An Optimal Fair Job Assignment Problem
We study the problem of how to allocate a set of indivisible objects like jobs or houses and an amount of money among a group of people as fairly and as efficiently as possible. A particular constraint for such an allocation is that every person should be assigned with the same number of objects in his or her bundle. The preferences of people depend on the bundle of objects and the quantity of money they take. We propose a solution to this problem, called a perfectly fair allocation. It is shown that every perfectly fair allocation is efficient and envy-free, income-fair and furthermore gives every person a maximal satisfaction. Then we establish a necessary and sufficient condition for the existence of a perfectly fair allocation. It is shown that there exists a perfectly fair allocation if and only if an associated linear program problem has a solution. As a result, we also provide a finite method of computing a perfectly fair allocation.
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