I have a next model.
Let two balls be colored in different colors: $\color{red}{\text{red}}$ and $\color{blue}{\text{blue}}$.
Colors have intensities, from $0$ upward. Suppose, the maximum intensity of $\color{red}{\text{red}}$ color is $i_r$, and for $\color{blue}{\text{blue}}$ it is $i_b$. And the initial intensities are picked according to $\sim \text{Uni}[0, i_r]$ and $\sim \text{Uni}[0, i_b]$.
The evolution of the model is as follows:
- denote, at each step the intensities to be $c_r$ and $c_b$ (like positive and negative numbers), then each ball is independently colored according to $\sim \text{Uni}[-c_b, c_r]$, where negative numbers are treated as intensities of $\color{blue}{\text{blue}}$.
- if both balls are colored in the same color, we stop.
Example:
In example we started with two balls, one colored $\color{blue}{\text{blue}}$ with intensity $0.25$ (depicted as negative number) and one colored $\color{red}{\text{red}}$ with intensity $0.75$.
Next, we picked new intensities and possibly new colors for both balls. Let say the previously $\color{blue}{\text{blue}}$ ball became a $\color{red}{\text{red}}$ with intensity $0.49$. And previously colored $\color{red}{\text{red}}$ is now $\color{blue}{\text{blue}}$ one with intensity $0.16$.
At last, both balls were colored $\color{red}{\text{red}}$, one with intensity $0.08$ and one with $0.31$.
I'm interested in calculating the expected number of steps until both balls are colored in the same color as a function of $i_r$ and $i_b$. Or, since wlog, we can assume $i_r = 1$, only $i_b$.
What I have thought:
Let blue intensity be $x$ and red intensity be $1-x$, then the probability to color both balls at the same step by the same color is $1 - 2x + 2x^2$. The expression is minimized at $x = \frac{1}{2}$.
Therefore, with probability of at least $\frac{1}{2}$ we are immediately done. Hence, the upper bound on $\mathbb{E}$ is $2$ - the number of fair coin tosses before the Heads are tossed.
But what is the approach to get a precise number for expected steps?