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During a research related to economy of land ownership, I ran into an interesting probability problem:

There are N citizens and N land-plots.

A. Initially, each land-plot is given to a citizen selected at random.

B. Then, each land is sold with probability $q$. In case it is sold, the buyer is selected at random.

C. Then, each citizen that had land before B, and has no land now, is given back ONE OF the land-plots he had before step B, selected at random. This returning is repeated until all citizens that had land before B have some land now.

This process (steps B-C) is repeated.

The question is: how will the expected number of citizens with no land ("landless") change as a function of time?

I ran some simulations, and found out that it decreases like $\frac{1}{ A t + B}$, where t is time, and A, B are linear regression coefficients.

However, I would like to find a closed formula.

Currently, the only thing I managed to find is the expected number of landless at time $0$, since the probability to be landless is the probability to get no land at the initial division: $(1 - \frac{1}{N}) ^ N$.

I have no idea how to continue from here. Any help will be appreciated!

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Do you consider in the "landless" people also those who have history of not owning a land in the past (but who may, currently own one)? – T. Eskin Apr 24 '12 at 19:08
When you say "each citizen that had land before, and has no land now, is given back his original land". I'm interpreting this as meaning: if they have lost some (but not all) of their land, nothing happens, but if they lose all of their land, they get back everything they have lost. Is that correct? – Emil Apr 24 '12 at 20:35
I consider as "landless" people that currently own no land, each time before step B begins. Emil, your interpretation is correct. The idea is to make sure eventually each citizen has land. I want to calculate how fast this will happen. – Erel Segal-Halevi Apr 25 '12 at 3:44
BTW, here is the simulation I used: – Erel Segal-Halevi Apr 25 '12 at 3:46
"... their original land" meaning land that they had at the end of step A? So somebody who had no land after step A, acquired some later, and then lost that land, won't get it back? – Robert Israel Apr 25 '12 at 3:52
up vote 1 down vote accepted

I found the answer!

If $M_t$ is the number of landless citizens in time t, then:

$$E[M_{t+1}] = \frac {M_t} {1 + \left( \frac1{e-1} + \frac{\sqrt{N/M_t}-1}{\sqrt{2}} \right)^{-1}}$$

The proof is about 2 pages long, I hope to publish it when it's ready.

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