By: Ole Peters and Alexander Adamou

In: Lecture notes and blog posts

Date: 2019

Introduction

The lecture notes and blog posts introduce and develop ergodicity theory, in the process challenging the foundations of economic theory, as well as providing mathematical explanations for inequality, cooperation and how financial markets behave.

The math at many places is well beyond me, but some of the examples are simple enough to model and explore in Excel as an alternative route to understanding the implications.

It would be great if over time a more user friendly version emerges with simpler formulas (if possible), more applications, a broader discussion of its implications and/or more explicit directions on how and where to apply ergodicity theory.

Key Takeaways

• To understand the properties of a system you can measure:
  • One performance in time (averaging one continuing sample).
  • Group performance (averaging multiple simultaneous samples).
• In so-called ergodic systems, both of these measurements are the same.
  • One individual, sufficiently long, sample is equal to the group average.
  • Each individual explores all possible states in a short enough time (for example, time series of individual coin flips).
• But, many systems we care about are non-ergodic:
  • What happens on average is not representative of what happens to one person over time.
  • The group average differs systematically from the individual’s performance.
• This is driven by the rate of change in these systems.
  • Change is not mechanical, there is not necessarily an equilibrium state that is reached over time.
  • Instead of looking for a group average, you should study the transitions, the dynamics, the way things change in these systems.
• This has implications for our decision making in these situations:
  • Don’t rely on the group average: your individual experience is likely to be very different.
  • This is important because you may only have one shot (you can’t have multiple tries or average out your multiple experiences).
• This happens a lot.
  • We live in a world where many things we care about change non-mechanistically.
  • Many important “gambles” involve things that “multiply” rather than simply “add up”.
• Specifically, for bets involving multiplicative situations.
  • The group average often outperforms many individual performances.
  • The group average is skewed upwards by the extreme out-performance of the few.
• This multiplicative character of the world helps to explain mathematically:
  • Cooperation and formation of social structure
    • The long-term benefits of cooperation tend to outweigh the short-term gains of breaking it.
  • Inequality and immobility.
    • Tendency for vast majority of bets to end up below average performance.
  • Risk-aversion.
    • Individual behavior is influenced by the situation, not the idiosyncrasies of the decision maker.
• In short:
  • You live one life, not multiple lives in parallel universes.
    • You want to know what is likely to happen to you, not to the average person.
    • I don’t care what happens on average, what happens if I keep playing the “game”.
    • For instance, as you keep playing your “investment game”, is your wealth likely to grow or shrink?
  • You intuitively behave differently depending on the game you are playing.
    • Biases such as risk-aversion reflect a rational avoidance and intuitive understanding of non-ergodic risks.
    • Russian roulette may on average not be incredibly risky.
    • But you “know” that you don’t want to play it indefinitely.

Ergodic and non-ergodic processes

• Ergodic process:
  • At each moment in time, value is a new instance of a random variable.
  • An equilibrium state exists.
  • The end state is (almost) independent of the initial state.
  • Example: time series of individual coin tosses.
• Non-ergodic process:
  • Over time, values either becomes infinitely small or large.
  • There is no equilibrium state.
  • Example: nuclear explosion, development of wealth.

 Decision theory: choosing the best gamble

• Common strategy:
  • Evaluate multiple outcomes.
    • Look what happens “on average” over many samples.
  • Eliminate random “noise”.
  • Calculate average: ensemble average.
    • Simple average.
  • Calculate expectation value.
    • Probability weighted average.
  • This is not what happens in real-life.
    • It assumes that you can play the game many times in parallel.
  • In real life, you can only play the game once.
    • Serially.
    • Need to know the time average.
  • So there are two “types” of averages.
    • Ensemble average.
    • Time average.
  • They are not always (and typically not) the same.
  • And two “types” of growth rates.
    • Ensemble averaged growth rate.
    • Time averaged growth rate.
  • These growth rates can also be different.
  • Fundamental driver of the dynamics of inequality.
    • Group average can grow, yet many individuals don’t experience growth.
  • Mistakes are made when you apply the “wrong” average.
    • Making decisions based on ensemble averages and ignoring time average.
      • The average group performance (many samples) differs systematically from the individual experience (one sample).
      • The average group performance tends to outperform individual performance.
      • The group average is skewed upwards by extreme (out-)performance of a small number of individual performances.
    • Example – gamble:
      • 50% chance of a 40% loss.
      • 50% chance of a 50% gain.
    • Expected value:
      • 1.05 -> (0.5*0.6 + 0.5*1.5 = 1.05).
      • So, this is a good gamble.
      • Evidenced by average wealth accumulated over time (N = 5,000):
    • Time average growth rate
      • 0.94 -> ((0.4*0.6) ^.5)
      • This is not a good gamble.
      • Evidenced by the declining ratio of people that are able to maintain their wealth above the starting position over time (N = 5,000).

Conventional economic analysis

• Is too “ergodic”.
  • Assumes equilibriums exist.
• Mostly built on statistical mechanics.
  • Macroscopic properties derived from microscopic interactions.
• But: the economy grows, changes, and is a system mostly away from equilibrium.
• Misinterprets the role of time.
  • The role of time is very different in equilibrium vs. non-equilibrium situations.
    • Equilibrium: wait, allow for things to reach their end state.
    • Non-equilibrium: analyze the transitions, don’t assume there is an end state.
      • Dynamics are important: what happens when time passes.
• We have been trained (incorrectly) to start with expectation values.
  • Calculate average most likely outcome over many samples.
• But, ensemble averages are irrelevant.
  • You do not live your life as an ensemble of many yous who can average over their wealths.
• Also, people don’t evaluate gambles based on the expectation value.
  • Things are more complicated.
  • Different people do different things.
• Fix: introduction of “utility theory”:
  • Transform general expected value into individual expected utility.
• The value to an individual of a possible change in wealth depends on:
  • How much wealth he already has.
  • Psychological attitude to taking risks.
• Still relies on using expected value (ensemble average) for individuals.
  • Assumes that “humans optimize utility across the ensemble.”
  • Assumes risk preference is an idiosyncratic, individual thing.
• Corrects for this by adjusting for the individual’s arbitrary “risk-apetite”.
  • In many cases, by luck or otherwise, this correction is in the form of a log function.
  • Correct for a multiplicative world (ie, a “living”, multiplying world) in which we evolved.
• Behavior is determined by idiosyncrasy.
  • Draws attention to the psychology of the decision maker.
• Ultimate reliance on personal preferences is a theme that runs through all of economic theory.
  • Relies concept of additive repetition (mistaken).

Ergodicity economics

• Does not rely on expectation value (ensemble).
  • What happens on average is not necessarily relevant to what happens to a typical individual over time.
• Focus instead on the time-average growth (individual).
  • Simple theory of growth rate maximization.
• Humans optimize wealth over time.
  • Understand the dynamics of growth (additive, exponential, something else).
  • Derive appropriate growth rate of wealth.
  • Make the decision that optimizes your wealth.
• No adjustment needed for individual risk appetites.
  • Use time-averaged growth rate of wealth instead.
• Behavior is determined by the dynamics of the situation.
  • Draws attention to the situation of the decision maker.
• Relies on averages over time not over parallel worlds.
  • Relies on concept of multiplicative repetition.
  • Unless ensemble average = time average.
    • A single path when followed for a sufficiently long time will explore all there is to explore and reflect what happens in an ensemble.
    • Or when the quantity of interest really is an average of multiple (many) independent systems.
• Multiplicative growth analysis can help explain:
  • Contracts.
  • Inequality.
  • Cooperation and formation of social structure.
  • Behavior of financial markets.

Contracts

• In the expected value world, contracts can only arise if parties have:
  • Different attitudes to bearing risk.
  • Different access to information.
  • Different assessments of the riskiness.
  • Or, if one party deceives, coerces, or gulls the other into a bad decision.
• In the time paradigm world, contracts arise where:
  • Both parties gain.
  • When time-average growth rates increase for both sides.
• A difference in starting wealth can be sufficient:
  • Multiplicative => % of wealth => differing growth rates.

Inequality

• Range:
  • Minimum: everyone has the same wealth.
  • Maximum: one individual has all the wealth and everyone else has nothing.
• Ensemble growth rate may be at one level, but:
  • Only a small percentage of individuals achieves growth above the average growth rate.
  • Only small percentage of people achieve wealth above the expected value.
• Highlights importance of difference between time-average and expected value growth rates.
• You need many repetitions (a large sample) to achieve the “lucky” path.

Cooperation

• Cooperation:
  • Benefits multiply over time and grow exponentially.
• Breaking cooperation:
  • Short-term, one-off gains.
• Not captured under expected value framework:
  • The observation of widespread cooperation constitutes a conundrum.
• Cooperation reduces the magnitude of fluctuations.
  • Helps to generate faster growth.
• Cooperation strategies outperform.
  • Favors the formation of social structures.
• Even without effects of specialization.
  • Cooperation pays in the long run.
• In a literal sense, the whole is more than the sum of the parts.

Reallocation / taxes

• When individuals share only a fraction of their resources.
  • Taxation, redistribution.
• Fast reallocation from richer to poorer – two related effects:
  • More individuals’ time-growth rates get closer to expected value growth rate.
    • Less growth divergence.
  • Wealth distribution landscape becomes more stable.
    • Even as inequality can remain.
  • Move towards a a more ergodic system.
• Slow reallocation from poorer to richer.
  • Growth gaps widen.
  • No stable wealth distribution landscape.
    • Population splits into “above” and “below” the mean.
  • Non-ergodic system.
• Implications not only for inequality, but also for immobility.
  • Less likely to move from one population to the other.
• US historic data:
  • Wealths being driven apart.
  • Populations splitting into groups with positive and negative growth rates.
  • No convergence to a stable distribution of rescaled wealth.

Markets.

• Trade-offs between risks and rewards.
  • “Gambles”.
• Classical approach:
  • Risk, reward and leverage assumptions define efficient portfolio trend-line.
  • Individual risk preferences determine optimal portfolio.
• Finance theory has similar issue as economic theory:
  • Reliance on individual preferences:
    • Utility, risk.
• Time-averaged growth rate approach:
  • Allows you to calculate optimal leverage based on maximizing the wealth growth rate.
  • Allows you to disregard individual risk preferences.
• Has implications for:
  • Calculation of equity risk premium.
  • Setting of central bank interest rates.
    • Detecting fraud.