thebestaquaman

I was initially talking about the total amount earned throughout a career. As a student I lived off roughly 10k USD/year (in Norway). Today I spend quite a bit more, but if my goal in life was to "get rich", let's say I could live off 20 k per year. Making ≈200k and paying 40 % tax (the ceiling in Norway is around there), I would still net 140k/yr after "cost of living", which would bring me to around 5.6 mill (net) throughout a career.

Thank you so much for taking the time to read and understand what I mean! I honestly mean it, it means a lot to me.

What you say is true. What you're neglecting is that you need a random process to choose who will go first. Let's use your own example: If four players go around the board 100 times, there's a near 25% chance that a given player gets around first. As you correctly say (indirectly), you will asymptotically approach a 25% chance as you increase the number of rounds towards infinity.

What you seem to be forgetting is that there's a very easy way to skip the infinite number of rounds, and get directly to the 25% chance: By choosing randomly who goes first. Of course, you need to do that anyway in order to start the warm-up rounds at all, so what you are effectively doing is

First: Give every player a 25 % chance to start. Then: Spend an arbitrary amount of "warm-up" rounds to randomly choose a different player that gets to start the real game.

Of course, these are not independent random processes, so the player that wins the first selection has an advantage in the second selection. The overall probability that a given player starts the "real" game first then becomes identical to the probability that they start the "warm-up" first. An infinite number of warmup rounds is literally identical to a single dice roll in terms of the probability that a given player goes first. So what you're doing is one quick random selection, which you immediately throw out in favour of an infinitely time consuming random selection with the same distribution.

Speaking from Norway: I can agree that order-of-magnitude 10 million USD is earnable in a lifetime. A wage of 100 k USD / yr isn't absurdly high, and after 45 years that brings you to 4.5 million. So getting to like 10-30 million in a lifetime is still within what I would consider reasonably possible. At 100 mill. it's getting pretty absurd though.

The right to be a pacifist is upheld by the people willing to use violence to uphold it.

Pacifism, in its most extreme form, is simply an unsustainable ideology, because you cannot oppose an adversary that is committed to subduing you with violence unless you are willing to use violence to stop them.

That's not how standard deviations work though. The point is that if you are n players, the probability of any given player starting is 1/n. After an arbitrary number of dice throws, the probability that a given player is ahead remains 1/n, when you account for the throw that decided who would go first.

Let's put it this way: Would it be "more random" who goes first if you throw ten dice to decide instead of one? Of course not. But that's essentially what you're doing when you go "warm up" rounds. You're just throwing the dice more times, and letting whoever has the highest total go first. Clearly, the probability that any given player gets the highest total remains 1/n, regardless how many dice are thrown.

But who took the first roll was already chosen randomly. My argument is that who gets to the first square where they can buy something doesn't become any more random by going more laps. The probability of any given player getting to the first purchasable square is 100% determined by the random process that decides who gets to go first in the "warmup round".

I think you're misunderstanding something here?

Let's say you and a friend are playing: You can roll a dice or flip a coin to decide who goes first, and both of you have a 50/50 chance of going first, then you start playing. After the first throw, the player that starts will on average be ≈ 7 squares ahead of the second player, and can buy a property before the second player. Let's call this a "7 square advantage".

Alternatively, you play one or more "warm up" rounds. When you get around the first round, the player that started will on average still have a 7 square advantage, and can still buy the same property before the second player. In fact, you can do as many "warmup rounds" as you like, and the player that started will retain their 7 square advantage whenever the first "real round" starts.

The point is, this doesn't become "more random" by playing "warmup rounds" the probability that any of the two players reaches a given square first is determined the instant the coin flip that decided who would go first landed.

I commented this elsewhere, but feel obliged to copy it in here as well:

The player that goes first has the EXACT SAME statistical advantage, regardless how many round trips you do before allowing purchases. No matter how many times you roll the dice, each player will, on average, be ≈7 places in front of the person that rolls after them (not exactly 7, because there are rules for rolling again on matching dice etc.). This is true for the first roll of the dice, and it is true for the millionth roll. The distance between two consecutive players is on average equal to the mean number of places you move on a turn.

The idea though was so the first player to go doesn’t have an advantage

I... the player that goes first has the EXACT SAME statistical advantage, regardless how many round trips you do before allowing purchases. No matter how many times you roll the dice, each player will, on average, be ≈7 places in front of the person that rolls after them (not exactly 7, because there are rules for rolling again on matching dice etc.). This is true for the first roll of the dice, and it is true for the millionth roll. The distance between two consecutive players is on average equal to the mean number of places you move on a turn.

Nah, it's definitely a worker protection ting. Sure, it has religious origins, which is why it's specifically on Sundays, but that's not why we keep it in place. It's pretty widely supported that we should have one day per week where most shops are mandated to keep closed, and nobody sees a good reason to make it any other day than Sunday, so we've kept it on Sundays.

Fair enough, I guess my overall point was more that we've all at some point taken that first step of doing something that now feels mundane, but at the time felt like we were doing something very advanced.