Comment by akramachamarei

Allow me to propose a model for this score-based ordering with fuzziness. (Perhaps we can call this problem probabilistic rasterization.)

The final output of an execution of the system, given a static, complete set of applicants is a particular ordering of applicants. Since lottery is involved, there are multiple acceptable orderings for a given input set. The question is to define a set of criteria to classify acceptable orderings, and a desired probability distribution of orderings, which can be satisfied by an algorithm for a maximal proportion of inputs.

For example, given a set of applicants A with score function F, we notate an ordering relation R(x,y) such that, given a limited number of seats, applicant y will be admitted before applicant x. For shorthand, x < y means R(x,y).

Possible acceptance criteria for an ordering R may include:

(1) Given some d in the codomain of F (presumably a group), FOR ALL x,y in A, if F(x) + d ≤ F(y), then x < y

Possible criteria for the distribution of orderings may include:

(1) FOR ALL x,y in A, if F(x) = F(y) then P(x < y) = P(x > y)