Ralph L. Keeney.

A utility model for product positioning online

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MAY 2 2 75



Ralph L. Keeney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilien

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-75






Ralph L. Kecney

International Institute for Applied Systems Analysis

Laxenburg, Austria

Gary L. Lilian

Massachusetts Institute of Technology

Cambridge, Massachusetts

March 1975

WP #777-/5









MAY 2 2 1975

■:ECElv£D 1

A model and procedure are proposed to help design and position consumer
durable and industrial products. These products are characterized on the
one hand by little data of repeat-purchase variety but on the other by a high
level of consistency between product preference and purchase behavior. The
procedure is based on utility theoretic concepts for assessing preference
and inferring probable behavior. Several numerical examples are included.



Firms continually face positioning and design issues related to their
products. The product design issue is essentially: "What physical and
psychological characteristics would I like my (new or revamped) product to
have? [to maximize profit, market share, or more generally, the firm's ex-
pected utility]." This product design question is relevant to both new and
existing products. The characteristics that can be controlled include price,
several dimensions of use or quality, packaging, flavor (perhaps), etc. The
firm generally has explicit control over physical quantities (price, taste,
objective performance) and implicit control, through message and communica-
tions design, over the psychological quantities (a young-swinging beer, e.g.).

How should products be designed and how can those designs be changed
during a product's life cycle to meet a firm's objectives? There have been
two modes of analysis to help answer these questions. One approach is that
of the psychometricians. In particular, making use of similarity data, Stefflre
[13] developed "perceptual maps" of market structures. He suggested introduc-
ing brands in areas of the map which were relatively vacant. Other work has
followed; a summary of the multidimensional scaling literature is available
in Green and Carmone [4] and Green and Rao [5] .

Another approach to the design question is provided by marketing model
builders. Many attempts have been made to establish functional forms which
relate product attributes, such as price, to output measures such as market
share. Kotler [10] provides a review of much of the literature.

A more recent, promising approach is one taken by Urban [14]. Urban
uses psychometric techniques to map the space of consumers' attitudes toward
existing brands as well as an "ideal" brand. He postulates that the farther
a brand is from the ideal point, the lower its market share should be. The


procedure, v/hich models the trial and repeat-purchase processes separately,
assumes that a brand's long run market share is a parameterized quadratic
function of its distance from the ideal point. That function is then cali-
brated using actual market share data from the market. The m.odel was shown
to predict ultimate market share for a number of new brands quite well. But,
due to data requirements, the model's use would seem to be lim.ited to frequent-
ly purchased products. Virtually all work in these areas has been limited to
frequently purchased products, mainly due to the greater availability of data.
Thus, the problem of analytically positioning consumer durable products and
most industrial products has been largely ignored.

A main difference between consumers purchasing packaged foods, and com-
panies purchasing minicomputers is that industrial purchasing behavior con-
forms more closely to prescribed criteria (see Webster and Wind [15]). Thus
one may "prefer" Krispy Krakers to "Whole Oats" and purchase Whole Oats any-
way due to "variety seeking," availability, or other random behavior. The
same inconsistency is not likely to occur in the purchase of a new minicom-
puter for one's firm or in the selection by parents of a medical treatment
for a baby's congenital defect.

The methodology suggested is designed to be used for precisely those
situations in which little data (of a repeat-purchase variety) are likely
to be available, but where the customers (individuals or firms) purchase
consistently with stated or inferred preferences. The approach is based on
utility theory (see Raiffa [H] for a discussion of the basic concepts of
utility theory) . It assumes customers have a von Neumann-Morgenstern utility
function defined over the product variables - that is, customers are expected
utility maximizers and "act as they should." Utility theory has been used in
the past mainly in a normative or prescriptive sense — telling decision


makors what they should do in given circumstances. The market situations con-
sidered here are, by definition, those in which individuals do what they should.

Hauser [6] is currently developing a structure for models of choice between
finite alternatives. His structure of the analytical process of choice includes:
(1) observation of behavior and measurement; (2) reduction and abstraction —
reducing the n'ar.iber of product dimensions to a few, "independent" ones and
labelling them; (3) compaction, developing brand preference measures; (4)
probability of choice, relating preference to purchase behavior, and (5) aggre-
gation, transforming probability of purchase measures to market share measures.

The methods developed here suggest augmenting the observation step, by
measuring attitudes in face-to-face interviews, through methods of direct
utility function assessment as in [1] , [8] , [9] . The compaction operation then
uses utility theory and the assumption of a von Neumann-Morgenstern utility
function obviates the probability-of -choice step. Aggregation is performed by
taking explicit account of consumer heterogeneity throughout the procedure.

The paper is organized as follows: Section 1 presents the formal struc-
ture of the model and introduces notation. Section 2 develops a general frame-
work of analysis which is applied to a simple, hypothetical example in Section 3.

1. The Model and Notation

Consider a well-defined product class with N firms F, , ..., F in a single,

^ IN

specific market. Let m. be the market share of F., and assume that each firm

X 1

makes a single product.

Let the set of attributes X , X , ... completely characterize a product,
where X could be price, X could be reliability ratings, etc. These attributes
would be attained by factor analysis of a series of well-defined product ratings


or perhaps non-metric scaling procedures, given a set of similarity judgments.

Both methods have proven effective though neither has established "superiority."

(Green, [3]) The output of these procedures, then, would be a reduced set of

product characteristics, X, , . . . X . A snecific level of X . is x . so a product
' 1' J - 3D

is completely described by x = (x , ... x ) . The product of firm F. will


denoted by x = (x , . . . , x ) . A no-product purchase x = (x , . . . x )

1 J 1 u

could be included for completeness.

Customers will be designated by C . . . , C . It v/ill be assuined that each
^ *^ 1 K

customer C has a von Neumann-Morgenstern utility function u (x/A) , where C 's

utility function is specified by the set of parameters X = (X ,...,A ). Assuming

each customer buys a product, utility theory suggests he should buy the product

of firm F. such that his utility is maximized.

Since viewing the problem from the firm's point of view will require the
same methodology regardless of the firm, let us take the viewpoint of firm F .
Firm F has certain objectives which could include maximizing market share,
maximizing profit, and so on. We postulcite that the objective function of F
is also specified by a von Neumann-Morgenstern utility function v over market
share m , profit and/or other variables.

There are uncertainties here for both firms and customers. The firm wants
to know utility functions for all customers in the market. This information about
customer heterogeneity will be expressed in the form of a probability distribu-
tion P, (M over the parameters A_ describing a random customer's utility function.
Thus, customers' utility functions are likely to differ so >_ does not take on
a single value, but, rather, is expressed as a probability distribution. Any
firm will not have perfect knowledge about the "true" P, (^) and will, in general,
attempt to estimate the distribution, entailing some error. Thus, we might con-


sider parameterizing the distribution to give P , (X/9) where 9 = (0 ,...,S )

X/0 — 1 T

indicates the uncertainties of F about the true X. We quantify that uncertain-
ty by the probability distribution P (9) .

6 —

Consumers in general will differ in their knov/ledge or attitude about the
characteristics X of firm F.'s product. From F 's perspective, this hetero-
geneity can be described by the parameterized probability distributions

P {x / ), i=l,...,N, where = ( r '^ r ■ • • i ) .

To summarize here, our model contains (from F 's view):

1. An objective function v , which is a utility function, kno'.vn with cer-

2. A distribution of utility functions u (x/A) which vary across the

heterogeneous customer population quantified by i^ (A/G) . The proba-

A/o — —

bility distribution Pn(9) quantifies this uncertainty.
6 —

3. A set of distributions of product-perceptions, P^ (x /i ), i=l,...,N,
also varying across the heterogeneous customer-population. The proba-
bility distribution P ((f) quantifies this uncertainty.

General Model Structure

In this section, we first consider the decision an individual consumer
must make and how his decisions are inputs to the decision-making processes
of the firm. Then we focus on how firms can use the model for product posi-
tioning decisions.


2. 1 Consumer Decisions

The consumer must decide v;hich if any of the products in the market to
buy given that he will buy at most one. Thus we explicitly consider the case
of a consumer not purchasing any product. Another possibility would have been
to define consumers as those who will buy one product and then formally include
uncertainty about the number of consumers in the market.

Our model does not explicitly include individual consumer uncertainty

about product characteristics. Rather, F. is uncertain both about the set


of consumer utility functions and the set of consumer product perceptions.

Explicit inclusion of consumer uncertainty would needlessly complicate the

croblem. We will assume that C has a utility function u, (x) and his choices

k k —

are not to buy a product and receive x or to buy the product of F. and re-
ceive x , i=l,2,...N. He will choose the option x* giving him the highest
utility where x* is defined by

(1) u (x*) = max {Uj^(2i^) >
i=0,l, . . .N

In the case of uncertainty, the consumer should choose the product of

firm F. which maximizes his expected utility. If P (x) represents the judg-
i X

ment of consun\er C about firm F.'s jjroduct, the option x_* should be chosen
such that

E[u, (x*)] = max Ju, (x) P^ (x ) dx ,
'^- i=0,l N '^ ^ -

where E [u (x*) ] is the expected utility of the product of firm F. (where F
k — 1 o

designates the no product option) for individual C .


2. 2 Firm Decisions Under Certainty

Under certainty, firm F should maximize its market utility v . Here we
assume the distribution of utility functions is known and that customers do
not vary in their perception of brand characteristics (that is 9 and ^ , i=l,
2,...,N, are known). The condition of certainty could be used as a first cut,
as less inform.ation is required here to attain a product design decision. Firm
F has a product with characteristics x and the population utility functions
are represented by u (x/^) where P (X) represents the population heterogeneity.
From (1) it follows that a consumer with parameters A_ will choose firm Ft ' s
product if

(2) \^^/h) > \^^/h) ' h=0,l,..., N, h?^l,

where we neglect ties in terms of equally desireable alternatives. The pro-
portion of consumers m, who choose x is

(3) m^ = /p. {X)d\_,

where At is that set of A's such that (2) holds. Neglecting ties, A , A, ,...,A^,
1 o 1 N

are mutually exclusive and collectively exhaustive so N

y m. = 1 as reauired.

1 -a

Suppose F IS considering changing its product position from x to 5< .
Then a new m can be determined exactly as m was using (2) and (3) .

Suppose a new firm with utility function v is trying to enter a volume
inelastic market with a maximum profit product (x) ; that is, say,
V = m • G • (s - C(x) ) - d(x) where s is unit product selling price, C is
the unit cost of the product (x) , G is the market volume, m is its market


share, and d(x) is the fixed development cost (plant, R&D, etc.) associated
with (x) . Tlien v is the profit associated v;ith the new product. There will
likely be a set Q of alternate product positions, generically denoted by x,
that the new firm could attain. Given the existing products in the market,
for each possible x there is a set A(x) (perhaps null) defined by


A(x) E {A: u, (x/X) > u (x /X) , i=0, 1, . . . , N}
— k — k — —

The best decision for this firm is to choose x in Q to maximize

(5)v(x) =

/ P (A_)dA


(G • (s-C(x)


where the first terra in brackets in (5) represents the market share of product (x)
and v(x) is the profit associated with the product. Under "nice" conditions
it may be possible to simply differentiate v(x) with respect to x to determine
the product position to maximize profit.

2.3 Firm Decisions Uncer Uncertainty

The problems under uncertainty are parallel but more complicated than
those with certainty. The two sources of uncertainty are the firms' imper-
fect knowledge about preference (or utility) heterogeneity, characterized by 9_,
and imperfect knowledge about perceptual heterogeneity, characterized by
^, i=0,l,...,N.

We v/ill assume here, for simplicity, that perceptual heterogeneity and
utility heterogeneity are independent within individuals. Fix the utility
heterogeneity parameters as A_ and select a customer with utility function
u{x/A) at random. That customer's selection will be from the set of product


characteristic distributions P (^1 /^ ) » i=0,l,...,N. The product purchased
should be x* such that

E(u(x*/_X,^) = max E(u(x )A,


Online LibraryRalph L. KeeneyA utility model for product positioning → online text (page 1 of 2)