Responsive preferences

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.

Suppose there are four items: ${\displaystyle w,x,y,z}$. 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 ${\displaystyle \{w,z\},\{x,y\}}$; we do not know which of them the person prefers.

The RS extension of the ranking ${\displaystyle w\prec x\prec y\prec z}$ 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.

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