I agree with most of your comment, but this part is misleading:
> 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
> 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 ...
It should be fairly anti-symmetric, since the dynamics of losing are similar to those of winning. So instead of exponential looking, it's shaped like exp(rank) - exp(-rank). (just confirmed this with a quick simulation)
With participants able to go into debt indefinitely, the process becomes a random walk.
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.
Is that what "skewed" means? To my mind a widening bell curve which remained symmetric (which is what we're talking about, right?) would not be "skewed".
Over time, the average participant would have either far more coins in credit than they began with or be in debt for far more coins than they started with.
Whether you have what statistician call "skewed" or not, you have what most people would call skewed, unequal, extreme.
"Varied" or "widely distributed" seem like better terms. To my mind, "skew" is what we have in the original problem (with a zero-bound) - it only describes asymmetric distributions.
Removing the boundary doesn't really make sense -- if you allow people to go into negative dollars but still be able to give a dollar each tick, then there's no longer a finite amount of money in the system.
You're kidding, right? No way a generic observation about a markov chain (which doesn't even model interest) implies that "uncapped debt reduces wealth mobility".
So, with typical debt, interest rates are higher for debt than for savings. Does seem that a) removing the cap, and b) introducing interest on savings < debt would trap some people in poverty?
> I agree with most of your comment, but this part is misleading:
>> 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).
> 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