The intuition for why this happens is that the losses of the majority are made up by the big gains of a minority. You can even see that after two iterations:
Case 1: two heads -- 225% wealth Case 2: heads, tails -- 90% wealth Case 3: tails, heads -- 90% wealth Case 4: two tails -- 36% wealth
So, most people are losing money but a single lucky guy makes enough money that the expected value is still positive.
If you run a simulation with a low sample size, you might not see this effect because the "lucky guy" case is rare (and gets more rare with more iterations).
(The question can be reformulated such that this could be the case.)
> that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!
is indeed untrue. Thanks!
The reason for this is that the very rare outliers will accumulate truly enormous wealth, while the rest of the players will never lose more than they start off with. You can see this in the first graph after say 250 rounds - the richest player has around 10,000x his initial wealth, even as the other 1000(or so) have very little - so on average they have 10x what they started with. And the reason no one is rich after 1000 rounds is simply that there were not enough samples. If there were say a billion samples, you might expect one to come in with say a trillion times its starting wealth, overcompensating for the other 999,999,999 losers - on average.
No, they don't. If 100 people starts with $100 (i.e. a total wealth of $10,000), and half of them win as expected, the total wealth at the end of first round will be $10,500.
Granted it won't be a 5% increase for the next round, and eventually collective wealth will also approach 0, but that's the point. 100 people playing 1 round is not the same as 1 person playing 100 rounds.
The interesting part is that they expect to lose money over time even though they expect to gain money if they only play once. But that holds for both the individuals and the collective.
A less confusing game with the same mechanism is "flip a coin, if it's heads I give you 1000x your initial investment, if it's tails I wipe you out completely" - you expect to make a massive gain on any given turn, but if you play enough rounds you'll eventually lose everything.
Round 1: 100x$100 (total $10000) -> 50x$60 + 50x$150 (total $10500)
Gain of $500 total
Round 2: 50x$60 ($3000) -> 25x$90 + 25x$36 ($3150)
50x$150 ($7500) -> 25x$225 + 25x$90 ($7875)
Gain of $150+$375=$575.
Round 3: 25x$36 ($900) -> half $54 and half $21.60 ($945 total) And so on.
Each round, the collective wealth goes up 5%, no matter how many rounds you run.
Think of it this way: If everyone's individual wealth approaches zero, why would the total go up?
Just run the following simulation in a python REPL:
import random
POPULATION = 100
INITIAL_MONEY = 1000
ROUNDS = 10000
wallets = [INITIAL_MONEY for _ in range(POPULATION)]
for iteration in range(ROUNDS):
for person in range(POPULATION):
if random.random() > 0.5:
wallets[person] = wallets[person] * 1.5
else:
wallets[person] = wallets[person] * 0.6
print(f"{sum(wallets)=}")And this isn't true.
I could agree with that.
(However, if you add "in expectation" it will be true for one round and for each round no matter how many rounds you run.)
However you agree that it's true that "in one round, the collective wealth goes up 5% in expectation", right?
(I agree that "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds ALMOST SURELY". The last bit is important though. Otherwise it's like saying that if you flip a coin enough times you will get heads - which is not exactly true.)
Just like you cannot be absolutely certain to get heads if you flip a coin enough times even though it will happen almost surely.
Also, "5% per round" assumes an infinite number of people. If you run this with 100 people, by the 3rd round, the expected group sizes are non-integers, so you will diverge from the "infinite people" case.
The collective wealth calculation just makes it easier to visualise.
I agree the distinction is not at all between individual and collective, but rather between one-off and repeated games.
If you change the rules to, say, +100% and -90%, they lose even when population > 20 * rounds. I'm sure there is a statistical formula that can be used to estimate the optimal number of rounds.
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
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.
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 wealth could grow while the individual wealths drop, and the answer is that it does not.
The only way it looks like the collective makes a gain is if you try to say that x_i(t) = x_i(t-1)*1.05, but that's a simplification that doesn't hold in either the individual or the collective case.
But if you forget the part about the supposed misalignment between individual and collective, then the idea that the expected value of 1 iteration can be positive while the expected value of repeated iteration can be negative is very fascinating!