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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.

Let $B(p,w)=\{z\in {\mathbb{R}}^L:p\cdot z\le w\}$ be the set allowed by the budget constraint. A demand correspondence is understood to choose only feasible bundles: $x(p,w)\subseteq B(p,w)$. Define the direct revealed preference relation byThe indirect revealed preference relation is its reflexive transitive closure. Thus $yRz$ precisely when there is a finite chain $y=y_0,y_1,\dots ,y_m=z$ whose successive comparisons satisfy $y_j{R}_D{y}_{j+1}$. Chains of length zero are included. This makes $R$ a preorder on all of ${\mathbb{R}}^L$, including alternatives that are never chosen. The associated strict preference isIf a convention defines indirect revelation using only positive-length chains, one must add the diagonal before the intersection assertion in part (c) can hold: every complete weak preference relation contains the diagonal.