Excursions - one, two, three.

One Ocean, One Index – a 'Composite Essay' on Opportunities and Limits.

Excursion One: An innocent average?

The ocean-health index is calculated as a “weighted arithmetic average”. That “weighting” does not look too much as problem. It makes the averaging a bit more complex as taking a simple “arithmetic average”. The “weighting” allows to account for some features that are a bit more important as other. The more important assets get “a bit more” weight and thus they determine the average a bit more as the less important assets. Setting the weights may give some room for tweaking the average, but can be understood easily and therefore “tweaking or cheating” can be made evident.

The mathematics of an "arithmetic average" look even more innocent and non-problematic; likely it is the most frequently used methods to calculate averages. An "arithmetic average" makes good sense, if the same feature is measured several times, and each measurement has a small random error. An "arithmetic average" also makes a good sense if no preferences shall be made among measurements. If preferences shall be made then weighting measurements is a transparent approach to set these preferences. Pushing these considerations further: facing the intrinsic complexity to balance different assets using the arithmetic average is like taking the approach "one asset one vote". And, on the other hand, attributing different weights to different assets can help to reflect social or political choices without excluding a "minority asset"; thus it is like an "affirmative action". Thus, considering the averaging method from a political angle the "weighted arithmetic average" looks much like as "applying first principles".

Nevertheless, these apparently simple averaging is not an innocent choice. It applies a specific “normative frame[s]” [2] embedded into the index and thus applied to the management of the assets. In a nutshell: The difficulty with arithmetic averaging is just that no preference is made. This “normative frame”, the implicit assumption behind arithmetic averaging, may effect the usefulness of the index as a management tool.

Using an “arithmetical average” to score a set of assets implies the assumption: assets can replace each other and the same score is calculated. Thus, “unconstrained substitution possibilities” exist among assets to obtain the same average score. In the term “unconstrained substitution possibilities” the notion “substitution” means that under-performance for one asset can be balanced by better-performance for another asset; “unconstrained” means that under-performance for one asset is not limited by a lower boundary; and “possibility” means that better-performance for any asset may balance under-performance of any asset. These assumptions are quite radical, indeed, and offer a wide range of management choices.

Using a “weighted arithmetic average” does not alter qualitatively the assumption of “unconstrained substitution possibilities”. Using a “weighted arithmetic average” modifies the “cost” of the substitution: performance for an asset with low weight has to improve much to balance a minor drop of performance of an asset with a high weight.

Excursion Two: A radical 'normative frame'?

Let's illustrate - by an example - “unconstrained substitution possibilities among various assets”: Assume first a shopping list of ten items for a tasty dinner; assume further getting these items in different quality and quantity, but so that, and this is the third assumption, the average “palatableness” of the dinner is the same. Evidently, a good starter may make good for an mediocre desert, or a good wine (or beer) compensates for…; but unconstrained substitution possibilities among the various parts of the dinner and same 'palatableness'? Common sense suggests that this may not work. However, consider a hypothetical "palatableness index" that is defined as the weighted arithmetic average of the quality and quantity of the items purchased for the dinner". In terms of that index, a dinner should have the same "average tastiness" as long as the score of the "palatableness index" is the same.

Evidently, “unconstrained substitution possibilities among various assets”, if it works, would be a framework for “a manager’s dream”. Such a framework would maximise the number of operational alternatives to amalgamate assets. In reality, "unconstrained substitution possibilities among various assets" is an exceptional case. It is rather "the real-world's manager's headache" that amalgamating assets is limited by their mutual substitution potential. The substitution potential may be limited for ecological, technical reasons or social preferences or economic viability to name but the most obvious. It is implicit for the application of ocean-health index to managerial or political choices that different assets can substitute each other, at least to some degree. It is implicit also, that assuming the full substitution of assets is problematic. Thus, how to describe these limitations by an appropriate mathematical method.

Excursion three: Mathematics for strong, weak or intermediate sustainability ?

The concepts of "strong sustainability" and "weak sustainability" can be used to compare different options to substitute assets. The "strong sustainability" concept constraints substitution options; all assets shall be kept above an asset-specific critical level. Under the "weak sustainability" concept, the substitution between assets is unconstrained.

In mathematical terms, the concept of “unconstrained substitution” is implicit to the ocean-health index calculated by a [weighted] arithmetic mean. Experienced managers of marine resources will be aware of limitations to substitution of assets, and thus will not accept any 'blind' averaging. However implementing that awareness in a competitive environment is fraught with difficulties, and therefore mathematical methods to describe “intermediate levels of substitution” may be appreciated as management tool.

The mathematical methods to describe "intermediate levels of substitution" are available [d]. Aggregation of scores for individual assets into a composite score under conditions of constraint or limited substitution can be described using ‘generalized averages' [e]. Arithmetic, geometric or harmonic averages are as special cases of the ‘generalized average'.

Obviously, intermediate levels of substitution of assets may be achieved for many real-world situations. Evidently, for many real-world situations it will be difficult to determine "what are boundaries to substitution?" Manifestly, any intermediate level of substitution of assets will depend on the specific ecological-human intersections of the respective human-ocean system. Nevertheless, whatever appears “obvious”, “evident” or “manifest”, it will be hard and tedious work to narrow the range of substitution possibilities. Therefore one may argue that "strong sustainability" should be applied across the board to guide management choices, and be it only for the sake of simplicity.

[2], [d], [e],  for references see "One Ocean, One Index – a 'Composite Essay' on Opportunities and Limits"