Abstract
This paper discusses the complexity and computability of the ‘Deissenberg Problem’ of finding the ‘best’ point in the efficient set. It is shown that the problem is computably undecidable and that the efficient set of the problem is algorithmically complex.
A modest contribution to the Festschrift in Honour of Christophe Deissenberg, edited by Herbert Dawid and Jasmina Arifovic, forthcoming in the Springer Series on Dynamic Modeling and Econometrics in Economics and Finance. The ‘Post’, in the Post-Turing Thesis, above, is a reference to Emil Post.
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Notes
- 1.
In 1983 I borrowed a book by a then well-known author from Christoph Deissenberg; unfortunately—or, with hindsight, fortunately—it was taken, without my knowledge, by a ‘then’ friend, from my home library; it was never returned. At some point in 1984, Christophe, in his usual diffident, civilized, way, gently reminded me of having borrowed a book from him. It was then that I had the courage to tell him what had happened—but I offered to buy a new copy of it. He—again, with the utmost politeness—declined the offer, saying that it was a copy presented by the author! I have, since then, familiarized myself with the contents of that book and I prepared a first draft of this contribution as a generalization and formalization of Chaps. 5, 6 and 7 of its contents. The ‘generalizations’ were based on my reading and knowledge of Arrow (1974), March and Simon (1958; 1993), in addition to many of Simon’s contribution, from 1947 to his Raffaele Mattioli Lectures of 1997. Fortunately for me, the author whose book I was guilty of borrowing from Christophe, did not refer to, or use, Arrow (ibid), March and Simon (ibid) and, of course, any of Simon’s important work on Organisations, particularly in Chap. 7 of the book by the subsequently ‘discredited’ author. This version is on entirely different topics.
- 2.
To the best of my knowledge, this paper is not listed in Deissenberg’s 2017 CV, which was kindly provided by one of the editors; but that does not make it any the less important.
- 3.
The idea, and its implementation, by Frisch, in a quantitative model of the Indian economy, goes back, at least, to the early 1950s (see, Goodwin 1992, pp. 22–24). The section in Frisch’s Nobel Prize Lecture, refer to The preference function, p. 23, ff., in Frisch (1970); the whole of Frisch (1972) is on the construction of an iterative mechanism to elicit, by way of structured questions—subject to modification in the light of experiences—by an econometrician (as defined ‘classically’ by Frisch), to elicit answers by politicians, to determine their preference functions. Deissenberg’s eminently realistic assumption of ‘partially conflicting goals’ (op.cit, p. 1) by a multiplicity of agents reflects Frisch’s considerations of eliciting and revising politicians’ preference functions, iteratively, till ‘some sort of consistency’ is achieved.
- 4.
He was just over 30 years of age, when the first draft of Deissenberg (1977) was originally prepared. The substance of that paper of the mid-1970s has retained its freshness, relevance and topicality for the ensuing four decades, and some.
- 5.
In the sense of efficient being from a well-defined maximum set. In a choice situation, it is as feasible to choose efficiently as to assume some form of Zorn’s Lemma (or an unrestricted Axiom of Choice).
- 6.
The model in Arrow-Debreu (op.cit) is over the real numbers; by the way, for many reasons, Scarf’s algorithm is not constructive—but it is possible to construct an algorithm over the reals so that Arrow’s assertion is justified. The computations and optimisations in all of Deissenberg’s models, to the best of my knowledge, are over the real numbers, to which the same comment applies (cf. Feferman 2013, Chap. 3 and Fenstad 1980).
- 7.
This is the ‘suggested’ bibliographic entry—but the paper itself appeared as a chapter in volume 2 of Concepts and Tools of Computer-Assisted Analysis, Birkhauser Verlag 1977, Basel, edited by H. Bossel. Eden’s extensive review of the chapter version in the book, which is ‘literally’ the same as above.
- 8.
This is the case in Scarf’s algorithmic method of finding an approximation to a general economic equilibrium of Arrow-Debreu type (cf., Scarf 1973, p. 52).
- 9.
Eden does go on, on the next page of his enlightening review article (p. 353; italics added):
- 10.
The two reading heads need not move simultaneously; they can move, such that one is a computable function of the other.
- 11.
Algorithms can be more general than computable functions (See Gurevich 2012).
- 12.
Except Gandy (1980), and a few others, who referred to Turing’s version of it as the Turing Theorem (actually as Theorem T, on p. 124, Gandy, ibid).
- 13.
As will be made clear in the next section, these are identified, in the Deissenberg Problem, with the feasible and efficient sets, respectively.
- 14.
I am particularly indebted to item 3, in the anonymous referee’s comments, for helping me clarify this point.
- 15.
For simplicity, and in the interests of conciseness, we shall assume a reader is familiar with the framework and assumptions of Putnam (op.cit).
- 16.
The questions by the model-builder and the answers by, say, the agent—who may be a political or economic decision-maker, are, at best, rational numbers, which can be coded (for example by Gödel numbering) in terms of positive integers. The rational numbers are, in any case, enumerable (cf., for example, Hardy 1908 [1960], p. 1, Example 4). However, as pointed out in footnote 4, above, it is not too difficult to do the same exercise for real number domain and range, as is the case in the case of Deissenberg (1977)—and, in fact, all of the computational examples in the Deissenberg Oeuvre.
- 17.
I daresay that it was also a starting point for many of Christoph Deissenberg’s rich speculations in the decision sciences.
- 18.
The referee (anonymous) points out that, delineating decidable and undecidable sets is in itself an (algorithmically) undecidable problem; this is, strictly speaking, incorrect. The correctness of the assertion depends on the structure of the set(s) under consideration. I have always maintained that Simon worked with sets that were complete (as above) and, therefore all of them were recursive.
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Vela Velupillai, K. (2021). Oracle and Interactive Computations, Post-turing Thesis and Man–Machine Interactions. In: Dawid, H., Arifovic, J. (eds) Dynamic Analysis in Complex Economic Environments. Dynamic Modeling and Econometrics in Economics and Finance, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-030-52970-3_10
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