Saturday, June 15, 2019

Learning The Origin Of "Duality"

Yesterday I learned that the person who first used the term "duality" in connection with linear programming, indeed with anything in economics, was John von Neumann in a private conversation with George Dantzig in 1947, the "father of linear programming."  That was the year Dantzig published his paper showing the simplex method for solving linear programming problems, bot their primals and their  duals.  Von Neumann wrote a paper on it the same year but did not publish it, with it only appearing in his Collected Papers in 1963, 6 years after he died. 

It is not really surprising that it would be von Neumann as there are deep links between oprimizing programming and game theory.  This would be seen in 1951 when David Gale, Harold Kuhn, and Albert Tucker invented nonlinear programming.  Kuhn and Tucker would, of course, prove the Kuhn-Tucker theorem that deeply involves duality.  1951 would also be the year that Tucker's student, John Nash, completed his PhD thesis with his ultimately Nobel-Prize-winning paper on non-cooperative game theory.  An irony is that while seeing the first prisoner's dilemma experiments at RAND soon after, Nash would become disillusioned with game theory and economics, it would be Tucker who would coin the term "prisoner's dilemma" in an address to the American Psychology Association.

While the term was introduced into linear programming by von Neumann, the concept itself had already been floating around in economics, with its first appearance being in 1886 when Antonelli first derived an indirect utility function.  It would be further developed by Harold Hotelling in a famous paper in 1932 with "Hotelling's Lemma," and the 1942 paper by Roy in which he introduced his famous identity.

While I had previously known bits of this, what I learned yesterday came from an ongoing project of mine to read the entire 3rd edition of the New Palgrave Dictionary of Economics, all 20 volumes of it, nearly 15,000 pages and about 3800 entries.  I am doing that because I am one of the coeditors of the fourth  edition, along with Matias Vernengo and Esteban Perez.  I have learned a great deal from this reading, which has now gotten me through entries for the letter D.  I have a ways to go, but I am genuinely impressed by the job that Steve Durlauf and Larry Blume did editing the third edition.

The original Palgrave Dictionary of Economics was published in 1893.  The current one still has entries from that original one by people like Edgeworth and Wicksteed.  Some of these oldies are really quite fascinating.  But the cumulative effect of reading all of it is really beginning to kind of blow my mind.

Oh, and returning to the original topic, for anyone who does not know, it has long been viewed by many as being one of the greatest mistakes and wrongs that when the Nobel Prize was given for linear programming, Dantzig was not one of the recipients, who were Leonid Kantorovich, the only actual Soviet economist to win a Nobel, and Tjalling Koopmans. There  was room for Dantzig (especially with von Neumann dead), but somehow he got left out.  But then, hey, they never gave it to Joan Robinson either.

Barkley Rosser


Anonymous said...

Cue Egmont in 5, 4, 3...

AXEC / E.K-H said...

The Palgrave Dictionary ― a comprehensive collection of False-Hero-Memorials
Comment on Barkley Rosser on ‘Learning The Origin Of “Duality”’

Barkley Rosser reports: “Yesterday I learned that the person who first used the term ‘duality’ in connection with linear programming, indeed with anything in economics, was John von Neumann in a private conversation with George Dantzig in 1947, the ‘father of linear programming.’ That was the year Dantzig published his paper showing the simplex method for solving linear programming problems, bot their primals and their duals. … It is not really surprising that it would be von Neumann as there are deep links between oprimizing programming and game theory.”

Yes, indeed, and the deep link is in the Walrasians axioms. The hardcore propositions of microeconomics are given with: “HC1 economic agents have preferences over outcomes; HC2 agents individually optimize subject to constraints; HC3 agent choice is manifest in interrelated markets; HC4 agents have full relevant knowledge; HC5 observable outcomes are coordinated, and must be discussed with reference to equilibrium states.” (Weintraub)

This axiom set contains a lot of NONENTITIES, a fact that did not escape the mathematicians: “Walras approached Poincaré for his approval. ... But Poincaré was devoutly committed to applied mathematics and did not fail to notice that utility is a nonmeasurable magnitude. ... He also wondered about the premises of Walras’s mathematics: It might be reasonable, as a first approximation, to regard men as completely self-interested, but the assumption of perfect foreknowledge ‘perhaps requires a certain reserve’.” (Porter)

The next mathematician to realize that Walrasian mathematics was not up to standards was von Neumann: “You know, Oskar, if those books are unearthed sometime a few hundred years hence, people will not believe they were written in our time. Rather, they will think that they are about contemporary with Newton, so primitive is their mathematics. Economics is simply still a million miles away from the state in which an advanced science is, such as physics.”

The two central issues the mathematicians addressed where HC2, i.e. constrained optimization, and HC5, i.e. equilibrium. The end result was General Equilibrium Theory which is known today to be a failure.#1 The mathematicians fixed the economists’ mathematical blunders#2, #3 but von Neumann uncritically bought into the silly behavioral assumptions of mainstream economics.#4 So, the project of the proper formalization of economics had one fatal drawback, von Neumann left the foundational assumptions untouched: “But this [establishing the analytic mother-structure] required one very crucial maneuver that was nowhere stated explicitly: namely, that the model of Walrasian general equilibrium was the root structure from which all further work in economics would eventuate.” (Weintraub)

Because of this, the NONENTITIES HC2 and HC5 are still with us from choice theory to game theory. Strictly speaking, von Neumann messed up the Paradigm Shift and this is where things stand to this day: “There is another alternative: to formulate a completely new research program and conceptual approach. As we have seen, this is often spoken of, but there is still no indication of what it might mean.” (Ingrao et al.)

See part 2

AXEC / E.K-H said...

Part 2

Barkley Rosser does not fail to mention that economists have developed the habit to award themselves faux Nobels for scientific failure and that “it has long been viewed by many as being one of the greatest mistakes and wrongs that when the Nobel Prize was given for linear programming, Dantzig was not one of the recipients, …” Indeed, False-Hero-Memorials are the real problem of the cargo cult science economics.

Egmont Kakarot-Handtke

#1 “At long last, it can be said that the history of general theory from Walras to Arrow-Debreu has been a journey down a blind alley, and it is historians of economic thought who seem to have finally hammered down the nails in this coffin. It has been a dead alley because the most rigorous solution of the existence problem by Arrow and Debreu turns general theory into a mathematical puzzle applied to a virtual economy that can be imagined but could not possibly exist, while the extremely relevant ‘stability problem’ has never been solved either rigorously or sloppily. General theory is simply a research program that has run into the sands.” (Blaug)

#2 “The so-called ‘mathematical’ economists in the narrower sense ― Walras, Pareto, Fisher, Cassel, and hosts of other later ones ― especially, have completely failed even to see the task that was before them. Professor Hicks has to be added to this list, which is regrettable because he wrote several years after decisive work had been done ― in principle ― by J. von Neumann and A. Wald.” (Morgenstern)

#3 “Consequently, it was the von Neumann perspective that shaped general equilibrium theory and game theory, and thus reconstituted economic theory. Thus, David Hilbert was the spiritual grandfather of this new economics. (Weintraub) Interestingly, both von Neumann and Einstein got/took their advanced math from Hilbert and the Göttingen School.

#4 “In any event, it seems that Morgenstern finally convinced von Neumann that they must proceed tactically by means of the conciliatory move of phrasing the payoffs in terms of an entity called ‘utility’, but one that von Neumann would demonstrate was cardinal ― in other words, for all practical purposes indistinguishable from money . . .” (Mirowski) said...

Gosh, took you awhile, Egmont.

Your analysis of this is seriously flawed. While linear programming can be stuck into a Walrasian framework, it is independent of it. Keep in mind that when Kantorovich first developed it, he was doing so for a centrally planned command socialist economy, that of the USSR. The decisionmaker is that entity, not a bunch of individual agents, and there are no markets. So of your pet Weintraubian axioms (not the actual axioms of GE theory), only H2 and H4 are relevant, but rewritten for a central planner.

As for your quotes, the one from Blaug is irrelevant. The one from MOrgenstern is hilarious. Even if one dumps non-constructivist Hilbertian math, one can still do linear programming, another irrelevant point. The final comment is correct, but also irrelevant.

AXEC / E.K-H said...

Barkley Rosser

When the classical economists for a moment stopped writing their silly propaganda pamphlets and watched what was going on in the triumphant sciences they could not fail to notice that calculus was a core element of analysis: “Already Maupertuis considered his minimum principle as proof that the world, where among many virtual movements the one leading to maximum effect with minimum effort is realized, is the ‘best of all worlds’ and work of a purposeful creator. Euler made a similar remark: ‘Since the construction of the whole world is the most eminent and since it originated from the wisest creator, nothing is found in the world which would not show a maximum or minimum characteristic.’ ...” (von Bertalanffy)

Economists had the presence of mind to adopt optimization as a foundational principle and eventually it became HC2 of the Walrasian axiom set. This is how economics became marginalistic and remained so to this day: “most of what I and many others do is sorta-kinda neoclassical because it takes the maximization-and-equilibrium world as a starting point ...” (Krugman)

Constrained optimization is simple if the assumption of a well-behaved production function is added. In the course of time, mathematicians found solutions for more complicated problems of production, transport, warfare and the like. This is where Dantzig and Kantorovich came in.

However, while the mathematicians developed solutions for real-world problems, economists applied constrained optimization to human behavior. This is the foundational idiotism of microfoundations: “Throughout its history, the idea of some ‘Fundamental Assumption’, some basic ‘Economic Principle’ about human conduct, from which much or most of economics can ultimately be deduced, has been deeply rooted in the procedure of economic theory. Some such notion is still, in many quarters, dominant at the present time. For example, it has recently been stated that the task of economics is ‘to display the structure and working of the economic cosmos as an outgrowth of the maximum principle’.” (Hutchison)

The point is: no way leads from a behavioral assumption (optimization or otherwise) to the understanding of how the economy works. Standard economics is based on behavioral axioms and this is not a solid enough foundation: “… if we wish to place economic science upon a solid basis, we must make it completely independent of psychological assumptions and philosophical hypotheses.” (Slutzky)

So, the mathematicians have done a fine job by solving all kinds of complex optimization problems. This, though, does not alter the fact microfoundations HC1 to HC5 are proto-scientific garbage and have to be replaced by macrofoundations. This is called a Paradigm Shift which, as everyone knows by now, has been messed up by the waffling economist Keynes. Economists are simply too stupid for the elementary mathematics that underlies macroeconomics.#1, #2

Economists claim since Adam Smith that the free market economy is self-regulating and self-optimizing if left to itself. In reality, it is just the opposite: the real part of the economy is kept on life-support by the government, the monetary/financial part is kept on life-support by the Central Bank.#3

Economics is a failed science and the Bank of Sweden fake Nobel is the most fraudulent of all False-Hero-Memorials.#3

Egmont Kakarot-Handtke

#1 Are economics professors really that incompetent? Yes!

#2 Econ 101: Economists flunk the intelligence test at the first hurdle

#3 The Levy/Kalecki Profit Equation is false said...

Sorry, but a lot of the people yiou label as "economists" were or are really mathematicians, e.g. von Neumann, Nash, Kantorovich. To the extent what they did id not good because of a failure to allow for a macrofoundatiion and that amounts to recognizing your silly profit accounting, your comments are just irrelevant and silly, Egmont.

I really did find it amusing to see you sneer at von Neumann and Einstein for using Hilbertian math, two of the smartest people of the 20th century, if not all time. But you know better than they did. (Einstein was also the father of something used in economics and finance, notably the mathematical concept of Brownian motion, which also happens to be what he got his Nobel Prize for, you know, those "fake hero" prizes).

AXEC / E.K-H said...

Barkley Rosser

You say: “Sorry, but a lot of the people you label as ‘economists’ were or are really mathematicians, e.g. von Neumann, Nash, Kantorovich.”

No. My point is that it was mathematicians and not economists who cleared up the mathematical mess of economists and did all the heavy lifting that resulted in General Equilibrium Theory. See Mirowski’s More Heat Than Light for the finer points, especially how mathematically trained engineers who dabbled in economics desperately tried to make economists aware that they applied calculus incorrectly. To no avail, of course.

It is mathematicians who deserve most or all of the credit for solving the tricky optimization problems that were implicit in the Walrasian axiom HC2. Solving optimization problems is mathematics and, strictly speaking, not economics. Economics is about how the economic system works. The simple fact of the matter is that General Equilibrium Theory, which is based on HC2, i.e. constrained optimization, and HC5, i.e. equilibrium, does NOT explain how the economy works. This showpiece of the mathematicians’ ingenuity has NO economic content at all.

To award an economics “Nobel” to a mathematical achievement has to be seen as an attempt to give the cargo cult science economics the appearance of genuine science.

Von Neumann’s contribution has to be seen in the context of the ideological/physical war capitalism/USA vs communism/Russia and the involvement of mathematicians/physicists in the development of the atom bomb. Von Neumann recommended himself to the US military with his famous statement: “If you say why not bomb them tomorrow, I say why not today? If you say today at five o’clock, I say why not one o’clock?” (Wikiquote)

The US military needed the most advanced mathematics available on the planet, so von Neumann was sent to Göttingen: “In 1926 von Neumann went to Göttingen on a Rockefeller fellowship to work as Hilbert’s assistant. Göttingen was not only one of the centers of mathematics but it also was a mecca of theoretical physics; thus in Göttingen von Neumann could familiarize himself with the latest developments concerning quantum mechanics. Hilbert himself gave lectures on the mathematical foundations of quantum mechanics in the academic year 1926-1927. Von Neumann attended these lectures, and working out the lecture notes taken during those lectures led to a joint publication and eventually to von Neumann’s three ground breaking papers on the mathematical foundations of quantum mechanics that served as the basis of his book.”#1

Not to forget, Game Theory is not so much about parlor games or the problems of incarcerated mobsters or strategic economic behavior as about war games.

What von Neumann learned from Hilbert was the crucial importance of axiomatization: “When we assemble the facts of a definite, more-or-less comprehensive field of knowledge, we soon notice that these facts are capable of being ordered. This ordering always comes about with the help of a certain framework of concepts .... The framework of concepts is nothing other than the theory of the field of knowledge. ... If we consider a particular theory more closely, we always see that a few distinguished propositions of the field of knowledge underlie the construction of the framework of concepts, and these propositions then suffice by themselves for the construction, in accordance with logical principles, of the entire framework. ... The procedure of the axiomatic method, as it is expressed here, amounts to a deepening of the foundations of the individual domains of knowledge — a deepening that is necessary for every edifice that one wishes to expand and to build higher while preserving its stability.” (Hilbert)

Note that the Theory of Games is axiomatized.

See part 2

AXEC / E.K-H said...

Part 2

You say: “To the extent what they [the mathematicians] did id not good because of a failure to allow for a macrofoundatiions and that amounts to recognizing your silly profit accounting, your comments are just irrelevant and silly, Egmont.”

What I actually say is that ECONOMISTS are too stupid for the elementary mathematics that underlies macroeconomics. This is a provable fact.

Since Keynes, neither Post-Keynesians nor New Keynesians nor Anti-Keynesians have realized that I=S is false and has to be replaced by Q=I−S with Q representing macroeconomic profit. What Hilbert called “the framework of concepts” is false in economics to this day.

Needless to emphasize that you neither understand the problem of the proper axiomatization of economics nor the macroeconomic Profit Law. So, you are the ideal person to work on a new edition of the Palgrave collection of False-Hero-Memorials of the cargo cult science economics.

Economics is a failed/fake science and the Palgrave Dictionary and you are part of it. Economics needs a Paradigm Shift from false Walrasian microfoundations and false Keynesian macrofoundations to true macrofoundations.

Egmont Kakarot-Handtke

#1 PDFs.semanticscholar said...

My only comment on all this is the idea that the 1926 Rockefeller grant to von Neumann to work with Hilbert was motivated by support for his hardline anti-Communist views is just rank silly nonsense. His political views were unknown at that time, and all that only became important after WW II, long after he has arrived in the US, which was quite some time after he worked with Hilbert.

Yes, von Neumann's work redid quantum mechanics, but I note that his first paper on game theory, in which he also first used a fixed point theorem, and which also has duality inherent in it, was published in 1928 in German coming out of all that.

AXEC / E.K-H said...

Barkley Rosser

Let’s return to the main point: “According to George Dantzig, the duality theorem for linear optimization was conjectured by John von Neumann immediately after Dantzig presented the linear programming problem. Von Neumann noted that he was using information from his game theory, and conjectured that two person zero sum matrix game was equivalent to linear programming.”#1

Von Neumann identified the Walrasian approach as given with the verbalized axioms HC1 to HC5 as petitio principii (TOG p. 15), that is as methodologically unacceptable.#2 So, he introduced a new set of axioms: “An exact and exhaustive elaboration of these ideas requires the use of the axiomatic method.” (TOG p. 19) The formulation of a new set (TOG p. 24 f.) amounts to a Paradigm Shift.

Note, in particular, von Neumann’s general remark on axiomatization: “The axioms should not be too numerous, their system is to be as simple and transparent as possible, and each axiom should have an immediate intuitive meaning by which its appropriateness may be judged directly.” (TOG p. 25)

Generally speaking, von Neumann’s approach is (i) behavioral and (ii) bottom-up. In this, it resembles the Walrasian approach. Both may be put together under the label microfoundations. Now, the correct approach to economics is (i) structural and (ii) top-down. For good methodological reasons, economics has to be macrofounded.

Game theory is predicated on von Neumann’s specification of utility (TOG p. 15) and this leads to the concept of zero-sum games: “In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant’s gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants.”#3 Note that there is a quite subtle transition from utility (ordinal) to profit (cardinal): “In 1944, John von Neumann and Oskar Morgenstern proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n+1 players; the (n+1)th player representing the global profit or loss.”#3

And exactly here is the lethal blunder: global profit or loss cannot be determined within the framework of game theory.

Here is the axiomatically correct proof. The elementary production-consumption economy is defined with this set of macroeconomic axioms: (A0) The economy consists of the household and the business sector which, in turn, consists initially of one giant fully integrated firm. (A1) Yw=WL wage income Yw is equal to wage rate W times working hours. L, (A2) O=RL output O is equal to productivity R times working hours L, (A3) C=PX consumption expenditure C is equal to price P times quantity bought/sold X.

Note that this set satisfies von Neumann’s general criteria of proper axiomatization as quoted above.

Under the conditions of market clearing X=O and budget balancing C=Yw in each period, the price as the dependent variable is given by P=W/R. This is the macroeconomic Law of Supply and Demand.

The focus is here on the nominal/monetary balances. For the time being, real balances are excluded, i.e. it holds X=O. The condition of budget balancing, i.e. C=Yw, is now skipped. The monetary saving/dissaving of the household sector is defined as S≡Yw−C. The monetary profit/loss of the business sector is defined as Q≡C−Yw. Ergo Q+S=0 or Q=−S.

The balances add up to zero. The mirror image of household sector saving S is business sector loss (-Q). The mirror image of household sector dissaving (-S) is business sector profit Q. Q=−S is the elementary version of the macroeconomic Profit Law.

See part 2

AXEC / E.K-H said...

Part 2

From Q+S=0 one could conclude that economics is a zero-sum game between the business sector and the household sector. However, this is only the first half of the truth. From the fact that profit comes from dissaving, it follows that the household sector accumulates debt (overdrafts at the central bank) while the business sector accumulates money (deposits at the central bank). Now, debt has to be eventually repaid and this inverts the whole game: profit Q in period t is exactly annihilated by loss (-Q) in period t+1. The mirror image is dissaving (-S) in t and saving S in t+1. The result of these interlocked zero-sum games is zero profit over all periods.

Economics is an intertemporal zero-sum game and the n+1th player is the future. This is the reason why capitalism will eventually break down and NOT Marx’s phony Revolution of the Proletariat. The intertemporal zero-sum theorem fully replaces folk psychology/sociology. Note that the intertemporal zero-sum theorem holds for a monetary economy that is for capitalism AND communism AND anything in-between. Which in turn proves that political economics has NEVER been anything else than proto-scientific garbage of the worst sort.

Egmont Kakarot-Handtke

#1 Duality (optimization)

#2 Petitio principii — economists’ biggest methodological mistake

#3 Zero-sum game

#4 The Levy/Kalecki Profit Equation is false