Suppose there are four items:. A person states that he ranks the items according to the following total order: . Assuming the items are independent goods, one can deduce that: But, one cannot deduce anything about the bundles ; we do not know which of them the person prefers. The RS extension of the ranking is a partial order on the bundles of items, that includes all relations that can be deduced from the item-ranking and the independence assumption.
Definitions
Let be a set of objects and a total order on. The RS extension of is a partial order on. It can be defined in several equivalent ways.
Responsive set (RS)
The original RS extension is constructed as follows. For every bundle, every item and every item, take the following relations:
The SD extension is defined not only on discrete bundles but also on fractional bundles. Informally, a bundle Y is SD-preferred to a bundle X if, for each item z, the bundle Y contains at least as many objects, that are at least as good as z, as the bundle X. Formally, iff, for every item : where is the fraction of item in the bundle. If the bundles are discrete, the definition has a simpler form. iff, for every item :
The AU extension is based on the notion of an additive utility function. Many different utility functions are compatible with a given ordering. For example, the order is compatible with the following utility functions: Assuming the items are independent, the utility function on bundles is additive, so the utility of a bundle is the sum of the utilities of its items, for example: The bundle has less utility than according to both utility functions. Moreover, for every utility function compatible with the above ranking: In contrast, the utility of the bundle can be either less or more than the utility of. This motivates the following definition: iff, for every additive utility function compatible with :
Equivalence
implies.
and are equivalent.
implies. Proof: If, then there is an injection such that, for all,. Therefore, for every utility function compatible with,. Therefore, if is additive, then.
It is known that and are equivalent, see e.g.
Therefore, the four extensions and and and are all equivalent.
A total order on bundles is called responsive if it is contains the responsive-set-extension of some total order on items. I.e., it contains all the relations that are implied by the underlying ordering of the items, and adds some more relations that are not implied nor contradicted. Responsiveness is implied by additivity, but not vice versa:
If a total order is additive then by definition it contains the AU extension, which is equivalent to, so it is responsive.
On the other hand, a total order may responsive but not additive: it may contain the AU extension which is consistent with all additive functions, but may also contain other relations that are inconsistent with a single additive function.
For example, suppose there are four items with. Responsiveness constrains only the relation between bundles of the same size with one item replaced, or bundles of different sizes where the small is contained in the large. It nothing about bundles of different sizes that are not subsets of each other. So, for example, a responsive order can have both and. But this is incompatible with additivity: there is no additive function for which while.
