M.9.1 Ergodicity from Zero

Basic Prerequisites: none; arithmetic and the idea of a percentage change.

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1. Intuition & Practical Objective

This page builds the whole topic with no prior probability or finance. The objective is one idea: “the average of many people’s outcomes at one time” and “the average of one person’s outcomes over time” are different things, and for money that grows by compounding they are not just different - they point in opposite directions.

Start with the dumbest possible question: if a bet has positive expected value, should you take it? Everyone “knows” yes. The trap is the word expected. Expected for whom, and when?

Imagine a coin game. Heads: your money is multiplied by $1.5$. Tails: multiplied by $0.6$. It costs nothing to play and you may play as many times as you like.

Do the arithmetic the textbook way - the ensemble average across many players after one round:

$$\mathbb{E}[W_1/W_0]=\frac{1}{2}(1.5)+\frac{1}{2}(0.6)=0.75+0.30=\mathrm{1.05.}$$

A $+5\%$ expected gain per round. Positive edge. Play it, right?

Now play it once, as one person, and repeat. Half the time you multiply by $1.5$; half the time by $0.6$. Over $N$ rounds your money is

$$W_N=W_0(1.5{)}^H(0.6{)}^T,H+T=N.$$

Take logs and divide by $N$ - the time average growth rate of your actual money:

$$g=\underset{N\to \infty }{\lim }\frac{1}{N}\ln \frac{W_N}{W_0}=\frac{1}{2}\ln 1.5+\frac{1}{2}\ln 0.6\approx -\mathrm{0.0527.}$$

That is $-5.27\%$ per round: a guaranteed, near-sure march toward zero. Same game. Positive ensemble return, negative individual experience. That gap is ergodicity breaking.

Three “aha”s:

1. The ensemble average is dominated by a vanishing tail of lucky paths. One path with many heads carries so much wealth that it can pull the group mean arbitrarily high - while almost every individual is ruined. The mean is real; it just isn’t typical.
2. Compounding is symmetric in logs, not in money. $+50\%$ then $-40\%$ leaves you at $0.9$, not $1.0$: $1.5\times 0.6=0.9$. Money is not a conserved quantity you can average - it multiplies.
3. “Expected” hides a horizon. $\mathbb{E}[W_N]=1.05^N\to \infty$ describes the ensemble at time $N$. Your $W_N$ describes one path up to time $N$. When they disagree, the one that describes your life is the second.

Takeaway. Before trusting any “expected return”, ask whether it is an ensemble average (of many paths, at one time) or a time average (of your one path, over time). For compounding wealth these are different numbers, and the time average is the one you live.

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2. Mathematical Ground Truth & Derivations

The two averages.

Let $x(t)$ be a stochastic process.

• Ensemble average (across realisations, at fixed $t$): $⟨x(t)⟩=\int xd\mathbb{P}(x)$.
• Time average (along one realisation): $\overline{x}={\lim }_{T\to \infty }\frac{1}{T}{\int }_0^{T}x(t)dt$.

The process is ergodic (for the mean) iff $⟨x(t)⟩=\overline{x}$ almost surely. The ergodic theorem says: if the process is stationary and ergodic, the time average converges to the ensemble average. For an additive quantity - a sum of independent increments - this is the law of large numbers, and it holds.

Why money spoils it. Wealth compounds: $W_N=W_0{\prod }_{i=1}^N(1+R_i)$. The additive object is $\ln W_N=\ln W_0+{\sum }_i\ln (1+R_i)$ - that sum is additive, so it obeys the law of large numbers:

$$\frac{1}{N}\ln \frac{W_N}{W_0}\stackrel{\text{a.s.}}{\to }\mathbb{E}[\ln (1+R)]=g\text{(the time-average growth rate).}$$

But the ensemble average of wealth is a different animal, governed by the arithmetic return:

$$\mathbb{E}[W_N/W_0]=\prod _i\mathbb{E}[1+R_i]=\mathbb{E}[1+R{]}^N=e^{N\ln \mathbb{E}[1+R]}.$$

So the ensemble grows at rate $\ln \mathbb{E}[1+R]$ while the typical path grows at rate $\mathbb{E}[\ln (1+R)]$, and Jensen’s inequality forces

$$\mathbb{E}[\ln (1+R)]\le \ln \mathbb{E}[1+R],$$

with equality iff $R$ is constant. The gap is exactly the “volatility drag” of page 03. The process $W_t$ is thus non-ergodic whenever returns are random.

The coin-toss numbers ($1.5/0.6$, $p=\frac{1}{2}$):

Quantity	Value	Meaning
Arithmetic (ensemble) return	$+5.00\%$/round	what the group mean does
Log (time-average) growth $g$	$-5.27\%$/round	what almost every individual does
Ensemble mean $W_{100}/W_0$	$1.05^{100}=131.5$	pulled up by lucky paths
Median $W_{100}/W_0$	$e^{100g}=0.00515$	the typical path, ruined

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3. Computational Implementation - simulating the paradox

Stdlib only. We simulate many traders playing the coin game and compare the ensemble to the typical trader - and count how many are wiped out.

The ensemble mean ($131.5\times$) and the typical outcome ($0.005\times$) differ by a factor of $\sim 25,000$, and 54% of traders are effectively wiped out within 100 rounds - in a game the textbook calls a $+5\%$ edge. This is the entire lesson in one screen.

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4. Failure Modes & First-Principles Breakdowns

1. Reading “expected value $=1.05$” as ”$1.05$ happens.” It is a cross-sectional mean, dragged up by exponentially rare winners. The typical outcome is the median, and the median is governed by $\mathbb{E}[\ln (1+R)]$.
2. Averaging percentages instead of multiplying ratios. The arithmetic mean of $+50\%$ and $-40\%$ is $+5\%$, but the compounded sequence yields $0.9$. Always compound; never add returns across time.
3. Believing positive edge $\Rightarrow$ safe to bet everything. As $N\to \infty$ a path that bets everything is ruined with probability one whenever the log-growth is negative - the “bold” criterion (maximise $\mathbb{E}[W_N]$) is exactly the one that ruins you.
4. Assuming time averages exist. For wealth, the time average of $W_t$ itself is often $0$ or $\infty$; only the time average of $\ln W_t$ is well-behaved. The well-behaved object is the growth rate - which is why the whole folder is written in logs.

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5. References

• Peters, Ole: The Ergodicity Problem in Economics, Nature Physics 15, 1216–1221 (2019)
• Peters & Gell-Mann: Evaluating Gambles Using Dynamics, Chaos 26, 023103 (2016)
• Thorp, Edward O.: The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market (2006)
• Kelly, J. L.: A New Interpretation of Information Rate, BSTJ 35:917–926 (1956)

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6. Connected Graph Bridges

• Next: 02 · Ensemble vs Time Averages · Index Hub
• Theory behind it: Probability & Measure Theory (law of large numbers, Jensen)
• Where the growth rate is used: 04 · Kelly Criterion · Mean–Variance Optimization