Effect of choice set size on choice
probabilities: An extended logit model
Dipak C. JAIN *
Frank M. BASS * *
The multinomial
logit model in the context
of the Lute choice axiom implies that the logarithm
of the ratio of choice probabilities
(the log odds ratio) of two objects depends only on the attributes
of the two objects and not on the attributes
of other objects in the
choice set. Noting
this weakness
of the Lute choice axiom and the conventional
logit model, Batsell and Polking
(1985) prove the
existence of a unique set of additive
numbers
that equate to the log odds ratio and that depend on the objects in the choice set. They
observed
that the Lute model spawned
multiattribute
extensions
and applications
and call for such extensions
and applications
of
their generalization
of the Lute model. We provide
such an extension
here and show analytically
that it is possible to stay within the
multinomial
logit family while avoiding
the HA restriction.
1. Introduction
The Lute choice axiom (Lute, 1959) states that the ratio of choice probabilities of two objects
doesnot depend upon the number or type of objects in the choice set. Although there are many
instanceswhen this very strong assumption seems to hold, there are many others when it does
not. The assumption has often been characterized verbally as the assumption of ‘independence
of irrelevant alternatives’ even though a better and more descriptive phrase might be ‘independenceamong alternatives’. Perhaps the primary application areas of the multinomial logit model
in the context of the Lute choice axiom has been in assessingthe choice probability of a new
brand and the consequent effect of the introduction of the new brand into the choice set on the
choice probabilities of existing brands. Batsell and Poll&g (1985) note three potential problems
with the Lute model in evaluating new alternatives. They refer to these problems as: the
similarity (or substitutability) problem, the dominance problem, and the regularity problem. The
Lute axiom implies that a new alternative will draw proportionally from existing alternatives,
but if the new alternative is very similar to an existing alternative a more reasonable expectation
might be that it would draw disproportionately from the similar alternative. The dominance
problem arises when one brand dominates the other when only the two are available, but when
neither dominates a third brand when paired with it. Because the Lute axiom indicates that the
ratio of choice probabilities does not depend on the choice set, the dominance phenomenon
representsa violation of the axiom. The Lute axiom requires that the introduction of a new item
* Kellogg Graduate
School of Management,
Northwestern
University,
Evanston,
IL 60208, U.S.A.
* * School of Management,
The University
of Texas at Dallas, Richardson,
TX 75083-0688,
U.S.A.
The authors wish to thank Yu-Min
Chen and Ram C. Rao of the University
of Texas at Dallas for their many
Comments
from Pradeep Chintagunta
and Vaman Shenoy are also greatly appreciated.
We are very grateful
IJRM reviewers for their valuable comments.
The usual disclaimer
applies.
Intern. J. of Research
North-Holland
0167-8116/89/$3.50
in Marketing
0 1989,
Elsevier
6 (1989)
Science
l-11
Publishers
B.V. (North-Holland)
helpful suggestions.
to three anonymous
D. C. Jain,
2
F. M. Bass / Effect
of choice
set size on choice probabilities
into the choice set will never increase the choice probability
for an item already in the set, but
there may be instances in practice when this regularity condition will be violated. i
Batsell and Polking have developed a generalization
of the Lute choice axiom that permits the
ratio of choice probabilities
to depend on the choice set thus avoiding the IIA (independence
of
irrelevant
alternatives)
limitations
of the Lute choice axiom. As a consequence,
the three
problems mentioned by Batsell and Polking can be dealt within the context of the Batsell and
Poll&g generalization.
They prove the existence of a unique set of additive numbers that equate
to the log odds ratio of choice probabilities
and that depend on the objects in the choice set. In
addition,
they show how, with experimental
data, these numbers may be estimated using
ordinary least squares (after performing
a simple log transformation
due to Nakanishi
and
Cooper, 1974) at the individual level or some level of aggregation.
Batsell and Polking observe that the Lute model spawned multiattribute
extensions and
applications
and call for such extensions and applications
of their generalization
of the Lute
model. We provide such an extension here and show analytically
that it is possible to stay within
the multinomial
logit family while avoiding the IIA restriction.
The rest of the paper is organized as follows. In section 2, we discuss briefly the previous
research dealing with the limitations
of the Lute-based
models. We present the extended logit
model in section 3. An example is illustrated
in section 4. Section 5 contains a discussion of
applications.
Conclusions can be found in section 6.
2. Previous
research
Lute’s choice axiom, though a major contribution
to the tradition of probabilistic
prediction
of choice, suffers from certain limitations
as discussed in the introduction.
Attempts have been
made by researchers in various disciplines to develop choice models that are free from these
discussed limitations.
For example, Hausman and Wise (1978) proposed a conditional
probit
model which allows for correlation among the error components of the random utility model and
therefore, does not suffer from the IIA property. This model, therefore, allows for the variation
in tastes for the attributes of the alternatives across individuals.
Consequently,
the scale value of
each alternative in a choice set depends not only on the attributes of that alternative, but also on
the attributes of other alternatives in the choice set. However, the conditional
probit model is
computationally
not very efficient (Currim,
1982). Another model that circumvents
the IIA
property is the Nested Multinomial
Logit (NMNL)
due to McFadden
(1981). The procedure for
estimating the parameters of this model requires that the choice process follow a tree structure
(Maddala, 1983). The limitation
is that when the number of levels in the tree structure and the
number of alternatives are large, then estimating the nested logit model is extremely difficult.
Two other very elegant probabilistic
models of individual
choice behavior developed by
mathematical
psychologists are the Elimination
by Aspects (EBA) model due to Tversky (1972)
and the Hierarchical
Elimination
Model (HEM) due to Tversky and Sattath (1979). In the EBA
model, each alternative is considered to be a collection of some measurable aspects. At each
.
’ We have not come
example of Huber,
across
Payne
any empirical
evidence
and Puto (1982).
on the violation
of regularity
in practice
other
than
the results
of the laboratory
D.C. Jain,
F.M.
Bass / Effect
of choice
set size on choice
probabilities
3
stage of the choice process, an aspect is first selected which eliminates
all alternatives
not
possessing the selected aspect from the choice process. This phenomenon
continues until only
one alternative is left in the choice set. The EBA model, therefore, explicitly takes into account
the ‘similarity’ aspect of alternatives in the choice set. This model, though theoretically
very well
grounded, has not been very popular with the marketing researchers in an applied context due to
methodological
and implementation
problems. This model requires 2” - 2 parameters to be
estimated for a choice set with n alternatives.
In the HEM model, Tversky and Sattath (1979) impose a tree structure on the set of
measurable aspects that reduces the number of parameters from 2” - 2 to 2n - 2. Though there
is a significant reduction in the number of parameters in the HEM model as compared to the
EBA model, still there are some problems in implementing
the HEM model (see Batsell and
Polking, 1985, p. 179). Furthermore,
one reason for the models like EBA and HEM not being
used by marketing researchers is that within the framework
of these models, it is difficult to
aggregate individual choice behavior to study market behavior.
Batsell and Polking propose a new model which is a generalization
of the Lute choice model.
Their model, like the EBA or HEM, does not explicitly model an individual’s
choice process but
‘simply scales the effects the competing products have on each other’s market shares’. Consequently, their model can be used to study the competitive
structure of a product market. One
additional advantage of their model is that it does not impose any a priori structure on the choice
sets like a tree-like structure. It also does not require a complete knowledge about the aspects
that govern choice as in the’case of EBA models. We discuss this formulation
below.
2.1. The Batsell and Polk@
Batsell and Polking
generalization
have generalized
the Lute
choice axiom by proving
that if
(2.1)
there exist unique numbers
aij such that
(2.2)
where A is the set of available alternatives. For example, if A = { i, j, k, I) then A - { i, j} =
{k, r}, and therefore, I= (8, {k}, {I}, {k, l}}. We can write # as
(2.3)
where c$ is the natural logarithm of the ratio of the market share of alternative i to the share of
alternative j. a(; (ali) measures the effect of alternative
k (1) on the competitive
nature of
alternatives i and j when the alternative k (r) is included in the choice set { i, j}. Similarly, afj
measures the effect of the pair { k, I} on the competition
between alternatives i and j. In other
words, aI”, and afj measure separately the individual effects due to alternatives
k and 1 and afj.
measures the joint effect of k and 1. The model in (2.3) is analogous to an analysis of variance
model with interactive effects. Other sophisticated analysis of variance models (orthogonal
main
effects and fractional factorial designs) have been used by researchers to study consumer choice
4
D. C. Jain,
F. M. Bass / Effect
behavior under controlled situations
1983). 2
We now propose a multiattribute
provides insights into the competitive
3. The extended
of choice
(Louviere
set size on choice probabilities
and Hensher,
1983; Louviere
extension of the Batsell and Poking
nature of alternatives in a choice set.
and Woodworth,
model
that
also
logit model
3.1. The classical logit model
Under the
on the utility
in turn, are
alternative i
classical multinomial
logit model the probability
of choosing an alternative depends
of that alternative relative to the utilities of all the available alternatives. Utilities,
functions of the attributes of alternatives.
The conventional
model of utility of
is the random utility model:
U(i)
= V(i)
+ e(i).
The scale values V(i) are deterministic
and depend on the attributes of alternatives
terms are stochastic. The scale values are assumed to have a linear form:
V(i) = iB,X,,,
(3.1)
and the e(i)
(3.2)
where Xi, is the observed value of attribute I for alternative i and B, is the importance weight of
attribute 1. The development
of the choice model on the basis of U(i) follows from the basic
principle that the individual
will choose an alternative
i if the perceived utility, U(i), is the
largest of all such values. Assuming that the e( i)‘s are independently
and identically distributed
extreme value random variables (McFadden,
1974, pp. 110-111) it can be shown that
p,(i)
= e’(‘)/Ce’(j),
P-3)
where A is the choice set containing alternatives i and j, and the summation in j extends to all j
in A. It is clear from (3.3) that for any two alternatives i and j, we can write
p,(
Taking
j)/p,
logarithms
(
j)
=
eVi)/eW)
=
eVi)-
on both sides of the equality
wm/e4(Al = w - W).
(3.4)
v(j).
in equation
(3.4), we have
(3.5)
Equation (3.5) states that the log odds ratio of two choice objects depends only on the utilities of
the two objects and is independent
of the choice set. These utilities and the corresponding
attribute evaluations of the objects do not depend on the choice set. The weaknesses of the
classical logit model stem from the limitations
implied by (3.5) and if these weaknesses are to be
* The contribution
of these studies is that any choice experiment
can be viewed as 2N factorial
design where N is the number
of
alternatives.
Consequently,
the analysis of variance
model would consist of main effects and the interaction
terms. The main effects
represent
the independent
scale values for each of the alternatives
and the interaction
terms are modifications
in the main effect
values depending
on the presence
or absence of other alternatives
in the set. Testing
the statistical
significance
of the interaction
terms corresponds
to a test of the IIA propert).
This is similar to the interpretations
of the terms (Y,,‘s in the Batsell and Polking
model. Under the Batsell and Polking
framework,
if the interaction
terms are zero, then /I,; = a$ implying
that the IIA property
holds.
D. C. Jain,
F.M.
Bass / Effect
of choice
set size on choice
probabilities
5
overcome, a way must be found to relax the independence
condition.
The Batsell and Polking
proof provides a way to do this and at the same time it provides a framework for incorporating
attributes into utility thus allowing the development
of an extended logit model.
3.2. The extended
logit model
The Batsell and Polking development,
as indicated in (2.3), provides a way around the IIA
restriction of the Lute axiom by permitting
the ratio of choice probabilities
and hence the
underlying probabilities
themselves, to vary with the presence or absence of other elements in the
choice set. In the classical logit model indicated in (3.3) it has traditionally
been assumed that the
scale values of utilities, the V(i), of objects does not vary with the choice set. Thus the classical
logit model is consistent with the Lute choice axiom. The logit model has many advantages, both
conceptual and computational.
Therefore, if a way can be found to retain the logit model, while
relaxing the IIA restriction, much will be gained. From a conceptual viewpoint it is desirable to
retain the notion that the utilities of objects depend on the attributes of objects. However, if
consumers evaluate utilities of objects relative to the availability
of other objects the utilities will
vary with choice sets.
We show here that it is possible to stay within the multinomial
logit family, while avoiding the
IIA restriction, by allowing the scale value for an alternative to depend on attributes of other
alternatives or on the structure of the choice set. Utilizing the random utility model of equations
(3.1) and (3.2), we propose that the error term can be further decomposed into additional terms
such that the remaining error is due only to unobserved
attributes of the alternatives
in the
choice set. Let T = {i, j, k, s} be a set of alternatives. For each nonempty subset A c T and for
i, j E A we propose that, for an individual
having choice set A, his utility for brand i can be
written as
L
V,(i)
= c
I=1
L
BIX,,+
c B:ZkXi,
I=1
L
L
+ c
I=1
B”ZsXii,+
c
BFsZksXjl,
(3.6)
I=1
where
1
0
ifk,sEA,
otherwise,
and where
B,Y =
the change in the importance
weight associated with the Ith attribute,
yth alternative is present in the choice set, I = 1, 2,. . . , L, y E A.
when the
The expressions above show that V,(i) depends on the structure of the choice set within the
random utility model.
In (3.6), unlike (3.9, the utilities depend on the choice set. The attributes of the choice objects
in the set are allowed by (3.6) to influence the utility of each choice object. In those cases in
which the Lute axiom is violated it would appear that consumers are evaluating objects on the
basis of a comparison of attributes among the choice objects in the set. The extended logit model
provides more than just a basis for fitting attribute evaluations to utilities when the Lute axiom
does not hold in that it provides a conceptual way to tie utility variation with choice sets to the
attributes of choice objects in the set. We begin by retaining the logit model, but we permit the
log odds ratio, and hence the utilities, to vary with the choice set.
D.C. Jain,
6
F.M.
Bass / Effect
of choice
set size on choice probabilities
Hence,
Therefore,
i B,Xi,+
I=1
p,“, =
;
B;Z,Xi,+
I=1
; B,Xj,+
I=1
;
B;ZsXi,+
5 B:Z,X,,
I=1
5 B:“Z,,Xi,
I=1
I=1
+ ;
B;Z,xj,+
I=1
i
6 B,k”Z,,X,,
I=1
.
(3.8)
i
Define
qj, = xi, - xj,.
Then, equation
(3.9)
(3.8) can be written
as
(3.10)
For example,
if A = {i, j, k}, then Z, = 1, Z, = 0, Z,, = 0, and (3.10) reduces to
,8; = i
BIFj,+
=
6
(3.11)
5 B:x.jl
I=1
I=1
[(Bl+
(3.12)
B,k)Yjl]*
I=1
Similarly,
if A = { i, j, k, s}, then we get
p;=
i
(3.13)
[(B,+B:+B;+B:~)Y,~~].
I=1
3.3. Comparison
with Batsell and Polking framework
If A = {i, j, k, s}, then equation
lq=i
c
I=1
[ IC{A-(i,
B:Fjl
j))
L
=
B:Yjl
IC{A-(i,
=
=
j))
C
IC{A-(i,
(3.13) can be written
as
1
(3.14)
ix I=1
[afj]3
(3.15)
A)
where
L
af, = c B:y.j,
I=1
and BIB = B,.
(3.16)
D.C. Jain,
F.M.
Bass / Effect
of choice
set size on choice probabilities
7
Interestingly,
equation (3.15) is identical to (2.2) of the Batsell and Polking generalization.
Hence, there exists a one to one correspondence
between each term in (3.10) and (2.3). Batsell
and Polking point out certain relationships
among the alphas. There exist similar relationships
among the parameters in our multiattribute
model (see Appendix A).
3.4. Interpretation
of the parameters
If A = { i, j},
then from (3.10) we get
p,‘: = ;
B,yj,=
ix:.,
(3.17)
I=1
and if A = {i, j, k}, then we know from (3.12) that
p$=
i (B,+B:)Y;~,=(Y$+cQ~,
(3.18)
I=1
where k can indicate the presence of any third object in the choice set. Now if B;k, the change in
the importance associated with the lth attribute when k is present in A, is zero for each 1, then
afj = 0 and
p,$ = i
BJjr=
cxfj,
I=1
implying that Lute choice axiom holds. Therefore, for a choice set A = { i, j, k}, testing that the
IIA property holds is equivalent to testing B,k = 0 for all 1. Similarly, we can interpret and test
the parameters in the higher order models.
Batsell and Polking have shown how to relate the effects of other objects in the choice set on
the log odds ratio for any pair of objects. We have shown how to tie these effects to the attributes
of the objects.
4. An example
When attribute evaluations are available for each subject and for each choice object along with
the log odds ratio for each subject for each choice set it is possible to estimate the importance
weights for individual
subjects. These weights will, of course, vary with the choice set.
Batsell and Polking conducted an experiment with a student subject in which the subject chose
a snack from among each of the possible subsets of two or more choices from: (1) Yodels, (2)
Twinkies, (3) Devil Dogs, (4) Milky Way, and (5) Three Musketeers. In this way, they were able
to obtain estimates of p,$ for all i, j, for all choice sets A for this subject and consequently they
were able to estimate the (Y parameters which equate to the log odds ratio for each choice set.
They define a second-order solution as one in which CX$is sufficiently
close to c$~ for all i, j, for
all choice sets A such that additional
(Y’S may be ignored. A third-order
solution exists when
p( = afj + ar; f or all i, j, k, for all choice sets A and a fourth-order
solution is consistent with
equation (2.3). They found that a third-order
solution was sufficient for their data.
We now show how the results of this experiment can be interpreted in an attribute framework.
We restrict ourselves to the third order solution of Batsell and Polking and illustrate
the
relationship between the estimated alphas and the attribute importance weights B,.
8
D.C. Jain,
Table
F.M.
Bass / Effect
of choice
set size on choice probabilities
1
Parameter
Estimate
t-value
R2
;:
- 0.4498
- 0.4645
- 0.4428
0.6040
- 9.6
- 9.04
- 9.09
1.76
0.9485
0.9423
0.9430
0.3831
0.6283
14.23
0.9759
B5
a An outlier
Define
noted
by Batsell
a
and Polking.
a simple dummy
variable
Xi, = X,, = X,, = 1
We then have, using equation
and
attribute:
cake vs. candy, i.e.,
X,, = X,, = 0.
(3.Q
where Z, = 1 if k E A, and i, j = 1, 2, 3, 4, 5. Consider choice sets of the form { i, j } . There are
ten pairs {i, j} and, therefore, there are ten values of a:.. Using the relationship
CYyj
= B,( xi, - xj1)7
(4.2)
these ten values may be regressed on the ten values of Xi, - Xjl to yield ii = 0.4433 with a
t-value of 4.86 and R2 = 0.7243. This result indicates that when only a pair of snacks are
available and when one of the snacks in the pair is a ‘cake’ the log odds ratio is increased by the
presence of the ‘cake’ attribute. Now in order to study the effect of including a new alternative
say k in the choice set, we consider the values of c$~. It is clear that with five alternatives we only
have six estimated values of afj. For example, if k = 1, we consider the values c&, a)&, &, &,
a&, and &. To establish relationship
between c$ and B,k( Xi, - Xi,) we regress the 6 values of
a> for each k on the values of Xi, - Xji. 3 The results are presented in table 1.
The results in table 1 are generally consistent with those of Batsell and Polking for their third
order solution. They found that the presence of Yodels, or of Twinkies,
in the choice set
involving one object with the cake attribute and one with the candy attribute, diminished
the
probability
of choosing cake. For Devil Dogs, however, they found that when Devil Dogs was
introduced
into a set with Twinkies and Three Musketeers, it took proportionally
more share
from Three Musketeers than from Twinkies. We also find that the fit for the third order effect of
Devil Dogs is significantly
lower than for the other objects.
Our results indicate that when three snacks are available, the presence of another snack in the
choice set with the ‘cake’ attribute diminishes the log odds ratio and the presence of another
choice with the ‘candy’ attribute increases the log odds ratio. Therefore, the introduction
of a
sixth snack into the choice set would result in the new item drawing share disproportionately
3 Note that in the Batsell and Polking model the values of a)j are not estimated
only from three alternative
sets of the form {l, 2, 3),
(1,3,4},
(1,2,5},
{l, 3,4}, (1,3,5},
(1,4,5).
The four alternative
sets and the five alternative
sets also contain
information
about
the effects of adding alternative
1. In the snack data there are 75 choice probabilities
for the 26 subsets (ten sets of size 2, ten sets of
size 3, five sets of size 4, and one set of size 5).‘In estimating
the second-order
model, Batsell and Polking
used data from all 26
subsets. In fact, the same total number
of observations
were used for estimating
the different
models.
However,
the number
of
parameters
being estimated
in each model were different.
D. C. Jain,
F.M.
Bass / Effect
of choice set size on choice probabilities
from the ‘cake’ snacks if it has the ‘cake’ attribute
snacks if it has the ‘candy’ attribute. 4
5. Discussion
and disproportionately
9
from
the ‘candy’
of applications
A common application
of logit models is in the evaluation of choice probabilities
for new
products. If the utility of the existing products and the new product can be elicited, the logit
model provides a basis for estimating
the choice probabilities
for each of the products.
Unfortunately,
however, in the traditional
approach the IIA limitation
is encountered
and the
new brand will draw its share from the other offerings on a proportional
basis. In the traditional
approach it is possible to relate attributes and utilities (or choice probabilities)
without reference
to choice sets. However, if choice probabilities
vary with choice sets and the new brand draws its
share from existing alternatives on other than a proportional
basis, the traditional
approach fails.
In the application of our approach in overcoming the limitation
of the traditional
approach one
could elicit choice probabilities
from consumers in a way that would permit these probabilities
to
vary with the choice set. Using a constant sum scale, for example, one could have consumers
allocate a fixed number of chips to the alternatives when different alternatives
were available.
The allocations could then be converted to choice probabilities
that would depend on the choice
sets. It would then be possible to apply the Batsell and Polking method to estimate the higher
order effects and, in turn, our method to relate these effects to the attributes of objects in the
different choice sets. In this way it is possible to determine the influence of particular attributes
on the non-proportional
choice structure.
One difference of our approach in terms of estimation is that in the extended logit model the
number of parameters to be estimated is L times the number of parameters in the Batsell and
Polking framework. Therefore, the degrees of freedom available for estimation in our approach
will diminish with the number of attributes. Possible restrictions and limitations
on the ability to
estimate at the individual
level, however, may be mitigated by pooling observations over groups
of individuals and by assuming that the attribute parameters are homogeneous
over the group.
6. Conclusions
We have developed a multiattribute
extension of the Batsell and Polking generalization
of the
Lute choice axiom. This extension is an extended logit model. We have established a one-to-one
correspondence between the parameters of the extended logit model and those of the Batsell and
Polking model. The extended logit model has been applied to individual level experimental
data
provided by Batsell and Polking. Aggregate and pooled data may also be analyzed within the
framework we suggest.
4 This procedure
of estimating
B, and B,k is an indirect
approach
because it first requires
an estimate
of C$ and of;. Alternatively,
we can estimate the B,‘s and B/ ‘s simultaneously
with regression
directly
just as Batsell and Poking
did, using their data. That is,
given the availability
of choice probabilities
for each set and hence the log odds ratios for all pairs and all sets and given attribute
values for each object, it is possible to regress the log odds ratios for all pairs in all sets on the attributes
to obtain estimates
of the
B,‘s and LI/ ‘s.
10
D.C. Jain,
Appendix
A. Relationship
F.M.
Bass / Effect
of choice
set size on choice probabilities
among the alphas
The parameters q,. in the Batsell and Polking framework possesscertain relationship among
them [see proposition 9 in Batsell and Polking (1985)]. We want to show here that there exist
similar relationships among the parameters of the multiattribute model. We consider the
following cases:
Case I:
afj+Dlfi=O
for1c
{A-{i,
j}}.
(A-1)
Let A = {i, j, k}; then I = {a, {k}}. Therefore, for (Al) to hold, we must have
$.+a$=0
and a&+aTi=O.
Using our formulation in (3.17), we have
a$ = i B,(Xt,I=1
Therefore,
x,,).
a$= 4:B,(xj,-xi,)=
- iB,(xi,-xjJ=
-a$.,
I=1
I=1
implying c$ + afj = 0. Similarly, we can show that cut.+ afi = 0.
Case2: If In{i,
j, k} =fl, then
af, + a;k + aLi + a; + a;: + a$ = 0.
(A-2)
Consider I = 8; then I n { i, j, k} = fl and, for (A.2) to hold, we must show that
~~j+~~~+~~i+~~j+~~~+~~i=O
or
&+c&+&=O.
Again from expression (3.18) we have
afj=
;B,(xi/-xj[),
I=1
$k=
iB,(X,,-X,/),
I=1
ati = i
B,(X,,-
Xi,).
I=7
Adding the above three terms, the result follows.
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