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Comment by wmorgan | Hacker News Reader

This is the St. Petersburg paradox with an extra variable.

In SPP, EV approaches infinity as the bank's resources approach infinity. Put bounds on the bank's resources, and you find that even with trillions of dollars your EV is less than $50.

Here, not only are we assuming that the bank's resources are infinite, we're also assuming that the population is large enough that there are always enough lucky players to compensate for the unlucky ones. Put bounds on the size of the population, and you see that everyone goes bust in all but a tiny fraction of cases. Put bounds on the bank, and even that tiny fraction can't compensate for all the losers, and EV is negative.

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I don't think this is SPP. SPP highlights in the difference between the mathematically optimal choice and the choice chosen in practice. It is a difference between theory and practice. The problem proposed in this article occurs even in theory alone. Non-ergodicity means there is a mismatch between "the average of all possibilities in the next time-step" and "the long-term trend of one datapoint".

If we put bounds on the bank in SPP, the first coin toss would still have positive EV. In the new ergodicity problem, even with bounds on the bank, it is unclear whether the "first" coin toss is worth taking.

> If we put bounds on the bank in SPP, the first coin toss would still have positive EV.

Not if the bank starts with $0. 0 is just as valid a bound as a trillion. You can’t calculate the EV without knowing how much the bank has, and once you know that, you realize the naive calculation for EV is wrong.

the St Petersburg paradox is also a problem of ergodicity, since every single player loses with probability 1 over time, even though the "space average" is net positive. No need to invoke messy reality to solve the paradox.

the exponential example is just much more useful, since there are plenty of systems easily described by compound growth

I guess this is my point of disagreement with the article:

> 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.

Wealth averaged over many systems doesn’t grow by 5%. It shrinks just like the average. The EV calculation is just wrong. For any finite starting wealth between the players and the bank, there is a number of iterations where the EV turns negative.

If you say, well let the starting wealth be infinite, I say, okay? If you have infinite dollars there are a lot of tricks for making infinite more dollars. It doesn’t work in the real world.

It also made me think of the St. Petersburg paradox and I think the authors did too, as they call their toss “Peter”

He called it "Peters", after himself, which is off-putting.

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