Indirect revealed preference ($R=(R_D\cup \Delta {)}^{\ast }$)

The weak indirect revealed preference is the reflexive closure of the transitive closure of a relation of direct revealed comparisons. Equivalently, it consists of finite chains, including chains of length zero. The diagonal convention is essential if the relation is to be an intersection of complete weak preference relations on all alternatives.

The congruence axiom says that an affordable alternative indirectly revealed at least as desirable as a chosen alternative must also be chosen. For a chosen bundle $x$ and an affordable unchosen bundle $y$, the axiom rules out $yRx$; because $xRy$, every strict-part-preserving completion ranks $x$ strictly above $y$.