We mathematicians have as much intuition as anyone else. I think of this as a random walk on a very large but finite state space with positive n-step transition probabilities between all states for all large enough n. It's an aperiodic irreducible Markov chain. So we should have ergodicity. My intuition now tells me that this means every state is positive recurrent, i.e. for each person x should see x hoard all of the money at some step. Given enough time, a very long time, we should see everyone eventually have all of the money and eventually have none of it. The intermediate steps may look very skewed, but we should see all steps happen.
I'm a bit wary about concluding that just because there's positive probability between all states that we should see all states. At least for infinite state spaces, we know this fails (Pólya's theorem on random walks in dimension 3 or higher). But my intuition tells me that the finite state space should mean that all states are positive recurrent.
Edit: Aha, here's the relevant theorem which confirms my intuition: in a finite-state Markov chain, every closed class is recurrent:
Untangling the jargon, if you have a random process in which you can go from any state to any other state with positive probability after a certain number of steps, and there are only finitely many states, then the probability of getting to any state as the number of steps goes to infinity is 1.
In spite all the states have a probability 1 to be reached infinitely many times, the frequency of each kind of state in the chain is not equal. Some states will have a much higher frequency than the others.
In this case, the states where everyone has a number of coins that is between average-2 and average+2 will be extremely infrequent. They will be reached, but you probably have to wait a lot of time to see them.
But there will be some family of cases that appear very frequently after enough time. I'm not sure if the most common case is
1) Some of the persons has 99% of the money and the rest have a tiny amount of money
2) A few persons have almost all the money.
3) There is some kind of heavy tail distribution were everyone is expected to have some money. If you order them by money you will see something like a wiggly line.
I vote for 1), but I'm not sure at all.
This is similar to the typical thermodynamics problems. Imagine that you have 100 boxes in a line, were you can distribute 100000 "atoms" (or coins). In each step the coins can go to neighbor box at random. This is not the same problem, so it has a different solution.
But it's also ergodic and you can see all kind of weird distribution of the coins/atoms if toy wait enough time.
If you wait enough time you can see for example that all the coins/atom went to the leftmost box and all the other are empty. But this is not common at all.
Most of the time all the boxes will have a number of coins/atoms that is close to the average. (Not exactly the average obviously.)
But this is a different problem with the same state space (100 boxes/persons, 10000 atoms/coins). The rules in the Markov chain are different, so the expected frequency of the states is different.
In the thermodynamic problem, the most frequent states are when the atoms are almost evenly distributed. In the problem of the article, I suspect that the most frequent state is when someone has almost all the money.
I'm a bit wary about concluding that just because there's positive probability between all states that we should see all states. At least for infinite state spaces, we know this fails (Pólya's theorem on random walks in dimension 3 or higher). But my intuition tells me that the finite state space should mean that all states are positive recurrent.
Edit: Aha, here's the relevant theorem which confirms my intuition: in a finite-state Markov chain, every closed class is recurrent:
https://math.stackexchange.com/questions/543051/markov-chain...
Untangling the jargon, if you have a random process in which you can go from any state to any other state with positive probability after a certain number of steps, and there are only finitely many states, then the probability of getting to any state as the number of steps goes to infinity is 1.