Fair division of a single homogeneous resource

The Envy-freeness rule says that the resource should be allocated such that no agent envies another agent. 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:

${\displaystyle \forall i:y_{i}=t/n}$

The utilitarian rule says that the sum of utilities should be maximized. Therefore, the utilitarian allocation is:

${\displaystyle y=\arg \max _{y}\sum _{i}u_{i}(y_{i})}$

The egalitarian rule says that the utilities of all agents should be equal. Therefore, we would like to select an allocation that satisfies:

${\displaystyle \forall i,j:u_{i}(y_{i})=u_{j}(y_{j})}$

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:

${\displaystyle y_{i}>0\implies u_{i}(y_{i})=\min _{j}u_{j}(y_{j})}$

Equivalently, the egalitarian allocation maximizes the minimum utility:

${\displaystyle y=\arg \max _{y}\min _{i}u_{i}(y_{i})}$

The utilitarian and egalitarian rules may lead to the same allocation or to different allocations, depending on the utility functions. Some examples are illustrated below.

Suppose all agents have the same utility function, ${\displaystyle u}$, but each agent ${\displaystyle i}$ has a different initial endowment, ${\displaystyle x_{i}}$. So the utility of each agent ${\displaystyle i}$ is given by:

${\displaystyle u_{i}(y_{i})=u(x_{i}+y_{i})}$

If ${\displaystyle u}$ is a concave function, representing diminishing returns, then the utilitarian and egalitarian allocations are the same - trying to equalize the endowments of the agents. For example, if there are 3 agents with initial endowments ${\displaystyle x=2,4,9}$ and the total amount is ${\displaystyle t=8}$, then both rules recommend the allocation ${\displaystyle y=5,3,0}$, since it both pushes towards equal utilities (as much as possible) and maximizes the sum of utilities.

In contrast, if ${\displaystyle u}$ 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: ${\displaystyle y=0,0,9}$.[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.

Suppose there is a common utility function ${\displaystyle u}$, but each agent has a different coefficient ${\displaystyle a_{i}}$ representing this agent's productivity. So the utility of each agent ${\displaystyle i}$ is given by:

${\displaystyle u_{i}(y_{i})=a_{i}\cdot u(y_{i})}$

Here, the utilitarian and egalitarian approaches are diametrically opposed.[1]^: 46–47

• The egalitarian allocation gives more resources to the less productive agents, in order to compensate them and let them reach a high utility level:

  ${\displaystyle a_{i}>a_{j}\implies y_{j}>y_{i}}$

• The utilitarian allocation gives more resources to the more productive agents, since they will use the resources better:

  ${\displaystyle a_{i}>a_{j}\implies y_{i}>y_{j}}$