A real-world motivation for this question is given below. But first let me recall what the Jefferson-D'Hondt “greatest divisors” method of proportional allotment (often used in electoral systems to elect an assembly) means in mathematical terms:

**The D'Hondt method:** Given a (sufficiently general) point $P$ on the $(n-1)$-dimensional simplex, viꝫ. $P = (p_1,\ldots,p_n)$ with $p_i$ nonnegative and summing to $1$ (the “votes”), and given $N \geq 1$ (the number of “seats” to allot), we define a point $\tau_N(P)$ on the “$N$-discrete $(n-1)$-dimensional simplex”, that is, an $n$-tuple $(m_1,\ldots,m_n)$ of natural numbers with sum $N$, as follows: $\tau_N(P)$ is the $n$-tuple $(m_1,\ldots,m_n)$ such that the $m_i = \lfloor t p_i\rfloor$ have sum exactly $N$, for some positive real $t$. In more geometric terms¹, consider the half-ray $\{tP : t>0\}$ through $P$ in $\mathbb{R}^n$ and see which cube $\prod_{i=1}^n [m_i,m_i+1]$ with $m_1+\cdots+m_n=N$ it intersects (even more visually: we are looking at the arrangement of unit cubes corresponding to the $N$-discrete simplex, and the division of the simplex given by $\tau_N$ corresponds to looking at this arrangement from the origin). See the Wikipedia article linked above for more computational descriptions. This $\tau_N(P)$ is well-defined except for cases of “ties” in the votes, which are contained in a finite set of hyperplanes in the simplex, and since I will be considering probabilities in my question, these are irrelevant here.

The D'Hondt method has the following interesting property, for which I do not know a reference in the literature (but here is a proof that was given to me long ago on `sci.math.research`

): if $P$ is drawn uniformly on the simplex, then $\tau_N(P)$ follows a uniform distribution on the $N$-dimensional simplex. (In slightly handwavy terms: “if you don't know anything about the result of the vote, you don't know anything about the attribution of the seats”.) This is what leads me to hope that the following question might not be too untrackable:

**General question:** Suppose $P$ is drawn from a Dirichlet distribution on the simplex, what distribution does $\tau_N(P)$ follow on the $N$-discrete simplex? What are the expected values of its coordinates?

Now since this question might turn out to be too difficult for explicit computations, here is a simpler form that I hope is made more manageable by considering Dirichlet distributions where all parameters are $1$ except one which is integer:

**Hopefully simpler version:** Let $1\leq k\leq n$, and consider the map $\pi\colon (p_1,\ldots,p_n) \mapsto (p_1,\ldots,p_{n-k}, p_{n-k+1}+\cdots+p_n)$ from the $(n-1)$-dimensional simplex to the $(n-k)$-dimensional simplex (which sums the last $k$ coordinates), so that $\pi(P)$ follows a Dirichlet distribution with parameters $(1,\ldots,1,k)$ if $P$ is uniform: what is the expected value of (the last coordinate of) $\tau_N(\pi(P))$ if $P$ is uniform?

Note that if we call $\pi_N$ the analogous map $(m_1,\ldots,m_n) \mapsto (m_1,\ldots,m_{n-k}, m_{n-k+1}+\cdots+m_n)$ on the $N$-discrete simplices, then $\pi_N(\tau(P))$ has expected value $(\frac{N}{n},\ldots,\frac{N}{n},\frac{kN}{n})$ since $\tau(P)$ is uniform by the property remarked above. The expected value of the last coordinate of $\tau_N(\pi(P))$ is greater than $\frac{kN}{n}$ and the question is “by how much?”.

**Real-life motivation:** European parliament elections are happening in 2024, and France allots its seats by the D'Hondt method. One of the questions that come up in the political debate is how many seats a certain set of $k$ political parties might gain if they run under a single list instead of $k$ separate lists.

**Experimental results:** By sampling points on the unit simplex and guessing the denominators, I have arrived at the following results for expected value of the last coordinate of $\tau_N(\pi(P))$ minus $\frac{kN}{n}$:

for $(n,k)=(3,2)$ we find $\frac{N}{6(N+1)}$,

for $(n,k)=(4,3)$ we find $\frac{N}{4(N+1)}$,

for $(n,k)=(4,2)$ we find $\frac{N(3N+5)}{12(N+1)(N+2)}$,

for $(n,k)=(5,4)$ we find $\frac{N(9N^2+17N+7)}{30(N+1)^3}$,

for $(n,k)=(5,3)$ we find $\frac{N(24N^3+114N^2+177N+89)}{60(N+1)^2(N+2)^2}$,

for $(n,k)=(5,2)$ we find $\frac{N(3N^2+13N+13)}{10(N+1)(N+2)(N+3)}$

(in other words, the quantity above is the expected advantage that $k$ party lists out of $n$ have in grouping together if there are $N$ seats to fill and the result of the election is random).

**Integral expression:** [added 2023-05-11] I should mention that the distribution of $\tau_N(P)$ in answer to the “general question” is proportional to the following integral expression (for $(m_1,\ldots,m_n)$ an $n$-tuple of natural numbers with sum $N$):
$$
\sum_{j=1}^n m_j \int_{\prod_{i\neq j}[m_i,m_i+1]}
\frac{x_1^{\alpha_1-1}\cdots x_n^{\alpha_n-1}}{(x_1+\cdots+x_n)^{\alpha_1+\cdots+\alpha_n}}
dx_1\cdots dx_n
$$
(a straightforward but tedious translation of the description of the D'Hondt method as projecting cubes to the simplex: the sum over $j$ ranges over the facets of the cube $\prod_{i=1}^n[m_i,m_{i+1}]$); here, $x_j$ in the integral should be interpreted as being identically $m_j$, and $dx_j$ should be omitted. This is the integral which has been used (laboriously) to compute the “experimental results” indicated above.

- Yet another description consists of noting that (again, for sufficiently general $P$), the ray $\{tP\}$ cuts the walls of the subdivision of space into unit cubes in a pattern that forms a Sturmian sequence, and the D'Hondt method counts the letters of each kind within the $N$ first letters in this sequence.