The Room with the Revolver and other Non-Ergodic Systems

A hundred people walk into a room. Each picks up a revolver with one bullet in six chambers. Each spins the cylinder, puts the gun against their temple and pulls the trigger once. Then they leave the room. About eighty-three of them walk out. The others leave in a body bag. The math says so.

Now picture one person walking into that same room a hundred days in a row. Same gun, same bullet, fresh spin each morning. The probability that this one person is still alive on day one hundred is roughly 0.0000012 percent. ​Survival across a hundred independent days means multiplying that 5/6 by itself a hundred times. (5/6)^100 ≈ 0.0000000121 or about 1 in 82.8 million. So basically zero.

Most decisions worth making have one of these two shapes hiding inside them. The decision maker who can't tell which one he is facing is playing a dangerous game.

The path that covers the work

This is ergodicity. The word comes from the Greek ergon (work) and hodos (path). Roughly, the path that covers the work. A system is ergodic when one trajectory through time, given long enough, eventually visits all the states the system can occupy. The average of one path over time (time average) and the average of all paths at one moment (ensemble average) converge. The concept was first introduced by Boltzmann (1896), when he worked on ideal gases.

Let's say you sit in the burgundy-colored room in the Kunsthistorisches Museum Vienna looking at the Raphael paintings. The room is closed and no airflow present, then one perfume molecule gets introduced. ​I might have been sitting in this room when an Italian girl walked past me, her perfume still lingering in the air, sparking this example When tracked long enough, the time average of this one molecule wandering through the room would be equivalent to the ensemble average of all air molecules in the room. One molecule across time equals all molecules across space.

Non-Ergodic Systems

Most of the systems we actually live inside are not like the perfume molecule. They are more like the second person, walking back into the same room every morning.

A system is non-ergodic when the time average and the ensemble average pull apart. The clearest way to feel the difference is to ask what makes the perfume molecule example work. The molecule cannot stop existing. Whatever corner of the room it drifts into, it can drift back out. Every state is reversible. Nothing it does on Tuesday closes off where it can be on Wednesday. The path is free to keep covering the work, day after day, and given enough time it visits everything.

The revolver breaks that condition, because here is a state you can enter and not leave. Taleb (2001) calls this an absorbing barrier. Once the firing pin finds the chamber with the bullet, the experiment ends for that path. You cannot average over a future that no longer exists. Once you entered the absorbing barrier, there is no getting out of it.

The expected value calculation that says "on average, you survive" is computing over a population that includes both the living and the dead. But you do not get to be the average. You only get your one path. Seventeen people in a body bag do not benefit from the eighty-three who walked out. The "average outcome" is a bookkeeping fiction.

This is also why the second scenario collapses so brutally. The same five-out-of-six odds that look survivable across a hundred people become almost certain death across a hundred days, because the path has to clear the barrier every single time. Each morning the clock resets on the gun but not on the probability of survival. ​The technical term is repeated exposure, another feature of non-ergodicity.

In the classical expected-value framework, the one that underwrites most of decision theory, ergodicity is always quietly assumed. Peters (2019) was the first to point this out clearly, in his paper The ergodicity problem in economics, published in Nature. The expected-value framework computes what a population of parallel selves would receive on average and treats that number as a guide for the single self actually making the decision. When the system is ergodic, this works. When it isn't, the expected value can be positive while the time-average outcome is ruin. A coin flip that pays +50% on heads and -40% on tails has a positive expected value per round and a path to zero if you keep playing.

The Italian author and guest on my podcast Entertaining Ideas, Luca Dellanna, translated the same machinery for a wider audience, working through what non-ergodicity looks like inside ordinary decisions about work, money, health and relationships (Dellanna, 2024).

Two views of the same game

Now consider venture capital. A fund spreads money across thirty companies. Most fail. A few break even. One returns the fund and then some. From the fund's vantage point, the failures are part of the math. They are how you find the one that pays for everything. The fund averages across an ensemble. The math works because the ensemble exists.

Now stand in the founder's shoes. She has one company and only one run with it. If hers is the one that fails, she does not get to average her failure with the partial successes of twenty-nine other founders. She is on a single trajectory. Her absorbing state is bankruptcy, as well as the time she will not get back, the debt she signed personally and maybe even a spouse who left after years of being kept waiting.

The same activity, viewed from one floor up, becomes ergodic. Viewed from inside, it is not. The fund is in a parallel game. The founder is in a sequential one. They sit across the table from each other and they are playing different games while looking at the same pitch deck. ​Worth holding onto when reading any advice from people one floor up. The floor changes the math.

This pattern shows up everywhere. A surgeon's career-long success rate matters to the hospital. To you, lying on the table this Tuesday morning, it doesn't. The oxygen line lies across your upper lip. The two small soft prongs in your nostrils, smelling like new plastic with a faint chemical sweetness underneath. Feeling the cold of the anaesthetic agent that climbs up the inside of your arm, tasting metal under the tongue, before the room loses its corners. Only this Tuesday matters.

How am I exposed?

Most of decision theory was built for the room with a hundred people in it. It computes expected value across the population and hands the number to whoever is making the choice, regardless of which side of the table that person sits on. The framework works for the fund or the hospital because they actually live in the ensemble, as they get to average it out.

The founder does or the patient does not and quite often you do not. You get one trajectory and every absorbing barrier on that trajectory is the end of the experiment for you, even if the experiment continues for everyone else.

So every time someone says "the expected return is positive" or "on average the strategy wins" you have to stop and ask yourself: How am I exposed to this? Ergodic or non-ergodic?

References

Boltzmann, L. (1896). Vorlesungen über Gastheorie. Leipzig: J. A. Barth.

Dellanna, L. (2024). Ergodicity: Definition, examples, and implications, even outside economics. Luca Dellanna.

Peters, O. (2019). The ergodicity problem in economics. Nature Physics, 15(12), 1216–1221.

Taleb, N. N. (2001). Fooled by randomness: The hidden role of chance in life and in the markets. New York: Texere.