Question about an ergodicity in economics paper Disclaimer: I'm not well read in economics 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 of the paper 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 then goes on to say that this renders the paper garbage. 

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?

EDIT #1:

One of the comments was where did the expectaion equation derive from. 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!!!

EDIT #2

I'm going to post an amended version this on the Statistics Exchange page too.

EDIT #3

Answer from Cross Validation Stack
OP's Python code is computing one path in time and is not averaging over the paths. So it is not the value shown in the plot. In fact what the OP's Python code computes is $(1+\Delta x_H)^h(1-\Delta x_T)^t$ where $h$ denotes the number of heads and $t$ the number of tails tosses. It is closer to the red line which is the most likely path where $h=t$.
$$\mathbf E[x_{t+1}|x_t] = \big[(1+\Delta x_H)p_H+(1-\Delta x_T)p_L\big]x_t=(1+p_H\Delta x_H-p_L\Delta x_T)x_t=:ex_t,$$
and
$$\mathbf E[x_{t+1}|x_{t-1}]=\mathbf E[\mathbf E[x_{t+1}|x_t]|x_{t-1}]=e^2x_{t-1}.$$
So $\mathbf E[x_t|x_t]=e^tx_0$ exactly as shown in the paper plot. This can also be derived directly through the generating function of the expected $\frac{x_t}{x_0}$ which is, due to the independence of each toss, simply the product of all the dot product of payoff and its associated probability, i.e., $e^t$.