Egalitarian equivalence (EE) is a criterion of fair division. In an egalitarian-equivalent division, there exists a certain "reference bundle"
Z
Z
The EE fairness principle is usually combined with Pareto efficiency. A PEEEA is an allocation that is both Pareto efficient and egalitarian-equivalent.
A set of resources are divided among several agents such that every agent
i
Xi
i
\succeqi
X\simiY
X\succeqiY\succeqiX
An allocation is called egalitarian-equivalent if there exists a bundle
Z
i
Xi\simiZ
An allocation is called PEEEA if it is both Pareto-efficient and egalitarian-equivalent.
The EE criterion was introduced by Elisha Pazner and David Schmeidler in 1978.[1] [2]
Previously, the main fairness criterion in economics has been envy-freeness (EF). EF has the merit that it is an ordinal criterion --- it can be defined based only on individual preference-relations; it does not need to compare utilities of different agents, or to assume that the agents' utility functions are normalized. However, EF might be incompatible with Pareto efficiency (PE). In particular, in a standard economy with production, there may be no allocation which is both PE and EF.[3]
EE, like EF, is an ordinal criterion --- it can be defined based only on individual preference-relations. However, it is always compatible with PE --- a PEEEA (PE and EE Allocation) always exists, even in production economies. Pazner and Schmeidler informally describe a PEEEA as follows:
"Consider the case where there are two consumers and two commodities (but note that every step in the argument carries over to any number of agents and commodities...). Suppose that each consumer is given precisely half the total endowments. This egalitarian distribution will in general not be PE. Consider the ray in commodity space that goes from the origin through the vector of aggregate endowments. The egalitarian distribution is represented by each man being given the same bundle along this ray.
If the egalitarian distribution is not PE, then (by monotonicity and continuity of preferences) moving each man slightly up along the ray yields distributions of utilities that are still feasible, since the starting utility distribution is in the interior of the utility possibility set. In particular, if we simultaneously move each man up along the commodity ray in precisely the same manner, we eventually shall hit a utility distribution that lies on the utility possibility frontier. This means that there exists a Pareto-efficient allocation that is equivalent from the viewpoint of each consumer to the hypothetical (nonfeasible) distribution along the ray that would give to each consumer the same bundle (which, by being strictly greater than the egalitarian distribution of the aggregate endowments, is itself not feasible). This PE allocation is thus equivalent to the egalitarian distribution in the hypothetical (larger than the original) economy...
The resulting set of allocations is what we call the set of Pareto-efficient and egalitarian-equivalent allocations (PEEEA). It is a restriction of the Pareto set of the economy to those allocations having the specified equity property that their underlying utility levels distribution could have been generated by some egalitarian economy.".
As a special case, assume that there is a finite number of homogeneous divisible goods. Let
W
r\in[0,1]
rW
r
W
Suppose the preference-relation of each agent
i
Vi
Vi(rW)=r
i
Vi(Xi)=r
Note that the maximin principle depends on numeric utility. Therefore, it cannot be used directly with ordinal preference-relations. The EE principle is ordinal, and it suggests a particular way to calibrate the utilities so that they can be used with the maximin principle.
In the special case in which
W
Herve Moulin describes this special case of the EE rule as follows:
"The EE solution equalizes across agents the utilities measured along the "numeraire" of the commodity bundle to be divided. In other words, this solution gives to each participant an allocation that he or she views as equivalent (with his or her own preferences) to the same share of the pie, where the "pie" stands for the resources to be divided and a share is a homothetic reduction of the pie --- this is the same fraction of the total available amount of each commodity".
The following example is based on.
The question is how to divide the 100 units of capacity in each road among the 100 agents? Here are some possible solutions.
(x=1,y=1)
r>1
40r+30r
30r+30r
r\leq100/70=30/21 ≈ 1.43
(x=30/21,y=30/21)
40/21 ≈ 1.90
(x=30/21,y=40/21)
Consider now the following variant on the above example. The utilities of the AB and BC agents are as above, but the utility of the AC agents when getting x units of AB and y units of BC is now (x+y)/2. Note that it is normalized such that their utility from having a unit of each resource is 1.
r>1
40r+30x
30r+30y
x+y=2r
r\leq200/130=60/39 ≈ 1.54
(x=60/39,y=60/39)
40r+30x
30s+30y
x+y=r+s
70r+60s\leq200
r=100/70=30/21 ≈ 1.43,s=100/60=35/21 ≈ 1.67
(x=30/21,y=35/21)
30/21 ≈ 1.43
35/21 ≈ 1.67
32.5/21 ≈ 1.54
To summarize: in this example, a divider who believes in the importance of egalitarian-equivalence must choose between equitability and envy-freeness.
When there are two agents, the set of PEEE allocations contains the set of PEEF allocations. The advantage of PEEEA is that they exist even when there are no PEEFA.[1]
However, with three or more agents, the set of PE allocations that are both EE and EF might be empty. This is the case both in exchange economies with homogeneous divisible resources[4] and in economies with indivisibilities.[5]
In the special case in which the reference bundle contains a constant fraction of each good, the PEEEA rule has some more desirable properties:
1/n
However, it is lacking some other desirable properties:
In some settings, the PEEEA rule is equivalent to the Kalai-Smorodinsky bargaining solution.