__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"

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