The Ergodicity Problem in Economics
A Critical Review of Peters (2019)
William Fayers
Nature Physics 15, 1216–1221
Aims & Objectives
- Understand the paper’s core argument about non-ergodicity.
- Evaluate the mathematical framework and coin-toss model.
- Assess the Copenhagen experiment’s empirical evidence.
- Examine the economics community’s response and rebuttals.
- Deliver a balanced verdict on the paper’s contribution.
Paper Overview & Context
Author
Ole Peters
London Mathematical Laboratory
Journal
Nature Physics
Perspective article — not primary research
Published
December 2019
Volume 15, pp. 1216–1221
Field
Physics → Economics
Cross-disciplinary critique
Self-citation
~37% of references
10+ of 27 refs are Peters / LML
Programme
Ergodicity Economics
Ongoing since approximately 2011
Core Argument — Ergodicity Explained
An observable is ergodic if its time average equals its ensemble average.
Peters’ Coin-toss Gamble
- Heads: gain 50% of wealth.
- Tails: lose 40% of wealth.
Stake everything, each round.
The Contradiction
- +5.0% ensemble average — looks profitable.
- −5.3% time-average growth — leads to ruin.
Wealth under multiplicative dynamics is non-ergodic: the time average does not equal the expectation value.
The Copenhagen Experiment
Design
- 18 subjects, about $150 in real money.
- 312 choices across additive and multiplicative environments.
- Fit parameter : 0 = linear, 1 = log utility.
Result
All 18 subjects shifted toward log utility in the multiplicative condition.
Methodological Concerns
- Sample size of 18 is very small.
- Stakes were low and lab-based rather than real-world.
- Outcomes were applied at the end, not sequentially — which weakens the time-average argument.
Citation Concern
Cited as a preprint (arXiv:1906.04652). Later published in PLOS Computational Biology (2021) after significant revisions.
Strength — Pedagogical Clarity
Clear Writing
Remarkably accessible for a paper spanning physics, mathematics, and economics.
Powerful Example
The coin-toss gamble makes non-ergodicity immediately concrete and memorable.
Important Question
Whether economic models properly handle the passage of time is genuinely worth asking.
Intellectual Honesty
Peters acknowledges his predecessors — the Kelly criterion is not hidden away.
“To his credit, Peters acknowledges his intellectual predecessors — he doesn’t pretend the Kelly criterion doesn’t exist.”
Weakness 1 — Mischaracterisation of EU
Peters Claims
- Expected utility implicitly assumes ergodicity.
- It requires “parallel universes.”
- Bernoulli’s 1738 paper “contains an error.”
Doctor, Wakker & Wang (2020)
- Expected utility is a representation theorem for preference axioms.
- It does not require ensembles, multiverses, or time averaging.
- Peters’ own Equation 8 itself requires the ergodicity assumption.
The paper’s central critique is directed at a strawman version of expected utility — a serious problem for a thesis that depends on expected utility being wrong.
Source: Doctor, Wakker & Wang (2020), Nature Physics 16, 1168.
Weakness 2 — Scope & Novelty
Historical Lineage
| Year | Who | Contribution |
|---|---|---|
| 1870 | Whitworth | Geometric mean |
| 1956 | Kelly | Growth-rate maximisation |
| 1959 | Latané | Growth-optimal portfolios |
| 2019 | Peters | Ergodicity framing |
Scope Overreach
Peters extrapolates from a specific critique to claim that “the entire field of economics” has drifted wrong — Doctor, Wakker & Wang call this the ubiquity fallacy.
Novelty Question
Toda (2023) argues that ergodicity economics has not produced any falsifiable prediction that differs from standard log-utility theory.
Weakness 3 — Thin Empirical Evidence
18
subjects
modern studies use far larger samples
$150
lab money
not real economic stakes
END
applied once
not sequentially realised
If outcomes are only applied at the end, there is no time average to test. The experiment therefore does not test the mechanism Peters says matters.
Source: peer reviewer concerns discussed in Meder et al. (2021), PLOS Computational Biology.
Balanced Assessment
What Peters Gets Right
- Time dynamics genuinely matter in economic models.
- Ensemble average is not the same as time average.
- The coin-toss example is pedagogically brilliant.
- The paper opened useful cross-disciplinary dialogue.
Where Peters Goes Wrong
- Expected utility does not assume ergodicity.
- The Kelly criterion predates this framing by more than 60 years.
- The scope is overstated — one issue does not show that all economics is wrong.
- The empirical evidence is far too thin for the paper’s strongest claims.
Verdict: a genuinely valuable pedagogical contribution wrapped in an overstated polemic. The insight is real; the revolution is not.
Conclusions & Future Work
Key Conclusions
- The mathematical insight about non-ergodicity is sound.
- The claim of a “pernicious error” in all economics is not supported.
- Expected utility does not assume ergodicity, and the growth-rate prescription is not new.
Future Directions
- Larger experiments with sequential, dynamic stakes.
- Serious engagement with the existing dynamic expected-utility literature.
- Tests of whether ergodicity-economics predictions differ from log-utility theory.
“The most constructive path is collaboration, not competition.” — Doctor, Wakker & Wang (2020)
References
- Peters, O. (2019). The ergodicity problem in economics. Nature Physics 15, 1216–1221.
- Doctor, J. N., Wakker, P. P. & Wang, T. T. (2020). Economists’ views on the ergodicity problem. Nature Physics 16, 1168.
- Toda, A. A. (2023). ‘Ergodicity Economics’ is Pseudoscience. arXiv:2306.03275.
- Meder, D. et al. (2021). Ergodicity-breaking reveals time optimal decision making in humans. PLOS Computational Biology 17(9), e1009217.
- Kelly, J. L. (1956). A New Interpretation of Information Rate. Bell System Technical Journal 35(4), 917–926.
References (cont.)
- von Neumann, J. & Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press.
- Bernoulli, D. (1738/1954). Exposition of a New Theory on the Measurement of Risk. Econometrica 22(1), 23–36.
- Peters, O. & Gell-Mann, M. (2016). Evaluating gambles using dynamics. Chaos 26(2), 023103.
- Latané, H. A. (1959). Criteria for Choice Among Risky Ventures. Journal of Political Economy 67(2), 144–155.
- Whitworth, W. A. (1870). Choice and Chance. Deighton, Bell.