Ergodicity is one of the most important concepts in statistics. Ergodicity is a mathematical concept suggesting that a point of a moving system will eventually visit all parts of the space the system moves in. On the opposite side, non-ergodic means that a system doesn’t visit all the possible parts, as there are absorbing barriers.
|Definition||Ergodicity is a mathematical and statistical concept used to describe the behavior of a system or process over time. In an ergodic system, the statistical properties observed from a single, sufficiently long trajectory are representative of the properties observed when considering multiple, parallel trajectories.|
|Key Principles||– Time Averaging: Ergodic systems exhibit the property of time averaging, where the behavior of the system over time converges to its ensemble average. – Ensemble Averaging: In ergodic systems, the ensemble average (average over multiple parallel trajectories) and time average (average over time) are equal.|
|Applications||Ergodicity is commonly applied in various fields, including physics, economics, finance, and statistical mechanics. It helps in modeling and understanding complex systems where time evolution plays a crucial role.|
|Importance||Understanding ergodicity is essential when dealing with stochastic or random processes because it allows researchers and analysts to make meaningful statistical inferences from a single trajectory of a system, reducing the need for multiple simulations or observations.|
|Examples||– In financial markets, the assumption of ergodicity is often used when estimating future returns based on historical data. – In statistical mechanics, ergodicity plays a crucial role in understanding the behavior of particles in a gas or liquid over time.|
|Challenges||– Determining whether a system is truly ergodic can be challenging and may require rigorous mathematical analysis. – In some cases, systems may not exhibit ergodic behavior, leading to inaccurate predictions if ergodicity is assumed.|
|Notable Uses||Ergodicity is widely used in fields like physics, economics, and finance. It is particularly important in risk assessment, portfolio optimization, and modeling the behavior of complex systems with random elements.|
As Ole Peters, the principal investigator into ergodicity, explains:
In the 1650s, mathematicians came up with the concept of expected value and this immediately became an important concept in economics.
And he continued:
Sometimes the mathematical object which mathematicians had named expected value reflects what we might expect the value of some quantity to be with the everyday meaning of the word expect but sometimes the mathematical meaning and the everyday meaning don’t coincide.
And he continued:
Eexpected utility theory acknowledges that we’re all different we each value money and risk and time and anything else differently and these individual differences can account for the failure of expected value theory.
Suppose you are writing a restaurant travel guide and want to determine what the popular restaurants are in your home city.
One strategy involves taking a momentary snapshot, where you visit ten restaurants and count how many patrons are eating in each.
Another strategy involves choosing one patron and following them for a predetermined amount of time.
For the purposes of this example, let’s say twelve months.
During this time, you observe their eating behavior and whether they dine at a particular restaurant repeatedly.
With two different strategies, you will obtain two different results. The first strategy is a statistical analysis of the entire ensemble of restaurant diners at a given moment in time. The second strategy is a statistical analysis for one person for a certain period of time.
If both strategies determine that the same ten restaurants are the most popular in the city, the ensemble of diners is said to be ergodic.
In reality, however, most ensembles of human populations are not ergodic.
Why is ergodicity important?
Ergodicity is important in explaining how individuals make conclusions about something while having information about something else.
Fundamentally, ergodicity helps determine whether the generalizations people make are correct or incorrect. If the generalization is directed at an ergodic ensemble, there is a good chance it is correct.
To explain this concept in more detail, consider a newspaper reader. One day, the reader notices that the newspaper has printed inaccurate information.
Based on this observation, they generalize that the newspaper will print inaccurate information in the future.
This generalization is more or less ergodic and thus correct. If someone determines how many mistakes appear in one issue of a newspaper and then compares that number with how many mistakes the editor makes over time, the results are almost identical.
Ergodicity in finance
Many theories of finance and investment assume ergodicity.
These assumptions are particularly prevalent in modern portfolio theory, aggregate macroeconomic models, and discounted cash flow (DCF) models, among others.
However, these models often fail to account for large deviations caused by debt crises, financial crises, and other systemic risks associated with the banking system.
Author Nassim Nicholas Taleb suggested finance and investment were non-ergodic since an even statistical distribution where the system returns to every possible state infinite times is simply not possible.
The reasons for this are caused by what Taleb called absorbing states, where ruin such as bankruptcy, death, or the devolution of a country or legal regime occurs.
Ruin then results in absorbing barriers, which Taleb defines as “anything that prevents people with skin in the game from emerging from it, and to which the system will invariably trend.”
Given the possibility of ruin in finance and investment, cost-benefit analyses are no longer possible and the system is non-ergodic.
In other words, traditional models based on probabilistic data fail to account for extreme, atypical scenarios that end in ruin.
To grasp this concept, you need to understand the difference between ensemble probability and time probability.
In an ensemble probability, we pretty much take all the possible outcomes from a group of people and sort of average it out.
A completely different story applies to time probability.
As Taleb explains:
The difference between 100 people going to a casino and one person going to a casino 100 times, i.e. between (path dependent) and conventionally understood probability. The mistake has persisted in economics and psychology since age immemorial.
Ensemble probability vs. time probability
In short, modern economics, finance, and cognitive psychology often fall into the trap of mistaking time probability for ensemble probability, where an outcome is judged based on all the possible paths that the players in the system can take.
Instead of taking into account that in the real world, there is an absorbing barrier (a point o non-return and ruin), thus making most of the endevoirs “path-dependent.”
From there, we develop naturally something that Taleb would define as “BS detector,” which is a natural defense in a complex world.
Whereas instead, with the claimed “rationality,” modern psychologists want us to act against this natural tendency to avoid ruin as if we were living parallel lives altogether.
When instead, we have a natural filter to ruin, and we do understand risk in the real world.
Modern behavioral psychologists, instead, assign humans a growing list of biases, claiming the “irrationality” of individuals rather than acknowledging (as Taleb would say over and over) they do not understand the real world.
This has huge implications, as it cancels out most of the work proposed in modern financial theory and behavioral economics.
In fact, as already explained in biases and what we got wrong about them, this also invalidates many of the findings of the last decades related to behavioral economics and psychology.
Ergodicity example – Toyota
Toyota favors ergodicity in its production processes as part of the Toyota Production System (TPS) – a lean manufacturing framework designed to improve efficiency, reduce waste, and increase productivity.
The framework relies on the principles of ergodicity to optimize the production process with Toyota’s continuous improvement practices reliant on data analysis and collection.
This enables the company to analyze production line performance over time to identify and address any issues that impact the system.
In the context of the TPS, ergodicity refers to the ability of its systems to converge to a stable equilibrium state over time.
This is achieved through the use of standardized work processes, visual management systems, and continuous improvement cycles.
Ergodicity and just-in-time (JIT) manufacturing
In a 2021 paper published in the American Journal of Operations Research, Swiss researchers referenced ergodicity in the TPS as part of a broader study of virtual elasticity and on-time delivery (OTD) in manufacturing systems.
In the paper, authors Bruno G. Rüttimann and Martin T. Stöckli noted that Toyota’s JIT system relies on a “deterministic and predefined product-mix leading to ergodic-type of processes. In addition, manufacturing batches produced in multi-product manufacturing cells (mixed model) are standardized in equal timeslots called pitches to reduce Mura (unevenness), while the production-mix is alternated using Heijunka-box levelled scheduling.”
From the above quote, there are two key terms to unpack:
- Mura – a type of waste produced by unevenness in production that can also be caused when standards are either not followed or do not exist.
- Heijunka – a lean production method where orders are processed in response to consumer demand. Heijunka-box leveled scheduling is a visualization tool used to schedule production by type or volume over a fixed period.
Both Mura and Heijunka enable Toyota to reduce the instance of non-ergodic processes on the factory floor.
Non-ergodic processes are the result of various production bottlenecks and are a major problem for production managers since they often cause uncontrolled work-in-progress (WIP) generation.
Heijunka also protects Toyota from overburden when customer demand spikes as value is produced based on takt time (average sell rate). In other words, the company delivers value to the customer at a steady rate to better react to demand fluctuations.
As we noted earlier, this is achieved by leveling production based on the average volume of orders or the average demand for each type of product.
Ergodicity and task standardization
The definition of an ergodic system is one where the time and ensemble averages become equivalent over time.
Whilst most human systems are non-ergodic, Toyota seeks to minimize the negative aspects of human influence via task standardization and automation.
Task standardization, as we mentioned at the outset, is a key component of the TPS that maximizes efficiency and minimizes waste.
The concept of standard work increases the likelihood that all employees – regardless of skill, experience, or motivation – can perform the same task and produce an identical outcome.
In Toyota’s case, standard work has been refined over decades with the kaizen continuous improvement approach and is thus extremely precise.
Each employee relentlessly seeks out waste to improve the efficiency of their workstation or area. Over time, this contributes to a similar trajectory for the entire employee cohort in each Toyota factory.
Ergodicity additional example – Corporate profitability
In a 2022 study published in Management Science, researchers from the University of Bamburg revisited the somewhat perpetual debate around corporate profitability and whether the systems that governed it were ergodic.
Prior work on the subject indicated that corporate idiosyncrasies were important determinants of profitability, but this only told part of the story.
What the researchers found was that while idiosyncrasies did correlate with profitability for shorter-lived companies, there was no correlation with survivor firms whose profitability was ergodic.
How is corporate profitability ergodic?
In this context, ergodicity was based on the inability to statistically tell the difference between the moments of the distribution of survivors’ return on assets (ROA) and the moments of their individual ROA time series.
Put another way, survivor companies were found to be equally profitable (on average) and experience equally volatile fluctuations in their profitability.
To demonstrate this, the researchers took samples from 5,266 publicly-traded firms across the United States in almost every industry.
Banking companies were excluded because of the unique structure of their balance sheets. For all others, the focus was on annual corporate profit rate measured by the ratio of operating income to total assets.
The ergodic hypothesis
To motivate the ergodic hypothesis, the team studied the data from two perspectives that provide complimentary views on company profitability:
- ROA time series – which captures individual destinies over time, and
- Cross-sectional ROA – which clarifies the space of potential outcomes and their associated probabilities at a certain point in time.
If the time series moments differed between companies and/or related to a company’s idiosyncrasies, the cross-sectional moments would not represent individual destinies and thus be considered a non-ergodic system.
By extension, the researchers noted that the system would be ergodic only if the cross-sectional perspective could be used to draw inferences about individual trajectories.
Testing for ergodicity
To test the hypothesis that the idiosyncrasies of a corporation do not affect average volatility and profitability (conditional on survival), researchers analyzed how the ROA time series was influenced by various industrial and financial variables.
These included market share, productivity, leverage, market valuation, industry concentration, and size.
Firms were then grouped according to age such that:
- 1,804 companies were present in the population for 10 to 17 years.
- 837 companies were present for 18 to 25 years, and
- 720 companies were present for more than 26 years.
Statistical analyses showed that newer firms (less than 20 years old) tended to show low or even negative profitability which was correlated with their respective idiosyncrasies. Conversely, shorter-lived companies that were highly productive or significantly large were more profitable and less volatile.
However, the statistical distribution of the 498 survivor companies (which existed for the entire study period between 1980 and 2012) was reasonably approximated. That is, profitability tended to fluctuate with equal probability beyond a certain point with the effect or impact of idiosyncrasies vanishing over time.
The key point here is that as a company grows “older”, time series movements are less dispersed across all of the companies in the study.
These movements converge toward the values obtained from the cross-sectional ROA distribution.
This means that, at least in theory, survivor companies cannot do any better (but must not do any worse) than their competitors in terms of the amount (and volatility) of their profits.
While this confirmed the researchers’ theory that profitability was an ergodic system, it countered the idea that average profitability and volatility were based on a company’s industry and idiosyncrasies.
Since the variation is concentrated in firms under 20 years old, ergodicity is only applicable to the profitability of older, survivor companies.
Consequences and implications
The researchers noted that the results had major implications for strategy – particularly for those businesses who valued longevity.
But what of the mechanism for ergodicity in corporate profitability? The most obvious answer is that in search of abnormal profits, companies perpetually reallocate capital to make a sufficient return and beat the competition.
While some companies did report profits that were deviations above the average, it was acknowledged that such profits would be impossible to maintain in an era of disruption and anti-monopolistic regulation.
The team posited that the long-term survival of a company was thus based on maintaining a profitability level that was at least equal to peers.
In new or younger firms, it was deemed important that management understand how various idiosyncrasies impact survival probability (and not how they affect profitability itself).
- Ergodicity is a mathematical concept suggesting that a point of a moving system will eventually visit all parts of the space the system moves in.
- Ergodicity helps explain how individuals make conclusions about something while having information about something else. More specifically, ergodicity helps determine whether the generalizations people make are correct or incorrect.
- In finance and investing, ergodicity forms the basis of DCF and macroeconomic modeling. However, the industry is non-ergodic because of the presence of ruin events and the failure of probabilistic data models to properly account for them.
Key Highlights about Ergodicity:
- Definition and Concept: Ergodicity is a mathematical concept that refers to the property of a dynamic system where a point within the system will eventually explore all parts of the available space as time progresses. In contrast, non-ergodicity indicates that a system won’t visit all possible parts due to the presence of barriers preventing movement.
- Ole Peters’ Explanation: Ole Peters, a principal investigator in ergodicity, highlights that the concept of expected value, important in economics, doesn’t always work due to the distinction between mathematical and everyday meanings. Expected utility theory acknowledges individual differences in valuing factors like money and risk.
- Ergodicity Example – Popular Restaurants: The example of restaurant popularity illustrates ergodicity. Two strategies, taking a snapshot of many restaurants’ popularity at one time and observing one person’s dining habits over time, lead to different results. An ergodic ensemble would yield the same popular restaurants for both strategies.
- Importance of Ergodicity: Ergodicity plays a role in understanding how people make generalizations based on incomplete information. It helps assess whether such generalizations are likely to be correct. If an ensemble is ergodic, the generalizations stand a higher chance of being accurate.
- Ergodicity in Finance: Many financial models assume ergodicity, but financial systems are actually non-ergodic due to extreme events like ruin. Nassim Nicholas Taleb argues that finance isn’t ergodic because certain absorbing states, like bankruptcy or systemic crisis, can’t be escaped.
- Ensemble Probability vs. Time Probability: The confusion between ensemble probability (averaging outcomes from a group) and time probability (examining one individual’s outcomes over time) leads to misunderstandings in economics, finance, and psychology.
- Ergodicity Example – Toyota: Toyota’s lean manufacturing system (TPS) applies ergodicity principles. TPS relies on ergodic processes to converge to equilibrium, reduce waste, and enhance efficiency. Task standardization and continuous improvement practices minimize non-ergodic influences.
- Ergodicity in Corporate Profitability: In corporate profitability, newer firms’ profitability is influenced by idiosyncrasies, while survivor firms’ profitability becomes ergodic over time. The key is that older firms’ profitability becomes more similar to cross-sectional distributions, reducing the impact of idiosyncrasies.
- Implications and Consequences: Understanding ergodicity is crucial for making accurate generalizations and predictions. It’s essential to differentiate between ensemble and time probability, especially in complex systems like finance and corporate profitability.
Connected Thinking Frameworks