Disclaimer: I'm not well read in economics or statitics for that matter but am mathematically literate (PhD in Mech Eng)

Recently I was sent a paper from Nature Physics, "The ergodicity problem in economics" by Ole Peters.

I was discussing this with a friend who also isn't an economist but has more than just a passing interest.

He made the claim that in figure 2 the graph is wrong i.e. that the expectation values (the blue line) are wrong in that they should be plotting a plot of 0.9^(number of rounds/2), which would have a negative slope (he derives this from equation 2 in the paper. He then goes on to say that this renders the paper garbage.

I'm not sure of my friends derivation but if you run the model proposed in equation 2 of the paper then you do indeed get an ever decreasng return. I used the following Python code to run the model...

from matplotlib import pyplot as plt
import numpy as np

wealth     = 1000
num_tosses = 1000
h_t        = np.random.randint(1, high=1000, size=num_tosses)
x          = []
acc_wealth = []

for idx, toss in enumerate(h_t):

    if toss % 2 == 0:
        wealth = wealth + (0.5*wealth)
        wealth = wealth - (0.4*wealth)


fig = plt.figure()
plt.plot(x, acc_wealth, "o-", lw = 2)

A typical plot of accumulated wealth v toss number....

enter image description here

This is clearly not what we're seeing in figure 2 of the paper...

enter image description here

Here the blue line is the accumulated wealth (log scale on the y axis) v toss number. It is increasing!!!

Now obvioulsy this paper was published in Nature Physics which currently has an impact factor of 22.8 so I find it hard to believe that this would have slipped past peer review.

So my question, is the paper wrong or is my friend?

  • $\begingroup$ Your blue line (a realised series) corresponds to the red one in Figure 2, not the blue one (an expected series). The point is precisely that the one cannot substitute the other because the process is non-ergodic. $\endgroup$
    – luchonacho
    Feb 4, 2020 at 14:59

1 Answer 1


If your friend is talking about the expected wealth he's wrong.

The point of the figure is that the expected value of gambling is very much not indicative of what happens typically for longer trajectories, so your simulation just backs that up.

We can already see this pattern if we consider gambling twice. With probability one half, we gain 50% in value, the other outcome is losing 40%. Starting with wealth $1$ if we win twice, we have $2.25$. But the other possible outcomes are $0.9$ if we win once and $0.36$ if we lose twice. The expected value is $1.1025$, larger than $1$. We lose money 3 times out of 4, so typically we lose, but because we win big enough if we do win, the expected change in wealth is positive.

In general we have $$E_{\text{Gamble}}[W_{t+1} | W_{t} = w]= w + 0.5 \cdot 0.5 \cdot w - 0.5 \cdot 0.4 \cdot w = 1.05w$$

Iterating that forward gives $$E_{\text{Gamble}}[W_{h} | W_{0} = 1] = 1.05^h$$ or

$$\log_{10} E[.] = h \log_{10}1.05$$ That's the blue line in the plot.

  • $\begingroup$ I didn't read the paper but CloseToC's good point is very related to the gambler's ruin concept. I forget the exact formulae, but even with probabilties, p and q, favoring you in a game where you win with prob p and lose with prob q, the chance of ruin is quite high. The absorbing barrier at wealth becoming zero of course causes the asymmetric-ness in the outcome. $\endgroup$
    – mlofton
    Dec 12, 2019 at 22:54

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