one example on game playing with some details, how this average step was calculated? I have one example in my TA notes:
We have a two players games. two players will be agree on number $n$. game start with $x=2$. first player select by Coin flipping and start game. after that each of players arbitrary do one of the following: $x=x^2$ or $x=x^3$.
for example if "David" start game and choose $x^3$ then $x$ will be equals to $8$. after playing of each players that condition $x>n$ occurred, the game will be stopped and the last player announced as winner. at average this game ended after $O(\log \log n)$ steps.
This is very nice example, but how we can detect this average order can be achieved?

$x$ is replaced with $x^2$ or $x^3$.
That is, $\log x$ is replaced with $2\log x$ or $3\log x$.
That is, $\log \log x$ is replaced with $\log 2+\log\log x$ or $\log 3+\log\log x$. So for $\log\log x$, we have a simple addition of random vars. In particular, the stopping time is between $\approx\frac{\log\log n-\log\log 2}{\log3}$ and $\approx\frac{\log\log n-\log\log 2}{\log2}$.
