Joining up the vertices
In 1958 Roger Penrose (by whom I later had the privilege of being supervised at Oxford, and who now includes a knighthood and the Order of Merit among many other distinctions), together with his father Lionel, published a paper – significantly, in a journal of psychology – describing the “impossible triangle” I show here. He had discovered it independently of a forerunner devised by the Swedish artist Oscar Reutersvärd, which embodied much the same idea using shaded cubes (later seen on Swedish postage stamps), while the Dutch artist M. C. Escher did much to popularise the concept in a series of prints showing endless staircases etc. Among numerous variants were a fork with three round prongs at one end, but two square ones at the other (the term “tribar” has been confusingly applied to both this and the triangle). Squares and more elaborate structures also exploit the same fundamental principle: a credible two-dimensional representation of an object that cannot exist in three dimensions. Meetings of Penrose’s research group were enlivened by a three-dimensional model of an object that cannot exist in four dimensions. But there is a purity to the impossible triangle which has the bare minimum to establish the principle: you need three components from which to draw different conclusions by examining any two in isolation.
There is plainly (imagine the height of the people walking the impossible staircase) a close connection between the impossible triangle and instances of failure in normally transitive relations, as for example the “is greater than” property: if A>B, and B>C, then you can infer that A>C in logic. But sometimes this breaks down in real life (or in nursery games, such as scissors–paper–stone): for example you may know a group of amateur squash players where A (generally) beats B, B beats C and C beats A (perhaps because A can’t return a particular serve that C has perfected, although he is otherwise a weaker player than A and B): such games are ill adapted to knock-out tournaments, and leave us with a Jarnac-ian sense of unfairness.
But what perhaps isn’t so well known is that the concept illuminates much of what happens in structured finance. Instead of looking at vast charts of money flowing between distantly connected companies, mostly resident in tax havens, a great many schemes can be quite simply reduced to a principle which is far better explained by the impossible triangle. Indeed the most successful schemes exploit the simplest paradigms. A principal-deductible structure, converting capital into income? The payments by one company are clearly income, and so deductible; the lump sum received by an affiliate is clearly capital; the third party making payments in and out is clearly trading. A double-dip tax deduction? One tax authority looks at its part of the transaction and sees pure debt, while the other, from an orthogonal perspective, can see only pure equity.
Nor are we confined to tax avoidance: deposits at undercapitalised banks exploit the depositor’s confidence that government will support the banks, the government pretends that it won’t, while banks can’t afford to pay the proper rate for deposits that are subordinated if government is to be believed.
Yet these inconsistent characterisations defy common sense, and lead to results many will recognise as abhorrent. How does this arise? How is it that tax authorities cannot show the joined-up thinking politicians advocate? How can they be satisfied that the problem is with the other authority’s approach?
At the root of the problem is the requirement that each player can look only at his part of the puzzle, ignoring the third component. Time and again, when we perceive the nonsense that results, we can only impute to these isolationists the insistent refrain Mme de Merteuil provided to her thinly disguised Valmont (letter cxli): “Ce n’est pas ma faute.”
Until politicians grasp this and take a three-dimensional, Gestalt view of the commercial world, don’t expect finance not to continue to exploit the anomalies arising from the cognitive errors. Only a proper general anti-avoidance rule, of far greater breadth than the current restricted anti-abuse rule, can possibly deal with avoidance that exploits these principles, which are far more powerful than the exploitation of drafting errors which HMRC do attend to from time to time (although usually after many animals have already bolted from the stable). Only a government that is prepared to accept the implications of looking at the whole picture of banking will have a response to the continued enrichment of bank staff through exploitation of the government subsidy by which under-remunerated deposits are obtained.
The continued reliance on piecemeal, two-dimensional approaches to such challenges is another example of the trahison des clercs.
I haven’t commented on the intellectual property issues arising from representations of the impossible triangle, but I can’t resist posting this link to the Österreichische Patentamt. Gedanken gut geschützt!