Uncomputability and undecidability in economic theory

Economic theory, game theory and mathematical statistics have all increasingly become algorithmic sciences. Computable Economics, Algorithmic Game Theory [Noam Nisan, Tim Roiughgarden, Éva Tardos, Vijay V. Vazirani (Eds.), Algorithmic Game Theory, Cambridge University Press, Cambridge, 2007] and Algorithmic Statistics [Péter Gács, John T. Tromp, Paul M.B. Vitányi, Algorithmic statistics, IEEE Transactions on Information Theory 47 (6) (2001) 2443–2463] are frontier research subjects. All of them, each in its own way, are underpinned by (classical) recursion theory – and its applied branches, say computational complexity theory or algorithmic information theory – and, occasionally, proof theory. These research paradigms have posed newmathematical and metamathematical questions and, inadvertently, undermined the traditional mathematical foundations of economic theory. A concise, but partial, pathway into these new frontiers is the subject matter of this paper. Interpreting the core of mathematical economic theory to be defined by General Equilibrium Theory and Game Theory, a general – but concise – analysis of the computable and decidable content of the implications of these two areas are discussed. Issues at the frontiers of macroeconomics, now dominated by Recursive Macroeconomic Theory (The qualification ‘recursive’ here has nothing to do with ‘recursion theory’. Instead, this is a reference to the mathematical formalizations of the rational economic agent’s intertemporal optimization problems, in terms of Markov Decision Processes, (Kalman) Filtering and Dynamic Programming, where a kind of ‘recursion’ is invoked in the solution methods. The metaphor of the rational economic agent as a ‘signal processor’ underpins the recursive macroeconomic paradigm.), are also tackled, albeit ultra briefly. The point of view adopted is that of classical recursion theory and varieties of constructive mathematics. 2009 Elsevier Inc. All rights reserved. 1. A mathematical and metamathematical preamble Distinguished pure mathematicians, applied mathematicians, philosophers and physicists have, with the innocence of integrity and the objectivity of their respective disciplines, observing the mathematical practice and analytical assumptions of economists, have emulated the ‘little child’ in Hans Christian Andersen’s evocative tale to exclaim similar obvious verities, from the point of view of algorithmic mathematics. I have in mind the ‘innocent’, but obviously potent, observations made by Rabin [35], . All rights reserved. on Physics and Computation, 7th International Conference on Unconventional Computation, Vienna, for clarifying many obscure issues in an earlier version of this paper. They also contributed with deep and ed considerably to improving the paper. Alas, they are not responsible for the remaining obscurities. K. Vela Velupillai / Applied Mathematics and Computation 215 (2009) 1404–1416 1405 Putnam [34], Osborne [29], Schwartz [41], Smale [42], Shafer and Vovk [39] and Ruelle [36], each tackling an important core issue in mathematical economics and finding it less than adequate from a serious mathematical and computable point of view – in addition to being contrived, even from the point of view of common sense economics. Decidability in games, uncomputability in rational choice, inappropriateness of real analysis in the modelling of financial market dynamics, the gratuitous assumption of (topological) fix point formalizations in equilibrium economic theory, the question of the algorithmic solvability of supply-demand (diophantine) equation systems, finance theory without probability (but with an algorithmically underpinned theory of martingales), are some of the issues these ‘innocent’ pioneers raised, against the naked economic theoretic emperor (see the Prologue in Ref. [32]). I hasten to add that there were pioneers even within the ‘citadel’ of economic theory. Their contributions have been discussed and documented in various of my writings over the past 20 years or so and, therefore, I shall not rehash that part of the story here. Suffice it to mention just the more obvious pioneers who emerged from within the ‘citadel’: Peter Albin, Kenneth Arrow, Douglas Bridges, Alain Lewis, Herbert Scarf and Herbert Simon. Albin, Arrow, Lewis, Scarf and Simon considered seriously, to a greater and lesser extent, the issue of modelling economic behaviour, both in the case of individually rational and in cases of strategically rational interactions, the place of formal computability considerations and their implications. Bridges and Scarf were early contributors to what may be called ‘constructive economics’, complementing the ‘computable economics’ of the former contributors. Scarf, of course, straddled both divides, without – surprisingly – providing a unifying underpinning in what I have come to call’algorithmic economics’. Economic theory, at every level and at almost all frontiers – be it microeconomics or macroeconomics, game theory or IO – is now almost irreversibly dominated by computational, numerical and experimental considerations. Curiously, though, none of the frontier emphasis from any one of these three points of view – computational, numerical or experimental – is underpinned by the natural algorithmic mathematics of either computability theory or constructive analysis. In particular, the much vaunted field of computable general equilibrium theory, with explicit claims that it is based on constructive and computable foundations is neither the one, nor the other. Similarly, Newclassical Economics, the dominant strand in Macroeconomics, has as its formal core so-called Recursive Macroeconomic Theory. The dominance of computational and numerical analysis, powerfully underpinned by serious approximation theory, is totally devoid of computable or constructive foundations. The reasons for this paradoxical lack of interest in computability or constructivity considerations, even while almost the whole of economic theory is almost completely dominated by numerical, computational and experimental considerations, are quite easy to discern: the reliance of every kind of mathematical economics on real analysis for formalization. I shall not go into too many details of this ‘conjecture’ in this paper, but once again the interested reader is referred to [51,53] for more comprehensive discussions and formal analysis (but see also Section 3, below). Two distinguished pioneers of economic theory and, appropriately, national income accounting, Kenneth Arrow and Richard Stone (in collaboration with Alan Brown) – who also happened to be Nobel Laureates – almost delineated the subject matter of what I have come to call Computable Economics. The former conjectured, more than two decades ago, as a frontier research strategy for the mathematical economic theorist, that: ‘‘The next step in analysis, I would conjecture, is a more consistent assumption of computability in the formulation of economic hypothesis. This is likely to have its own difficulties because, of course, not everything is computable, and there will be in this sense an inherently unpredictable element in rational behavior.” [1] 2 In addition to the themes Ruelle broached in this ‘Gibbs Lecture’, the first four, chapter 9 and the last four chapters of his elegant new book [37] are also relevant for the philosophical underpinnings of this paper. Although the two Ruelle references are not directly related to the subject matter of this paper, I include them because the mathematical themes of these works are deeply relevant to my approach here. 3 Discerning scholars would notice that I have not included the absolutely pioneering work of Louis Bachelier in this list (cf. [9,12] for easily accessible English versions of Bachelier’s Théorie de la Spéculation). This is only because he did not raise issues of computability, decidability and constructivity, that he could not possibly have done at the time he wrote, even though Hilbert’s famous ‘Paris Lecture’ was only five months away from when Bachelier defended his remarkable doctoral dissertation – also in Paris. 4 By ‘Uncomputability’ I mean both that arising from (classical) recursion theoretic considerations, and from those due to formal non-constructivities (in any sense of constructive mathematics). 5 The absence of any detailed discussion of honest priorities from within the ‘citadel’ in this paper is also for reasons of space constraints. 6 Douglas Bridges is, of course, a distinguished mathematician who has made fundamental contributions – both at the research frontiers and at the level of cultured pedagogy – to constructive analysis, computability theory and their interdependence, too. However, I consider his contributions to ‘constructive economics’ to be at least as pioneering as Alain Lewis’s to ‘computable economics’. Alas, neither the one nor the other seems to have made the slightest difference to the orthodox, routine, practice of the mathematical economist. 7 In this paper I shall not discuss the place of computational complexity theory in economics, which has an almost equally distinguished ancestry. I provide a fairly full discussion of the role of computational complexity theory, from the point of view of algorithmic economics in [54]. 8 By this I aim to refer to classical numerical analysis, which has only in recent years shown tendencies of merging with computability theory – for example through the work of Steve Smale and his many collaborators (cf. for example [2]). To the best of my knowledge the foundational work in computable analysis and constructive analysis was never properly integrated with classical numerical analysis. 9 With the notable exception of the writings of the above mentioned pioneers, none of whom work – or worked – in any of these three areas, as conceived and understood these days. For excellent expositions of numerical and computational methods in economics, particularly macroeconomics, see [4,18,23]. 10 A complete and detailed analysis of the false claims – from the point of view of computability and constructivity – of the proponents and practitioners of CGE modelling is given in my recent paper devoted explicitly to the topic (cf. [52]). 1406 K. Vela Velupillai / Applied Mathematics and Computation 215 (2009) 1404–1416 Richard Stone (together with Alan Brown), speaking as an applied economist, grappling with the conundrums of adapting an economic theory formulated in terms of a mathematics alien to the digital computer and to the nature of the data, confessed his own credo in characteristically perceptive terms: ‘‘Our approach is quantitative because economic life is largely concerned with quantities. We use [digital] computers because they are the best means that exist for answering the questions we ask. It is our responsibility to formulate the questions and get together the data which the computer needs to answer them.” [3, p. viii] Economic analysis, as practised by the mathematical economist – whether as a microeconomist or a macroeconomist, or even as a game theorist or an IO theorist – continues, with princely unconcern for these conjectures and conundrums, to be mired in, and underpinned by, conventional real analysis (see also Ref. [6]). Therefore, it is a ‘cheap’ exercise to extract, discover and display varieties of uncomputabilities, undecidabilities and non-constructivities in the citadel of economic theory. Anyone with a modicum of expertise in recursion theory, constructive analysis or even nonstandard analysis in its constructive modes, would find, in any reading from these more algorithmically oriented perspectives, the citadel of economic theory, game theory and IO replete with uncomputabilities, undecidabilities and non-constructivities – even elements of incompleteness. Against this ‘potted’ background of pioneering innocence and core issues, the rest of this paper is structured as follows. Some of the key results on uncomputability and undecidability, mostly derived by this author, are summarized in a fairly merciless telegraphic form (with adequate and detailed references to sources) in the next section. In Section 3 some remarks on the mathematical underpinnings of these ‘negative’ results are discussed and, again, stated in the usual telegraphic form. The concluding section suggests a framework for invoking my ‘version’ of unconventional computation models for mathematical models of the economy. 2. Uncomputability and undecidability in economic theory Although many of the results described in this section may appear to have been obtained’cheaply’ – in the sense mentioned above – my own reasons for having worked with the aim of locating uncomputabilities, non-constructivities and undecidabilities in core areas of economic theory have always been a combination of intellectual curiosity – along the lines conjectured by Arrow, above –and the desire to make the subject meaningfully quantitative – in the sense suggested by Brown and Stone [3]. In the process an explicit research strategy has also emerged, on the strategy of making economic theory consistently algorithmic. The most convincing and admirably transparent example of this research strategy is the one adopted by Michael Rabin to transform the celebrated Gale-Stewart Game to an Algorithmic Game and, then, to characterise its effective content. A full discussion of this particular episode in the development of Computable Economics is given in [48] and [50, chapter 7]. However, the various subsections below simply report some of the results I have obtained, on uncomputability, non-constructivity and undecidability in economic theory, without, in each case, describing the background motivation, the precise research and proof strategy that was developed to obtain the result and the full extent of the implications for Computable Economics. 11 Maury Osborne, with the clarity that can only come from a rank outsider to the internal paradoxes of the dissonance between economic theory and applied economics, noted pungently: ‘‘There are numerous other paradoxical beliefs of this society [of economists], consequent to the difference between discrete numbers. . . in which data is recorded, whereas the theoreticians of this society tend to think in terms of real numbers. . . .No matter how hard I looked, I never could see any actual real [economic] data that showed that [these solid, smooth, lines of economic theory] . . . actually could be observed in nature. . . . At this point a beady eyed Chicken Little might . . . say, ‘Look here, you can’t have solid lines on that picture because there is always a smallest unit of money . . . and in addition there is always a unit of something that you buy. . .[I]n any event we should have just whole numbers of some sort on [the supply-demand] diagram on both axes. The lines should be dotted. . . . Then our mathematician Zero will have an objection on the grounds that if we are going to have dotted lines instead of solid lines on the curve then there does not exist any such thing as a slope, or a derivative, or a logarithmic derivative either. . . .. _ If you think in terms of solid lines while the practice is in terms of dots and little steps up and down, this misbelief on your part is worth, I would say conservatively, to the governors of the exchange, at least eighty million dollars per year. [29, pp. 16–34]. 12 Prefaced, elegantly and appositely, with a typically telling observation by Samuel Johnson: ‘‘Nothing amuses more harmlessly than computation, and nothing is oftener applicable to real business or speculative enquiries. A thousand stories which the ignorant tell, and believe, die away at once when the computist takes them in his grip” [3, p. vii] Surely, this is simply a more literary expression of that famous credo of Leibniz: ‘‘. . .[W]hen a controversy arises, disputation will no more be needed between two philosophers than between two computers. It will suffice that, pen in hand, they sit down . . . and say to each other: Let us calculate.” [19]. 13 An acute observation by one of the referees requires at least a nodding mention. The referee wondered why the paper did not consider the famous ‘Socialist Calculation Debate’, emerging, initially, from careless remarks by Pareto about the computing capabilities of a decentralised market. This issue later – in the 1920s and 1930s, revisited by one of the protagonists as late as 1967 – became a full-blooded debated about the feasibility of a decentralised planning system, an oxymoron if ever there was one. However, the reason I am not considering the debate in this paper is twin-pronged: firstly, it was, essentially, about analog computing (although Oskar Lange muddied the issue in his revisit to the problem in 1967 in the Dobb Festschrift); secondly, it is less about computability than computational complexity. For reasons of space, I have had to refrain from any serious consideration of any kind of complexity issue – whether of the computational or algorithmic variety. K. Vela Velupillai / Applied Mathematics and Computation 215 (2009) 1404–1416 1407 2.1. Undecidability (and Uncomputability) of maximizing choice All of mathematical economics and every kind of orthodox game theory rely on some form of formalized notion of individually ‘rational behaviour’ Two key results that I derived more than two decades ago, are the following, stated as theorems: Theorem 1. Rational economic agents in the sense of economic theory are equivalent to suitably indexed Turing Machines; i.e, decision processes implemented by rational economic agents – viz., choice behaviour – is equivalent to the computing behaviour of a suitably indexed Turing Machine. Put another way, this theorem states that the process of rational choice by an economic agent is equivalent to the computing activity of a suitably programmed Turing Machine. Proof. Essentially by construction from first principles (no non-constructive assumptions are invoked). See [49]. h An essential, but mathematically trivial, implication of this Theorem is the following result. Theorem 2. Rational choice, understood as maximizing choice, is undecidable. Proof. The procedure is to show, again by construction, that preference ordering is effectively undecidable. See [50, Section 3.3] for the details. h Remark 3. These kinds of results are the reasons for the introduction of formalized concepts of bounded rationality and satisficing by Herbert Simon. Current practice, particularly in varieties of experimental game theory, to identify boundedly rational choice with the computing activities of a Finite Automaton are completely contrary to the theoretical constructs and cognitive underpinnings of Herbert Simon’s framework. The key mistake in current practice is to divorce the definition of bounded rationality from that of satisficing. Simon’s framework does not refer to the orthodox maximizing paradigm; it refers to the recursion theorist’s and the combinatorial optimizer’s framework of decision procedures. 2.2. Computable and decidable paradoxes of excess demand function 2.2.1. Algorithmic undecidability of a computable general equilibrium The excess demand function plays a crucial role in all aspects of computable general equilibrium theory and, indeed, in the foundation of microeconomics. Its significance in computable general equilibrium theory is due to the crucial role it plays in what has come to be called Uzawa’s Equivalence Theorem (cf. [44], §11.4) – the equivalence between a Walrasian Equilibrium Existence Theorem (WEET) and the Brouwer Fixed Point Theorem. The finesse in one half of the equivalence theorem, i.e., that WEET implies the Brouwer fix point theorem, is to show the feasibility of devising a continuous excess demand function, XðpÞ, satisfying Walras’ Law (and homogeneity), from an arbitrary continuous function, say f ð Þ : S ! S, where S is the unit simplex in R , such that the equilibrium price vector, p , implied by XðpÞ is also the fix point for f ð Þ, from which it is ‘constructed’. I am concerned, firstly, with the recursion theoretic status of XðpÞ. Is this function computable for arbitrary p 2 S? Obviously, if it is, then there is no need to use the alleged constructive procedure to determine the Brouwer fix point (or any of the other usual topological fix points that are invoked in general equilibrium theory and CGE Modelling) to locate the economic equilibrium implied by WEET. The key step in proceeding from a given, arbitrary, f ð Þ : S ! S to an excess demand function XðpÞ is the definition of an appropriate scalar: