Fair division of a single homogeneous resource is one of the simplest settings in fair division problems.
There is a single resource that should be divided between several people.
The challenge is that each person derives a different utility from each amount of the resource.
Hence, there are several conflicting principles for deciding how the resource should be divided.
A primary conflict is between efficiency and equality.
Efficiency is represented by the utilitarian rule, which maximizes the sum of utilities; equality is represented by the egalitarian rule, which maximizes the minimum utility.
For example:[1]: 44 In any case, the society has to decide how to divide the resource among the agents: it has to find a vector
In the case of a single homogeneous resource, it always selects the allocation that gives each agent the same amount of the resource, regardless of their utility function: The utilitarian rule says that the sum of utilities should be maximized.
Therefore, the utilitarian allocation is: The egalitarian rule says that the utilities of all agents should be equal.
Therefore, we would like to select an allocation that satisfies: However, such allocation may not exist, since the ranges of the utility functions might not overlap (see example below).
To ensure that a solution exists, we allow different utility levels, but require that agents with utility levels above the minimum receive no resources: Equivalently, the egalitarian allocation maximizes the minimum utility: The utilitarian and egalitarian rules may lead to the same allocation or to different allocations, depending on the utility functions.
is a concave function, representing diminishing returns, then the utilitarian and egalitarian allocations are the same - trying to equalize the endowments of the agents.
is a convex function, representing increasing returns, then the egalitarian allocation still pushes towards equality, but the utilitarian allocation now gives all the endowment to the richest agent:
[1]: 45 This makes sense, for example, when the resource is a scarce medication: it may be socially best to give all medication to the patient with the highest chances of curing.
is given by: Here, the utilitarian and egalitarian approaches are diametrically opposed.