Earlier, I asked a question about a series of competitions:
A series of matches are held between n identical competitors. Each is won by one of the n with equal probability (no ties). I'm looking for a probabilistic description of the outcome when looking at the first player to win 1, 2, ... matches.
Now I am able to ask my full question. In this scenario, what is the expected time until all players have been in the lead at least once? (Say, with probability at least 1/2 -- though other measures of central tendency are probably fine if they're easier to work with.)
For $n=1$ this takes only one match. For $n=2$ this is just a 1-D random walk (essentially, start at 1 and see how many steps you take until you become negative); I find that after 9 matches both players have been in the lead with probability 65/128 > 50%. Even with $n=3$ it gets hard to find an exact answer because of the exponential blow-up (it is at least several hundred).
So what I'm really looking for is an asymptotic, since my interests lie in $n$ much larger than 3.