In
fair division
Fair division is the problem in game theory of dividing a set of resources among several people who have an entitlement to them so that each person receives their due share. That problem arises in various real-world settings such as division of inh ...
, a topic in
economics
Economics () is the social science that studies the production, distribution, and consumption of goods and services.
Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analyzes ...
, 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
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value u(x_i) of one i ...
functions, while weak additivity is applicable to
ordinal utility In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to a ...
functions.
Weak additivity is often a realistic assumption when dividing up
goods
In economics, goods are items that satisfy human wants
and provide utility, for example, to a consumer making a purchase of a satisfying product. A common distinction is made between goods which are transferable, and services, which are not t ...
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 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.
*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
In economics, a complementary good is a good whose appeal increases with the popularity of its complement. Technically, it displays a negative cross elasticity of demand and that demand for it increases when the price of another good decreases. I ...
.
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.
See also
*
Responsive set extension#Responsiveness
References
Utility function types
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