  This paper develops a novel characterization for random utility models (RUM),
which turns out to be a dual representation of the characterization by Kitamura
and Stoye (2018, ECMA). For a given family of budgets and its "patch"
representation \'a la Kitamura and Stoye, we construct a matrix $\Xi$ of which
each row vector indicates the structure of possible revealed preference
relations in each subfamily of budgets. Then, it is shown that a stochastic
demand system on the patches of budget lines, say $\pi$, is consistent with a
RUM, if and only if $\Xi\pi \geq \mathbb{1}$, where the RHS is the vector of
$1$'s. In addition to providing a concise quantifier-free characterization,
especially when $\pi$ is inconsistent with RUMs, the vector $\Xi\pi$ also
contains information concerning (1) sub-families of budgets in which cyclical
choices must occur with positive probabilities, and (2) the maximal possible
weights on rational choice patterns in a population. The notion of Chv\'atal
rank of polytopes and the duality theorem in linear programming play key roles
to obtain these results.
