Ashby's Law of Requisite Variety

[spacebeam]

Ashby's Law of Requisite Variety

[jchassoul]

Cumulative adaptation

The existence of different available strategies allows us to consider a separate superior domain of processing, a "meta-level" above the mechanics of switch handling from where the various available strategies can be randomly chosen.

Suppose N events each have a probability p of success, and the probabilities are independent. An example would occur if N wheels bore letters A and B on the rim, with A's occupying the spun and allowed to come to rest; those that stop at an A count as successes. Let us compare three ways of compounding these minor successes to a Grand Success, which we assume, occurs only when every wheel is stopped at an A.

Case 1: All N wheels are spun; if all show an A, Success is recorded and the trials ended; otherwise all are spun again, and so on till 'all A's' come up at one spin.

Case 2: The first wheel is spun; if it stops at an A it is left there; otherwise it is spun again. When it eventually stops at an A the second wheel is spun similarly; and so on down the line of N wheels, one at a time, till all show A's.

Case 3: All N wheels are spun; those that show an A are left to continue showing it, and those that show a B are spun again. When further A's occur they also are left alone. So the number spun gets fewer and fewer, until all are at A's.

The conclusion that Case 1 is very different from Cases 2 and 3, does not depend closely on the particular values of p and N.

Comparison of the three Cases soon shows why Cases 2 and 3 can arrive at Success so much sooner than Case 1: they can benefit by partial successes, which 1 cannot. Suppose, for instance, that, under Case 1, a spin gave 999 A's and 1 B. This is very near complete Success; yet it counts for nothing, and all the A's have to be thrown back into the melting-pot. In Case 3, however, only one wheel would remain to be spun; while Case 2 would perhaps get a good run of A's at the left-hand end and could thus benefit from it.

The examples show the great, the very great, reduction in time taken that occurs when the final Success can be reached by stages, in which partial successes can be conserved and accumulated.

We can draw, then, the following conclusion. A compound event that is impossible if the components have to occur simultaneously may be readily achievable if they can occur in sequence or independently.