This, is, by the way, somewhat similar to how Google's original PageRank[2] worked: Each page has some "worthiness capital", and at each point in time, it randomly gives one "worthiness coin" to one of the outgoing links. Give each page some starting capital (e.g, 1,000,000 coins), run the game for a very long time, and rank pages by the capital they managed to collect. There is a big difference, though - each page is not equivalent to a person in the 5-person game, but to a state in the 23751-state space; also, PageRank had damping factors, sink escapes, and other changes, but the basic idea is the same, and was actually (at least) 20 years old at the time that Page and Brin applied it to links of the "href" kind.

[0] https://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theor...

[1] for a suitable definition of easily.

[2] https://en.wikipedia.org/wiki/PageRank

> This means that every state is recurrent, i.e. will happen if you wait long enough.

It doesn't mean that the expected frequency of every state in the chain is equal. Moreover, the frequencies are very different.

The interesting part of the experiment is not that each state is reachable. The state where someone has all the money - 1 is reachable. The state where all have the same amount of money again is reachable. But if you leave these people alone in a room for some time and open the door unexpectedly, you will almost sure find out that they have very different amount of money.

I ran an numerical experiment and I get that the average distribution of money is for 5 persons with 5 coins each is:

11.04 6.38 4.03 2.44 1.11

i.e. if you enter the room at random, the one with more money will have on average 11.04 coins, not almost 24 coins

If you count the number of times they have

11±1 6±1 4±1 3±1 1±1

coins, you get a 21% frequency.

If you count the times someone has 24 coins, you get only a 0.00125% frequency. If you count how many times someone has at least 90% of the coins you get 0.0131% frequency (10x more).

This is difficult to compare, because each classification has a different amount of states, but nevertheless, it's clear that an almost linear distribution is more common than a very concentrated distribution.

For 10 persons with 10 coins I get that the average is

28.77 19.11 14.25 10.98 8.53 6.57 4.93 3.53 2.27 1.06

See also: https://news.ycombinator.com/item?id=14730315

> The point is not that some people become rich and never lose their top position. This runs infinitely and will contain every possible sequence of good and bad luck for every person.

> The richest will become the poorest, everyone will experience every rank, and so on.

> The interesting thing is that this simple simulation arrives at a stationary distribution with a skewed, exponential shape. This is due to the boundary at zero wealth ...

The result should be each participant being up or down from 0 at a rate of proportionated sqrt of t, the number of iterations of the system. Implying a wealth distribution which would become ever-more skewed over time.

Whether you have what statistician call "skewed" or not, you have what most people would call skewed, unequal, extreme.

>> This means that every state is recurrent, i.e. will happen if you wait long enough.

> It doesn't mean that the expected frequency of every state in the chain is equal. Moreover, the frequencies are very different.

I do not find it misleading; I do not see how you equate "if you wait long enough it will happen" with "every state is equally likely" or "the average waiting time to get to any state is the same" (which may or may not be equal depending on your where you consider your starting point).

1) X had some money on the previous turn, giving a dollar to Y. Then Y has at least 1 dollar.

2) X had no money on the previous turn. Then by pigeonhole principle there exists a person Y who has at least 2 dollars. Then Y has at least 1 dollar on the current turn.

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?

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.

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.

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.

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).

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

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

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

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.

here's how that looks in a 100 person with 100 bucks game (10,000 money supply)

https://github.com/un1crom/givemoneyaway/blob/master/qqplots...

but no you cannot get back to that initial condition.

but in the real world you most definitely can start that way.

chess is like that too... the initial condition can never be regained.

Because both players have already taken two turns, a player can claim a draw after only 48 moves. If the movement of the knights was repeated 25 times, for instance, then either player could claim a draw. However, this is not possible at the beginning of the game, so simply moving the knights back does not actually restore the game state.

It's just built into the rules: 50 moves without a capture results in a stalemate (https://en.wikipedia.org/wiki/Fifty-move_rule). I guess you could say that the two moves 'go' into the implicit state counter that keeps track of the number of moves without capture.

Not true. The problem is in one-to-one correspondence with the non-negative solutions to

a + b + c + d + e = 25

which is in one-to-one correspondence with the positive solutions to

a + b + c + d + e = 30

which is a simple counting problem---(29 choose 4) = 23751.

In my opinion, you should really try to prove your math to yourself before answering with such an authoritative tone.

Why is my solution incorrect, though?

* Oh, I see why my solution is wrong. It's never the case that one person has all the money. I should take some of my own advice.

23751 - 5 = 23746

then do some statistical analysis to categorize the various initial conditions and their distributions.

then inject perturbations in the simulation like "death taxes" and other redistribution policies.

I built up a lot of mathematica code for anyone to play with.

https://github.com/un1crom/givemoneyaway

one of my favs was to start 1 person with all the money and see how long it takes for them to no longer be the richest person in the game.... you know like millions of ticks of the clock.

https://github.com/un1crom/givemoneyaway/blob/master/qqplots...

enjoy!

> This is due to the boundary at zero wealth which, we imagine, people don’t consider when they think about the problem quickly.

So I wanted to see if this is true. (I ran 5000 ticks, 45 people, 45 initial dollars each). I used my own program.

I removed the condition that you must have at least $1 to redistribute it (in other words I allowed negative wealth) and ended up with this distribution:

http://i.imgur.com/SctNJWi.png

vs with the >$0 redistribution condition:

http://i.imgur.com/wRVZxlO.png

While it affects the shape, it is far from the case that it looks like some brownian motion near the starting wealth. When sorted, it still has a basically unequally distributed shape.