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.[1]:sub.2.5

## Contents

* 1 Setting
* 2 Allocation rules
* 2.1 Envy-free
* 2.2 Utilitarian
* 2.3 Egalitarian
* 3 Examples
* 3.1 Common utility and unequal endowments
* 3.2 Constant utility ratios
* 4 Properties of allocation rules
* 5 See also
* 6 References

## Setting

In a certain society, there are:

* [math]\displaystyle{ t }[/math] units of some divisible resource.
* [math]\displaystyle{ n }[/math] agents with different "utilities".
* The utility of agent [math]\displaystyle{ i }[/math] is represented by a function [math]\displaystyle{ u_i }[/math]; when agent [math]\displaystyle{ i }[/math] receives [math]\displaystyle{ y_i }[/math] units of resource, he derives from it a utility of [math]\displaystyle{ u_i(y_i) }[/math].

This setting can have various interpretations. For example:[1]:44

* The resource is wood, the agents are builders, and the utility functions represent their productive power - [math]\displaystyle{ u_i(y_i) }[/math] is the number of buildings that agent [math]\displaystyle{ i }[/math] can build using [math]\displaystyle{ y_i }[/math] units of wood.
* The resource is a medication, the agents are patients, and the utility functions represent their chance of recovery - [math]\displaystyle{ u_i(y_i) }[/math] is the probability of agent [math]\displaystyle{ i }[/math] to recover by getting [math]\displaystyle{ y_i }[/math] 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 [math]\displaystyle{ y_1,\dots,y_n }[/math] such that: [math]\displaystyle{ y_1+\cdots+y_n = t }[/math]

## 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:

[math]\displaystyle{ \forall i: y_i = t/n }[/math]

### Utilitarian

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

[math]\displaystyle{ y = \arg\max_y \sum_i u_i(y_i) }[/math]

### Egalitarian

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

[math]\displaystyle{ \forall i,j: u_i(y_i) = u_j(y_j) }[/math]

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:

[math]\displaystyle{ y_i\gt 0 \implies u_i(y_i) = \min_j u_j(y_j) }[/math]

Equivalently, the egalitarian allocation maximizes the minimum utility:

[math]\displaystyle{ y = \arg\max_y \min_i u_i(y_i) }[/math]

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, [math]\displaystyle{ u }[/math], but each agent [math]\displaystyle{ i }[/math] has a different initial endowment, [math]\displaystyle{ x_i }[/math]. So the utility of each agent [math]\displaystyle{ i }[/math] is given by:

[math]\displaystyle{ u_i(y_i) = u(x_i + y_i) }[/math]

If [math]\displaystyle{ u }[/math] 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 [math]\displaystyle{ x=2,4,9 }[/math] and the total amount is [math]\displaystyle{ t=8 }[/math], then both rules recommend the allocation [math]\displaystyle{ y=5,3,0 }[/math], since it both pushes towards equal utilities (as much as possible) and maximizes the sum of utilities.

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

### Constant utility ratios

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

[math]\displaystyle{ u_i(y_i) = a_i\cdot u(y_i) }[/math]

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:

[math]\displaystyle{ a_i\gt a_j \implies y_j\gt y_i }[/math]
* The utilitarian allocation gives more resources to the more productive agents, since they will use the resources better:

[math]\displaystyle{ a_i\gt a_j \implies y_i\gt y_j }[/math]

## 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.[1]:47

## See also

* Utilitarianism
* Egalitarianism

## References

1. ↑ 1.0 1.1 1.2 1.3 1.4 Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231.

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