What percentage of people will lose money is also very important but does not seem to be written. What percentage?
Almost every person who plays the game will see their money decay towards zero as time goes to infinity.
The infamous coin toss
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Re: The infamous coin toss
#32Earlier quoted context omitted.
You can substitute $ for % in my comment if it helps. If you start with $100, your expected wealth after two throws is the average of $225, $90, $90 and $36.
That average is still greater than $100, because you haven’t yet hit the Kelly point beyond which the downside catastrophe dominates. Play it out a few more rounds and see where the average heads to. [Edit: delete bad math]
The expected value of this distribution goes up with every iteration, there is no such Kelly point. You could try this with
heads: double your money tails: lose all your money
in which case the expected value is always $1, as you have a 1/2^n chance of having $2^n dollars after n rounds, and 0 otherwise.
The point of discussing ergodicity here, however, is whether you can describe the behavior of the iterated distribution deterministically if you exclude a portion of that distribution which has measure zero.
Re: The infamous coin toss
#33Earlier quoted context omitted.
Absolutely. The individual is long-run guaranteed to be wiped out. But I disagree with the original author’s way of concluding that fact (ie, that it arises from “losing 5% per round”, which is just false).
I believe the entire point of the ergodicity question here is "If you apply this process n times, with n approaching infinity, obviously the result may depend on what point in the n-times iterated distribution you sample, but if you choose a volume of vanishingly small measure to exclude, can you make a single concrete statement about what the process is doing without taking an expected value over the different outco…
Great description of a framing I hadn’t considered before, thanks!
Re: The infamous coin toss
#34I think the part about the misalignment between the individual and the collective is basically wrong. In the given game, the collective loses money just as the individual does. The collective wealth is the summation of the individuals' wealths, and both the collective and individual wealths drop over time. Write a simulation and try it if you don't believe me. I did, because I couldn't work out how the collective wea…
I tried this and collective wealth does increase: https://imgur.com/a/Zc2aApj
The median person does get poorer over time though, a small percent get extrmeely wealthy: https://imgur.com/a/v2AVyrv https://imgur.com/a/7pPfKk6
Re: The infamous coin toss
#35I think the part about the misalignment between the individual and the collective is basically wrong. In the given game, the collective loses money just as the individual does. The collective wealth is the summation of the individuals' wealths, and both the collective and individual wealths drop over time. Write a simulation and try it if you don't believe me. I did, because I couldn't work out how the collective wea…
Any finite collective eventually loses almost surely, but the larger the collective, the longer it takes, and the greater the inequality in the meantime, before everyone loses. In the limit, an infinite collective exhibits the paradoxical behavior from the article: expected value goes up forever, but almost every individual eventually loses.
An ever smaller slice of huge winners is pulling up the average, but in any finite population this slice disappears in finite time. Only in an infinite population can it persist indefinitely.
All of this is fairly straightforward for a mature student of probability. If enough folks want, maybe I could write up the details to prove these things.
Re: The infamous coin toss
#36This 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 com…
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
Re: The infamous coin toss
#37Re: The infamous coin toss
#38But this is purely a result of the distribution of returns from a single toss. If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win. I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
> When I lose, I lose more than I win.
I don't see how this is true. Winning adds 0.5x of your current bankroll. Losing subtracts 0.4x of your current bankroll. When you win, you win more than you lose. The gamble has positive expected value.
Re: The infamous coin toss
#39But this is purely a result of the distribution of returns from a single toss. If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win. I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
Yeah, as soon as I realized that, I rolled my eyes at not immediately seeing that. When expressed as 150% and 60%, it became obvious to me.
Can you clarify? I think you might be mistaken, since a gamble with payoffs of 150% and 60% has positive expected value.
Re: The infamous coin toss
#40But this is purely a result of the distribution of returns from a single toss. If I win I get 1.5 times my money, if I lose I'm left with less than 1/1.5 times. When I lose, I lose more than I win. I agree it's counterintuitive how any individual (on average) will lose over time, yet the entire system grows.
> When I lose, I lose more than I win. I don't see how this is true. Winning adds 0.5x of your current bankroll. Losing subtracts 0.4x of your current bankroll. When you win, you win more than you lose. The gamble has positive expected value.
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