In
fair division, a topic in
economics, a
preference relation is weakly additive
if the following condition is met:
: If A is preferred to B, and C is preferred to D (and the contents of A and C do not overlap) then A together with C is preferable to B together with D.
Every
additive utility function is weakly-additive. However, additivity is applicable only to
cardinal utility functions, while weak additivity is applicable to
ordinal utility functions.
Weak additivity is often a realistic assumption when dividing up
goods between claimants, and simplifies the mathematics of certain fair division problems considerably. Some procedures in fair division do not need the value of goods to be additive and only require weak additivity. In particular the
adjusted winner procedure
Adjusted Winner (AW) is a procedure for envy-free item allocation. Given two agents and some goods, it returns a partition of the goods between the two agents with the following properties:
# Envy-freeness: Each agent believes that his share of th ...
only requires weak additivity.
Cases where weak additivity fails
Case where the assumptions might fail would be either
*The value of A and C together is the less than the sum of their values. For instance two versions of the same CD may not be as valuable to a person as the sum of the values of the individual CDs on their own. I.e, A and C are
substitute goods
In microeconomics, two goods are substitutes if the products could be used for the same purpose by the consumers. That is, a consumer perceives both goods as similar or comparable, so that having more of one good causes the consumer to desire less ...
.
*The values of B and D together may be more than their individual values added. For instance two matching bookends may be much more valuable than twice the value of an individual bookend. I.e, B and D are
complementary goods.
The use of money as compensation can often turn real cases like these into situations where the weak additivity condition is satisfied even if the values are not exactly additive.
The value of a type of goods, e.g. chairs, dependent on having some of those goods already is called the
marginal utility
In economics, utility is the satisfaction or benefit derived by consuming a product. The marginal utility of a Goods (economics), good or Service (economics), service describes how much pleasure or satisfaction is gained by consumers as a result o ...
.
See also
*
Responsive set extension#Responsiveness
References
Utility function types
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