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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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    Sorry for the confusion. I fixed step C to make it clearer. (The relevant model in the simulation is "jubilee-for-landless").2012-04-25

1 Answers 1

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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.