TITLE: What are the odds of a tie in a random election with k candidates?
QUESTION [8 upvotes]: 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.

REPLY [7 votes]: In general, the probability is
$$
\frac{1}{k^N}\sum_{j=2}^k{k \choose j}\sum_{m=0}^{\lfloor N/j\rfloor}{N \choose mj}\frac{(mj)!}{(m!)^j}\sum_{\substack{(a_1,\dots,a_{k-j})\\a_1+\dots+a_{k-j}=N-mj\\0\leq a_i<m}}{N-mj\choose a_1,\dots,a_{k-j}}
$$
where for some choices of $(N,k,m,j)$ the innermost sum is $0$. The variable $m$ is the same as in Robert Israel's answer. The winning candidates each receive $m$ votes. The variable $j$ is the number of winning candidates. Thus, we choose $j$ winning candidates out of the original $k$ candidates in ${k \choose j}$ ways, to each receive $m$ votes. Then we choose $mj$ voters out of the $N$ voters in ${N \choose mj}$ ways to vote for the winning candidates. Then we assign the $mj$ votes that go to the winning candidates to the $j$ winning candidates in $\frac{(mj)!}{(m!)^j}$ ways. Now, there are $N-mj$ votes left to go to the $k-j$ non-winning candidates, and each non-winning candidate receives a maximum of $m-1$ votes, thus the conditions on the innermost sum. The formula is also valid for $k>N$.