Complexity, ‘fog and moonlight’, prediction, and politics III – von Neumann and economics as a science

The two previous blogs in this series were:

Part I HERE.

Part II HERE.

All page references unless otherwise stated are to my essay, HERE.

Since the financial crisis, there has been a great deal of media and Westminster discussion about why so few people predicted it and what the problems are with economics and financial theory.

Absent from most of this discussion is the history of the subject and its intellectual origins. Economics is clearly a vital area of prediction for people in politics. I therefore will explore some intellectual history to provide context for contemporary discussions about ‘what is wrong with economics and what should be done about it’.

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It has often been argued that the ‘complexity’ of human behaviour renders precise mathematical treatment of economics impossible, or that the undoubted errors of modern economics in applying the tools of mathematical physics are evidence of the irredeemable hopelessness of the goal.

For example, Kant wrote in Critique of Judgement:

‘For it is quite certain that in terms of merely mechanical principles of nature we cannot even adequately become familiar with, much less explain, organized beings and how they are internally possible. So certain is this that we may boldly state that it is absurd for human beings even to attempt it, or to hope that perhaps some day another Newton might arise who would explain to us, in terms of natural laws unordered by any intention, how even a mere blade of grass is produced. Rather, we must absolutely deny that human beings have such insight.’

In the middle of the 20th Century, one of the great minds of the century turned to this question. John Von Neumann was one of the leading mathematicians of the 20th Century. He was also a major contributor to the mathematisation of quantum mechanics, created the field of ‘quantum logic’ (1936), worked as a consultant to the Manhattan Project and other wartime technological projects, and was one of the two most important creators of modern computer science and artificial intelligence (with Turing) which he developed partly for immediate problems he was working on (e.g. the hydrogen bomb and ICBMs) and partly to probe the general field of understanding complex nonlinear systems.  In an Endnote of my essay I discuss some of these things.

Von Neumann was regarded as an extraordinary phenomenon even by  the cleverest people in the world. The Nobel-winning physicist and mathematician Wigner said of von Neumann:

‘I have known a great many intelligent people in my life. I knew Planck, von Laue and Heisenberg. Paul Dirac was my brother in law; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. But none of them had a mind as quick and acute as Jansci von Neumann. I have often remarked this in the presence of those men and no one ever disputed me… Perhaps the consciousness of animals is more shadowy than ours and perhaps their perceptions are always dreamlike. On the opposite side, whenever I talked with the sharpest intellect whom I have known – with von Neumann – I always had the impression that only he was fully awake, that I was halfway in a dream.’

Von Neumann also had a big impact on economics. During breaks from pressing wartime business, he wrote ‘Theory of Games and Economic Behaviour’ (TGEB) with Morgenstern. This practically created the field of ‘game theory’ which one sees so many references to now. TGEB was one of the most influential books ever written on economics. (The movie The Beautiful Mind gave a false impression of Nash’s contribution.) In the Introduction, his explanation of some foundational issues concerning economics, mathematics, and prediction is clearer for non-specialists than any other thing I have seen on the subject and cuts through a vast amount of contemporary discussion which fogs the issues.

This documentary on von Neumann is also interesting:

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There are some snippets from pre-20th Century figures explaining concepts in terms recognisable through the prism of Game Theory. For example, Ampère wrote ‘Considerations sur la théorie mathématique du jeu’ in 1802 and credited Buffon’s 1777 essay on ‘moral arithmetic’ (Buffon figured out many elements that Darwin would later harmonise in his theory of evolution). Cournot discussed what would later be described as a specific example of a ‘Nash equilibrium’ viz duopoly in 1838.  The French mathematician Emile Borel also made contributions to early ideas.

However, Game Theory really was born with von Neumann. In December 1926, he presented the paper ‘Zur Theorie der Gesellschaftsspiele’ (On the Theory of Parlour Games, published in 1928, translated version here) while working on the Hilbert Programme [cf. Endnote on Computing] and quantum mechanics. The connection between the Hilbert Programme and the intellectual origins of Game Theory can perhaps first be traced in a 1912 lecture by one of the world’s leading mathematicians and founders of modern set theory, Zermelo, titled ‘On the Application of Set Theory to Chess’ which stated of its purpose:

‘… it is not dealing with the practical method for games, but rather is simply giving an answer to the following question: can the value of a particular feasible position in a game for one of the players be mathematically and objectively decided, or can it at least be defined without resorting to more subjective psychological concepts?’

He presented a theorem that chess is strictly determined: that is, either (i) white can force a win, or (ii) black can force a win, or (iii) both sides can force at least a draw. Which of these is the actual solution to chess remains unknown. (Cf. ‘Zermelo and the Early History of Game Theory’, by Schwalbe & Walker (1997), which argues that modern scholarship is full of errors about this paper. According to Leonard (2006), Zermelo’s paper was part of a general interest in the game of chess among intellectuals in the first third of the 20th century. Lasker (world chess champion 1897–1921) knew Zermelo and both were taught by Hilbert.)

Von Neumman later wrote:

‘[I]f the theory of Chess were really fully known there would be nothing left to play.  The theory would show which of the three possibilities … actually holds, and accordingly the play would be decided before it starts…  But our proof, which guarantees the validity of one (and only one) of these three alternatives, gives no practically usable method to determine the true one. This relative, human difficulty necessitates the use of those incomplete, heuristic methods of playing, which constitute ‘good’ Chess; and without it there would be no element of ‘struggle’ and ‘surprise’ in that game.’ (p.125)

Elsewhere, he said:

‘Chess is not a game. Chess is a well-defined computation. You may not be able to work out the answers, but in theory there must be a solution, a right procedure in any position. Now, real games are not like that at all. Real life is not like that. Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do. And that is what games are about in my theory.’

Von Neumman’s 1928 paper proved that there is a rational solution to every two-person zero-sum game. That is, in a rigorously defined game with precise payoffs, there is a mathematically rational strategy for both sides – an outcome which both parties cannot hope to improve upon. This introduced the concept of the minimax: choose a strategy that minimises the possible maximum loss.

Zero-sum games are those where the payoffs ‘sum’ to zero. For example, chess or Go are zero-sum games because the gain (+1) and the loss (-1) sum to zero; one person’s win is another’s loss. The famous Prisoners’ Dilemma is a non-zero-sum game because the payoffs do not sum to zero: it is possible for both players to make gains. In some games the payoffs to the players are symmetrical (e.g. Prisoners’ Dilemma); in others, the payoffs are asymmetrical (e.g. the Dictator or Ultimatum games). Sometimes the strategies can be completely stated without the need for probabilities (‘pure’ strategies); sometimes, probabilities have to be assigned for particular actions (‘mixed’ strategies).

While the optimal minimax strategy might be a ‘pure’ strategy, von Neumann showed it would often have to be a ‘mixed strategy’ and this means a spontaneous return of probability, even if the game itself does not involve probability.

‘Although … chance was eliminated from the games of strategy under consideration (by introducing expected values and eliminating ‘draws’), it has now made a spontaneous reappearance. Even if the rules of the game do not contain any elements of ‘hazard’ … in specifying the rules of behaviour for the players it becomes imperative to reconsider the element of ‘hazard’. The dependence on chance (the ‘statistical’ element) is such an intrinsic part of the game itself (if not of the world) that there is no need to introduce it artificially by way of the rules of the game itself: even if the formal rules contain no trace of it, it still will assert itself.’

In 1932, he gave a lecture titled ‘On Certain Equations of Economics and A Generalization of Brouwer’s Fixed-Point Theorem’. It was published in German in 1938 but not in English until 1945 when it was published as ‘A Model of General Economic Equilibrium’. This paper developed what is sometimes called von Neumann’s Expanding Economic Model and has been described as the most influential article in mathematical economics. It introduced the use of ‘fixed-point theorems’. (Brouwer’s ‘fixed point theorem’ in topology proved that, in crude terms, if you lay a map of the US on the ground anywhere in the US, one point on the map will lie precisely over the point it represents on the ground beneath.)

‘The mathematical proof is possible only by means of a generalisation of Brouwer’s Fix-Point Theorem, i.e. by the use of very fundamental topological facts… The connection with topology may be very surprising at first, but the author thinks that it is natural in problems of this kind. The immediate reason for this is the occurrence of a certain ‘minimum-maximum’ problem… It is closely related to another problem occurring in the theory of games.’

Von Neumann’s application of this topological proof to economics was very influential in post-war mathematical economics and in particular was used by Arrow and Debreu in their seminal 1954 paper on general equilibrium, perhaps the central paper in modern traditional economics.

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In the late 1930’s, von Neumann, based at the IAS in Princeton to which Gödel and Einstein also fled to escape the Nazis, met up with the economist Oskar Morgenstern who was deeply dissatisfied with the state of economics. In 1940, von Neumann began his collaboration on games with Morgenstern, while working on war business including the Manhattan Project and computers, that became The Theory of Games and Economic Behavior (TGEB). By December 1942, he had finished his work on this though it was not published until 1944.

In the Introduction of TGEB, von Neumann explained the real problems in applying mathematics to economics and why Kant was wrong.

‘It is not that there exists any fundamental reason why mathematics should not be used in economics.  The arguments often heard that because of the human element, of the psychological factors etc., or because there is – allegedly – no measurement of important factors, mathematics will find no application, can all be dismissed as utterly mistaken.  Almost all these objections have been made, or might have been made, many centuries ago in fields where mathematics is now the chief instrument of analysis [e.g. physics in the 16th Century or chemistry and biology in the 18th]…

‘As to the lack of measurement of the most important factors, the example of the theory of heat is most instructive; before the development of the mathematical theory the possibilities of quantitative measurements were less favorable there than they are now in economics.  The precise measurements of the quantity and quality of heat (energy and temperature) were the outcome and not the antecedents of the mathematical theory…

‘The reason why mathematics has not been more successful in economics must be found elsewhere… To begin with, the economic problems were not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are. There is no point using exact methods where there is no clarity in the concepts and issues to which they are applied. [Emphasis added] Consequently the initial task is to clarify the knowledge of the matter by further careful descriptive work. But even in those parts of economics where the descriptive problem has been handled more satisfactorily, mathematical tools have seldom been used appropriately. They were either inadequately handled … or they led to mere translations from a literary form of expression into symbols…

‘Next, the empirical background of economic science is definitely inadequate. Our knowledge of the relevant facts of economics is incomparably smaller than that commanded in physics at the time when mathematization of that subject was achieved.  Indeed, the decisive break which came in physics in the seventeenth century … was possible only because of previous developments in astronomy. It was backed by several millennia of systematic, scientific, astronomical observation, culminating in an observer of unparalleled calibre, Tycho de Brahe. Nothing of this sort has occurred in economics. It would have been absurd in physics to expect Kepler and Newton without Tycho – and there is no reason to hope for an easier development in economics…

‘Very frequently the proofs [in economics] are lacking because a mathematical treatment has been attempted in fields which are so vast and so complicated that for a long time to come – until much more empirical knowledge is acquired – there is hardly any reason at all to expect progress more mathematico. The fact that these fields have been attacked in this way … indicates how much the attendant difficulties are being underestimated. They are enormous and we are now in no way equipped for them.

‘[We will need] changes in mathematical technique – in fact, in mathematics itself…  It must not be forgotten that these changes may be very considerable. The decisive phase of the application of mathematics to physics – Newton’s creation of a rational discipline of mechanics – brought about, and can hardly be separated from, the discovery of the infinitesimal calculus…

‘The importance of the social phenomena, the wealth and multiplicity of their manifestations, and the complexity of their structure, are at least equal to those in physics.  It is therefore to be expected – or feared – that mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this field… A fortiori, it is unlikely that a mere repetition of the tricks which served us so well in physics will do for the social phenomena too.  The probability is very slim indeed, since … we encounter in our discussions some mathematical problems which are quite different from those which occur in physical science.’

Von Neumann therefore exhorted economists to humility and the task of ‘careful, patient description’, a ‘task of vast proportions’. He stressed that economics could not attack the ‘big’ questions – much more modesty is needed to establish an exact theory for very simple problems, and build on those foundations.

‘The everyday work of the research physicist is … concerned with special problems which are “mature”… Unifications of fields which were formerly divided and far apart may alternate with this type of work. However, such fortunate occurrences are rare and happen only after each field has been thoroughly explored. Considering the fact that economics is much more difficult, much less understood, and undoubtedly in a much earlier stage of its evolution as a science than physics, one should clearly not expect more than a development of the above type in economics either…

‘The great progress in every science came when, in the study of problems which were modest as compared with ultimate aims, methods were developed which could be extended further and further. The free fall is a very trivial physical example, but it was the study of this exceedingly simple fact and its comparison with astronomical material which brought forth mechanics. It seems to us that the same standard of modesty should be applied in economics… The sound procedure is to obtain first utmost precision and mastery in a limited field, and then to proceed to another, somewhat wider one, and so on.’

Von Neumann therefore aims in TGEB at ‘the behavior of the individual and the simplest forms of exchange’ with the hope that this can be extended to more complex situations.

‘Economists frequently point to much larger, more ‘burning’ questions…  The experience of … physics indicates that this impatience merely delays progress, including that of the treatment of the ‘burning’ questions. There is no reason to assume the existence of shortcuts…

‘It is a well-known phenomenon in many branches of the exact and physical sciences that very great numbers are often easier to handle than those of medium size. An almost exact theory of a gas, containing about 1025 freely moving particles, is incomparably easier than that of the solar system, made up of 9 major bodies… This is … due to the excellent possibility of applying the laws of statistics and probabilities in the first case.

‘This analogy, however, is far from perfect for our problem. The theory of mechanics for 2,3,4,… bodies is well known, and in its general theoretical …. form is the foundation of the statistical theory for great numbers. For the social exchange economy – i.e. for the equivalent ‘games of strategy’ – the theory of 2,3,4… participants was heretofore lacking. It is this need that … our subsequent investigations will endeavor to satisfy. In other words, only after the theory for moderate numbers of participants has been satisfactorily developed will it be possible to decide whether extremely great numbers of participants simplify the situation.’

[This last bit has changed slightly as I forgot to include a few things.]

While some of von Neumann’s ideas were extremely influential on economics, his general warning here about the right approach to the use of mathematics was not widely heeded.

Most economists initially ignored von Neumann’s ideas.  Martin Shubik, a Princeton mathematician, recounted the scene he found:

‘The contrast of attitudes between the economics department and mathematics department was stamped on my mind… The former projected an atmosphere of dull-business-as-usual conservatism… The latter was electric with ideas… When von Neumann gave his seminar on his growth model, with a few exceptions, the serried ranks of Princeton economists could scarce forebear to yawn.’

However, a small but influential number, including mathematicians at the RAND Corporation (the first recognisable modern ‘think tank’) led by John Williams, applied it to nuclear strategy as well as economics. For example, Albert Wohlstetter published his Selection and Use of Strategic Air Bases (RAND, R-266, sometimes referred to as The Basing Study) in 1954. Williams persuaded the RAND Board and the infamous SAC General Curtis LeMay to develop a social science division at RAND that could include economists and psychologists to explore the practical potential of Game Theory further. He also hired von Neumann as a consultant; when the latter said he was too busy, Williams told him he only wanted the time it took von Neumann to shave in the morning. (Kubrick’s Dr Strangelove satirised RAND’s use of game theory.)

In the 1990’s, the movie A Beautiful Mind brought John Nash into pop culture, giving the misleading impression that he was the principle developer of Game Theory. Nash’s fame rests principally on work he did in 1950-1 that became known as ‘the Nash Equilibrium’. In Non-Cooperative Games (1950), he wrote:

‘[TGEB] contains a theory of n-person games of a type which we would call cooperative. This theory is based on an analysis of the interrelationships of the various coalitions which can be formed by the players of the game. Our theory, in contradistinction, is based on the absence of coalitions in that it is assumed each participant acts independently, without collaboration or communication with any of the others… [I have proved] that a finite non-cooperative game always has at least one equilibrium point.’

Von Neumann remarked of Nash’s results, ‘That’s trivial you know. It’s just a fixed point theorem.’ Nash himself said that von Neumann was a ‘European gentleman’ but was not impressed by his results.

In 1949-50, Merrill Flood, another RAND researcher, began experimenting with staff at RAND (and his own children) playing various games. Nash’s results prompted Flood to create what became known as the ‘Prisoners’ Dilemma’ game, the most famous and studied game in Game Theory. It was initially known as ‘a non-cooperative pair’ and the name ‘Prisoners’ Dilemma’ was given it by Tucker later in 1950 when he had to think of a way of explaining the concept to his psychology class at Stanford and hit on an anecdote putting the payoff matrix in the form of two prisoners in separate cells considering the pros and cons of ratting on each other.

The game was discussed and played at RAND without publishing. Flood wrote up the results in 1952 as an internal RAND memo accompanied by the real-time comments of the players. In 1958, Flood published the results formally (Some Experimental Games). Flood concluded that ‘there was no tendency to seek as the final solution … the Nash equilibrium point.’ Prisoners’ Dilemma has been called ‘the E. coli of social psychology’ by Axelrod, so popular has it become in so many different fields. Many studies of Iterated Prisoners’ Dilemma games have shown that generally neither human nor evolved genetic algorithm players converge on the Nash equilibrium but choose to cooperate far more than Nash’s theory predicts.

Section 7 of my essay discusses some recent breakthroughs, particularly the paper by Press & Dyson. This is also a good example of how mathematicians can invade fields. Dyson’s professional fields are maths and physics. He was persuaded to look at the Prisoners’ Dilemma. He very quickly saw that there was a previously unseen class of strategies that has opened up a whole new field for exploration. This article HERE is a good summary of recent developments.

Von Neumann’s brief forays into economics were very much a minor sideline for him but there is no doubt of his influence. Despite von Neumann’s reservations about neoclassical economics, Paul Samuelson admitted that, ‘He darted briefly into our domain, and it has never been the same since.’

In 1987, the Santa Fe Institute, founded by Gell Mann and others, organised a ten day meeting to discuss economics. On one side, they invited leading economists such as Kenneth Arrow and Larry Summers; on the other side, they invited physicists, biologists, and computer scientists, such as Nobel-winning Philip Anderson and John Holland (inventor of genetic algorithms). When the economists explained their assumptions, Phil Anderson said to them, ‘You guys really believe that?

One physicist later described the meeting as like visiting Cuba – the cars are all from the 1950’s so on one hand you admire them for keeping them going, but on the other hand they are old technology; similarly the economists were ingeniously using 19th Century maths and physics on very out-of-date models. The physicists were shocked at how the economists were content with simplifying assumptions that were obviously contradicted by reality, and they were surprised at the way the economists seemed unconcerned about how poor their predictions were.

Twenty-seven years later, this problem is more acute. Some economists are listening to the physicists about fundamental problems with the field. Some are angrily rejecting the physicists’ incursions into their field.

Von Neumann explained the scientifically accurate approach to economics and mathematics. [Inserted later. I mean – the first part of his comments above that discusses maths, prediction, models, and economics and physics. As far as I know, nobody seriously disputes these comments – i.e. that Kant and the general argument that ‘maths cannot make inroads into economics’ are wrong. The later comments about building up economic theories from theories of 2, 3, 4 agents etc is a separate topic. See comments.] In other blogs in this series I will explore some of the history of economic thinking as part of a description of the problem for politicians and other decision-makers who need to make predictions.

Please leave corrections and comments below.