In utility theory, the responsive set (RS) extension is an extension of a preference-relation on individual items, to a partial preference-relation of item-bundles.
A person states that he ranks the items according to the following total order: (i.e., z is his best item, then y, then x, then w).
Assuming the items are independent goods, one can deduce that: But, one cannot deduce anything about the bundles
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.
a total order on
The RS extension of
It can be defined in several equivalent ways.
[1] The original RS extension[2]: 44–48 is constructed as follows.
, take the following relations: The RS extension is the transitive closure of these relations.
The PD extension is based on a pairing of the items in one bundle with the items in the other bundle.
if-and-only-if there exists an Injective function
The SD extension (named after stochastic dominance) is defined not only on discrete bundles but also on fractional bundles (bundles that contains fractions of items).
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.
If the bundles are discrete, the definition has a simpler form.
: The AU extension is based on the notion of an additive utility function.
Many different utility functions are compatible with a given ordering.
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
according to both utility functions.
Moreover, for every utility function
compatible with the above ranking: In contrast, the utility of the bundle
This motivates the following definition:
iff, for every additive utility function
A total order on bundles is called responsive[4]: 287–288 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.
Similarly, a utility function on bundles is called responsive if it induces a responsive order.
To be more explicit,[5] a utility function u is responsive if for every bundle X and every two items y,z that are not in X:
Responsiveness is implied by additivity, but not vice versa: For example,[6] 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 says nothing about bundles of different sizes that are not subsets of each other.