I examine a multinomial distribution with parameters $\\vec{p} = [p_1, p_2, \\ldots, p_s]$. I denote the cells by $C = \{ c_1, c_2, \\ldots, c_s\}$. In this setting $p_{[i]}$ is the $i^{\\text{th}}$ highest probability. That is $p_{[1]} \\geq p_{[2]} \\geq \ldots \\geq p_{[s]}$. Of course, $\\sum_{i = 1}^s p_i = \\sum_{i = 1}^s p_{[i]} = 1$.

We have $r$ observations from this multinomial distribution and want to determine the cell $c_{[1]}$ (i.e., the cell that has the highest probability of appearing in some observation). To do that, we use the following simple rule: we select the cell with the highest number of observations (breaking ties arbitrarily). That is, if $\\vec{N} = [N_1, N_2, \ldots, N_s]$ are the number of observations per cell for the $r$ total observations, then we select the cell $c_k$ for the $k$ that gives the maximum $N_k$.

The rule described gives the correct cell with probability:

$$ \\Pi(r) = \\sum_{l = 1}^s \\frac{1}{l} \\sum_{n = 1}^r \\sum_{{\\cal L}} \\underset{\\sum_{i \\in \\bar{{\\cal L}}} k_i + l \\cdot n = r}{\sum_{0 \\leq k_i \\leq n - 1, i \\in \\bar{{\\cal L}}}} \\left[\\frac{r!}{(n!)^l \\cdot \\prod_{j \\in \\bar{{\\cal L}}} k_j!} \\cdot \\prod_{z \\in {\\cal L}} p_z^n \\cdot \\prod_{w \\in \\bar{{\\cal L}}} p_w^{k_w}\\right] $$

where $l$ is the number of winners (cells with most number of ``votes''), $n$ is the number of votes for each winner, and ${\\cal L}$ are subsets of $\{1, 2, \\ldots, s\}$ including $[1]$ ($ = k$ maximizing $N_k$) with $|{\\cal L}| = l$. Finally, $\bar{\\cal L} = \{1, 2, \\ldots, s\} - {\\cal L}$.

Two questions:

- can $\\Pi(r)$ be expressed in a simpler way?
- is there any work that describes how I prove that $\\Pi(r)$ is non-decreasing on $r$?