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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
by jstanley 3y ago
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.
- selcuka 3y agoAh, 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.
- pclmulqdq 3y agoThe 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.
- selcuka 3y agoNo, 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)=}")
- jstanley 3y agoTry 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.
- nhinck 3y agoThe claim was: "Each round, the collective wealth goes up 5%, no matter how many rounds you run. " And this isn't true.
- kgwgk 3y agoDo 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.)
- nhinck 3y agoFor a finite population and starting wealth, the collective wealth will tend to 0 given enough rounds.
- kgwgk 3y agoSo 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.)
- nhinck 3y agoYeah 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.