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Comment by jordigh | Hacker News Reader

Oops, right. I'm counting the number of states wrong. But the general point stands, you'll see everyone hit the maximum amount of money possible for one individual.

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I wonder what their point is though? Like, are we supposed to be surprised with the shape of the distribution?

Also, while I agree this is theoretically possible to "see everyone hit the maximum amount of money possible for one individual"... given enough individuals the possibility of someone ending up with nearly all the money is close to zero. I've run a simulation with 500 individuals (here: https://github.com/subroutines/DollaBillz ) and it seems like the distribution of money reaches a steady-state. What should we make of that?

Think of it as "entropy mostly increases". It's theoretically possible that a bowl of soup left to itself could become hot in one corner and cold everywhere else, but it's vanishingly improbable compared to a mostly uniform temperature distribution.

I don't understand this example. Are you claiming the laws of thermodynamics only work with a very high probability?

The second law of thermodynamics is intrinsically a statistical law.

I'm not a physicist and I probably could have phrased that a little better. The example of a bowl of soup randomly and unevenly changing temperature implies to me that the physical system behaves randomly which I do not believe it does.

Entropy can decrease, just with vanishingly small probability: https://en.wikipedia.org/wiki/Fluctuation_theorem

You don't have a detailed knowledge about the state of every molecule in the bowl soup and its environment, so you have to rely on statistical mechanics. And the current macroscopic state that you observe could correspond to a particular microscopical state that evolves in surprising ways. But the probability is so low that it just doesn't happen.

To make things worse, quantum mechanics is intrinsically random (at least as far as we know) and having a detailed knowledge of the state of the system allowing for certain prediction of its evolution is impossible in principle (and not just in practice).

Yes, the second law of thermodynamics is a statistical law. It is physically possible that all the particles in an ideal gas container will be moving east all at the same time, for example (it doesn't break any law of mechanics), or that a temperature fluctuation will spontaneously appear in a material in thermal equilibrium. It's just that the probability for such spontaneous entropy decreases happening is astronomically small.

Ok right. It just seemed like the article and jordigh were making contrasting points. Although neither make an explicit broad claim (in which case, what are we doing here other than running an arbitrary MCMC and then stating a Markov property), the article is saying hey look, rich people get richer, while jordigh is saying, nah "you should eventually see each person hoard all of the money in turn.".

So which is it, probably? I think this might be more complex than "everyone will hord all the money eventually", given that the most probable number of sign changes in a 1D random walk is zero.

I'm disinclined to attribute them with having much of any point at all. Their discussion is too brief, and their presentation style is more clickbait thance science. They tout a counterintuitive result (It's really not, at least not for anyone used to thinking about data or state changes, certainly not once you think about what the 4th or 5th state change looks like) And then it ends the headline with the overused, even in for content farm clickbait, "you'll never guess what happens next". This tells me they're less interested in discussing an interesting result than getting eyeballs on the page via a shallow observation passed off as insight. If there's a deeper point to be made, they haven't made it.

Thank you. Finally someone said it. This was not the least bit counterintuitive to anyone familiar with randomness.

We're they trying to imply that people should assume the distribution would remain uniform?. That strikes me as more "insane" than intuitive.

It's simply a function of how the tokens are being exchanged that would almost require some actors get the short straw, and since this is happening 100 times per round, that randomness ensure some actors get progressively shorter straws over time, because randomness is neither fair, not even. It's randomness.

No, they're making a simple digital model that conclusively disproves the notion that 'because disparity in wealth exists, meritocracy exists'. Knowingly or not, that's the point being made.

If you want an even more insanely staggered distribution, let people with resources increase their chances of being given a dollar. This establishes that even in the complete absence of merit, a striking distribution will arise and stably persist, and it's created out of the mechanical behavior constraint of 'possibly being broke and having no dollar to give'. That alone creates what looks like a 'meritocracy' with 'high performers'.

Add literally any form of actual merit, and this gets more extreme still. The funny thing is, the merit-less distribution is not wildly out of line with what people think the distribution should be WITH merit.

I feel like there's some weird fundamental misunderstanding going on here.

The way I interpreted the solution was that every individual will at some point have the max possible amount of money (which is roughly $100).

I don't think jordigh is trying to imply that a single individual will have a majority of the total wealth

Everyone starts with $100, and there are 100 people, so the max possible would be $100 * 100 - 1

(minus 1 because you give away $1 on every turn).

Assuredly, but this is the same as with a deck of cards. If you apply a shuffle that you know creates a closed markov chain (say, moving the top card to a random position within the deck), then you will reach a steady state. But, given enough shuffling, you'll also return to a new deck order

A dose of realism would have anyone with >$x be able to reduce the amount or frequency of giving away money. Or be able to spend some of their wealth on 'marketing' and increase the likelihood of being the recipient of any particular dollar being given away.

Not to demonize accumulation of wealth but just to model reality where rules can sometimes be changed by those with enough wealth accumulated.

True, but I think Grue3's remark is more about the fact that this Markov chain is not irreducible than the number of states being wrong. Because you can go from the state where one person has all the money to the other states but not the other way around.

This doesn't matter all that much -- I think the Markov chain becomes irreducible if you chop off the states in which one person has all the money.

Why would that not matter? If you ignore some states that are possible is that not changing the problem?

In this case, the unreachable states don't partition the space in a meaningful manner, nor does the system start in an unreachable state, so the only effect is that those states occur with probability zero instead of with very low probability in the stationary distribution. There are also very few (N) of those states compared to the total number of states (about N^(2N) if I'm estimating right), so removing them has very little effect on the overall statistics.

For a more interesting example, suppose that on each round, everyone with money picked a random person and gave them 2 dollars (and take N even). Now you'd have a Markov chain that is reducible in a nontrivial manner -- think about the parity constraints.

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