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Consider 2 urns (or bags), $U_1$ and $U_2$ with $n_1$ and $n_2$ balls. The balls can have $k$ different colors and we know the initial distribution of balls in both urns. Thus, we can calculate $$P_1 = 1 - \sum_{i=1}^k p_{1i}^2,$$ $$P_2 = 1 - \sum_{i=1}^k p_{2i}^2,$$ and $$P_3 = 1 - \sum_{i=1}^k p_{1i} p_{2i}$$ (with $p_{1i}$ and $p_{2i}$ as relative frequencies of balls with color $i$ in $U_1$ and $U_2$). $P_1$ and $P_2$ represent the probability for drawing randomly 2 different-colored balls from $U_1$ and $U_2$. $P_3$ is the probability for getting 2 different-colored balls after drawing 1 ball from each urn.

Now we randomly take a proportion of $m_{12}$ balls from $U_1$ and put it into $U_2$ and do the same (simultaneously) vice versa with $m_{21}$ balls from $U_2$.

What I'm interested in now is how $P_3$ changes after each round of random allocation.

Here are my thoughts ... We can calculate $P_1^{t_1}$ (and similar $P_2^{t_1}$),

$$P_1^{t_1} = \frac {\big( 1 - m_{12} \big)^2 P_1^{t_0} + 2 \big( 1 - m_{12} \big)m_{21}P_3^{t_0} + m_{21}^2 P_2^{t_0}}{\big( \big( 1 - m_{12} \big) + m_{21} \big)^2},$$ The superscripts $^{t_1}$ refer to the probabilities after 1 exchange of balls and thus refer to the system state that follows $^{t_0}$. It is obvious that $P_1^{t_\infty}$ and $P_2^{t_\infty}$ converges towards a common limit $x$ and that $P_3^{t_\infty}$ will converge towards $x$, too.

But which formula does the decay of $P_3$ follow? And how can $x$ be calculated?

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  • $\begingroup$ My intuition (but this needs some careful thinking) is that for $m_{12}$ and $m_{21}$ positive, after a sufficiently long time each ball is in each urn with probability $1/2$ and thus computing $P_3$ simply becomes computing the probability that considering all balls together, you don't pick the same colour twice. If this is true the easiest way to obtain this is probably through a coupling, and this will provide an upper bound for the convergence rate of $P_3$ in the right metric. Can you detail a bit how you obtained the formula for $P_1^{t_1}$? $\endgroup$ – Gâteau-Gallois Aug 12 at 19:38
  • $\begingroup$ Btw you implicitey assumed that $t_1$ is the time after the first random allocation, but never specified it. $\endgroup$ – Gâteau-Gallois Aug 12 at 19:41
  • $\begingroup$ @Gâteau-Gallois since there's too few space to completely reply to your question, I added an answer (which shouldn't be understood as an answer to my question) ... $\endgroup$ – Anti Aug 12 at 20:15
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Gallois, thanks a lot for your reply. I think it could be thought that way. And: Thanks a lot for you hint with $t_1$ and $t_0$. I edited my post.

So, how did I came up with $P_1^{t_1}$? I remove $m_{12} \times n_1$ balls from $U_1$ and thus $\big( 1 - m_{12} \big)$ balls remain in the urn. The probability for drawing 2 different balls from $U_1$ remains $P_1^{t_2}$. However, $U_1$ receives $m_{21} \times n_2$ balls from $U_2$. If I put the balls from $U_2$ into the bag with the remaining balls from $U_1$ and draw randomly 2 balls, they can be either:

  1. both originally (at $t_0$) from $U_1$ (see above),
  2. both brought from $U_2$ in the last exchange round or
  3. 1 could have been previously in $U_1$ and 1 could have been previously in $U_2$

Since we know the proportions for all 3 occurences as well as the probabilites of drawing different colors in all 3 scenaria, we can just calculate an expected values for $P_1^{t_1}$ by applying a formula for weighted averages (at least that's what I've thought).

That's the whole idea behind my question: I want to calculate the changes in $P_1$ and $P_2$ for given $m_{12}$ and $m_{21}$ without knowing the values of $p_{1i}$ and $p_{2i}$. And it seems to be only possible to calculate that changes with the introduction of $P_3$.

BTW - I just see that I also assumed $n_1 = n_2$ in the formula (see question) of my original post, so that I haven't added the population sizes to it.

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  • $\begingroup$ Ok thanks for the details, I think your computation is valid. What is your precise question now ? You want to compute $P_3^{\infty}$ to obtain both $P_1^{\infty}$ and $P_2^{\infty}$ ? With the decay rate as well ? $\endgroup$ – Gâteau-Gallois Aug 13 at 8:30
  • $\begingroup$ @Gâteau-Gallois I'm especially looking for a formula to calculate $P_3^{t_1}$ using the available measures ($P_1^{t_0}$, $P_1^{t_1}$, $P_2^{t_0}$, $P_2^{t_1}$ and $P_3^{t_0}$). However, I guess that this formula should contain a term giving the limit $x$ (my idea was to first derive $x$ and thus maybe come up more easily with the remainding formula for $P_3^{t_1}$). $\endgroup$ – Anti Aug 13 at 9:23
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I think I found an answer to my own question yesterday before I fell asleep ...

This is how the two urns look after 1 exchange of balls:

The 2 urns after 1 exchange of balls

Now let's first define

  • $\alpha:=m_{12} \times n_1^{t_0}$
  • $\beta:=m_{21} \times n_2^{t_0}$
  • $\gamma:= \big( 1 - m_{12} \big) \times n_1^{t_0}$
  • $\delta:= \big( 1 - m_{21} \big) \times n_2^{t_0}$

If we draw 2 balls randomly from both urns at $t_1$ we would get balls from

  • $U_1^{t_0}$ and $U_2^{t_0}$ with a probability of $\frac {\gamma}{\beta + \gamma} \times \frac {\delta}{\alpha + \delta}$ which represents the measure $P_3^{t_0}$.
  • $U_2^{t_0}$ and $U_2^{t_0}$ with a probability of $\frac {\beta}{\beta + \gamma} \times \frac {\delta}{\alpha + \delta}$ which represents the measure $P_2^{t_0}$.
  • $U_1^{t_0}$ and $U_1^{t_0}$ with a probability of $\frac {\gamma}{\beta + \gamma} \times \frac {\alpha}{\alpha + \delta}$ which represents the measure $P_1^{t_0}$.
  • $U_2^{t_0}$ and $U_1^{t_0}$ with a probability of $\frac {\beta}{\beta + \gamma} \times \frac {\alpha}{\alpha + \delta}$ which represents the measure $P_3^{t_0}$.

Thus:

$$P_3^{t_1} = \frac {\gamma}{\beta + \gamma} \times \frac {\delta}{\alpha + \delta} \times P_3^{t_0} + \\ + \frac {\beta}{\beta + \gamma} \times \frac {\delta}{\alpha + \delta} \times P_2^{t_0} + \\ + \frac {\gamma}{\beta + \gamma} \times \frac {\alpha}{\alpha + \delta} \times P_1^{t_0} + \\ + \frac {\beta}{\beta + \gamma} \times \frac {\alpha}{\alpha + \delta} \times P_3^{t_0}$$

Does it sound logical?

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