´ 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
j
either variable and h 1,1 5 1 j 5 1, . . . ,n then, for all j [ h1, . . . ,nj,
j a b
h x, y 5 x y x . 0, y . 0
39
j
for some nonzero constants a, b. Eq. 39 is, of course, the same as 7. Accordingly, we will write
a b
hx, y 5 h x, y 5 x y . 40
j
Proof. Eqs. 25 and 32 imply that
h mx ,ny
h mx ,ny
1 1
1 n
n n
]]]]] ]]]]]
K , . . . ,
n n
j
1
O
h mx ,ny
O
h mx ,ny
2
k k
k k
k k
k 51 k 51
h x , y h x , y
1 1
1 n
n n
]]]] ]]]]
5 K , . . . ,
j 5 1, . . . ,n.
n n
j
1
O
h x , y
O
h x , y
2
k k
k k
k k
k 51 k 51
Thus it follows from 36 that h
mx ,ny h x , y
j j
j j
j j
]]]]] ]]]]
5 j 5 1, . . . ,n,
n n
O
h mx ,ny
O
h x , y
k k
k k
k k
k 51 k 51
that is,
n
O
h mx ,ny
k k
k
h mx ,ny
j j
j k 51
]]]] ]]]]]
5 5 Qx , . . . ,x ,y , . . . ,y ,
m,n j 5 1, . . . ,n.
n 1
n 1
n
h x , y
j j
j
O
h x , y
k k
k k 51
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 253
On the right end Q carries no subscript because the middle term does not depend on j. On the other hand, the term on the left does not depend upon x , y l ± j . Writing the
l l
same string of equations for different j’s, the middle term and thus Q remains the same but j varies on the left. So Q cannot depend on any x , y and we get
j j
h mx ,ny
j j
j
˜ ]]]] 5 Qm,n j 5 1, . . . ,n,
h x , y
j j
j
that is, ˜
h mx,ny 5 Qm,n h x,y j 5 1, . . . ,n.
j j
´ The general locally bounded nonconstant solution of this equation see e.g. Aczel, 1987
a b
˜ is given by Q
m,n 5 m n and by 39 if h 1,1 5 1. This concludes the proof of
j
Theorem 1. From 22 and 39 we have
a b
a b
F x , y , . . . ,x , y 5 K x y , . . . ,x y j 5 1, . . . ,n. 41
j 1
1 n
n j
1 1
n n
Notice that, while we used 32 to get 41, conversely, the 41 form of ‘simple scalability’ ii in Section 1 and the ‘conjoint Weber type II’, that is i, imply
homogeneity of degree 0 32, iii:
a b a
b a
b a b
K m n x y , . . . ,m n x y 5 F mx ,ny , . . . ,mx ,ny
j 1
1 n
n j
1 1
n n
a b
a b
5 F x ,y , . . . ,x ,y 5 K x y , . . . ,x y j 5 1, . . . ,n.
j 1
1 n
n j
1 1
n n
Now, substituting 41 into 37 and applying 35 and 36, we get
m ac
bd
i i
a b
a b
a b
a b
P
x y
ij ij
F [K x y , . . . ,x y , . . . ,K x y
, . . . ,x y
]
j 11
11 1n
1n j
m1 m1
mn mn
i 51
]]]] ]]]]]]]]]]]]]]]]
5
n n
m ac
bd a
b a
b a
b a
b
i i
OP
x y
O
F [K x y , . . . ,x y , . . . ,K x y
, . . . ,x y
]
ik ik
k 11
11 1n
1n k
m1 m1
mn mn
i 51 k 51
k 51
j 5 1, . . . ,n, 42
where cf. 40 Ft , . . . ,t 5 hF t , . . . ,t ,F t , . . . ,t ]
1 m
1 1
m 2
1 m
a b
5 F t , . . . ,t F t , . . . ,t .
43
1 1
m 2
1 m
b
Since on the right hand side of 42 every x figures with exponent a, multiplied by y
il il
a b
to give x y , also the products of powers of x and y on the left hand side can only be
il il
il il
a b
x y or its powers. Thus we have to have
il il
m m
d 5 c i 5 1, . . . ,m, d 5
O
d 5
O
c 5 c ± 0 44
i i
i i
i 51 i 51
and, by 35,
´ 254
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
m m
c
i
G z , . . . ,z 5 G z , . . . ,z 5 Gz , . . . ,z 5
P
z , c 5
O
c ± 0, 45
1 1
m 2
1 m
1 m
i i
i 51 i 51
cf. 2 which defines G.
a b
In view of d 5 c , with the notation z 5 x y i 5 1, . . . ,m; j 5 1, . . . ,n, 42
i i
ij ij
ij
becomes simpler:
m c
i
P
z
ij
F [K z , . . . ,z , . . . ,K z , . . . ,z ]
j 11
1n j
m1 mn
i 51
]]] ]]]]]]]]]]]]
5 j 5 1, . . . ,n.
46
n n
m c
i
OP
z
O
F [K z , . . . ,z , . . . ,K z , . . . ,z ]
ik k
11 1n
k m1
mn i 51
k 51 k 51
´ The following has been proved in Aczel et al., 1997.
m
Lemma. Eqs. 30–32,36,46 and c 5
o c ± 0 are satisfied and
Ft, . . . ,t is
i 51 i
continuous and strictly monotonic if, and only if, the function w, given by
1 c
wt 5 Ft, . . . ,t 47
is the inverse of a continuous , strictly monotonic function c satisfying
1 ]
0 inf cz ,
, sup cz
48 n
0,z ,` 0,z ,`
and K is of the form
j
K z , . . . ,z 5 c[z Lz , . . . ,z ] j 5 1, . . . ,n,
49
j 1
n j
1 n
where w 5 Lz , . . . ,z is the unique solution of
1 n
n
O
cz w 5 1,
j j 51
thus
n
O
cz Lz , . . . ,z 5 1, 50
j 1
n j 51
and L is homogeneous of degree 21:
21
L lz , . . . ,lz 5 l
Lz , . . . ,z . 51
1 n
1 n
We reformulate Eq. 46 of this Lemma as follows. Up to notation, 46 is the same as 42 [even 44 d 5 c ; i 5 1, . . . ,m followed from 42]. We put both sides of 42
i i
into K , apply 44 and 32, and choose
j m
a b
c
i
F t , . . . ,t F t , . . . ,t 5 Ft , . . . ,t 5
P
wt . 52
1 1
m 2
1 m
1 m
i i 51
This tranforms 46, that is 42, into
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 255
m m
a b
c a
b c
i i
K
P
x y , . . . ,
P
x y
F G
j i 1
i 1 in
in i 51
i 51 m
m a
b a
b c
a b
a b
c
i i
5 K
P
w[K x y , . . . ,x y ] , . . . ,
P
w[K x y , . . . ,x y ]
S D
j 1
i 1 i 1
in in
n i 1
i 1 in
in i 51
i 51
j 5 1, . . . ,n. 53
We trust that the reader noticed that the strange looking Eq. 53 is really 37 with 41,52,45 taken into account, the latter originating from 23 and 24 through 35 and
44. ´
It may seem that we chose 52 arbitrarily. But it has been proved in Aczel et al., 1997 Appendix B that, in 46 for n . 2 this is the only possible
F, while for n 5 2 the general
F is given by
m c
i
Ft , . . . ,t 5 Tt , . . . ,t
P
wt
1 m
1 m
i i 51
where T is an arbitrary function satisfying T1 2 t , . . . ,1 2 t 5 Tt , . . . ,t
t [ ]0,1[; i 5 1, . . . ,m 54
1 m
1 m
i
for T ; 1 we get 52 again, cf. also Remark 8.
Thus we have proved the following.
Theorem 2. The general solution of the system 30,31,32,36,53 of functional equations
21
with continuous strictly monotonic w is given by 49, where c 5 w
has to satisfy 48 and L given by 50 has the property 51. For n 5 2 we get the same solution if, in 53
m c
i
each
P
w[K ] term is multiplied by a T which satisfies 54 but is otherwise
i 51 j
arbitrary. From the Lemma and from 41 we get
a a
a b
a b
F x , y , . . . ,x , y 5 c[x y Lx y , . . . ,x y ] j 5 1, . . . ,n,
55
j 1
1 n
n j
j 1
1 n
n
which, by 16 and 17, gives explicit expressions for Pe :pE and P e :E i 5
j i
j
1, . . . ,m; j 5 1, . . . ,n. All these results can be summarized, as they concern the probabilities and scales, in
the following main theorem
Theorem 3. Assume the following. [A]: R, P, v, w, n and R, P , v , w , n i 51, . . . ,m;
i i
i
m .1 are nonatomic two-scale n-families of choice probabilities with the same generating system F , . . . ,F i.e. 16,17 and 27,28 hold . [B]: There exist functions
1 n
˜ ˜ H , . . . ,H , v,w,
F ,F , M ,M , N, injective functions K , . . . ,K , in the sense 36 and
1 n
1 2
1 2
1 n
locally bounded nonconstant functions h , . . . ,h , G ,G with
1 n
1 2
h 1, 1 5 . . . 5 h 1, 1 5 G 1, . . . ,1 5 G 1, . . . ,1 5 1
1 n
1 2
´ 256
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
such that x∞G x, . . . ,x is nonconstant and Eqs. 18–26 are satisfied. Then the
1
following statements hold. a There exist nonzero real numbers a,b, a continuous strictly monotonic function
c which satisfies 48 and a function L satisfying 50 and 51 such that, for all
E 5 he , . . . ,e j [ S,
1 n
a b
a b
a b
Pe :E 5 cve we L[ve we , . . . ,ve we ],
56
j j
j 1
1 n
n a
b a
b a
b
P e :E 5 cv e w e L[v e w e , . . . ,v e w e ]
57
i j
i j
i j
i 1
i 1
i n
i n
i 5 1, . . . ,m; j 5 1, . . . ,n. b Furthermore
, there exist real numbers c , . . . ,c with nonzero sum such that
1 n
m m
c c
i i
v e 5
P
v e , we 5
P
w e , j 5 1, . . . ,n.
j i
j j
i j
i 51 i 51
We have determined also the auxiliary functions:
Corollary. Assumptions [A] and [B] imply also the following:
c For the functions N, M ,M , G ,G , h , . . . ,h , K , . . . ,K we have N
l ; 1
1 2
1 2
1 n
1 n
l . 0,
m c
i
M t , . . . ,t 5 M t , . . . ,t 5
P
t t , . . . ,t . 0,
1 1
m 2
1 m
i 51 i
1 m
m c
i
G z , . . . ,z 5 G z , . . . ,z 5
P
z ,
1 1
m 2
1 m
i i 51
a b
h x, y 5 x y x, y . 0; j 5 1, . . . ,n,
j
and K z , . . . ,z 5
c[z Lz , . . . ,z ] z . 0; j 5 1, . . . ,n.
j 1
n j
1 n
j
d Finally , the functions H j 5 1, . . . ,n are given by 33, where 55 holds for
j
F , . . . ,F and for F and F we have
1 n
1 2
m a
b c
i
F t , . . . ,t F t , . . . ,t 5
P
wt , 58
1 1
m 2
1 m
i i 51
if n . 2 and
m a
b c
i
F t , . . . ,t F t , . . . ,t 5 Tt , . . . ,t
P
wt , 59
1 1
m 2
1 m
1 m
i i 51
if n 5 2. Here w is the inverse function of c, t [ ]0,1[ j 5 1, . . . ,m and T is an
j
arbitrary function satisfying 54. Somewhat intricate calculations show that the converse of Theorem
3 and Corollary holds too:
´ J
. Aczel et al. Mathematical Social Sciences 39 2000 241 –262 257
Proposition. If a, b, c, d hold then also [A], [B], in particular Eqs. 16–28 are satisfied.
Remark 7. One may ask the following question. If an experiment has been done, in which various choice probabilities P i 5 1, . . . ,m have been measured, how can it be
i
ascertained whether there exist appropriate functions c and L, and real numbers a, b
such that 57 hold. Theorem 3 shows that in principle this could be done by recognizing from the experimental data that each probability depends only upon the corresponding
scale values, more exactly upon their power products, and that the P , . . . ,P can be
1 m
combined aggregated into a probability which has the same property. This aggregation should be implemented in two steps: first taking a function of P e :E , . . . , P e :E
1 j
m j
the same function for j 5 1, . . . ,n, then combining the results into what will be the aggregated probability [this is done with the functions
F and K cf. 43,33,41,40,19]. A
j
method of finding a, b, c and L may be based again on noticing that P e :E depends
i j
a b
only upon v e w e k 5 1, . . . ,n, that is
i k
i k
a b
a b
P e :E 5 K [v e w e , . . . ,v e w e ] 5 K z , . . . ,z ,
i j
j i
1 i
1 i
n i
n j
1 n
which yields a, b. One could possibly also notice that, with an external function w, the
a b
expression z 5 ve we may be extracted from w + K same j so:
j j
j j
w[K z , . . . ,z ] 5 z Lz , . . . ,z j 5 1, . . . ,n.
j 1
n j
1 n
The requirement, that 57 should hold, gives w[K z , . . . ,z ]
j 1
n 21
]]]]] c 5 w
and Lz , . . . ,z 5 .
1 n
z
j
If the last expression is the same for all j [ h1, . . . ,nj and L is homogeneous of degree 21 then we have the representations 57,56 and we know also what a, b,
c and L in it are.
The above process may require, however, too much guesswork. It will therefore probably be as or, possibly, more feasible to test these ideas by attempting to fit data
with the representation 57. We only touch on the relevant model fitting issues here; as indicated earlier, related issues are discussed in detail in Townsend and Landon 1982,
Ashby et al. 1994, Estes 1997, Bamber and van Santen 2000, and in Myung et al. 2000. In the present paper also in this Remark we have assumed and continue to
assume that the scales v , w i 5 1, . . . ,m are known; the model fitting issues are
i i
similar, though more complex, when the scales have also to be estimated from data. We also continue to assume that only the P i 5 1, . . . ,m are observable, i.e., we want to fit
i
the data with a representation that satisfies 57 and thus can be aggregated into a P as represented in 56. Again, the model fitting issues are similar, though more complex,
when P is also observable. First we consider the case where we have data only for one set E. Note that 57
implies that, for every e , e in E and for i 5 1, . . . ,m,
j k
´ 258
J . Aczel et al. Mathematical Social Sciences 39 2000 241 –262
21 a
b
c [P e :E ]
v e w e
i j
i j
i j
]]]] ]]]]
5 .
21 a
b
c [P e :E ]
v e w e
i k
i k
i k
Thus one should search for the ‘best fitting’ function c and parameters a, b, using
traditional and newly developing techniques. For instance, one might compare the fit of ct 5 t corresponding to a two-scale Luce model with exponents to that of ct 5 t 1
1 2
˜ ˜
t . Once one has the ‘best fitting’ function
c, the estimate L of L can be taken to be the solution of
n
˜ ˜
O
c [z Lz , . . . ,z ] 5 1.
j 1
n j 51
Our task is easier when we have data for at least two sets E, E9 with two or more common elements, say e , e . Then, for 57 to hold, we must have
j k
21 21
c [P e :E ]
c [P e :E9]
i j
i j
]]]] ]]]]
5 .
21 21
c [P e :E ]
c [P e :E9]
i k
i k
When ct 5 t, we have an observable property of the representation. In general, we can
first determine the function c that gives the ‘best fit’ to equality for these and similar
ratios, then turn to the estimation of the exponents a, b and of the function L.
Remark 8. On the other hand, as mentioned, while 58, 57 and 43 determine
F, even ignoring T there is some arbitrariness left for
F and F : even if F t , . . . ,t 5
1 2
1 1
m m
m A
B
i i
P
t , F t , . . . ,t 5
P
t , we have only aA 1 bB 5 c i 5 1, . . . ,m as
i 51 i
2 1
m i 51
i i
i i
restriction. Notice, however, that already in the calculations in Section 1 we used the
a b
function F 5 F F for aggregation, rather than the individual functions F ,F .
1 2
1 2
Actually, we could write 37 as cf. 53 K h [G x , . . . ,x
,G y , . . . , y ], . . . ,h [G x , . . . ,x
,G y , . . . , y ]
j 1
1 11
m1 2
11 m1
n 1
1n mn
2 1n
mn
5 K [ FK [h x ,y , . . . ,h x ,y ], . . . ,K [h x ,y , . . . ,h x ,y ],
j 1
1 11
11 n
1n 1n
1 1
m1 m1
n mn
mn
: FK [h x ,y , . . . ,h x ,y ], . . . ,K [h x ,y , . . . ,h x ,y ]]
n 1
11 11
n 1n
1n n
1 m1
m1 n
mn mn
j 5 1, . . . ,n, 60
which has the advantage of containing F rather than F and F , although it does not
1 2
look simpler than 37. But the symmetry in having both F and F as counterparts of
1 2
G and G may be attractive, even though it turns out eventually that G 5 G but in
1 2
1 2
general F ± F . Ultimately the choice between 37 and 60 is a matter of taste.
1 2
Remark 9. The above Proposition shows that without adding further assumptions to [A] and [B] one cannot restrict
F , F , T and c, L more closely than by 58, 59, 54 and
1 2
´ 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’
