Dual Random Utility Maximisation

A Menu-dependent dRUM (mdRUM) is a stochastic choice rule p for which the following
is true: there exists a triple (r 1 , r 2 , ˜α) where r 1 and r 2 are rankings and ˜α :
2 X \∅ → (0, 1) is a function that for each menu A assigns a probability ˜α (A) to r 1
(and 1 − ˜α (A) to r 2 ), such that, for all A, p (a, A) = p ′ (a, A) where p ′ is the dRUM
generated by (r 1 , r 2 , ˜α (A)).
An mdRUM obviously loses, compared to a dRUM, all the properties that pertain
to the specific magnitudes of choice probabilities, which depend on the menu in an
unrestricted way. Hence in particular an mdRUM fails Regularity. But it is precisely
this feature that gives the model its descriptive power.
It is easy to check, however, that discarding alternatives from a menu still preserves
the possibility and the certainty of the event in which an alternative is chosen, when
moving to sub-menus: in this sense it preserves the ‘mode’ of choice: 8
Modal Regularity: Let a ∈ B ⊂ A. (i) If p (a, A) > 0 then p (a, B) > 0. (ii) If p (a, A) = 1
then p (a, B) = 1.
For the second key property we consider a notion of impact related to modality.
Let’s say that b modally impacts a in A if
p (a, A) > 0 and p (a, A ∪ {b}) = 0
or p (a, A) = 1 and p (a, A ∪ {b}) ∈ (0, 1)
That is, b modally impacts a if adding b transforms the choice of a from possible (including
certain) to impossible or from certain to merely possible.
Impact Consistency: Let b /∈ A. If b does not modally impact a for all a ∈ A then
p (b, A ∪ {b}) = 0.
Impact Consistency is the modal version of a property that is implied automatically
in the menu-independent model, and is in fact satisfied by any RUM (if b is chosen
with positive probability in a menu, then it obviously impacts some alternative in any
sub-menu).
8 ‘Mode’ and ‘modal’ are meant here and elsewhere in ther logical and not statistical meaning.
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