Dual Random Utility Maximisation

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{a, b, c} {a, b} {a, c} {b, c}
a
1
3
1
2
1
2
−
b
1
3
1
2
−
1
2
c
1
3
−
1
2
1
2
Table 8:
This stochastic choice function cannot be a mdRUM since in the set {a, b, c} there
are three alternatives, which cannot be all at the top of two linear orders.
C.2: Modal Regularity (i), Impact Consistency, but not Modal Regularity (ii)
{a, b, c} {a, b} {a, c} {b, c}
a 1
1
2
1 −
b 0
1
2
−
1
2
c 0 − 0
1
2
Table 9:
This stochastic choice function cannot be a mdRUM since on the one hand p (a, {a, b, c}) =
1 implies that a is the highest ranked alternative in both linear orders, while on the
other p (b, {a, b}) > 0 would require b to rank above a in at least one ranking.
C.3: Modal Regularity (ii), Impact Consistency, but not Modal Regularity (i)
{a, b, c} {a, b} {a, c} {b, c}
a
1
2
0 1 −
b
1
2
1 −
1
2
c 0 − 0
1
2
Table 10:
This stochastic choice function cannot be a mdRUM since p (b, {a, b}) = 1 requires
b to be ranked before a in both rankings, while p (a, {a, b, c}) > 0 would require a to
rank above b in at least one ranking.
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