In fair division

Fair division

Fair division, also known as the cake-cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount...

, a set of preferences is weakly additive
if the following condition is met:

If A is larger than B, and C is larger than D (and pieces A and C do not overlap) then A together with C is preferable to B together with D.

Weak additivity is often a realistic assumption when dividing up goods between claimants, and simplifies the mathematics of certain fair division problems considerably.

Use of weak additivity

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 procedure

In problems of fair division, the adjusted winner procedure is used to partition a bundle of goods between two players in such a way as to minimize envy and maximize efficiency and equitability...

 only requires weak additivity.

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.
• 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.

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

Marginal utility

In economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...

.