I'm not a mathematician but I working on a problem that feels like it an example of a more general kind of problem and I'm hoping that someone might be able to point me in the right direction.

The problem is trying to find a convex combination of different ways of ranking n items that satisfy certain constraints. Let me be more concrete: Suppose we have $n$ individuals with endowments $b_1 > b_2 > b_3 \ldots b_n$. There are positions $1 \dots n$, each of which gives a value of $v_1 > v_2 > v_3 \ldots v_n \ge 0$ that the individuals can be assigned to.

I want to create a non-deterministic algorithm for assigning individuals to positions that is in some sense "equitable." so that for each individual, their expected proportional value from their positions is equal to their proportional endowment. E.g., suppose we have some method that assigns individual $i$ to position with 1 with probability $p_{i1}$, to position 2 with probability $p_{i2}$ and so on, I want it so that the algorithm generates an allocation such that:

$\forall i, \frac{\sum_{j=1}^n p_{ij} v_j}{\sum_{j=1}^n v_j} = \frac{b_i}{\sum_{j=1}^n b_j} $

### Some observations / thoughts:

For this to work even in the n=2 case, we need to constrain $b_1$ to that it isn't proportionally larger than $v_1$ (otherwise even always placing individual 1 at position 1 wouldn't be enough).

I originally thought this could be framed as a linear programming problem, where the goal is to find weights for each of the $n!$ possible orderings. Maybe this would work, but it would be computationally infeasible.

A particularly nice approach might be one that sequentially assigns positions by having each remaining individual "buy" probability shares with their budget and then draw a winner. Unfortunately, this doesn't have the equitable property above, but I was thinking that perhaps if we thought of the allocation as happening repeatedly, we could give the "losers" (payoff from position too small) a bigger endowment, taken from the "winners" in such a way that expected payoff converges to the equitable outcome.

Anyway, thanks for reading this far and I appreciate any comments, suggestions, answers etc.