I kept telling myself I wouldn’t write this post because it doesn’t matter. But I’ve seen one too many smart people fall into the trap of thinking “ergodicity” is a useful concept.

Ergodicity economics is one of those rare theories that somehow manages to be both unfalsifiable and false.

I originally wrote that sentence as a joke, then I deleted it, then I re-wrote it because I realized it’s actually true. Ergodicity economics is sufficiently vague in general that it can’t be falsified, but it is commonly interpreted as making specific falsifiable claims that are, in fact, false.

Contents

• Contents
• What is ergodicity?
• The concept of ergodicity doesn’t do anything useful
• The ergodic principle is an unfalsifiable metaphysical claim
• …also it’s false
• Ergodicity is even worse than that
• Mathematical problems for ergodicity
• Conclusion
• Notes

What is ergodicity?

Taking a definition from Taylor Pearson:

A way to identify an ergodic situation is to ask do I get the same result if I:

1. look at one individual’s trajectory across time¹
2. look at a bunch of individual’s trajectories at a single point in time

If yes: ergodic.

If not: non-ergodic.

Ole Peters, the physicist² who invented ergodicity economics, gave³ a precise mathematical definition which says essentially the same thing.

(Ergodicity is confusing and hard to define without using math, so I appreciate Pearson for figuring out a clean definition. I tried to come up with a definition myself but my version was worse.)

The ergodic principle states that you should follow a strategy that produces ergodic outcomes.

An illustrative example: I offer you a bet. You choose how much money to wager, then I flip a fair coin. If the coin lands heads, I triple your money. If it lands tails, you lose your wager.

You maximize expected earnings by betting your entire net worth. Should you do that?

If you make many bets in a row, you will eventually lose at least one of them, and you will end up with $0.⁴ But if many people make this bet simultaneously, then the average person will make a profit. The across-time outcome is not the same as the across-individuals outcome; therefore, this strategy is non-ergodic. Thus, the ergodic principle says you shouldn’t bet your entire net worth on this coin flip.

Ole Peters proposed the ergodic principle as a replacement for expected utility theory, which is a “foundational assumption in mathematical economics”.

Replacing a “foundational assumption” sounds like a tall order. Does ergodicity live up to Peters’ aspirations?

Resoundingly, no.

The concept of ergodicity doesn’t do anything useful

Ergodicity proponents like to talk about Russian Roulette (1, 2). They say Russian Roulette is non-ergodic: if six people play, 5/6 are alive at the end. If you play six times in a row, you are definitely dead. The across-individual outcome is different from the time-series outcome. That’s why Russian Roulette is a bad idea, you see.

I am not sure what this is supposed to prove. Is there someone out there who believes it’s a good idea to play Russian Roulette, but then you invoke the concept of ergodicity, and this person realizes no, Russian Roulette is bad actually? Why do you need this fancy word to explain why people shouldn’t play Russian Roulette?⁵

Or let’s look at an example in finance, since that’s where ergodicity is supposed to be useful. Take the gamble I proposed in the previous section: if a coin lands heads, you triple your money. If it lands tails, you lose any money you put in.

According to ergodicity economics, you shouldn’t wager all your money because the result would be non-ergodic. Instead, they say, you should bet according to the Kelly criterion, which is the strategy that maximizes the geometric growth rate of your money. Maximizing geometric growth is ergodic, therefore it’s good.

I can get behind the Kelly criterion.⁶ I can get behind maximizing the geometric growth rate. But that’s not a new concept, and ergodicity isn’t adding anything new.

According to standard expected utility theory, if you have logarithmic utility of money, then you should maximize the geometric growth rate (or, equivalently, you should use the Kelly criterion). Exepected utility theory already gives a good answer. What’s the purpose of introducing the concept of ergodicity?

I get the impression that Ole Peters thinks economists are stupider than they are. Quoting Peters (2019)³:

To make economic decisions, I often want to know how fast my personal fortune grows under different scenarios. This requires determining what happens over time in some model of wealth. But by wrongly assuming ergodicity, wealth is often replaced with its expectation value before growth is computed. Because wealth is not ergodic, nonsensical predictions arise.

No? Economics don’t falsely assume that all situations are ergodic, and they don’t say everyone should maximize expected wealth. Standard economic theory says you should maximize expected utility of wealth, for some utility function (and the choice of utility function depends on your risk tolerance). For a logarithmic utility function, maximizing expected utility = maximizing geometric growth. Which is the same as what Peters says to do.

(Peters’ caricature of economists reminds me of Scott Alexander’s “person who’s never read any economics, criticizing economists”.)

The ergodic principle is an unfalsifiable metaphysical claim

So, in practical situations, the ergodic principle is equivalent to “maximize expected log(wealth)”. But Peters says ergodicity economics is superior to expected utility theory. Why?

Peters’ justification is metaphysical, not practical. He has wordy explanations³⁷ for why ergodicity is metaphysically superior to utility maximization, in spite of producing identical results. He says the reason you should use the Kelly criterion is because it’s ergodic, not because it maximizes a logarithmic utility function.

His wordy explanations are mostly wrong, but I don’t want to get into the weeds of metaphysics. (Ford & Kay (2022)⁸ provides some analysis if you’re interested, under the headings “Psychology Is Fundamental to Decision Making” and “The Purpose of a Decision Theory”; see also Toda (2023)⁹.) My big question is, why should I care about which theory is metaphysically superior? If the ergodic principle behaves identically to maximizing the logarithm of wealth, then the alleged superiority of the ergodic principle is unfalsifiable.

…also it’s false

The ergodic principle recommends maximizing geometric growth. Logarithmic utility recommends maximizing geometric growth. But not everyone wants to maximize geometric growth.

A geometric-growth-maximizer is abnormally risk-tolerant. Historically, an investor would have maximized geometric growth by investing in stocks with 2:1 to 3:1 leverage. Most people are not comfortable with that level of risk.

Ergodicity economics recommends that everyone should pursue the same strategy of maximizing geometric growth. That’s too risky for most people. More generally, not everyone should pursue the same strategy because not everyone has the same risk tolerance. Therefore, the ergodic principle is false.

Expected utility theory is not so restrictive. Logarithmic utility also implies a high degree of risk tolerance, but most people don’t have logarithmic utility of wealth. Most people are better modeled as having more risk-averse utility functions than that.

(Gordon Irlam reviewed research on risk aversion and concluded that most investors are perhaps 2x to 3x more risk-averse than a geometric-growth-maximizer.)

Some people (not limited to ergodicity proponents) claim that everyone should maximize geometric growth, or everyone should use the Kelly criterion (which is equivalent). This is wrong for the same reason that the ergodic principle is wrong: not everyone has the same risk tolerance.

Paul Samuelson, “the Nobel laureate whose mathematical analysis provided the foundation on which modern economics is built”¹⁰, was apparently as bothered by this misconception as I am, because he wrote a short refutation using only one-syllable words¹¹. An excerpt:

He who acts in N plays to make his mean log of wealth as big as it can be made will, with odds that go to one as N soars, beat me who acts to meet my own tastes for risk.

Who doubts that? What we do doubt is that it should make us change our views on gains and losses¹² — should taint our tastes for risk.

To be clear is to be found out. Know that life is not a game with a net stake of one when you beat your twin, and with net stake of nought when you do not. A win of ten is not the same as a win of two. Nor is a loss of two the same as a loss of three. How much you win by counts. How much you lose by counts.

As soon as we see this clear truth, we are back to our own tastes for risk.

(I recommend reading the whole thing, it’s only two pages long.)

For a more serious analysis, see Samuelson (1971)¹³. Quoting the abstract:

Because the outcomes of repeated investments or gambles involve products of variables, authorities have repeatedly been tempted to the belief that, in a long sequence, maximization of the expected value of terminal utility can be achieved or well-approximated by a strategy of maximizing at each stage the geometric mean of outcome (or its equivalent, the expected value of the logarithm of principal plus return). The law of large numbers or of the central limit theorem as applied to the logs can validate the conclusion that a maximum-geometric-mean strategy does indeed make it “virtually certain” that, in a “long” sequence, one will end with a higher terminal wealth and utility. However, this does not imply the false corollary that the geometric-mean strategy is optimal for any finite number of periods, however long, or that it becomes asymptotically a good approximation. […] The novel criterion of maximizing the expected average compound return, which asymptotically leads to maximizing of geometric mean, is shown to be arbitrary.

Ergodicity is even worse than that

If ergodicity just meant “use the Kelly criterion” / “maximize geometric mean”, then it would be pointless, and also false, but no worse than those other two principles.¹⁴

The ergodic principle is worse than that. It’s not equivalent to geometric mean maximization, and in fact there is no principled way to derive geometric mean maximization from ergodicity.¹⁵ There are infinitely many strategies that satisfy the ergodic principle, and ergodicity economics does not offer any reason why “maximize geometric mean” is the correct one.

Consider the following decision-making rule:

When offered any bet, wager $0.

This strategy is ergodic because the across-time outcome is the same as the across-individuals outcome:

1. If you make many $0 wagers in a row, you won’t gain or lose any money.
2. If many people make $0 wagers simultaneously, the average person won’t gain or lose any money.

This is a bad strategy. Even if I offer you an extremely favorable bet, this strategy says you shouldn’t wager any money. But ergodicity economics can’t say why you should use the Kelly criterion rather than wagering $0.

Let’s go back to Russian Roulette. They say Russian Roulette is bad because it’s non-ergodic. At the risk of being morbid, let me propose an alternative game. The rule of the game is that you load a revolver with six bullets and then shoot yourself in the head.

This game is ergodic! If six people play, the average player dies. If one person plays six times, that person dies. The two situations are equal. Therefore, according to ergodicity economics, you should play this game! Or at least, ergodicity economics has no explanation for why you shouldn’t play this game.

Expected utility theory has no difficulty explaining why it’s bad to shoot yourself with a revolver. It also has no problem explaining why you shouldn’t play Russian Roulette. Why introduce a new decision-making principle that’s less useful than the one we already have, and adds no value?

Mathematical problems for ergodicity

This section is adapted from a comment I wrote a year ago.

Ergodicity economics has more problems, but explaining these problems requires doing math.

Ole Peters defined a system as “ergodic” if there exists some transformation function \(f\) such that it satisfies the equation³

\begin{align} \displaystyle\lim\limits_{T \rightarrow \infty} \frac{1}{T} \int\limits_0^T f(W(t)) dt = \int f(W) P(W) dW \end{align}

where \(W\) is wealth (or any other thing you care about), \(t\) is time, \(P(W)\) is the probability density of \(W\), and \(T\) is the number of time steps.

In plain English, there must be some function such that the time average of the function output equals the function’s expected value.

(This definition is adapted from Birkhoff’s erodic theorem, a theorem in statistical dynamics where the concept of ergodicity originates, and where—unlike in economics—it is actually useful.¹⁶)

What function is \(f(W)\)? Ergodicity economics does not require you to use any particular function. When discussing multiplicative bets, Peters takes \(f(W) = \log(W)\). If you size your bets so as to maximize the geometric mean of wealth, then indeed you will satisfy the ergodic principle, because the time-limit of log(wealth) equals the expected value of log(wealth).

But as I discussed in the previous section, there is no clear reason why you have to use \(f(W) = \log(W)\). In fact, there are infinitely many transformation functions that satisfy the ergodic principle for multiplicative bets.

The function \(f(W) = 0\) is ergodic: its expected value is constant (because the EV is 0), and the finite-time average converges to the EV (because the finite-time average is 0). Ergodicity economics has no way to say that maximizing \(f(W) = \log(W)\) is better than maximizing \(f(W) = 0\).

Peters claims⁷ that

1. A rational agent faced with an additive bet (example: 50% chance of winning $2, 50% chance of losing $1) ought to maximize the expected value of \(f(W(t)) = W(t)\).
2. A rational agent faced with a multiplicative bet (example: 50% chance of a 10% return, 50% chance of a –5% return) ought to maximize the expected value of \(f(W(t)) = \log(W(t))\).

These assumptions do not follow from the ergodic principle, but I will temporarily accept them so I can show that ergodicity doesn’t work even when you assume away its inability to select a transformation function.

Consider a choice between two lotteries:

Lottery A: 50% chance of winning $200, 50% chance of losing $199.

Lottery B: 99% chance of multiplying your money by 100x, 1% chance of losing 0.0001% of your money.

Peters’ version of the ergodic principle cannot say which of these lotteries is better. It doesn’t evaluate them using the same units: Lottery A is evaluated in dollars, Lottery B is evaluated in growth rate of dollars.

If your theory can’t see that Lottery B is better, then your theory is insufficient.

There is no transformation function that satisfies Peters’ requirement of maximizing geometric growth rate for multiplicative bets (Lottery B) while also being ergodic for additive bets (Lottery A). Maximizing growth rate specifically requires using the exact function \(f(W(t)) = \log(W(t))\), which does not satisfy ergodicity for additive bets (expected value is not constant with respect to \(t\)).

In fact, multiplicative bets cannot be compared to any other type of bet, because \(\log(W(t))\) is only ergodic when \(W(t)\) grows at a constant long-run exponential rate.

In terms of Von Neumann-Morgenstern utility, the ergodic principle violates the axiom of completeness: there are pairs of bets where it is impossible to say which one is better (and it’s also impossible to say that they’re equal).

For a more thorough analysis, see Psychology Is Fundamental: The Limitations of Growth-Optimal Approaches to Decision Making under Uncertainty⁸. (This paper includes a similar proof of non-completeness, although our two proofs were derived independently.)

Conclusion

In summary:

• The way it’s used in practice, the ergodic principle is equivalent to logarithmic utility.
• There is no reason to prefer the ergodic principle over logarithmic utility.
• Logarithmic utility is often inappropriate. You could use a more- or less-risk-averse utility function instead. But ergodicity economics has no answer for how to handle different levels of risk aversion.
• The ergodic principle offers no reason why you should maximize geometric growth rate, rather than using some other ergodic decision criterion.
• The ergodic principle can only provide guidance in a narrow set of situations. In other situations, it has no way of comparing choices.

The concept of ergodicity is complicated enough that it feels like it’s providing useful insights. (Ah, yes, Russian Roulette is bad! Gambling away all your money is bad!) In practice, its main prediction is that people shouldn’t be risk neutral, and this is indeed true, so ergodicity initially looks like it makes useful predictions. But the theory falls apart when you look at it closely.

Ergodicity economics is nearly 15 years old and still hasn’t produced any useful results. I think it’s time we give up on it.

Posted on May 29, 2025

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