece is a function of its variety, which is the one given by the rules and reduced by the constraints at every move. Was there a need to know about requisite variety to design this game? Or any other game for that matter? Or was it necessary to know how to wage war? Certainly not. And yet, it’s all there:

It is the rule in war, if our forces are ten to the enemy’s one, to surround him; if five to one, to attack him; if twice as numerous, to divide our army into two.

Sun Tzu, The Art of War

Let’s leave the games now and come back to the relative nature of variety. The light signals in ships should comply with the International Regulations for Preventing Collisions at Sea (IRPCS). The agreed signals have a reduced variety to communicate the states of the ships but enough to ensure the required control. For example, if an observer sees one green light, she knows that another ship is passing from left to right. If she sees one red light, it passes right to left. There are lots of states – different angles of the course of the other ship – that are reduced into these two, but that serves the purpose well enough. Now, if she sees both red and green, that means that the ship is coming exactly towards her. That’s a dangerous situation. The reduction of variety, in this case, has to be very low.

The relativity of variety is not only related to the observer’s “powers of discrimination”, or those of the purpose of regulation. It could be dependent also on the context. Easop’s fable “The Fox and the Stork”comes to mind.

Fables, and stories in general, influence people and survive centuries. But is it that do you need a story instead of getting directly the moral of the story? Yes, it’s more interesting, there is this uncertainty element and all that. But there is something else. Stories are ambiguous and interpretable. They leave many things to be completed by the readers and listeners. To put it in different words, they have a much higher variety than morals and values.

That’s it for this part.

And here is the next.

One thought on “Variety, Part 1

  1. R H Atkin in his book “The Mathematical Structure of Human Affairs” converts chess moves into a mathematical language in the Chapter “The Game of Chess”.

    In the chapter “Space is Full of Holes” he writes:

    “The thesis of this book is that the heart of the scientific method and of all rational study of any human activity, lies in the process of identifying sets and of understanding the structural properties of relations between sets.

    The physicist is dedicated to identifying his sets under controlled conditions so as to eliminate spurious and non-essential phenomena. This naturally involved an interaction between himself and not-himself, between the observer and the observed, and ultimately this means that he must use his physical senses (my emphasis).

    Nature in in all her infinite goodness has provided us with a selection signals, or seeing-agents, for just this purpose – what the writers has previously referred to as base elements. And with such a signal the observer explores the world around hie, finding his sets and relations.”

    The above would seem to tie in with your observer powers of discrimination.

    I would also add that Goethe’s concept of the Organising Idea “forces” us to select those sets and elements that are relevant to that organising idea.

    If we substitute Atkin’s “physicist” with “systems practitioner” how would that affect how a systems practitioner identify and deal with problems? The requisite variety of a system could be said to have been constructed by agents with individual or group Organising Ideas and thus may not give a accurate picture of what is going – accurate in the context of the ongoing problem.

    Add to that the Organising Idea(s) of the systems practitioner and you may have more problems! This is all very obvious but it sometimes needs to be emphasised.

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