fair chore allocation with qualification


Background

There are $ n$ indivisible heterogenous bads (chores).
There are $ m$ agents.
The subjective utility functions of the agents are additive and identical. e.g $ \forall X,i,j\ V_i(X) = V_j(X)$ .
The agents have different entitlements over the chores.
Finally, there is an additional constraint: there are chores that some agents cannot be assigned to perform (i.e the qualification constraint)

The task

I would like to be able to:

  1. define some fairness criterion that matches the problem
  2. find an algorithm that provides an allocation that implements it.

I believe that due to the qualification constraint some standard fairness criteria, like $ EF1$ , may not exist. This is why I am also asking for a fairness criterion.