In 2011 I gave a 15-minute talk to a lay audience in London. The topic I had chosen was ergodicity breaking, and the challenge was clear: how do you get this across? I invented a coin-toss gamble, which has since become a go-to illustration of ergodicity breaking and a very intuitive way of explaining how ergodicity economics differs from other approaches to economics, and how its concepts may apply to problems unrelated to economics.

The gamble I’m about to describe, now sometimes called `the Peters coin toss,’ is discussed in detail in my 2016 paper with Murray Gell-Mann, there’s a youTube video about it on the ergodicity.tv channel, and my public talk from 2011 is also available online. In this post, I will present the basic setup, hint at its significance, and then mention a few generalizations.

Imagine I offer you the following gamble. I toss a fair coin, and if it comes up heads I’ll add 50% to your current wealth; if it comes up tails I will take away 40% of your current wealth. A fun thing to do in a lecture on the topic is to pause at this point and ask the audience if they’d like to take the gamble. Some will say yes, other no, and usually an interesting discussion of people’s motivations emerges. Often, the question comes up whether we’re allowed to repeat the gamble, and we will see that this leads naturally to the ergodicity problem.

The ergodicity problem, at least the part of it that is important to us, boils down to asking whether we get the same number when we average a fluctuating quantity over many different systems and when we average it over time. If we try this for the fluctuating wealth in the Peters coin toss the answer is no, and this has far-reaching consequences for economic theory.

Let’s start with averaging wealth, x_i(t) over an ensemble of many different systems. In our case this corresponds to N different players, each starting with x_i=\$100, say, and each tossing a coin independently. After the coins have been tossed, about half of the people will have thrown heads, and the other half tails. As the number of players goes to infinity, N\to\infty, the proportions of heads and tails will approach 1/2 exactly, and half the players will have $150, the other half $60. In this limit, we know what the ensemble average will be, namely \langle x(1)\rangle = 1/2\left(\$150 +\$60\right) = \$105. For historical reasons, this average is also called the expected value, and for the Peters coin toss, it grows by 5% in every round of the gamble so that

\langle x(t)\rangle = \$100\times 1.05^t .We can recover this average numerically by setting up a Monte Carlo simulation: let a large number, N, of agents play the game for T rounds, then average over N. You can see what happens for T=10 as you increase N in the app below.

Choose the number of players and hit “Simulate.” The red straight line shows the expected value for the first 10 rounds (starting at 1). The light grey lines are the wealth trajectories of individual players, and the solid blue line is the average over the ensemble.

To see that the gamble is not ergodic, let’s now find the average value of wealth in a single trajectory in the long-time limit (not in the large-ensemble limit). Here, as T grows, again the proportions of heads and tails converge to 1/2. But, crucially, a head and a tail experienced sequentially is different from two different agents experiencing them. Starting at x_1(0)=\$100, heads takes us to x_1(1)=\$150, and following this sequentially with tails, a 40% loss, takes us down to x_1(2)=\$90 — we have lost 10% over two rounds, or approximately 5% per round. Since we lose 5% per round, averaged over time, individual wealth is guaranteed to approach zero (or negative infinity on logarithmic scales) in the long-time limit T\to\infty. You can try this out too, in the app below, by simulating the game for different numbers of repetitions.

Choose the number of rounds to play and hit “Simulate.” The red straight line shows the expected value, the green straight line decays at the time-average growth rate. The light grey lines are the wealth trajectories of individual players, and the solid blue line is the wealth of the one individual we’re simulating (all starting at 1 as before).

We have thus arrived at the intriguing result that wealth averaged over many systems grows at 5% per round, but wealth averaged in one system over a long time shrinks at about 5% per round. Plotted on logarithmic vertical scales, this gives one of the iconic images of ergodicity economics, the featured image of this post (taken from my 2019 Nature Physics paper).

The significance of this ergodicity breaking cannot be overstated. First, all living processes, including economic growth processes are similar to the coin toss in the sense that they rely on self-reproduction. The number of rabbits, or viruses, or the dollars in your trading account, grow in a self-reproducing noisy multiplicative way (until some carrying capacity is reached), just like wealth in the Peters coin toss. Second, most mainstream economic decision theories are based on the concept of expected value, and all of that has to be questioned in the presence of ergodicity breaking. Third, one core problem of economics and politics is to address conflicts between an individual, for example a citizen, and a collective, for example a state. This is the question of societal organization, institutions, Rousseau’s social contract and so on. This problem can seem puzzling, and it often attracts naive answers, because the collective consists of individuals. How, then, can the interests of the individual be misaligned with those of the collective? One important answer is ergodicity breaking.

The coin toss is often the starting point for more detailed investigations. We may allow players to withhold some of their wealth and only subject a fraction of it to the coin toss dynamics. This gives us Kelly betting and optimal leverage. We can allow players to pool their resources, which leads to the ergodicity solution of the cooperation puzzle, and to the emergence of complexity. Or players may pool a proportion of their wealth, leading to reallocating geometric Brownian motion and an intriguing perspective on the dynamics of wealth inequality.

I’ll end here and encourage you to gamble away and meditate on the many curious stories this simple little coin toss has to tell us.

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