´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 259
48, 50, 51, respectively. One further restriction will complete the characterization of two-scale Luce models with exponents in the next section.
4. Characterization of the ‘two-scale Luce model with exponents’
In order to get 38, that is 9, characterized, we aim to obtain wt 5 t because then
cs 5 s, so that 55 reduces to
a b
a b
a b
F x , y , . . . ,x , y 5 x y Lx y , . . . ,x y j 5 1, . . . ,n. 61
j 1
1 n
n j
j 1
1 n
n
Summing from 1 to n we get, in view of 27,
n a
b a
b a
b
1 5
O
x y Lx y , . . . ,x y ,
k k
1 1
n n
k 51
that is, 1
a b
a b
]]] Lx y , . . . ,x y 5
,
n 1
1 n
n a
b
O
x y
k k
k 51
so that 61 indeed becomes 38:
a b
x y
j j
]]] F x , y , . . . ,x , y 5
, j 5 1, . . . ,n.
n j
1 1
n n
a b
O
x y
k k
k 51
If wt 5 t then Ft , . . . ,t given by 52 will be, cf. 45,
1 m
m c
i
Ft , . . . ,t 5
P
t 5 Gt , . . . ,t . 62
1 m
i 1
m i 51
Conversely, if 62 holds then, by 47 and 44,
1 ]
S c S c
i i
c
wt 5 Ft, . . . ,t 5 t 5 t.
[As we saw before Theorem 2, 52 and thus 62 is a possible choice for n 5 2 and the only choice for n . 2]. So we have the following.
Theorem 4. The choice probabilities in the ‘two-scale Luce model with exponents’, that is those given by 1 or equivalently by 9, and only these satisfy, in addition to the
conditions in Theorem 2 or Theorem 3, also F 5 G.
Remark 10. Till now we supposed that G x, . . . ,x is not constant, that is, see 45,
1 m
m
o c ± 0. Conversely,
o c 5 0 means that G x, . . . ,x is constant. But
i 51 i
i 51 i
1
G x , . . . ,x should not be constant, that is, c ± 0 should hold for at least one p. In
1 1
m p
this case, everything else in Theorem 4 being unchanged, one gets instead of 38 the equation
´ 260
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
a b
C x y
j j
j
]]] F x , y , . . . ,x , y 5
j 5 1, . . . ,n, 63
n j
1 1
n n
a b
O
C x y
k k
k k 51
where C , . . . ,C are positive constants the proof uses a method similar to that in
1 n
´ Section 4 of Aczel et al., 1997.
The representation in 63 has interesting psychological interpretations. The form indicates that the ordinal position of an option e in the set E affects its representation —
in particular, option e in position j has a ‘weighting’ parameter C placed in front of its
j j
a b
e.g. overall scale value ve we . Returning to the noise burst example in Section 1,
j j
one could think of the stimuli e j 5 1, . . . ,n, as being presented in temporal order,
j
with e being presented before e for j 5 1, . . . ,n 2 1. The weight C then might be
j j 11
j
interpreted as a bias to report the stimulus in position j as louder or softer than its scale values would indicate; alternatively, C might be interpreted as a memory effect, with
j
stimuli at different positions in the sequence not being equally represented relative to their scale values.
5. Conclusion: open problem
