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What is ergodicity?
Ergodicity economics deals with mathematical models where some quantity changes randomly over time. The “expected value” of such a process is the average over a large number of possible values at a point in time. The “time average” is the average over a long time in a single realization. We call the process ergodic if the expected value and the time average are identical. Click here for a 15-minute explainer video.
But the expected value will equal the time average only under very special conditions. When these are violated, we must keep in mind the physical meanings of expected value and time average. For instance, when we model the behavior of a gambler, what matters to that individual is its particular performance over time, whereas the expected value (the average over a large group of individuals) is irrelevant.
What is ergodicity economics?
“Ergodicity economics” is research into economics which puts the ergodicity question center-stage: is the expected value identical to the time average? When it is not, we speak of ergodicity breaking.
Ergodicity economics as an approach to economics comes as a diffuse collection of insights which different researchers have had at different points in time and space. For a more about its history, see Ergodicity economics — a history, or watch our introductory video.
At the London Mathematical Laboratory we collect the insights into ergodicity which researchers have had over the centuries, and, along with our own work, we piece them together into a coherent theory.
An example of ergodicity breaking
Perhaps the most striking illustration of ergodicity breaking and its relevance to economic theory which we have found is the infamous coin toss. This is a multiplicatively repeated gamble: toss a coin, and on heads your wealth increases by 50%, on tails it decreases by 40%. The expected value of your wealth under these dynamics increases by 5% per round. But ergodicity is broken here, which means that the expected value differs from what happens over time. In this case this manifests itself as loss of about 5% per round if the game is played for a long time.
How the example relates to economic theory
This relates to economic theory as follows: the first quantitative model for human behavior under risk is “expected-value theory,” and it goes back to the 17th-century mathematicians Blaise Pascal and Pierre de Fermat. The theory says that when people act under conditions of uncertainty, like the coin toss, they will try to maximize their expected wealth.
It was soon found that this model does not describe how people actually behave. In response, the physicist Daniel Bernoulli introduced expected-utility theory in 1738. Bernoulli suggested that people may maximize the expected value not of their wealth but of a non-linear function of their wealth. Expected-utility theory is much more general and flexible than expected-value theory, and can describe more types of behavior. However, it leaves the question of the origin of the non-linear functions open. Over the centuries, expected-utility theory has been significantly revised and generalized further. In its many forms and with its many descendant models it is still influential, both in economics and in fields which borrow from economics, such as neuroscience and machine learning.
Ergodicity economics offers one possible answer to the question “where do the nonlinear functions come from?” As the coin toss shows, what is guaranteed to happen over time is often not what happens in expectation. This is not something the inventors of expected value theory or of expected-utility theory were aware of. The ergodicity question — “does expected value reflect what happens over time?” — was first asked in the 1870s, and in physics, not economics. But if we delve into the mathematics, we find that what does happen over time in the coin toss is indeed what happens to the expected value of a non-linear function of wealth (namely, here, the logarithm).
In this way, ergodicity economics offers an interpretation of expected-utility theory.
Ergodicity economics, optimization, rationality
Significantly, we arrive at the ergodicity-economics interpretation via optimization. We ask what behavior is optimal over time, not in expectation, and we find that people optimize reasonably well according to this temporal view of optimality. By contrast, in economics, especially in behavioral economics, a different narrative has taken hold. Here, deviations in people’s behavior from old definitions of optimality are considered signs of human irrationality.
Ergodicity economics suggests that people are not as irrational as we may have been led to believe.
For example, when we look at the performance over time of cooperating entities, we find that cooperation is beneficial in simple performance terms, and the puzzling observation that many entities in nature share resources is unpuzzled. Similarly, we can resolve the St. Petersburg paradox by focusing on long-term performance, not on expected value, and so on.
Ongoing research
Our current research is concerned with any kind of ergodicity breaking in economic models (absorbing states are an example, as are Brian Arthur’s Polya urns), and we investigate the consequences both theoretically and experimentally.