´ 246
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
K lz , . . . ,lz 5 K z , . . . ,z j 5 1, . . . ,n.
j 1
n j
1 n
The homogeneity iii looks similar to i and will turn out to be related to it through ii.
In this paper we develop aggregation conditions which, with i, ii, iii and under some simple regularity conditions, imply a choice model which includes 1 as particular
case. Then we give conditions which characterize just 1. This characterization will also show why powers and not other functions figure in 1. This is related to considering v
and w as ratio scales.
2. Definitions, assumptions, equations
We will call choice probability and denote by Pe:E the probability of choosing selecting an element option e from an n-element 1 , n , ` subset E of a global set
R of options. We exclude elements which are impossible to choose. Thus, 0 , Pe:E , 1
11 and, of course,
O
Pe:E 5 1. 12
e [E
We now formally define this and further concepts required in this paper interspersed with remarks.
Definition 1. Let R be a set and S the set of its finite subsets, where both R and every E [S has at least two elements, and D5he:E u E [S, e[Ej7R3S. The pair R, P
consisting of the set R and of the function P: D →
]0,1[is a choice model. The function value Pe:E e [E, E [S is a choice probability, e is an option.
Remark 1. While here we required that R have at least two elements, a later assumption will imply that it has uncountably infinite elements see Definition 3, Remark 4.
Definition 2. A choice model, two functions v, w: R
→ ]0,`[ and two nonzero numbers
a, b form a two-scale Luce model with exponents if 1, that is
a b
v e we
]]]] Pe:E 5
a b
O
v d wd
d [E
holds. If a 5 b 5 1 then we have a two-scale Luce model. The functions v and w are the scales, ve, we are scale values.
Remark 2. Here and throughout the paper the scales v and w are given; otherwise not only could the exponents be immerged into v and w but the two scales could be replaced
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 247
a b
by a single scale u 5 v w and we would have the original one-scale Luce model cf. ´
Aczel et al., 1997 and 8 ue
]]] Pe:E 5
. 13
O
ud
d [E
Notice, however, that, if the choice probabilities can be represented in the form 1 with scales v, w, they can equally well be represented by scales
gv, dw with arbitrary constants
g .0, d .0. As we said in the Introduction, our purpose in this paper is to determine those
scale-dependent two-scale, see Definition 4 families of choice probabilities for which formulae like 10
Pe : E 5 F [ve ,we , . . . ,ve ,we ]
j j
1 1
n n
5 K [h ve ,we , . . . ,h ve ,we ] j 5 1, . . . ,n 14
j 1
1 1
n n
n
hold, where both the scales and the probabilities can be aggregated. This means that, in analogy to 2 and 3, equations of the form
v e 5 G v e , . . . ,v e , we 5 G w e , . . . ,w e ,
f g
f g
j 1
1 j
m j
j 2
1 j
m j
Pe : E 5 H P e : E , . . . ,P e : E , . . . ,P e : E , . . . ,P e : E ,
f g
j j
1 1
m 1
1 n
m n
j 5 1, . . . ,n are satisfied, where the component probabilities satisfy equations analo- gous to 14 with the same F j 5 1, . . . ,n:
j
P e : E 5 F [v e ,w e , . . . ,v e ,w e ]
i j
j i
1 i
1 i
n i
n
5 K [h v e ,w e , . . . ,h v e ,w e ]
j 1
i 1
i 1
n i
n i
n
i 5 1, . . . ,m; j 5 1, . . . ,n. 15
These equations, with assumptions linked to v , w and their combinations h v , w
i i
j i
i
being ratio scales, will lead to functional equations, from which we will endeavor to determine the functions h , K and thus F ; j 5 1, . . . ,n obtaining in passing also the
j j
j
general forms of G ,G and, under further restrictions, H j 5 1, . . . ,n. The variables in
1 2
j
these equations will go through the set R of positive numbers, because we will deal
11
with nonatomic scales in analogy to nonatomic Kolmogorov probability spaces, see e.g. ´
in Renyi, 1970. These are such that the scales can assume any positive value. We will make further assumptions in Definitions 3 and 4.
A nonatomic pair of scales is a triple R, v, w, where R is a set and v, w: R →
R
11 2
are scales functions such that for every pair x, y [ R there exists an e [ R for
11
which x 5 ve, y 5 we.
Remark 3. Nonatomicity has the following consequence. For each 2n-tuple
2n
x , y , . . . ,x , y [ R for which x , y ± x , y if j ± k that is, the same x , y
1 1
n n
11 j
j k
k j
j n
pair does not occur more than once there exist e , . . . ,e [ R such that x 5 ve ,
1 n
j j
´ 248
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
y 5 we j 5 1, . . . ,n. The bold exclusion was necessary because otherwise E may
j j
not have n distinct elements; in particular it may have just one element — but we ´
excluded singletons cf. 11. Therefore we will need here as in Aczel et al., 1997 a somewhat stronger condition, the n-nonatomicity which postulates the existence of
different distinct elements e ± e even if x 5 x , y 5 y thus ve 5 x 5 x 5 ve ,
j k
j k
j k
j j
k k
we 5 y 5 y 5 we j ± k. This is not unexpected: For instance, e ,e might be two
j j
k k
j k
different cars that a person considers equal in safety x 5 ve 5 ve 5 x and equal
j j
k k
in road handling second dimension y 5 we 5 we 5 y ; see also the example in the
j j
k k
Introduction about equal perceived intensity and duration for different tone bursts. We define now the n-nonatomicity formally.
Definition 3. An n-nonatomic pair of scales is a quadruple R, v, w, n, where R is a set,
v , w: R
→ R
are scales functions and n . 1 is an integer such that for every
11
2n-tuple x , y , . . . ,x , y of positive numbers there exist n distinct elements e , . . . ,e
1 1
n n
1 n
of R for which x 5 ve , y 5 we j 5 1, . . . ,n.
j j
j j
Remark 4. While the n-nonatomicity assumption clearly implies that the global set R is uncountably infinite, the choice is always done from finite subsets E of R as is
appropriate for applications in psychology and elsewhere. The papers Falmagne, 1981; Marley, 1982 discuss the use and misuse of nonatomicity in related problems without
explicitly defining this concept.
Remark 5. Of course, there may exist not only e ± e of the same set E with equal
j k
scale values but to a 2n-tuple of positive reals there may also exist two or more
n 2n
9 9
n-tuples e , . . . ,e , e , . . . ,e in R such that, for the same x , y , . . . ,x , y [ R
1 n
1 n
1 1
n n
11
9 9
simultaneously x 5 ve 5 ve , y 5 we 5 we j 5 1, . . . ,n hold. The last part of
j j
j j
j j
the next definition refers to this situation.
Definition 4. Let n . 1 be an integer. An n-nonatomic two-scale family of choice probabilities a two-scale model for short is a 5-tuple R, P, v, w, n, where R, P is a
choice model with only n element subsets E of R considered and R, v, w, n is an n-nonatomic pair of scales such that, whenever for two subsets E 5
he , . . . ,e j and
1 n
9 9
9 9
E9 5 he , . . . ,e j of R we have ve 5 ve and we 5 we j 5 1, . . . ,n then also
1 n
j j
j j
9
Pe :E 5 Pe :E9 j 5 1, . . . ,n.
j j
Remark 6. The last requirement, where n has an essential role, implies that there exist
2n
functions F :R →
]0,1[ such that
j 11
Pe : E 5 F ve ,we , . . . ,ve ,we j 5 1, . . . ,n
j j
1 1
n n
for every n-element subset E of R. We need the last assumption and n-nonatomicity so that equations like 14 make sense and that our later functional equations hold on
suitable domains. We will call F , . . . ,F the generating system of the choice
1 n
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 249
probability P. As already mentioned, we will suppose that P and P i 5 1, . . . ,m have
i
the same generating system. ´
We generalize 1, 2 and 3 to the following aggregation equations cf. Aczel et al., 1997:
Pe :E 5 F ve ,we , . . . ,ve ,we , 16
f g
j j
1 1
n n
P e :E 5 F v e ,w e , . . . ,v e ,w e , 17
f g
i j
j i
1 i
1 i
n i
n
v e 5 G v e , . . . ,v e , we 5 G w e , . . . ,w e ,
18
f g
f g
j 1
1 j
m j
j 2
1 j
m j
Pe :E 5 H P e :E , . . . ,P e :E , . . . ,P e :E , . . . ,P e :E , 19
f g
j j
1 1
m 1
1 n
m n
here i 5 1, . . . ,m and j 5 1, . . . ,n. Substituting 18 into 16 on one hand and 17 into 19 on the other gives, in view of the nonatomicity, a system of n functional equations.
The fact that the scales and the probabilities depending on them by 16 and 17 are invariant covariant under linear transformations ratio scales give 2n 1 2 others.
Finally 11 and 12 generate one more equation and n inequalities.
´ As in Aczel et al., 1997, however, there would be far too many solutions for
practical use and this system is far too weak to characterize the ‘‘two-scale Luce model with exponents’’ or even halfway reasonable generalizations. So two further spe-
cifications are needed. The first links H more closely to F , at the same time introducing
j j
two more scale values depending on the choice probabilities: ˜ve :E 5
F [P e :E, . . . ,P e :E],
j 1
1 j
m j
˜ we :E 5
F [P e :E, . . . ,P e :E] j 5 1, . . . ,n, 20
j 2
1 j
m j
´ cf. 4 and Aczel et al., 1997 to which 16 should also apply:
˜ ˜
˜ ˜
Pe :E 5 F [ve :E ,we :E , . . . ,ve :E ,we :E ] j 5 1, . . . ,n 21
j j
1 1
n n
the more specific Eq. 21 with 20 will replace 19. The second ‘simple scalability’ ii we know already see 14. We repeat it here to point out that it establishes a link
between the two original sets of scales within F :
j
Pe :E 5 F [ve ,we , . . . ,ve ,we ]
j j
1 1
n n
5 K h [ve ,we ], . . . ,h [ve ,we ] j 5 1, . . . ,n. 22
j 1
1 1
n n
n
We list now the assumptions linked to v , w and the combinations h v,w of v and w
i i
j
being ratio scales i 5 1, . . . ,m; j 5 1, . . . ,n. There exist functions M , M such that
1 2
G [ g v e , . . . ,g v e ] 5 M g , . . . ,g G [v e , . . . ,v e ],
23
1 1
1 1
m m
m 1
1 m
1 1
1 m
m
G [ d w e , . . . ,d w e ] 5 M d , . . . ,d G [w e , . . . ,w e ].
24
2 1
1 1
m m
m 2
1 m
2 1
1 m
m
Also
´ 250
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
F [ mve ,nwe , . . . ,mve ,nwe ] 5 F [ve ,we , . . . ,ve ,we ]
j 1
1 n
n j
1 1
n n
j 5 1, . . . ,n, ‘conjoint Weber type II’ Falmagne and Iverson, 1979, which translates for K and h
j j
into K h [
mve ,nwe ], . . . ,h [mve ,nwe ]
j 1
1 1
n n
n
5 K h [ve ,we ], . . . ,h [ve ,we ] j 5 1, . . . ,n. 25
j 1
1 1
n n
n
Furthermore there exists a function N such that K
lh [ve ,we ], . . . ,lh [ve ,we ]
j 1
1 1
n n
n
5 N lK h [ve ,we ], . . . ,h [ve ,we ] j 5 1, . . . ,n.
26
j 1
1 1
n n
n
holds. Finally, the obvious equation 12 and inequalities 11 translate, in view of 16, into
n
O
F [ve ,we , . . . ,ve ,we ] 5 1, 27
j 1
1 n
n j 51
0 , F [ve ,we , . . . ,ve ,we ] , 1 j 5 1, . . . ,n. 28
j 1
1 n
n
We introduce, for the sake of brevity, the notation permitted by the n-nonatomicity x 5 ve , y 5 we , z 5 h x , y ,
j j
j j
j j
j j
x 5 v e , y 5 w e i 5 1, . . . ,m; j 5 1, . . . ,n. 29
ij i
j ij
i j
The scale values and thus these variables are supposed to be positive. So, by 22, 27–29,
n
O
K z , . . . ,z 5 1, 30
j 1
n j 51
0 , K z , . . . ,z , 1 j 5 1, . . . ,n. 31
j 1
n
From 26,30 N l
; 1, thus z
z
1 n
]] ]]
K z , . . . ,z 5 K lz , . . . ,lz 5 K
, . . . , j 5 1, . . . ,n.
32
n n
j 1
n j
1 n
j
1
O
z
O
z
2
k k
k 51 k 51
Also, by 19–21, with s 5 P e :E [ R not all values in R
need to be assumed:
ij i
j 11
11
H s , . . . ,s , . . . ,s , . . . ,s
j 11
m1 1n
mn
5 F [ F s , . . . ,s ,F s , . . . ,s , . . . F s , . . . ,s ,F s , . . . ,s ]
j 1
11 m1
2 11
m1 1
1n mn
2 1n
mn
j 5 1, . . . ,n. 33
We conclude by stating our regularity assumptions. About G and G we suppose only
1 2
that they are locally bounded on an m-dimensional interval, no matter how small and
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 251
´ ´
that G x, . . . ,x is not constant. With 23,24 this gives cf. Aczel, 1987; Aczel et al.,
1
1997
m m
m c
d
i i
G x , . . . ,x 5 C
P
x ,
O
c ± 0, G y , . . . , y 5 D
P
y .
1 1
m i
i 2
1 m
i i 51
i 51 i 51
Also h , . . . ,h will be supposed locally bounded and nonconstant. We may assume,
1 n
without loss of generality, that G 1, . . . ,1 5 G 1, . . . ,1 5 1, h 1,1 5 1 j 5 1, . . . ,n.
34
1 2
j
So C 5 D 5 1, and we have
m m
m c
d
i i
G x , . . . ,x 5
P
x ,
O
c ± 0, G y , . . . , y 5
P
y , 35
1 1
m i
i 2
1 m
i i 51
i 51 i 51
´ In addition to these very weak conditions, we assume cf. Aczel et al., 1997 that K is
j
injective in the following sense:
n n
9 9
9
K z , . . . ,z 5 K z , . . . ,z j 5 1, . . . ,n and
O
z 5
O
z
F G
j 1
n j
1 n
k k
k 51 k 51
9
imply z 5 z
j 5 1, . . . ,n. 36
j j
Notice that 32 allows us to bring the sum of variables in K to 1, which we need for the
j
number of equations on both sides of the implication 36 be equal to n 2 1, remember 30.
Substituting 33 and 17 into 19 and 18 into 16, and equating the two expressions thus obtained for Pe :E j 5 1, . . . ,n we get, in terms of the notation 29,
j
the system of functional equations F [G x , . . . ,x
,G y , . . . , y , . . . ,G x , . . . ,x
,G y , . . . , y ]
j 1
11 m1
2 11
m1 1
1n mn
2 1n
mn
5 F F [F x ,y , . . . ,x ,y , . . . ,F x ,y , . . . ,x ,y ],
j 1
1 11
11 1n
1n 1
m1 m1
mn mn
F [F x ,y , . . . ,x ,y , . . . ,F x ,y , . . . ,x ,y ],
2 1
11 11
1n 1n
1 m1
m1 mn
mn
: F [F x ,y , . . . ,x ,y , . . . ,F x ,y , . . . ,x ,y ],
1 n
11 11
1n 1n
n m1
m1 mn
mn
F [F x ,y , . . . ,x ,y , . . . ,F x ,y , . . . ,x ,y ] j 5 1, . . . ,n,
2 n
11 11
1n 1n
n m1
m1 mn
mn
37 which looks at first sight quite formidable.
The question is, whether cf. 1 and 9
´ 252
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
a b
x y
j j
]]] F x , y , . . . ,x , y 5
j 5 1, . . . ,n 38
n j
1 1
n n
a b
O
x y
k k
k 51
with 35 and with aptly chosen F and F are the only solutions of the above equations.
1 2
The answer is negative but we determine the general solutions F , . . . ,F under these
1 n
assumptions and also the auxiliary functions G ,G , H , . . . ,H , F ,F , K , . . . ,K ,
1 2
1 n
1 2
1 n
h , . . . ,h and M ,M we have already N l
; 1. Then we characterize the ‘two-scale
1 n
1 2
Luce model with exponents’ 38 by one additional supposition.
3. Solutions Theorem 1. If 25, 32 36 hold, the h ’s are locally bounded but not constant in
