Consider an election with $N$ voters and $k$ candidates, where each voter votes randomly for one of the candidates. What are the odds of a tie?

Here "tie" means that multiple candidates get the highest number of votes, *not* necessarily that all candidates get the same number of votes.

For $k = 2$ the answer is given by ${N \choose {N/2}} \frac{1}{2^N}$.

I'm not sure how to generalize to $k > 2$. If a closed formula doesn't exist, can we still say something about the asymptotic behavior as $k$ grows?

Edit: asking for "asymptotic behavior" was ambiguous. The question I'm most interested in is: for $k \ll N$, what can we say about how the odds change if $k$ increases by $1$ (or, if it's a cleaner answer, if $k$ doubles). Perhaps it's incorrect to call this asymptotic behavior at all! And, unfortunately, I'm not sure what "$\ll$" should mean in this context. I suppose that's part of the question as well.