The ergodic axiom: Davidson versus Stiglitz and Lucas
Contemporary neoclassical economists proceed under the assumption that as concerns the economy there exists a predetermined reality that can be fully described by “unchanging objective conditional probability functions”. This is called the ”ergodic axiom”, and its current supporters include Joseph . Stiglitz and Robert Lucas. In the anti-scientism spirit of Keynes, Paul Davidson has long campaigned against the use of the ergodic axiom, but never so tellingly as in his most recent paper, Is economics a science? Should economics be rigorous? It appears in the current issue of the Real-World Economics. Here is one section from this paper.
The ergodic axiom
First, let us take up the ergodic – nonergodic stochastic process distinction. Paul Samuelson [1969] has written that if economists hope to move economics from “the realm of history” into “the realm of science” they must impose the “ergodic hypothesis” on their theory[1]. In other words Nobel Prize Winner Paul Samuelson has made the ergodic axiom the sine qua non for the scientific method in economics. Lucas and Sargent [1981] have also claimed the principle behind the ergodic axiom is the only scientific method of doing economics.
Following Samuelson’s lead, most economists (e.g., Cochrane, Stiglitz, Mankiw, M. Friedman, Scholes, etc) and economic textbook writers either implicitly or explicitly have assumed that observable economic events are generated by an ergodic stochastic process.
But not Keynes! Keynes [1936, p. 16] suggested the way to understand why classical economic theory (e,g., efficient market theory) is not relevant to the world of experience, when he noted that old economic thinkers were “like Euclidean geometers in a non-Euclidean world who discover that apparent parallel line collide, rebuke these lines for not keeping straight. Yet, in truth there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry. Something similar is required to-day in economics”. Keynes developed a theory that is more general than classical and mainstream economic theory because it is based on fewer restrictive fundamental axioms[2]. The fewer the number of underlying axioms, the more general the theory. The most important classical axiom Keynes eliminated in his general theory[3] is the ergodic axiom.
This ergodic axiom assumes the economic future is already predetermined[4] . The economy is governed by an existing ergodic stochastic process. One merely has to calculate probability distributions regarding future prices and output to draw significant and reliable statistical inferences [information] about the future. Once self-interested decision makers have reliable information about the future, their actions on free markets will optimally allocate resources into those activities that will have the highest possible future returns thereby assuring global prosperity.
In order to draw any statistical (probabilistic risk) inferences regarding any universe, however, one should draw a sample from that universe. Since drawing a sample from the future economic universe is impossible, the ergodic axiom presumes that the economic future is governed by an already existing unchanging ergodic stochastic process. Consequently, a sample drawn from the past is equivalent to a sample drawn from the future. In other words, calculating the probability distribution from past statistical data sample is presumed to be the same as calculating the risks from a sample drawn from the future.[5] This ergodic axiom is an essential foundation for all the complex risk management computer models developed by the “quants” on Wall Street. If the economy is nonergodic, however, then these computer models are weapons of math destruction [For deterministic models, the “ordering axiom” plays the same role as the ergodic axiom in stochastic models.]
For a technical explanation of the difference between ergodic and nonergodic stochastic processes one should read my book, The Keynes Solution: The Path To Global Economic Prosperity [Davidson (2009)] . For our discussion here we merely need note that, in essence, the ergodic axiom imposes the condition that the future is already predetermined by existing parameters (market fundamentals). Consequently the future can be reliably forecasted by analyzing past and current market data to obtain the probability distribution governing future events. In other words, if future events are assumed to be generated by an ergodic stochastic process (to use the language of mathematical statisticians), then the future is predetermined and can be discovered today by the proper statistical probability analysis of past and today’s data regarding market “fundamentals”. If the system is nonergodic, calculated past and current probability distributions do not provide any statistically reliable estimates regarding the probability of future events.
New Keynesians such as Stiglitz accept the ergodic axiom as the basis of the economic system but then add additional ad hoc assumptions to try to tame this presumed knowledge of the future approach to better reflect what they believe is reality. Stiglitz, for example, in his asymmetric information theory assumes that some market participants cannot make the proper statistical calculations because they do not perceive the correct information about the future. In other words, Stiglitz imposes the asymmetric information condition that there are some decision makers who act while lacking the correct information about the (presumed to exist today) probability distribution of future events. Consequently these decision makers (speculative fools?) misread the future and thereby mess up the beauty of the efficient market system.
Nobel prize winner Robert Lucas [1981, p. 287] has boasted that the mainstream theory axioms are “artificial, abstract, patently unreal”. Like Nobel Laureate Samuelson, Lucas insists such unreal assumptions are the only scientific method of doing economics. Lucas insists that “Progress in economic thinking means getting better and better abstract, analogue models, not better verbal observations about the real world” [Lucas, 1981, p. 276]. The rationale underlying this argument is that these unrealistic assumptions make the problem more tractable and, with the aid of a computer, the analyst can then predict the future. Never mind that the prediction might be disastrously wrong.
In the introduction to his book Against The Gods, a treatise that deals with the questions of relevance of risk management techniques on Wall Street, Peter L. Bernstein [ 1996, p. 6] writes:
“The story that I have to tell is marked all the way through by a persistent tension between those who assert that the best decisions are based on quantification and numbers, determined by the [statistical] patterns of the past, and those who based their decisions on more subjective degrees of belief about the uncertain future. This is a controversy that has never been resolved . . . to what degree should we rely on the patterns of the past to tell us what the future will be like?”
One would hope that the empirical evidence of the collapse of those “masters of the economic universe “ that have dominate Wall Street machinations for the last three decades has at least created doubt regarding the applicability of the ergodic axiom to our economic world. Even Alan Greenspan in testimony before Congress in October 2008 seems to be having second thoughts although he still has not completely changed his tune. Keynes’s ideas and Soros’s reflexivity concept support Bernstein’s latter group.
Samuelson, Lucas and others adopted the ergodic axiom because they want economics to be in the same class as the “hard sciences” such as physics or astronomy. For example the science of astronomy is based on the presumption of an ergodic stochastic process that governs the movement of all the heavenly bodies from the moment of the “Big Bang” to the day the universe ends. Accordingly probability analysis using past measurements of the movements of heavenly bodies permit astronomers to predict future solar eclipses within a few seconds of when they actually occur. Nothing Congress, the President of the United States, the United Nations, or environmentalists can do will alter the predetermined dates and time for future eclipses. For example, Congress cannot pass a law outlawing solar eclipses in order to provide more sunshine and thereby enhance crop production. In an ergodic world, all future events are already predetermined and beyond change by human action today. The future movement of the heavenly bodies can be known by anyone who has measured past movements and projected these movements into the future. There are no speculative fools, who suffering from asymmetric information, think Mars is going to crash into the earth.
George Soros has explained why the efficient market theory is not applicable to real world financial markets with a slightly different terminology than Keynes but conceptually in the same way. Soros (2008) wrote: “we must abandon the prevailing [efficient market] theory of market. behavior. ” Soros states that there is a direct connection “between market prices and the underlying realty [that] I [Soros] call reflexivity”.
What is this reflexivity? In a letter to the Editor published in the March 15-21, 1997 issue of The Economist Soros objects to Paul Samuelson insistence on requiring the ergodic axiom to make economics a science. Soros argues the ergodic hypothesis does not permit “the reflexive interaction between participants’ thinking and the actual state of affairs” that characterizes real world financial markets. In other words, the way people think about the market today can affect and alter the future path the market takes; the future is not predetermined. Soros’s concept of reflexivity, therefore, is the equivalent of Keynes’s rejection of the ergodic axiom[6]. Reflexivity means peoples thoughts and actions create the future, while mainstream economists presume the future has already been predetermined and can be discovered by analyzing today’s market fundamentals.
[1]. P. A. Samuelson,[1969] “Classical and Neoclassical Theory” in Monetary Theory, edited by R.W. Clower (Penguin Books,, London) p.12.
[2]. Keynes [1936, p. 3] stated that the classical economics fundamental axioms are applicable to a “special case….[that] happen[s] not to be those of the economic society in which we live with the result that its teaching is misleading and disastrous if we attempt to apply it to fact of experience”. This “special case” statement is even more applicable today, given the economic austerity discussions in Washington, the UK, Euroland, etc, and the export-led growth , i.e., mercantilist, policies pursued by nations such as China who are still enjoying an “economic miracle” in an otherwise depressed global economy.
[3]. Two other axioms that Keynes rejected are 1. Money is neutral (at least in the long run) so that changes in the quantity of money do not affect real outcomes, and 2. Gross substitution is ubiquitous and therefore liquid assets are good substitutes for real capital goods. (See Davidson , 2009).
[4]. Consequently, government action today can only delay, but not change the long run optimal solution already predetermined by free markets.
[5]. This is equivalent to thinking that drawing the sample of heights from a pygmy tribe in Africa is equivalent to drawing a sample of Swedish citizens’ height.
[6]. In place of the rejected ergodic axiom Keynes argued that when crucial economic decisions had to be made, decision makers could not merely assume that the future can be reduced to quantifiable risks calculated from already existing market data. Instead they depended on “animal spirits” since most animals do not know how to calculate the moments around the mean!
For decisions that involved potential large spending outflows or possible large income inflows that span a significant length of time, people “know” that they do not know what the future will be. They do know that for these important decisions, making a mistake about the future can be very costly and therefore sometimes putting off a commitment by maintaining liquidity today may be the most judicious decision possible.
You can read Davidson’s whole paper: Is economics a science? Should economics be rigorous?
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clarity in thought for this article as far as this old fellow (me) is concerned. Although argument from analogy is sometimes dangerous. the Euclidean/non-euclidean comparison seems apt. It is also true that the simple 2 dimension to 3 dimension helps even when the dimensionality is much higher …
thanks for posting this
I have an objection to the term “anti-scientism”. There is a considerable scientific literature devoted to non-ergodic processes. Expunging the ergodic axiom from economics does not require an anti-scientism stance. Far from it. The issue is how to alter economics to accommodate both the uncertainty that Davidson highlights and the creativity that our reaction to that uncertainty engenders. As Stuart Kauffman has said: the economy is in a constant state of becoming. This opens the door for evolutionary thinking, dynamics, complexity and other notions. All of which are solidly scientific.
Please read what I wrote again, Peter. You have interpreted it backwards from the meaning intended. Edward
Science is a great and necessary tool, but it must take a back seat to Wisdom which is a higher order way of of thinking (philosophy) and acting (policy) which supercedes, underlies and includes every one of Man’s other bodies of thought. Wisdom is OF the entirety of Life and Living, and a condensation of the Good from long time human IMMERSION in that Life and Living. Economics (and any other hard science or social science) is ABOUT a specific aspect of Life and Living, i.e. an abstract from it. Reality will always trump theory, and immersion, especially conclusions derived from long term and agreed upon observations from immersion in reality, will always trump theory. Wisdom, and policy’s alignment with it….an idea whose time has come.
Hmm. This is barely English. Let’s get some clarity into the discussion. A physical science defines its terms rigorously and the students of that science then set out to discover, through observation and experiment, consistencies in the relationships between different entities. Those consistent relationships can then be formulated as laws.
When, as seems to be the case, there is no agreement on the definition of fundamental concepts in economics such as “wealth” and “capital”, there is no possibility of developing an agreed body of fundamental theory.
Davidson’s article is a nice piece – but ergodicity is a difficult concept that many students of economics have problems with understanding. To understand real world ”non-routine” decisions and unforeseeable changes in behaviour, ergodic probability distributions are of no avail. In a world full of genuine uncertainty – where real historical time rules the roost – the probabilities that ruled the past are not those that will rule the future.
Time is what prevents everything from happening at once. To simply assume that economic processes are ergodic and concentrate on ensemble averages – and a fortiori in any relevant sense timeless – is not a sensible way for dealing with the kind of genuine uncertainty that permeates open systems such as economies.
When you assume the economic processes to be ergodic,ensemble and time averages are identical. Let me giva an example: Assume we have a market with an asset priced at 100 €. Then imagine the price first goes up by 50% and then later falls by 50%. The ensemble average for this asset would be 100 €- because we here envision two parallel universes (markets) where the assetprice falls in one universe (market) with 50% to 50 €, and in another universe (market) it goes up with 50% to 150 €, giving an average of 100 € ((150+50)/2). The time average for this asset would be 75 € – because we here envision one universe (market) where the assetprice first rises by 50% to 150 €, and then falls by 50% to 75 € (0.5*150).
From the ensemble perspective nothing really, on average, happens. From the time perspective lots of things really, on average, happen.
Assuming ergodicity there would have been no difference at all.
Just in case you think this is just an academic quibble without repercussion to our real lives, let me quote from an article of physicist and mathematician Ole Peters in the Santa Fe Institute Bulletin from 2009 – On Time and Risk – that makes it perfectly clear that the flaw in thinking about uncertainty in terms of “rational expectations” and ensemble averages has had real repercussions on the functioning of the financial system:
“In an investment context, the difference between ensemble averages and time averages is often small. It becomes important, however, when risks increase, when correlation hinders diversification, when leverage pumps up fluctuations, when money is made cheap, when capital requirements are relaxed. If reward structures—such as bonuses that reward gains but don’t punish losses, and also certain commission schemes—provide incentives for excessive risk, problems arise. This is especially true if the only limits to risk-taking derive from utility functions that express risk preference, instead of the objective argument of time irreversibility. In other words, using the ensemble average without sufficiently restrictive utility functions will lead to excessive risk-taking and eventual collapse. Sound familiar?”
Peter is right in indicating that by rejecting the ergodic axiom one does not mean that the economic discipline is anti-science!!
For example, most readers believe that the Darwin theory of evolution is a
“science”. But no one can predict what the next link in the evolutionary chain will be produced after human beings. Evolution is therefore governed by a nonergodic stochastic process –but it is still a science even if it can not predict the future species..
as for Lars — If the stochastic process is ergodic, then for for an infinite realizations, the time and space [what Lars calls the ensemble] averages will concide. An ensemble a is samples drawn at a fixed point of time drawn from a universe of realzations For finite realizations, the time and space stastistical averages tend to converge (with a probability of one) the more data one has.
Even in physics there are some processes that physicists recognize are governed by nonergodic stochastic processes.[ see A. M. Yaglom, An Introduction to Stationary Random Functions [1962, Prentice Hall]]
i do object to Ole Peters exposition quote where he talks about “when risks increase”. Nonergodic systems are not about increasing or decreasing risk in the sense of the probability distribution variances differing. It is about indicating that any probability distribution based on past data can not be reliably used to indicate the probability distribution governing any future outcome.. In other words even if (we could know) that the future probability distribution will have a smaller variance (“lower risks”) than the past calculated probability distribution, then the past distribution is not is not a reliable guide to future statoistical means and other moments around the means.
Peter is misleading himself when, not knowing his history, he conflates scientism (projecting the appearance of science on the non-scientific) with science. Adam Smith’s mentor David Hume transformed much science into scientism by basing his philosophy of science on knowledge of appearances rather than an understanding of how the direction of energy by different types of structure explains the appearances. The signals in a coded message such as a broadband signal often appear random until you decode them. As Einstein famously said, “God doesn’t play dice”; but Hume chose not to believe in a God he couldn’t see.
The application of ergodic maths to post-Humean economics is thus rooted in the very definition of ‘statistic’: “Any function of a number of random variables”. A “big bang” will give rise to turbulence, but its predominant motion is not random but like the expansion of a bubble, while its significant fundamental particles, like the vortices or whirlpools of the turbulence in which they originated, are polarised, not inert atoms in conflicting motion. People have been taught to believe (i.e. have a habitual, non-random, way of thinking) that if a commodity is readily available, its price will go down. So, monopolists non-randomly make it scarce and put up its price, while local cooperatives (like a fisherman’s one I saw on television last night) are now reacting to this non-randomly, getting a better return from selling the best produce at a fixed price and increased it by finding other markets for surplus and wastes.
Paul:
Re nonergodic processes in physics I would even say that MOST processes definitely are nonergodic.
Re Ole Peters I totally agree that what is important with the fact that real social and economic processes are nonergodic is the fact that uncertainty – not risk – rules the roost. That was something both Keynes and Knight basically said in their 1921 books. But I still think that Peters’ discussion is a good example of how thinking about uncertainty in terms of “rational expectations” and “ensemble averages” has had seriously bad repercussions on the financial system.
Lars:
there is a difference between the uncertainty concept developed by Keynes and one developed by Knight.
As I have pointed out ,Keynes’s concept of uncertainty involves a nonergodic stochastic process . On the other hand, Knight’s uncertainty — likeTaleb’s black swan– assumes an ergodic process. The difference is the for Knight (and Taleb) the uncertain outcome lies so far out in the tail of the unchanging (over time) probability distribution that it appears empirically to be [in Knight’s terminolgy] “unique”. In other words ,like Taleb’s black swan, the uncertain outcome already exists in the probability distribution but is so rarely observed that it may take severa lifetimes for one observation — making that observation “unique”.
In the latest edition of Taleb’s book , he was forced to concede that philosophically there is a difference between a nonergodic system and a black swan ergodic system –but then waves away the problem with the claim that the difference is irrelevent.
aul Davidson
Paul:
On the whole, I think you’re absolutely right on this. Knight’s uncertainty concept has an epistemological founding and Keynes’s definitely an ontological founding. Of course this also has repercussions on the issue of ergodicity in a strict methodological and mathematical-statistical sense. I think Keynes’s view is the most warranted of the two.
BUT – from a “practical” point of view I have to agree with Taleb. Because if there is no reliable information on the future, whether you talk of epistemological or ontological uncertainty, you can’t calculate probabilities.
The most interesting and far-reaching difference between the epoistemological and the ontological view is that if you subscribe to the former, knightian view – as Taleb and “black swan” theorists basically do – you open up for the mistaken belief that with better information and greater computer-power we somehow should always be able to calculate probabilities and describe the world as an ergodic universe. As both you and Keynes convincingly have argued, that is ontologically just not possible.
Lars —
your last sentence says it all. If you believe it is an ergodic system and epistomology is the only problem, then you should urge more transparency , better data collection, hiring more “quants” on Wall Street to generate “better” risk management computer problems, etc — and above all keep the government out of regulating financial markets — since all the government can do is foul up the outcome that the ergodic process is ready to deliver.
Long live Stiglitz and the call for transparency to end assymetric information — and permit all to know the epistromological solution for the ergodic process controlling the economy.
Or as Milton Friedman would say, those who make decisions “as if” they knew the ergodic stochastic process create an optimum market solution — while those who make mistakes in trying to figure out the ergodic process are. like the dinosaurs, doomed to fail and die off — leaving only the survival of the fitest for a free market economy. to prosper on. The proof is why all those 1% far cats CEO managers in the banking business receive such large salaries for their “correct” decisions involving financial assets —).
Alternatively, if the financial and economic system is non ergodic then there is a positive role for government to regulate what decision makers can do so as to prevent them from mass destruction of themselves and other innocent bystanders — and also for government to take positive action when the herd behavior of decision makers are causing the economy to run off the cliff.
So this distinction between ergodic and nonergodic is essential if we are to build institutional structures that make runing off the cliff almost impossible. — and for the government to be ready to take action when some innovative fool(s) discovers a way to get around institutional barriers and starts to run the economy off the cliff.
Paul Davidson
In his abstract of Book 1 of his Treatise, Hume says “It is sufficient, if I can make the learned world apprehend, that there is some difficulty in the case, and that whoever solves the difficulty must say something very new and extraordinary; as new as the difficulty itself”.
What I have been saying for many years is that what Shannon discovered about the workings of logic, and how he resolved Hume’s objections to woolly thinking about thought with his definition of information as in effect “news”, and how he enabled us to correct the corruption of non-ergodic “information” by ergodic “noise”, and how Wiener linked that to the correction of error in steering (c.f. “cybernetics”) or control, where probabilistic reasoning about the past is used to compensate for side-effects but advance information (as from a lookout) about a need to move sideways is used to avoid problems in the future: this “information science” is in fact something very new and extraordinary, and (taken with a physical understanding of the dynamics of communication and emotion), resolves the “difficulty” in Hume’s “mechanical” interpretation of human science.
Though economic horses seem loathe to drink of unfamiliar waters, they can no longer say they haven’t been taken to them.
Davidson writes: “Paul Samuelson [1969] has written that if economists hope to move economics from “the realm of history” into “the realm of science” they must impose the “ergodic hypothesis” on their theory. In other words Nobel Prize Winner Paul Samuelson has made the ergodic axiom the sine qua non for the scientific method in economics.”
Going to his source, it appears that Samuelson was describing (highly critically) his position when aged 20:
“Technically speaking, we theorists hoped not to introduce hysteresis phenomena into our model, as the Bible does when it says “We pass this way only once” and, in so saying, takes the subject out of the realm of science into the realm of genuine history.”
Samuelson then described models with ergodic properties as being oversimplified. This reads to me as if he acknowledges the importance of starting conditions and process. It may be hair-splitting, but his reference to the Bible is an example of hysteresis, not a definitive statement that all non-ergodic events. Consequently he may well see lessons to be learned from history. At least, he does not seem to be arguing for an acceptance of an ergodicity assumption.
The reasoning seems to have similarities with Lawson’s criticism of closed systems, but also can be likened to some criticisms of econometrics and the assumption of an underlying structure of which we have several observations. For example, E H Carr (2008, What is history? Harmondsworth: Penguin , p.171) talks of history being concerned with processes of change (i.e. changing structures) while economists “take cover” in econometrics.
To take a purely rhetorical perspective, and given that real world economic data (of all kinds) refer to the past, it could be argued that economists ARE historians. Moreover, we could consider how we would view historians if they insisted on trying to understand the past solely by means of ergodic frameworks. I suspect that we would criticise them for only looking under the lamp posts.
A delightful response, Stuart, especially that “looking under lamp-posts”!
I’m not sure what you mean by “all non-ergodic events.” I will assume you intended “all nonergodic events.”
I’ve come up with a different approach to Samuelson’s problem with hysteresis, which is to account for time in the logic (as in the time taken for computer logic to perform its operations) rather than in the universe of discourse. Thus the earth’s orbit represents the cyclically varying rate of change of the earth’s position, and the hysteresis the second differential: the rate of change of the rate of change, other things being equal. But on economic timescales what matters is that the earth stays in orbit, and taking time out of the
model by integrating time back into the model of position leaves the path as a near enough circuital model [NB topological loop rather than geometrical circle; c.f. the course rather than the position of a ship). The hysteresis then becomes an external rate of change of the model, similar to sideways positional drift when steering. When over time that has become big enough to matter, then this real world difference can be corrected (as Keynes’s proposals did by redirecting the circulation of money over time) or it will matter (as when correcting the model precipitately and acting according to the model causes us, as we are doing, to precipitately reduce the energy of the system to the lower level).
As David Hume pointed out, not only in our professional lives, but also in our private lives, we invariably introduce the assumption of the ‘uniformity of nature’ over time and over space, in order to reason by induction. The question of ergodicity or non-ergodicity is an old and fundamental one. Before Taleb, David Hume was writing about black swans – in 1739! And in contemporary statistical/econometric/machine learning literature, the question is addressed under the rubric of ‘overfitting’.
In short, attempts to confront the question can be found in the mainstream literature. For example,
Danielsson, J (2002) The emperor has no clothes: Limits to risk modelling, Journal of Banking & Finance, Volume 26, Issue 7, July 2002, Pages 1273-1296
http://www.sciencedirect.com/science/article/pii/S0378426602002637
DeMiguel, V, Garlappi, L, Uppal, R (2009) Optimal Versus Naive Diversification: How Inefficient is the 1/N Portfolio Strategy? Review of Financial Studies 22(5), pp. 1915–1953. http://rfs.oxfordjournals.org/content/22/5/1915.abstract
Danielsson, J (2011) Risk and Crises, http://www.voxeu.org/index.php?q=node/6118
An interesting article. To anyone with a background in the natural sciences, the very idea that something could be called a “science”, no matter how mathematically constructed and internally consistent, without meticulously deriving its claims from what actually happens in the real world is nonsense. Any discipline which simply asserts that something is the case without explaining how it has reached that conclusion and presenting its evidence should be ridiculed. Perhaps it is time that more thinking people from other disciplines held economics to account.
Paul, I would be really interested to know what, if any, defence has been given of the ergodic axiom in response to sceptical criticism along these lines.
On a minor point, I don’t think it is correct to say that probability analysis allows us to predict solar eclipses. The movements of the moon and Earth are governed by Newton’s laws of motion and are not probabilistic in nature.
Helen:
There has been no rerspose from the other side as to a defense of the erodic axiom. Of course, Paul Samuelson is dead and he can not respond. In writing Nobel Prize laureates John Hicks and Doug North has both endorsed my nonergodic view of economics. So has Bob Solow.
But when I try to get a response from the fresh water economists — I get none.
This may change. I am in the Chicago area and Jim Heckman is trying to get the Univ. of Chicago economists to invite me to give a seminar about ergodicity vs nonergodicity and Keynes’s theory of liquidity. So far, no invitation date has been forthcoming but If I get an invitation I will let people know the outcome of my going into the dragon’s den to tryto slay the dragon.
Paul Davidson
Best of luck, Paul. Helen and you might be amused by the practice c.1970 in the Royal Radar Establishment using Algol68 for database work. The database was initialised in such a way that if you didn’t actually put any data in part of it, when you accessed that bit the answer came up “FOOL”!
I hadn’t seen G R Steele defending Hume. What Hume was assuming, of course, was that induction tries to induce absolte truth rather than the best available hypothesis (all knowledge being subject to the possibility of error).
@ #15, apologies for a cock-up trying to work out what Stuart Birks was on about @14. Stuart had written: “not a definitive statement that all non-ergodic events.” I think I meant to assume he meant “about” rather than “that”.
Good to see Steve (@ #4) coming back to this important thread after such a long time.
“Science is a great and necessary too, but it must take a back seat to Wisdom, which is a higher way of thinking (philosophy) and acting (policy) … a condensation of the Good from long time human IMMERSION in that Life and Living”.
Steve, one of the lessons learned by Wisdom is that apparent motion can be ambiguous, as when one of two trains in a station starts and people on the other feel as if they were moving. Does the sky (and with it the sun) go round the earth, or the earth round the sun, turning on its axis? The followers of Aristotle believed for centuries that every motion needed a force to sustain it, but Newton was able to show that in frictionless space, a moving object will keep moving in its original direction unless caused not to. We see money used as if it is valuable, but isn’t it really of negative value: a mere record of debt where only apparently the banker is the creditor? (Wisdom in any case calls love of it the root of all evil). So is one immersed in reality, or in the APPEARANCES of reality?
Here the issue is whether knowledge of the past is a reliable guide to an uncertain future. If what probably happened in the past will equally probably determine the future, then Wisdom can change nothing. If not, then there is a role for wise government. But knowledge of what? Knowledge of what to do (which constrains us), or of what NOT to do, whatever it is we want to do? I seem to remember Wisdom proposing the latter, and certainly that’s what Shannon’s theory of communication and error-correction is all about.
Messages often have typos in them, but they are usually decipherable so long as there are relatively few of them, i.e. the “signal to noise” ratio is good enough. Vehicles stay on course only so long as you keep redirecting them as they start going off course.
See also my #8, #13 and #15; the other contributors don’t seem to have grasped this yet.
Here the issue is whether knowledge of the past is a reliable guide to an uncertain future. This is the historians territory because so much of what us taken for the “past” is very poor history.
We talk about experience and experience includes learning about what happened in the past. It can be a useful guide to decision-making in then present. When Bushn invaded Iraq he adopted a policy that gave resylts thgt were exactktthge oppsite f that intended. He brought the Shites to power in Iraq,who are natural allies ofnthe Shites in Iran. Any knowledg eo fhe religious and political history6 of th Middle East would haven cautionedn him against thye policy he adopted. We face thesame probleblem now in Iran. An axiom of statecraft is, drawn from experience, is never let a small power (Israel) determine the policy of a great power (USA). It is too risky for the world.
pardon mistakes on previous post. My set-up did not allow me to see the text I was typing.
Yes, Robert. Like Steve @ #4, you seem to be advocating wisdom! But what a pain this system is technically, with its non-expanding comment box and (one suspects) transmission errors adding to our unseen typos. Opening #20 by quoting Steve, I surely wrote “Science is a great and necessary tool” rather than some strange thing called a “too”!
i wonder why brain arthur (path dependence in econ) is not mentioned. arthur also said that marshall (in the ’30’s i think) more or less pointed out history dependent issues in econ (also in evidence when the full neo/classsical general equilibrium model (arrow etc.) is studied (eg sonnenchein ’72, mantel, debreu…). there are no general unique solutions, etc.
this situation is similar to newtonian mechanics (as is implied by saari’s mnodels. typically, newtonian mechanics is thought to be conservative, deterministic and essentially predictable (IE ergodic) but in practice trajectories are noncomputable often, or chaotic (effectively noncomputable). so a sample appears nonergodic (and since noone really can see the big picture (has the mind of god) all we see are samples. (whether there actually is an ergodic system out there which finite humans cannot appreciate is an open, metaphysical type question—-one can think of boltzmann, or keynes —in the long run, we are all dead, or for alternatives Tipler (physics of immortality).
my own view is essentially no matter what you do, one is assuming some ergodic process because any derivations are always tautologies, and form a closed system. but that is a definition. a chaotic attractor is still an equilibrium in this view, and i tend to think one can (following cantor) view even noncomputable evolutions as equilibria of some system with a new axiom of infinity (so your probability distribution sums to one). (but one might have to ask god about that, if one exists (god, probability distributions, and whether if you have a trajectory such as .99999… whether then one exists).
Interesting. If Brian Arthur was on about path dependence he was also on about positive rather than negative feedback, i.e. about increasing returns, effectively about making money by selling each copy of broadcastable information, as compound interest does. Given the size and dispositions of the major forces, Newtonian orbits are (in topology) predictably circuital even when (in algebraic geometry) the sideways deviation of individual trajectories is not. I think you capture the issue nicely when you assume “SOME ergodic process”, Ishi, but is the “process” referred to random errors in the measurement of (past) positions or in the systematic variations due to gravitational forces? Einstein believed “God does not play dice”.
If a chaotic attractor is an equilibrium, is an electric circuit a repulser of chaos?
as usual, i’m always on time—i like to respond to emails promptly. (of course, this is ‘ishi time’—sortuh like ‘ishi space’: you want directions? wave your hand and say ‘its over there somewhere, just follow the trails and don’t fall off the cliff’. ishi time is based on current ideas in cosmology, so that the ‘fundamental constants’ vary in time (eg http://www.arxiv.org/abs/hep-th/0208093). to be on time, set your watch to make sure its right. i know what time it is.)
as to whether god plays dice, that seems to be both an empirical and theoretical question. i haven’t seen god up the corner playing dice, so maybe einstein was right, unless i didnt recognize him or her. (don’t get shot at those dice games which happens). on a theoretical level, one can always look at fox news (or maybe bulletin of american math soc 1991), a paper by ornstein and weiss. many stochastic and deterrministic processes cannot be distinguished. (there is even a lit(t)erature on ‘schrodinger’s problem’ from way back which show or suggest U kant distinguish a stochastic process from a deterministic one. i’ve seen one economist cite ornstein and weiss, an italain. one can always look at gillespie showing schrodinger’s equation is not a markov process, or more recently its the square root of a markov process. seems irrational to me. ).
if you add a little noise to your dynamical system (ie turn it into a langevin eqution) you can get a system quasi-stationary state
only thing i know about electrical circuits is more or less Gabriel Kron (in Physical review in 1945 or so)—you can recast any dynamical system including schrodinger’s equation into the form of an electrical circuit system (people i knew did them for things like immunology or chemical reactions; its a cumbersome formalism—one of the mentors of that group got killed in a terror bombing –aharon katzir). this is all on wikipedia. (Kron has a paper on ‘non-reimannian geometry of rotating electrical machines’ wchich is on the right track, tho i prefer starting at a more microeconomic level, such as A Caticha on information geometry, which predates entropic gravity).
so after all, maybe the world is fat (uncle ‘tom fried, man’—NYT). or flat (using ishi geometry—-wherever you go, its always downhill if you’re walking or biking, except for life which is based on merit and hence grows exponentially, having stolen or reclaimed capitalism. )
—belly of the beast, inc. (dc—divided and conquered, damge control….)
I am relatively new to RWER and thus late to arrive at this thread in the “Minsky” archive, which seems not to have continued, and that is a shame. I like the thread because I’ve learned so much, critical thinking is employed, and the tension of disagreement is civilized.
I’m retired and elderly. I was trained in economics in one of the leading schools of neoclassical economics in the mid-60s, but never practiced the profession. I left in disgust, my mind muddled by a couple personal problems, combined with the seduction of activism in the streets, combined with divorcing myself from the seduction of the elegance of such as Cobb-Douglas production functions, Edgeworth Boxes, von Thunen location circles and IS-LM curves, I couldn’t find the handle to articulate my disgust, not clear at the time why I felt that way. I pursued a good career in several fields, always in the formulation and execution of public policy.
Today I read these three paragraphs from Paul Davidson’s article published in RWER Issue 59:
“The most important classical axiom Keynes eliminated in his general theory is the ergodic axiom.
“This ergodic axiom assumes the economic future is already predetermined. The economy is governed by an existing ergodic stochastic process. One merely has to calculate probability distributions regarding future prices and output to draw significant and reliable statistical inferences [information] about the future. Once self-interested decision makers have reliable information about the future, their actions on free markets will optimally allocate resources into those activities that will have the highest possible future returns thereby assuring global prosperity.
“Since drawing a sample from the future economic universe is impossible, the ergodic axiom presumes that the economic future is governed by an already existing unchanging ergodic stochastic process. Consequently, a sample drawn from the past is equivalent to a sample drawn from the future. In other words, calculating the probability distribution from past statistical data sample is presumed to be the same as calculating the risks from a sample drawn from the future.”
Adding the later comment showing the neoclassical logic of why the government is unnecessary, there it is, a tidy and articulate summary of the absurdity I felt in the mid-60s. I was a young man, working hard to begin a career in a field divorced from reality. No wonder! Thank you!
Can we keep this excellent thread going? It seems to me there are some loose ends here to try to tie up, even if it takes months and years to reach a point where, not necessarily striving to be a science, economics can present to ordinary people a coherent understanding of economic reality. Minsky, Davidson, and others have provided a great start.
And so I have a retirement hobby: joining the revolt against the neoclassical cover for the predatory corporate plutonomy.
Ergodic theory is interesting and somewhat basic or fundamental. (Fermi , pasta and ulam I think used the first computer after it was developed for atom bomb project, to test the ergodic theorem. It basically seemed to fail , and this was one origin of chaos theory, KAM, etc. I don’t know the status of the debate, but later people said actually the FPU expeiments did show ergodicity, but their computer did not have sufficient computational power to show this. This overlaps with James Yorke’s discssions of showing that real chaotic systems exist, and are not just illusions due to computer roundoff error .)
cosma shalizi’s blog Bactra or ‘three toed sloth’ has some good discussions of ergodic theory (though some of his research papers in physics/thermodynamics I don’t fund convicing or understand.)
I don’t really think this discussion of ergodic, versus something else—whatever it is, it seems never to be mentioned apart from ‘a role for government’—is that meaningful. The only issue is what kind of model you use—ie what language or dialect. Alot of things i read seem to say you can convert a stochastic process into a deterministic one or the reverse to ‘predict’ to some approximation the future. Or you can use words—‘tomorrow the sun will, or may, rise’. Most people seem to assume ergodicity for some time interval. For other intervals, you assume a different ergodic process. (In statistical thermodynamics, determinism arises from an ergodic process at zero temperature—the ‘deterministic limit’).
If my views are correct or partially so, it suggests to me that the English language is not a good way to talk about these issues. In fact, it biases the discussion (essentially the ‘linguistic relativity’ process of Sapir-Whorf.) By choice of vocabulary you are ‘choosing sides’. In pure general equilblrium theory (often seen as basis of capitalism or economic markets) at equilibrium, there is zero profit. So in that equilibrium state, its indistinguishable from communism. This top me is the point of the two fundemental theorems of GET (discussed by Paul Romer in an old paper , I think in J Econ Philosophy. Its also related if not equivalent to the SMD theorem) . You can sort of ‘prove anything’ and the economy can ‘do anything’—go to any equilibrium desired— just by changing your assumptions (or initial conditions). You throw a dart at a wall and hit any place (see old article if you throw a dart at the real number, can you hit a real number’).
A certain amount of current work in logic and math seems to basically modeling theorem proving as a stochastic process over some probabilkity distribution. (Tegmark’s ‘mathematical universe’ seems based on a similar idea for the physical universe.) Math equations sort of ‘compete’, and also consume and produce other math expressions. Some last because they are true (or at least last awhile until they prove false—-eg the various prime number polynomials, which seem to work for awhile (produce prime numbers as output) but then fail (produce composite numbers. ).
Probability distributions for income in general hold only for some range of incomes, and its basically impossible to tell whether the process that generates them is random (ergodic process) or instead deterministic (ie people operating according to marginal utility —though that can also be included in a random process using a ‘chemical potential’ —-ie people are not indistinguishable colliding particles, but instead have various ‘affinities’ and properties—but the calculation is analogous—-a bit like doing coin flips using pennies, nickels and dies, rather than just pennies.) . To model full income distribution usually you have to use a ‘mixture model’—several different probability distributions which are mixed (so you get some multimodal or other non-Gaussian, non-uniform distribution).
ps The last statement in OP states ‘reflextivity’ (a term I associate with G Soros) is the alternative. In math terms one models that with a nonlinear process like General Relativity —eg mass deytermines curvature of space, and curvature of space determines mass (a sort of godelian or self-referential statement). A nonlinear process is just a generalization of a linear one. Many physicists model things with Taylor’s series because they know the process is nonlinear. But commonly, for some application, they ‘truncate the series’ and drop all the nonlinear terms to get a linear process. In a sense this is like seeing everything is a nail because all you have is a hammer (under your streetlight).
In a letter to the editor published in the March 15-27 1997 issue of THE ECONOMIST magazine George Soros specifically states that his concept of reflexivity requires the rejection of the “ergodic axiom”
I just googled and found two papers with different formulations in mathematics of Soros’ reflexitivity concept—both apparently correct at a skim read .
Its exaclty as i said—Soros apparently only published 2 equations—a very simple variant of the lotka-volterra/Keene/Goodwin ‘class struggle model’.He doesn’t know ergodic theory (except maybe the 1930’s version–eg G Birkhoff, i think Weiner or Von Neumann also proved it, basically the same way Botzmann did (and the way you prove General equilibrium and similar results in biology—eg Fisher fundamental law of natural selection. First, you assume your conclusion. Then you start with some axioms, and derive your conclusion.
Although Gergescu_Roegescu when i read his ‘entropy law’ had some mathematical flaws, he got the correct conclusion. In standard GET there is no dynamics. The economy is always at an economy, a dollar is a dollar , and you will never find a dollar on the ground. P Samuelson via EMH also showed this. )
One of those papers on Soros’ equations is a determinstic model, the other a stochastic model (which leads to some sort of quasi-ergodic hypothesis). Soros has said he wasn’t good at math, but he was good at investing. Modern Ergodic theory (which i dought he has read—it comes from chaos theory) is a complex area, like investing.
I havent seen a mathematical formulation of reflexitivty, just conceptual discussions which indicate its a sort of learning procedure. (eg people learn from mistakes, or act in the world, which changes it, so they update or change their behavior. Genetic algorithms and neural nets sort of do this as do some glasses of Turing machines. Many of these can be placed in similar formalisms as dynamical systems (with a hamiltonian or conserved quantity analogous to energy). These can then be studied a la Botzmann/statistical mechanics using ergodic theorems. Often these are not strictly ergodic (they are non ergodic) but they switch between subsystems each of which is basically ergodic.
Its like a time series with breaks or transitions—it will appear apprximately stationary (eg an efficient market ) but then will have a discontuous transition to another regime with a different probability distribution. (I think Cosma Shalizi discusses this on his blog, but i forget where. He also mentioned you on it in at least one place, and discussed some technical objections to some of your uses of ergodicity. It would be interesting if these differences could be resolved to some degree of satisfaction).
Biological systems are not ergodic—they evolve (unless there is some sort of master equation of evolution noone knows). But they are commonly modeled using stochastic process, and the ergodic axiojm. ,
https://www.arxiv.org/abs/1608.04832
Yakovenko is another believer in ergodic axiom—saw him speak recently.
This is (or seems) so far from other economics theories/discussions i don’t know what to think. (and his approach has its own possibly severe limitations).It would be nice if there was one sortuh consensus theory around. (One could that the fact there isnt is good evidence that the world is non-ergodic—rocks arent trees, and economic theory doesnt equilibrate or mix.
“reflexitivty, just conceptual discussions which indicate its a sort of learning procedure. (eg people learn from mistakes, or act in the world, which changes it, so they update or change their behavior.”
this reminds me of something i studied a bit on my own during grad school in the 60s, called heuristic programming, which i thought then basically meant “learning by doing” in computer simulation talk
i think then it was a special sort of dynamic programming, an offshoot of operations research, but i don’t recall precisely and never practice it
wonder if this sort of thing has penetrated behavioral economics
https://en.wikipedia.org/wiki/Learning-by-doing_(economics)
thx
seems a neoclassical technical expression of how we all learn some of the stuff we know