# Fair division of a single homogeneous resource

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.$^{:sub.2.5}$

## Setting

In a certain society, there are:
* $t$ units of some divisible resource.
* $n$ agents with different "utilities".
* The utility of agent $i$ is represented by a function $u_{i}$; when agent $i$ receives $y_{i}$ units of resource, he derives from it a utility of $u_{i}(y_{i})$.
This setting can have various interpretations. For example:$^{:44}$
* The resource is wood, the agents are builders, and the utility functions represent their productive power - $u_{i}(y_{i})$ is the number of buildings that agent $i$ can build using $y_{i}$ units of wood.
* The resource is a medication, the agents are patients, and the utility functions represent their chance of recovery - $u_{i}(y_{i})$ is the probability of agent $i$ to recover by getting $y_{i}$ doses of the medication.
In any case, the society has to decide how to divide the resource among the agents: it has to find a vector $y_{1},\dots ,y_{n}$ such that: $y_{1}+\cdots +y_{n}=t$

## Allocation rules
### Envy-free

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

### Utilitarian

The utilitarian rule says that the sum of utilities should be maximized. Therefore, the utilitarian allocation is:
       $y=\arg \max _{y}\sum _{i}u_{i}(y_{i})$

### Egalitarian

The egalitarian rule says that the utilities of all agents should be equal. Therefore, we would like to select an allocation that satisfies:
       $\forall i,j:u_{i}(y_{i})=u_{j}(y_{j})$
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:
       $y_{i}>0\implies u_{i}(y_{i})=\min _{j}u_{j}(y_{j})$
Equivalently, the egalitarian allocation maximizes the minimum utility:
       $y=\arg \max _{y}\min _{i}u_{i}(y_{i})$
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.

## Examples
### Common utility and unequal endowments

Suppose all agents have the same utility function, $u$, but each agent $i$ has a different initial endowment, $x_{i}$. So the utility of each agent $i$ is given by:
       $u_{i}(y_{i})=u(x_{i}+y_{i})$
If $u$ 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 $x=2,4,9$ and the total amount is $t=8$, then both rules recommend the allocation $y=5,3,0$, since it both pushes towards equal utilities (as much as possible) and maximizes the sum of utilities.
In contrast, if $u$ 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: $y=0,0,9$.$^{:45}$ 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.

### Constant utility ratios

Suppose there is a common utility function $u$, but each agent has a different coefficient $a_{i}$ representing this agent's productivity. So the utility of each agent $i$ is given by:
       $u_{i}(y_{i})=a_{i}\cdot u(y_{i})$
Here, the utilitarian and egalitarian approaches are diametrically opposed.$^{:46–47}$
* The egalitarian allocation gives more resources to the less productive agents, in order to compensate them and let them reach a high utility level:
       $a_{i}>a_{j}\implies y_{j}>y_{i}$
* The utilitarian allocation gives more resources to the more productive agents, since they will use the resources better:
       $a_{i}>a_{j}\implies y_{i}>y_{j}$

## Properties of allocation rules

* Resource-monotonicity: the envy-free rule and the egalitarian rule are always resource-monotonic. The utilitarian rule is resource-monotonic when all utility functions are concave functions, representing diminishing returns; but, when some utility functions are convex functions, representing increasing returns, the utilitarian rule might be not resource-monotonic.$^{:47}$