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) else: wealth = wealth - (0.4*wealth) x.append(idx) acc_wealth.append(wealth) fig = plt.figure() plt.plot(x, acc_wealth, "o-", lw = 2)
A typical plot of accumulated wealth v toss number....
This is clearly not what we're seeing in figure 2 of the paper...
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?