Yeah, that's after 1 round. Both the individuals and the collective expect to gain if they only play 1 round.

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.

The collective also gains in round 2, and each subsequent round. From the intuition that you have about the first round, treat each group with the same amount of money separately, and you will see that money grows in every round. Example:

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.

No, it will approach zero like the individual one.

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)=}")

Try it with a larger population and a smaller number of rounds. It turns out the mean gain actually is positive, but the median gain is negative.

The claim was: "Each round, the collective wealth goes up 5%, no matter how many rounds you run. "

And this isn't true.

Do you mean that it's not true that "in one round, the collective wealth goes up 5%"?

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.)

For a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds.

So you say that it's false that "in one round, the collective wealth goes up 5%", right?

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.)

Yeah I would say "in one round, the collective wealth goes up 5% in expectation" but "for a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds" not almost, but absolutely.

Maybe you're not aware but almost surely is a technical term - which makes the statement correct.

Just like you cannot be absolutely certain to get heads if you flip a coin enough times even though it will happen almost surely.

I think that the effect is from the fact that you're sampling from all possible outcome history paths. To realise the positive expected value, your sample needs to include some of the increasingly rare possibilities where you gain enormously. This means that if your population is large compared to the number of branches (2^rounds) then you probably do sample that and the overall gain is positive. If your number of branches is large compared to your population, you probably don't sample the branches where you make lots of money, and your overall gain is negative.

The total goes up because not everyone loses wealth at each round. Wealth just gets more concentrated in the hands of the people who win the most.

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.

Ah, I get your point now. True, there is no need to consider multiple people, the expectation is the same for a single person, too.

The collective wealth calculation just makes it easier to visualise.