The Ergodicity Trap: Why a 90% Failure Rate Looks Different from the Inside

Population statistics work for venture funds. They will ruin you.

A senior engineer joins a Series B startup in late 2022. The recruiter walks her through the cap table on a Notion page. Her option grant, against a $400M post-money valuation, projects to a paper outcome of $2.4M on a clean exit. She has read the CB Insights piece. She knows the headline number. Ninety percent of startups fail.

She does the expected value math herself. Even at a 10% survival probability, weighting outcomes against bigger exits, the EV is positive. She accepts the offer.

The EV is real. It just does not belong to her.

The 90% failure rate is a property of the population of startups. Her trajectory is a property of one person across one timeline. Those are two different mathematical objects, and modern career decision-making constantly mistakes the first for the second.

This is the Ergodicity Trap. It is the error of using an ensemble average, which is the outcome of many people at one moment, to plan a time average, which is the outcome of one person across many moments. Whenever a system is multiplicative and contains a point of no return, the two averages diverge. Sometimes by orders of magnitude.

The ensemble belongs to the fund. The time average belongs to you.

The Two Averages Diverge

The formal definition first. An observable $x$ of a stochastic process is ergodic if its time average across one trajectory converges to its ensemble average across many trajectories at one moment.

$\lim_{T \to \infty} \frac{1}{T} \int_0^T x(\tau)\, d\tau = \lim_{N \to \infty} \frac{1}{N} \sum_{i=1}^N x_i$

The left side is one life, lived long. The right side is many lives, photographed at the same instant. Ergodicity says these two numbers agree. For additive processes like a fair-wage worker, they do. For multiplicative wealth dynamics under volatility, they do not. That breakdown has a name. Ergodicity breaking.

Take the canonical Peters and Gell-Mann gamble. Heads, your wealth multiplies by 1.5. Tails, by 0.6. A fifty-fifty coin. Compute the expected multiplier per round.

$E[m] = 0.5 \cdot 1.5 + 0.5 \cdot 0.6 = 1.05$

Here m is the multiplier on your wealth in a single round. The expected value is 1.05, meaning the system grows 5% per round on average. Played 100 rounds, expected wealth compounds by a factor of $1.05^{100} \approx 131$ . A venture fund running this gamble across thousands of bets at once will see roughly that return.

Now play it yourself, sequentially, with your one bank account.

After 100 flips you will, with overwhelming probability, see something close to 50 heads and 50 tails. Your wealth does not follow the arithmetic mean. It follows the geometric mean.

$\bar{m} = \sqrt{1.5 \cdot 0.6} = \sqrt{0.9} \approx 0.9487$

That number is less than one. Compounded across 100 rounds, the geometric multiplier shrinks your wealth by 99.48%. The system grew. You went broke.

Spell out the trajectory after $n$ rounds explicitly. With $n/2$ heads and $n/2$ tails:

$S_n = S_0 \cdot (1.5)^{n/2} \cdot (0.6)^{n/2} = S_0 \cdot 0.9^{\,n/2}$

Plug in $n = 100$ . A starting balance $S_0$ of $100,000 becomes $S_{100}$ of roughly $515. The fund running the same gamble across thousands of trajectories sits at $13.1M per slot, on paper. Two numbers, same coin, same hundred flips. Different mathematics.

This is not a paradox.

It is a structural property of multiplicative compounding under volatility. The fund holds a portfolio. You hold a single sequence. The portfolio enjoys the arithmetic mean because winning bets cross-subsidize losing ones in real time. You enjoy nothing of the sort. Your losing bets compound down the same base your winning bets need to compound up.

The general result, derived from Geometric Brownian Motion, is the volatility drag. The system follows the standard SDE:

$dx_t = \mu \, x_t \, dt + \sigma \, x_t \, dW_t$

where $x_t$ is wealth at time $t$ , $\mu$ is the drift (expected return rate), $\sigma$ is volatility (the standard deviation of the return process), and $W_t$ is a Wiener process, which is just continuous-time Brownian noise. Apply Itô’s lemma to $\ln x_t$ . The Itô correction subtracts a half-variance term from the drift in log space:

$d(\ln x_t) = \left(\mu - \frac{\sigma^2}{2}\right) dt + \sigma \, dW_t$

The time-average growth rate is the drift of the log process, because log-wealth grows linearly while wealth itself compounds geometrically.

$\bar{g} = \mu - \frac{\sigma^2}{2}$

The headline takeaway is brutal in its simplicity. Volatility is taxed at half its variance, and the tax is paid by the individual, not the system.

Plug in career-realistic numbers. Take a startup-equity-shaped trajectory with expected drift $\mu = 0.15$ (a 15% expected annual return on the option pool, which is generous) and volatility $\sigma = 0.6$ (high but defensible given the tail-shaped distribution of outcomes).

$\bar{g} = 0.15 - \frac{0.6^2}{2} = 0.15 - 0.18 = -0.03$

Negative. The expected return is positive 15% per year. The trajectory shrinks 3% per year. The fund running the same exposure across a hundred bets captures the 15%. The employee holding one position eats the −3%.

That gap is the volatility tax. It scales with the square of the volatility, which means the worse the variance, the more punishing the asymmetry between the system and the individual.

When $\sigma^2 / 2$ exceeds $\mu$ , the time average goes negative while the ensemble average stays positive. The system grows. Every individual in it, on average, gets poorer.

That is the trap.

Absorbing Barriers Make It Worse

The volatility drag alone is enough to kill most career bets. Then careers add a second wrinkle the textbook gamble does not have. Absorbing barriers.

An absorbing barrier is a state you cannot leave once you enter. Bankruptcy. A career-ending injury. A failed tenure clock. A mandatory up-or-out bounce. Once you cross it, the compounding stops, and not in a “you can recover” way. In a “the timeline ends” way.

The ensemble does not see absorbing barriers because the ensemble is, by construction, infinite. New entrants replace those who get absorbed. The fund replenishes its portfolio. The macro statistics keep growing. Your specific timeline does not.

Optimizing for an ensemble average in the presence of absorbing barriers is mathematically equivalent to optimizing for a reality you will not inhabit.

The ensemble belongs to the institution. The timeline belongs to you.

Where the Trap Eats Careers

Pick almost any high-variance career and the same structure shows up.

Startup Equity

The recruiter pitch is the ensemble. Total venture funding hit $425B in 2025. The technology sector is valued in the trillions. Average outcomes look fantastic.

Then look at the engineer’s actual surface. Carta’s data shows that average new equity packages dropped 37% between November 2022 and January 2024. The rate at which employees actually exercised their vested in-the-money options collapsed from 58% in late 2021 to 32.2% by Q4 2024. Roughly two thirds of employees with vested, valuable options walked away from them.

Why? Because exercising costs cash, the strike price has to clear an alternative minimum tax surface, the company is not liquid, and the employee is now reading TechCrunch headlines about down rounds. The theoretical option becomes a practical write-off.

The employee cannot diversify. They cannot work at fifty startups in parallel. They are bound to one trajectory, and that trajectory carries an absorbing barrier (the company dies) and a volatility drag (every down round resets their compounding base).

The fund holds a hundred companies and captures the arithmetic mean. The employee holds one and gets the geometric mean. The two numbers are not in the same zip code.

The Tenure Pyramid

Academia is the cleanest absorbing barrier in the labor market.

The American Association of University Professors reports that in fall 2023, only 31.8% of US faculty held tenure or tenure-track appointments. The other 68.2% sat on contingent or adjunct contracts. Outside the compounding mechanism. Then look at the reproduction rate.

A 2015 NSF analysis of US engineering PhDs found that the average tenured professor produces 7.8 PhDs across their career. A professor vacates one chair when they retire. The arithmetic does the rest. Roughly 12.8% of PhD graduates can land a tenure-track position. The other 87.2% have to be absorbed somewhere else.

The system advertises tenure. The system is structurally incapable of granting it to most participants.

A new PhD student looking at the average compensation of tenured faculty is staring at the wrong distribution. They are not joining the population of tenured professors. They are joining the population of PhD students, of whom most will hit an absorbing barrier within a decade and never compound a single year of tenured salary.

Sequence of Returns Risk

The 4% rule is the FIRE community’s version of the same mistake.

Bengen’s original 1994 analysis was careful. The popular version is not. The popular version reads the long-run average return of US equities, which sits somewhere around 7% annualized in real terms, and concludes that withdrawing 4% indefinitely is safe with margin to spare. Ensemble math.

The retiree does not experience the average. They experience a sequence. If the sequence opens with a brutal drawdown in years one through five, while the retiree is still pulling 4% of the original portfolio in dollar terms, the base capital collapses. The portfolio does not get to wait for the recovery. The dollar withdrawals have already been carved out of a smaller pie, and now the future compounding base is permanently stunted.

The market recovers. The portfolio does not.

This is exactly the volatility drag with an extra cruelty layered on top. Forced withdrawals during a drawdown convert paper losses into realized ones, which means the geometric base can never reach the arithmetic average no matter how many good years follow. A retiree two years into a 1973-style sequence is mathematically a different person from a retiree two years into a 1995-style sequence, even if both started with the same balance and the same long-run market.

Refugee and Immigrant Trajectories

The Ergodicity Trap also runs in reverse.

A 2017 NBER paper by Evans and Fitzgerald used the American Community Survey to follow synthetic cohorts of refugees over twenty years. Cross-sectional snapshots, the kind that drive most policy debates, show refugees underperforming native-born populations on employment, income, and language proficiency. The ensemble looks bleak.

The longitudinal data tells a completely different story. After six years in the country, refugees in their cohort were working at higher rates than the native-born comparison group. After twenty years, their economic integration matched or exceeded local benchmarks.

The ensemble snapshot, frozen in time, mistook the early phase of a long compounding curve for a permanent ceiling. The time-series, traced through one trajectory after another, recovered the truth.

For anyone reading this who is moving across borders right now, the practical lesson is uncomfortable but useful. Year one and year three look like failure when measured against locals at the same point in calendar time. They look like nothing of the kind when measured against your own compounding curve.

The trap cuts both ways. Sometimes it makes you take bets that ruin you. Sometimes it makes you abandon trajectories that would have compounded if you let them.

The Same Insight Across Four Surfaces

The startup employee, the PhD student, the FIRE retiree, and the new arrival are all running the same calculation. They all read a population number and treat it as a personal forecast.

The population number was never theirs.

Whenever the underlying process is multiplicative, whenever there is a path-dependent absorbing barrier, whenever the variance is large relative to the drift, the ensemble average and the time average decouple. In some careers they decouple by a factor of ten. In others, the ensemble average is positive while the time average is negative. The further the two diverge, the more dangerous it is to plan with the wrong one.

Cross-sectional data is the natural language of institutions. Funds, universities, leagues, and labor markets all speak it because it is the only language that scales to populations. Time-average data is the natural language of people. It is the language of one life, lived once, in physical time, with no reset button.

The mistake is treating the institutional language as if it were the personal one.

What the Counter-Argument Misses

The serious objection here comes from Expected Utility Theory. Doctor, Wakker, and Wang have argued, in Nature Physics and elsewhere, that the Peters gamble does not require any new physics. Concave utility functions already capture the rejection. A rational agent with logarithmic utility will refuse the multiplicative coin flip on its own terms, no time-averaging needed. Risk aversion does the work. Adding ergodicity is, in their reading, a redundant layer.

The argument is mathematically clean. It is also incomplete.

Concave utility is a parameterization of human psychology. It is fitted, not derived. We pick a curvature that matches observed behavior and call it preference. Ergodicity economics flips the causal arrow. It says people are not risk-averse because of an exogenous psychological constant. They are risk-averse because volatility physically destroys compounding wealth across time, and any species that did not evolve to feel that destruction in its gut would not be here making the argument.

Risk aversion is not a quirk of human cognition. It is an artifact of multiplicative dynamics in a single timeline.

The second part of the EUT defense is more practical. Modern financial structures, the argument goes, restore ergodicity through pooling. Limited liability, secondary markets, index funds, insurance. All of these convert individual time-series risk into shared ensemble risk. Fair point, where pooling is available.

The problem is that career time mostly cannot be pooled. You cannot insure ten years of failed partner track at McKinsey. You cannot diversify a torn ACL across a portfolio of bodies. You cannot share the opportunity cost of a stalled academic career with a cohort. The financial wrapper around the career can sometimes be pooled. The career itself cannot.

Pooling is real. It is also a partial defense.

What to Do About It

Three operational rules survive the math.

Optimize for the geometric mean, not the arithmetic mean. A bet that pays $10M with 10% probability and zeroes you out with 90% probability has a positive arithmetic EV and a geometric mean of zero. Take the bet only if you can take it many times in parallel, which is to say only if you are a fund. The Kelly criterion is the same insight in a different vocabulary. For a binary bet that pays net odds $b$ on a win with probability $p$ and loses your stake with probability $q = 1 - p$ , the geometric-mean-optimal fraction of capital to risk is:

$f^* = \frac{p \cdot b - q}{b}$

When the EV is positive but small, $f^*$ is small. When the downside grows or the win probability slips, $f^*$ collapses to zero, then turns negative. Negative is the formula’s way of saying take the other side of the bet. Bet sizing is geometric mean optimization with explicit downside caps. Career bets work the same way, even when you cannot put a clean number on them.

Treat absorbing barriers as binding constraints, not soft penalties. Survival supersedes performance. A career path that requires a non-trivial probability of bankruptcy or career-ending injury merely to achieve the average outcome is mathematically invalid for an individual. The right response is structural. Cash buffers before a market drawdown. Lateral mobility before an up-or-out cycle. Secondary liquidity before the company runs out of runway.

Manufacture ergodicity wherever you can. The angel investor across fifty startups holds the ensemble. The single founder does not. Convert your trajectory into a portfolio at every available margin. Take secondary liquidity when you can get it. Rotate through projects, employers, geographies. Build optionality not because variety is virtuous but because pooled trajectories converge to the ensemble average and unpooled ones do not.

The fundamental error of modern career planning is to treat one human life as a diversified portfolio. It is not.

The portfolio belongs to the institution. The market. The fund. The university. The league. The portfolio enjoys the arithmetic mean because it is, by definition, an ensemble.

You are an ensemble of one.

The One-Line Version

The headline is the fund’s number. Yours is the geometric mean.

Sources

The mathematical core of this piece comes from Peters and Gell-Mann’s 2016 Chaos paper on evaluating gambles using dynamics, and Peters’ 2019 Nature Physics essay on the ergodicity problem in economics. The startup equity figures are from Carta’s 2024 reporting and CB Insights data on aggregate startup failure. The academic pyramid math draws on the AAUP 2023 data snapshot and a 2015 NSF analysis of engineering PhD reproduction rates. Sequence of returns risk traces to Bengen 1994. The longitudinal refugee data comes from Evans and Fitzgerald’s 2017 NBER working paper on refugee outcomes in the United States. The Expected Utility critique is articulated in Doctor, Wakker, and Wang’s 2020 Nature Physics response on economists’ views of the ergodicity problem.