A preference relation is called a weakly additive , if the condition is met:
- If A is preferable to B, and C is preferable to D (A and C do not intersect), then a set of A and C is preferable to a set of B and D.
Any additive utility function is weakly additive. Additivity is applicable only to cardinalistic functions, while weak additivity is applicable to ordinalistic .
The assumption of weak additivity is often justified in fair sharing games. Some procedures, including the adjustable winner procedure, do not require additivity; its weakened version is sufficient. Such an assumption greatly facilitates the solution of problems in this area.
Lack of Additivity
Weak additivity may not be true if:
- The utility of a set of A and C is less than the sum of their utilities separately (that is, A and C are substitutes ).
- The utility of a set of B and D is greater than the sum of their utilities separately (i.e., B and D are complements ).
However, the lack of additivity does not prevent weak additivity in principle: it can be achieved by introducing monetary compensation.
See also
- Additive utility
Notes
- Steven J. Brams; Alan D. Taylor (1996). Fair division: from cake-cutting to dispute resolution. Cambridge University Press. ISBN 978-0-521-55644-6 .