Dual Random Utility Maximisation

More in general, the source of non-uniqueness when rankings have symmetric
probabilities is the following. Suppose that, given the rankings r 1 and r 2 , the set of
alternatives can be partitioned as X = A 1 ∪ ... ∪ A n such that if a ∈ A k , b ∈ A l and
k < l, then a ≻ i b for i = 1, 2. Now construct two new rankings r
1 ′ and r′ 2
in the following
manner. Fix A k , and for all a, b ∈ A k set a ≻ i b if and only if a ≻ j b, i ̸= j, while
for all other cases r
1 ′ and r′ 2 coincide with r 1 and r 2 (i.e. r
1 ′ and r′ 2
swap the subrankings
( )
within A k ). Then it is clear that the probabilities generated by r
1 ′ , r′ 2, 1 2
are the same as
)
1
those generated by
(r 1 , r 2, 2
. Apart from these “swaps”, the identification is unique.
7 Concluding remarks
We have argued that the dual Random Utility Model constitutes a parsimonious framework
to represent the variability of choice in many contexts of interest, both in the fixed
state probability version and in the new menu-dependent probability version.
The properties at the core of our analysis are simple and informative about how
choices react to the merging, expansion and contraction of menus and thus may provide
the theoretical basis for comparative statics analysis and for inferring behaviour
in larger menus from behaviour in smaller ones, or vice-versa.
The menu-dependent version of the model, while still quite restrictive (as demonstrated
by our characterisation), significantly extends the range of phenomena that the
model can encompass, because it does not need to satisfy the full Regularity property
but only a ‘modal’ version of it: as a consequence, for example the attraction and the
compromise effects and Luce and Raiffa’s ‘frog legs’ types of anomalies can be described
as dual random utility maximisation.
References
[1] Apesteguia, Jose and Miguel Angel Ballester (2016) “Single-crossing Random
Utility Models", mimeo
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