[ryebit]

Thank you! That's... well, beautiful isn't the right word, given nature of example... But I was hoping to find a simple non-ergodic example.

The situation the article explores was interesting, but made the jump to something mathematically complex before I sunk my teeth into the fundamental bit.

[aliceryhl]

As I understand it, the difference is basically whether it is possible to end up in a situation where you are out of the game going forward. E.g. with repeated bets, you eventually hit zero money, and then you are stuck at zero forever.

[inimino]

There could also be attractor states that reduce the risk. For example in this case (if the numbers are taken to be such that the EV is positive) then you can also end up with a lucky player getting rich. The chance of that player going bust goes down much lower than the chance for a new player starting with $1. So while individual players may tend to go bust reliably, the total pool of wealth can still grow beyond any set upper boundary. Over time each player tends to get fabulously rich or go bust, so the game is mostly one of whether an initial run of luck gets you out of the danger zone before running into zero.

If you added some effects on what kind of gambles are available to players at different levels, you can create several different attractor states.

Ergodicity is a nice property of models like molecules of gas bouncing around a room, which means that statistical mechanics is practical. If one percent of the molecules tended to end up with all the kinetic energy, while the other molecules gradually one by one reached a complete standstill, then statistical mechanics wouldn't work.

Since the very simple process shown in the article doesn't have this property, it means some familiar statistical tools can't be used naively with these models, or to extrapolate a little bit, to any model of any human activity that tends to these kinds of capturing, fixed-point, attractor outcomes.

[inimino]

Update: the first paragraph above was not quite right for the system described here. It's more about the EV calculation involving ever growing outcomes multiplied by ever shrinking probabilites, with the mean still growing without bound, while in any cohort of practical size you'll never see any of these outcomes. So it's not necessary for any player to go to zero to see the behavior. Of course there are also martingale systems that work the other way and they are also non-ergodic.

So in fact regardless of the initial run of luck, every player still goes to zero with probability one. The youtube video that another commenter linked to actually explains the 40% and 50% example much better.