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Fair division of a single homogeneous resource

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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 

Setting

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In a certain society, there are:

  • units of some divisible resource.
  • agents with different "utilities".
  • The utility of agent is represented by a function ; when agent receives units of resource, he derives from it a utility of .

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 - is the number of buildings that agent can build using units of wood.
  • The resource is a medication, the agents are patients, and the utility functions represent their chance of recovery - is the probability of agent to recover by getting 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 such that:

Allocation rules

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Envy-free

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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:

Utilitarian

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

Egalitarian

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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. Some examples are illustrated below.

Examples

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Common utility and unequal endowments

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Suppose all agents have the same utility function, , but each agent has a different initial endowment, . So the utility of each agent is given by:

If 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 and the total amount is , then both rules recommend the allocation , since it both pushes towards equal utilities (as much as possible) and maximizes the sum of utilities.

In contrast, if 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.

Constant utility ratios

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Suppose there is a common utility function , but each agent has a different coefficient representing this agent's productivity. So the utility of each agent is given by:

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

Properties of allocation rules

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See also

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References

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  1. ^ a b c d e Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231.