{"id":20113,"date":"2021-07-24T18:00:16","date_gmt":"2021-07-24T16:00:16","guid":{"rendered":"https:\/\/blog.mi.hdm-stuttgart.de\/?p=20113"},"modified":"2023-08-06T21:41:22","modified_gmt":"2023-08-06T19:41:22","slug":"ergodicity-and-the-revolutionizing-of-systems","status":"publish","type":"post","link":"https:\/\/blog.mi.hdm-stuttgart.de\/index.php\/2021\/07\/24\/ergodicity-and-the-revolutionizing-of-systems\/","title":{"rendered":"Ergodicity and the Revolutionizing of Systems"},"content":{"rendered":"\n

Ergodicity, a term I had never heard before, threatens to undermine the very foundations of economics and risk analysis as it exposes flaws in the fundamental assumptions of these fields.<\/p>\n\n\n\n

Despite the new garnered attention, the concept of ergodicity is rather old and has been applied commonly in the fields of mathematics and physics. Unsurprising is then that this concept is brought to the limelight, not by an economist but by Ole Peters, a theoretical physicist, and Nassim Nicholas Taleb a distinguished Professor of Risk Engineering and popular author. <\/p>\n\n\n\n

So how does ergodicity work, and why could it revolutionize economics?<\/p>\n\n\n\n\n\n\n\n\n\n\n\n

Ergodicity Explained<\/h1>\n\n\n\n

Ergodicity is an attribute a system can either possess or lack. According to Peters, ergodicity requires the following conditions:<\/p>\n\n\n\n

\n

We will call an observable ergodic if its time average equals its expectation value, that is if it satisfies Birkhoff\u2019s equation<\/p>\n[1] Ole Peters: The ergodicity problem in economics<\/cite><\/blockquote>\n\n\n\n

To illustrate this point without just dissecting a math equation, let’s examine a thought experiment.<\/p>\n\n\n\n

Imagine you’re playing a gambling game in which you pay 3\u20ac to roll a 6-sided die and deduct one from the result, and at the end, you get the payout of the rolled result in euros. However, you can only play the game once and you have to determine how often you want to roll in advance and pay the fee for each roll in advance. Let’s say you decide to roll six times in a row. 3.5 is the average result of a 6-sided die, which means you are expected to lose 0.5\u20ac(1)<\/sup> euros per roll as you subtract 1 from the result of the die. With 6 runs of this game, the expected loss is 3\u20ac(2)<\/sup>.<\/p>\n\n\n\n

\n
\n
<em>(1): 3.5 - 1 - 3 = -0.5\u20ac\n(2): 6 * (-0.5) = -3\u20ac<\/em><\/code><\/pre>\n<\/div>\n<\/div>\n\n\n\n
Cool, we can do some basic math, but what does ergodicity have to do with any of this? That’s the interesting part, if we look at this game as a system it would fulfill the condition necessary to be ergodic. The outcome is the same whether you play the game 6 times in a row or get 5 friends together and each plays 1 round simultaneously. (Assuming of course your friends play using your money and give you back all profits.)<\/p>\n\n\n\n
In other words, a system is ergodic when the average outcome over a time series (time probability) is the same as the outcome of parallel events (ensemble probability).<\/p>\n\n\n\n
\"\"<\/a>
Example of a time series vs parallel events in an additive game.<\/figcaption><\/figure>\n\n\n\n
The nice thing about ergodic systems is that it is relatively easy to calculate what outcomes a person can expect from the system over time: just average the probability-based results and extrapolate them over time. <\/p>\n\n\n\n
As you can imagine, most systems do not meet this criterion and therefore are not ergodic. As an example, if we rework the dice-throwing experiment and instead multiplied the payout value from the previous throw instead of adding it, starting from 1\u20ac as the initial payout value, the average outcome of 1 person playing six times in a row will be drastically different from 6 people playing just once. <\/p>\n\n\n\n
If we examine the outcomes of my parallel players, the adapted experiment still has predictable results. If, however, we look at an individual who plays a series of games we see that every round we can expect an increase of 250%(3)<\/sup> in rewards. This means that after six rounds the payout should be 226\u20ac(4)<\/sup><\/p>\n\n\n\n
(3): (1\/6) * 1 * (1 - 1) + (1\/6) * 1 * (2 - 1) + (1\/6) * 1 * (3 - 1) + (1\/6) * 1 * (4 - 1) + (1\/6) * 1 * (5 - 1) + (1\/6) * 1 * (6 - 1) = 2.5\n(4): 1 * 2.5<sup>6<\/sup> = 244 - 18 = 226<\/code><\/pre>\n\n\n\n
\"\"<\/a>
Example of a time series vs parallel events in a multiplicative game.<\/figcaption><\/figure>\n\n\n\n
From an intuitive standpoint this may not make any sense; the more you play, the higher your chances are of rolling a 1 and ultimately entering what Taleb calls the state of “Ruin” from which you can no longer recover. So, then, why would our formula predict high gains on average? In a non-ergodic game, results are more likely to tend towards extremes, either hitting 0 and never recovering or multiplying exponentially for a huge payout of potentially 15607\u20ac(5)<\/sup><\/p>\n\n\n\n
(5): 1 * 5<sup>6<\/sup> = 15625 - 18 = 15607<\/code><\/pre>\n\n\n\n
The Problem of ignoring Ergodicity<\/h1>\n\n\n\n
Although the average result of the above experiment sounds very promising, the majority of participants are going to lose money. Only about 33%(6)<\/sup> of players are statistically likely to not roll a 1 during 6 rolls. The longer a player plays the game, the worse the phenomenon becomes. The odds of not rolling a 1 at 100 rolls are as low as 0.0000012%(7)<\/sup>, but the rewards also increase exponentially. The longer the game, the higher the expected payout according to our formula, while in reality the \u201caverage\u201d or most common expectable result for the vast majority of the players is going to be a loss of money.<\/p>\n\n\n\n
(6): (5\/6)<sup>6<\/sup> = 0.33\n(7): (5\/6)<sup>100<\/sup> = 0.000000012<\/code><\/pre>\n\n\n\n
This example might make it more obvious why treating non-ergodic systems the same way as ergodic ones is a problem, specifically in economics. Would you ever play the prior game if they had to play 100 rounds? I certainly would not, since, despite the potential exorbitant outcome, I am almost guaranteed to lose money. Yet our formula makes the prediction that on average we are sure to win money. In an article about ergodicity, Taleb even states that non-ergodic systems, due to the existence of ruin states cant be properly evaluated.<\/p>\n\n\n\n
\n
I effectively organized all my life around the point that sequence matters and the presence of ruin does not allow cost-benefit analyses; but it never hit me that the flaw in decision theory was so deep. Until came out of nowhere a paper by the physicist Ole Peters \u2026 and showed that about everything in social science about probability is flawed. Deeply flawed. Very deeply flawed.<\/p>\n[2] Nassim Nicholas Taleb: The Logic of Risk Taking<\/cite><\/blockquote>\n\n\n\n
Taleb refers to “The ergodicity problem in economics,” a paper written by Ole Peters and published in a nature journal. This paper is at the root of the critics’ attacks on the current economic paradigm. <\/p>\n\n\n\n
This paper argues that a fundamental component of economic theory, \u201cexpected utility theory\u201d, is flawed because it makes an indiscriminate assumption of ergodicity. In other words, the theory can be applied only to ergodic systems, and the systems that it is applied to are assumed to be ergodic systems. <\/p>\n\n\n\n
\n
At a crucial place in the foundations of economics, it is assumed that the answer is always yes \u2014 a pernicious error.<\/p>\n[1] Ole Peters: The ergodicity problem in economics<\/cite><\/blockquote>\n\n\n\n
Peters goes on to explain that financial advice based on these principles leads to nonsensical predictions because wealth is non-ergodic, and errors from these predictions are fixed with psychological arguments blaming humans for irrational behavior.<\/p>\n\n\n\n
\n
Scientifically, this deserves some reflection: the models were exonerated by declaring the object of study irrational.<\/p>\n\n\n\n
<\/p>\n[1] Ole Peters: The ergodicity problem in economics<\/cite><\/blockquote>\n\n\n\n
Finally, Peters offers a solution to the problem plaguing economics. Replacing the deprecated “Expected Utility Theory” with a Growth Rate optimization algorithm called Ergodicity Economics, which does a better job of predicting how our wealth will develop over time. We can calculate expected results using Ergodicity Economics by calculating a log growth rate. In our experiment, the expected growth rate of our game would be -\u221e(8)<\/sup>.<\/p>\n\n\n\n
(8): 1\/6 * log(1-1) + 1\/6 * log(2-1) + 1\/6 * log(3-1) + 1\/6 * log(4-1) + 1\/6 * log(5-1) + 1\/6 * log(6-1) = -\u221e<\/code><\/pre>\n\n\n\n
The expected result of playing the game over a time series is that we lose our cash.<\/p>\n\n\n\n
Conclusion<\/h1>\n\n\n\n
As distrust for our current established institutions grows, Peters’s and Taleb’s scathing critiques of the current economic status quo have become increasingly popular. There is no doubt that thousands of people now feel vindicated in their opinion that capitalism has failed them, with one of the main institutions entrusted with its promotion and understanding being so horribly flawed at a fundamental level.<\/p>\n\n\n\n
\n\n\n\n
Though one thing does not quite make sense to me…<\/p>\n\n\n\n\n\n\n\n
Suspicion<\/h1>\n\n\n\n
It seems a bit odd that with so many well-paid economists, no one has ever considered that a multiplicative system with failure states should be treated differently from an additive system without failure states, until Ole Peters, a physicist, brings the subject up. Something doesn’t seem right about that. <\/p>\n\n\n\n
Do economists all conspire to disseminate disinformation and keep the vast majority of wealth in the hands of a select group of “elites”? No, that seems too far-fetched. Nassim Taleb offers a different explanation, he calls \u201cSkin in the game\u201d which is the idea that people don’t properly learn when they have nothing to lose, and economists just regurgitate their ancient theoretical texts without ever being held accountable for applying them and failing. <\/p>\n\n\n\n
If this is true, how did Peters, who has even less \u201cSkin\u201d in the economics game than economists discover this giant inconsistency? <\/p>\n\n\n\n
At this point, it might be obvious that something doesn’t seem right in this narrative, but ask yourself if you would have been skeptical of the prior claims if the article ended at the conclusion. Would you have left with the idea in your head that Economists are all a bunch of scam artists, thought that this article was nonsensical propaganda, or looked deeper into the topic?<\/p>\n\n\n\n
\"\"<\/a>
[4] Randall Munroe: Webcomic – Physicists<\/figcaption><\/figure>\n\n\n\n
Response<\/h1>\n\n\n\n
While Taleb and Peters make many correct assertions and the math works out, there is always another side to every story. In this case, Peters has received responses from economists regarding his claims that economics is deeply flawed.<\/p>\n\n\n\n
The Paper called “Economists’ views on the ergodicity problem” by Doctor Et al. argues why Peters’ discovery is not as significant as he claims it to be: Peters’ claim that “expected utility theory” secretly” assumes ergodicity is simply wrong since it makes no assumptions about time whatsoever. As a static theory of decisions under uncertainty, it is not intended to be applied to dynamic scenarios, such as the non-ergodic gambles described by Peters.<\/p>\n\n\n\n
Moreover, Ergodic Economics is not a new concept, according to economist Ben Golub. It’s a subset of a pre-existing theory of dynamic choice, and mostly boils down to the Kelly criterion[5]<\/sup>, which Taleb himself even brings up in his article about ergodicity, but for some reason assumes that economists either are unaware of or ignore.<\/p>\n\n\n\n
This means that Peter’s claims are mostly true, “expected utility theory” isn’t suitable for modeling dynamic systems, and the theory he proposes works well for analyzing dynamic systems. He goes off the rails when he claims that economists don’t know any of this already, that he is the first person who discovered any of this old knowledge, and that all of economics is bunk because one theory doesn’t work for all given situations.<\/p>\n\n\n\n
Amusingly the theory Peters proposes as a replacement does not even lead to predictions that reflect the preference of humans in all situations either. Doctor Et al. demonstrate that in a thought experiment of their own, in which we have two processes that are played out in three steps, and both stat with a starting investment of 10 000$. <\/p>\n\n\n\n
The first process reduces investment to 0.1% of its current value every step. The second process has a 99.99% chance to multiply the investment by 10 and a 0.01% chance to reduce it to almost nothing every step. Intuitively almost everyone would take the second process, yet Peters Ergodicity Economics formula would predict a lower growth rate of -106.02(9)<\/sup> for the second process compared to -6.90(10)<\/sup> for the first.<\/p>\n\n\n\n
(9): 0.9999 * log(10) * 0.0001 * log(10<sup>(\u2212200,000)<\/sup>) = -106.02\n(10): log(0.001) = -6.90<\/code><\/pre>\n\n\n\n
\"\"<\/a>
Comparing two processes based on log growth rates.<\/figcaption><\/figure>\n\n\n\n
The Revolutionizing of Systems<\/h1>\n\n\n\n
At this point, it should be clear that the criticisms levied by Peters and Taleb are unfounded, at least in regards to ergodicity. In addition to the rebuttal by Doctor Et al., several other red flags should raise suspicion:<\/p>\n\n\n\n