David E Bell.

A market share theorem online

. (page 1 of 1)
Online LibraryDavid E BellA market share theorem → online text (page 1 of 1)
Font size
QR-code for this ebook




JUN 2 4 74




David E. Bell*
Ralph L. Keeney=^
John D. C. Littlef

March, 1974





JUN 24 74


David E. Bell*
Ralph L. Keeney^
John D. C. Littlef

March, 1974

*International Institute for Applied System Analysis, Laxenburg, Austria
=|=M.I.T., Sloan School of Management & Operations Research Center

This work was supported in part by a grant from NABISCO, Inc. for research
in marketing science.


JUL 1 1974
^rt. 1. T. LiBKArtltS


Many marketing models use variants of the relationship: Market share
equals marketing effort divided by total marketing effort. Although the
relation can be assumed directly, certain insight is gained by deriving
it from more fundamental assumptions as follows. For a given customer
group, each competitive seller has a real valued "attraction" with the
properties: (1) attraction is non-negative, (2) two sellers with equal
attraction have equal market share, (3) the market share for a given seller
will be affected in the same manner if the attraction of any other seller is
increased by a fixed amount.

A theorem proven states that if the relation between share and
attraction satisfies the above assumptions, then share equals attraction
divided by total attraction. Insofar as marketing factors can be assembled
into an attraction function that satisifies the assumptions, the theorem
provides a method for modeling market share.


1 . Introduction

Marketing model builders frequently use relationships of the form
(us)/(us + them) to express the effects of "us" variables on purchase
probability and market share. For example, Hlavac and Little [1]
hypothesize that the probability a car buyer will purchase his car at a
given dealer is the ratio of the dealer's attractiveness (which depends on
various dealer characteristics) to the sum of the same quantities over
all dealers. Urban [2], in his new product model SPRINTER, makes the
sales rate of a brand in a store depend on the ratio of a function of
certain brand variables to the sum of such functions across brands. Kuehn
and Weiss [3] make use of (us)/(us + them) formulations in a marketing game
model, as does Kotler [4] in a market simulation. Mills [5] and Friedman [6]
employ models of this form in game-theoretic analyses of competition.
Urban [7] and Lambin [8] fit similar models to empirical data. Urban to a
product sold in supermarkets and Lambin to a gasoline market.

In all these cases the result of the formulation is to bring a
competitive effect into the model by simple normalization. That is, a
quantity, let us call it attraction , is defined that relates only to
marketing actions and uncontrolled variables of a specific selling entity.
Then, by adding attractions over sellers and using the sum as a denominator,
a market share is obtained for each seller. The result is a competitive
model, since any seller's market share depends on the actions of every other
seller. Time lags, market segmentation or other phenomena may subsequently
be added so as better to represent other market features.

This approach to competition solves a dilemma for the model builder.
Suppose he believes, for example, that salesmen affect sales. He can draw
up a relation between sales and sales effort and try to calibrate it with
field data. However, competitive actions clearly affect what happens and
the model builder seems to need a new relationship for each possible level
of activity of each competitor. The problem has suddenly become very com-
plicated. Yet, it seems plausible that the salesmen's efforts can be viewed
as enhancing the seller's position with the customers on some absolute scale.
This can then interact with the effects created by other sellers measures on
comparable absolute scales. The linear normalization offers a way to repre-
sent the interaction.

Normalized attraction models of this type can be postulated directly,
but it is of interest to examine them more closely and ask what basic assump-
tions can be used to derive them. We shall demonstrate that under certain
conditions such a normalization is mathematically required.

The present paper deals with share, whereas sales are also a needed
output in most marketing models. A common approach is to relate total
market sales to total marketing effort, thereby breaking the model building
task into the two parts. However, only the first part will be studied here.
It should also be pointed out that there are other approaches to modeling
competitive interaction. For one such see Little [9].

2. Problem Definition

Given a finite set S = {s,,...,s } of sellers which includes all

I n

sellers from whom a given customer group makes its purchases, suppose that
for each seller s.eS an "attraction" value a(s.) is calculated. We suppose the
competitive situation can be completely determined by the vector of attractions

a = (a(s ), a(s„),...,a(s^)) = (a, ,a,,. . . ,a^) .

That is, the market share m(s.) of a seller is fully determined by a.

Attraction may be a function of the seller's advertising expenditure
and effectiveness, the price of his product, the reputation of the company,
the service given during and after purchase, location of retail stores and
much more. Indeed, the attraction of an individual seller can, if we wish,
be a function of these qualities for all the other sellers, or

a(s.) = 0.(q^,...q^;p^,...p^; ) ,

where q. may be quality of service of seller j, p. might indicate seller j's


price, and so on. However, one would hope that most of a seller's attraction

would be the result of his own actions and most model builders have treated
it this way.

Since, by definition, attraction completely determines market share,
it can be said that

m(s.) = f.(a^) , i = 1,... ,n ,

for some function f. where m(s.) is the market share of seller i. Clearly,

i=l ^


< m(s.) < 1 , i = 1 ,... ,n ,

but otherwise the functions f. are as yet arbitrary.

The aim here is to give conditions on the relationship between
attraction and market share which force the simple linear normalization model


f.(a) = — L
1 — n

j = l


3. Formal Development

The assumptions are:

Al ) The attraction vector is non-negative and non-zero,

a > and X! a . > .
i^l ^

A2) A seller with zero attraction has no market share.

a. = — m(s.) -

A3) Two sellers with equal attraction have equal market share,

a . = a . — ► m(s. ) = m(s .)
1 J 1 J

A4) The market share of a given seller will be affected in the same
manner if the attraction of any other seller is increased by a
fixed amount a. Mathematically,

f.(a + Ae^) - f.(a) , for j M ,
is independent of j, where e. is the jth unit vector.

Theorem . If a market share is assigned to each seller based only on the
attraction vector and in such a way that assumptions Al - A4 are satisfied,
then market share is given by

m(s.) = ■ , for i = 1 ,2,. .. ,n .

Z a(s.
j = l ^

Proof Since the vector a^ completely defines the vector (m(Si ), . . . ,m(s ))

then functions f, ,...,f exist such that
1 n

m(s.) = f.(a) , for all i = l,...,n



i = l ^


f.(a) > , for all i = l,...,n . (2)

Consider the set


CX = {a : ^T is constant and ^ a. = A for some A > 0}
' i = l ^

Let a^ , a^eCX, a ^ a , then it will be shown that

f^(a) = f^(a) ,

from which it may be concluded that f.(a_) is a function only of a. and ^ a.

Let a° = min(a^ , a) taken componentwise and e. be the j^h unit vector.
Then if b is defined as the smallest non-zero component of two vectors
(a - a , a - a ), some i and j exist such that we can define

-1 ^0 -1 -
a=a^+be., a for all k ,

we have

— k — =k =

a = a^ , a = a , for k ^ n-1 .

Thus, f 1 (a ) ~ f 1 (§_) as required, establishing the claim that the market

share m(s,) is constant over the set CX and hence depends only upon the

quantities a, and A. So, in general, we will express f.(a) in the form
fi{a.,A). By A3

f.(a,A) = fj(a,A)

so that

f. = f. , for all i,j - l,...,n. (3)

By A2

f.(0,A) ^ . (4)

Now suppose by contradiction that, for any fixed a and A,

f.(a,A) = A / a/A

Assume X > a/A; the case A < a/A being similar.
Consider two vectors a , a where

a. = , i = 1,...,k-l,

a . = a , i = k ,

a^ = a^. , i = k + l,...,n.

By (3)

and by (1 )

so that by (3) and (4)


a. = a/k , i = 1 ,... ,k

n n _

i=k+1 ^ i=k+1 ^

k k _

E u® - E fi(a) >

i=l ^ i=l ^

A = kf.{a/k,A) ,

f.(a/k,A) = A/k

Now consider a vector a with


a . = a/k , i = 1 , ... ,n-l

a = A - (n-l)a/k ,

where A > (n-1 )a/k




i=l ^ i=l ^ ^

= (n-l)A/k + f (A - (n-l)a/k)

> (n-l)A/k by (2) .

Hence, there is a contradiction if k and n can be chosen such that

(n-l)A/k > 1 ,

(n-l)a/k < A .
That is, if

(n-l)a/A < k < A(n-l) ,

a/A < k/n-1 < A . (5)

Obviously, (5) can be satisfied for some values of n and k. Hence,

f.(a,A) = a/A ,
and the theorem is proved.


4. Discussion

The key point of the mathematical analysis is that, subject to certain
basic assumptions relating the vector quantity, attraction, to the scalar
quantity, market share, mathematical consistency implies that market share
is a simple linear normalization of attraction. Let us look at the impli-
cations of the assumptions used.

Assumptions Al and A2 are rather inconsequential and made to simplify
the analysis. A2 states that sellers with zero attraction will have no
market share. Al requires attraction to be non-negative and says the attrac-
tion of at least one firm must be positive. Otherwise there would be no
active sellers in the market. Assumption A3 does have some substance. It
says that if two competing sellers have equal attraction, then they will
have an equal share of the market. If attraction were simply defined as
advertising, for instance, then one could argue against A3 in many cases.
Clearly, there are other factors which influence market share. Thus, A3
helps make clear to the model builder what he must include in his attraction
function to obtain a sensible result from the model.

A crucial assumption is A4. It states that if the attraction of a
competitor of s. increases by some amount A, then the new market share of s.
will not depend on which competitor made the increase. A4 does not say the market
share of s. would remain fixed. Intuitively, we would expect, in fact, a
drop in seller i's share if competitors increased their attraction. Is A4


We can think of two possible sources of deviations from A4: non-
linearty and asymmetry. Nonlinearity would be evidenced if adding an
increment to a small attraction produced a different effect (on others)
from adding the same amount to a large attraction. To some extent, how-
ever, this is a matter of the scale along which attraction is measured.
There is a clear advantage if attraction is additive in the sense of A4.

Asymmetry could arise if changes in attraction of one seller were
differentially effective on the customers of another. Aspects of asymmetry
can be formally considered in the linear normalization model by making
attraction of seller i partially dependent on some of the qualities of
seller j. However, in general, our assumptions do not accomodate asymmetry,
and, an extension of the theory would be required. In some situations
market segmentation would be sufficient to represent asymmetric effects.
Thus a marketing action may increase attractiveness more in one group than
another (for example, a sportier car may appeal more to younger people).
The algebra of market segmentation is described below.

To understand the implications of the theorem further, we present two
corollaries. However, either of them could be made as an assumption to re-
place A4. Then A4 would follow as a corollary.

CI: The market share of seller i depends only on his attraction a^ and
the sum of all attractions.

C2: If the attraction of seller i increases by an amount a and if the
attraction of seller j decreases by the same amount ^, while the
attraction of all other sellers s, , k ^ i,j, remains the same, then
the market share of sellers s, , k ?< i , j remains constant.


Corollary CI says that in considering the market share of seller i, one can
aggregate the other sellers together, take their aggregated attraction to be
the sum of their individual attractions, and then focus on seller i versus
the rest. Corollary C2 is similar in spirit but less encompassing. C2 is
local, whereas CI is global. One point worth noting is that A4 is an
assumption concerned with what happens when the total attraction, i.e., the
sum, increases. The alternatives CI and C2, on the other hand, concern the
reaction of the market when total attraction remains constant.

Considerations for Model Builders . The main point for model builders is
that a simple model which focuses on the attributes of a single seller, is
sufficiently rich to model a fully competitive market.

It is instructive to point out an appealing method that cannot be used
to deduce the normalization model. At first glance it appears that, since
market share is, by definition, the ratio of sales to total sales, it would
be sufficient to assume that sales are proportional to the seller's attrac-
tion function. Calculation of share immediately gives the normalization
model. However, this will only be valid in a totally non-competitive
market where the marketing activities of one seller do not influence the
sales of another. If, for example, the market is of fixed size in total
sales, individual sales cannot be linear with the attraction function.
Furthermore, sales cannot be independent of competitive attraction.


Notice we have not deduced any specific results about market behavior,
but rather some mathematical rules of the game. Thus, if someone asserts
an attraction function depending on, say, advertising and price, and it is
wrong, then the calculation of market shares will be wrong. Once attraction
is specified, however, we can answer such questions as what is the impact on
market share of incremental changes in price or advertising or any of the
other factors composing attraction.

Another interesting aspect of this model is the quantity A, the total
attraction of the sellers. One might construct a model of the size of the
market as a function of A. Combining this with the market share, one could
calculate for a given seller the total increase in his number of sales
generated by increases in attraction. Part of these new sales would be
due to an increased market size and part to an increased market share. In
fact, one could consider A^ , A^..., A^ to be the attractions of a number
of different product classes which compete with each other for consumers.
For instance, A^ may represent the total attraction of radios, A^ televison
sets, A3 stereo systems, and so on. One might postulate a different model
for computing the share of the electronic media market held by each of
these product classes. Combining this with our model for individual sellers
within a segment provides a more sophisticated competitive model.

Assumptions Al - A4 essentially make a(-) an unnormalized probability
function on the set of sellers. For an alternative axiomation that closely
parallels probability, see the Appendix. Market share, on the other hand,
satisfies all the axioms of probability theory and so, mathematically speak-
ing, is a probability function defined on the set of sellers. The statement
of the assumptions and results is in terms of market share, but the term "prob-
ability of purchase" could clearly be substituted without affecting the mathematical


development. Notice that the results refer to probability of purchase from
a seller given that a purchase will be made. In other words, the sum of
the purchase probabilities is presumed to be one. Obviously, the probability
of no purchase can be introduced as an extension of the model.

The fact that market share has the mathematical properties of a
probability can be helpful in various ways. For example, if several customer
groups or markets segments are identified, the concept of conditional market
share becomes useful. Let

C = {Ct,...,c ) - a set of r customer groups,
1 r

a(s.|c.) = attraction of seller s. within customer
group j,

p{c.) = proportion of total sales coming from customer

group c . .


Then assuming that Al - A4 hold for each customer group, the market share
of s. within customer group j is

i(s.|c.) = a(s.|c )/X! a(s, |c.).

and so the total market share is


i(s.) = X) m(s.|c.)p(c.)
1 j = i 1 J J


By partitioning the population into groups or segments a complex model can
be built up from simple elements. Different marketing variables, say, price,
promotion, advertising, and distribution, may impinge differently on different
segments, which may, in turn, respond differently. The responses would
define a relative attraction function which would then be assembled as shown
above. Thus, the adoption of a basic normalized attraction model does not
mean that all share expressions end up as simple ratios.



[1] T.E. Hlavac, Jr., and J.D.C. Little, "A Geographic Model of an Urban
Automobile Market," Proceedings of the Fourth International
Conference on Operational Research , D.B. Hertz and J. Melese, eds.,
Wiley-Interscience, New York, 1969, pp. 302-11.

[2] G.L. Urban, "SPRINTER Mod III: A Model for the Analysis of New
Frequently Purchased Consumer Products," Operations Research , 18 ,
pp. 805-54, (September 1970).

[3] A. A. Kuehn and D.L. Weiss, "Marketing Analysis Training Exercise,"
Behavioral Science , 10, pp. 51-67, (January 1965).

[4] P. Kotler, "Competitive Stragies for New Product Marketing Over the
Life Cycle," Management Science . U, B104-19, (December 1965).

[5] H.D. Mills, "A Study in Promotional Competition," Mathematical Models
and Methods in Marketing , F. Bass et al . , eds., Richard D. Irwin,
Homewood, Illinois, 1961, pp. 271-88.

[6] L. Friedman, "Game Theory in the Allocation of Advertising Expenditures,
Operations Research , 6, pp. 699-709, (September 1958).

[7] G.L. Urban, "An On-Line Technique for Estimating and Analyzing Complex
Models," Changing Marketing Systems , R. Moyer, eds., American Marketing
Association, 1968, pp. 322-27.

[8] J.J. Lambin, "A Computer On-Line Marketing Mix Model," Journal of
Marketing Research , 9, pp. 119-26, (May 1972).

[9] J.D.C. Little, "Brandaid II," SSM Working Paper 687/73, M.I.T.,
(November 1973).

[10] E. Parzen, Modern Probability Theory and Its Applications, Wiley, New
York, 1960, p. 18.



Attraction As An Unnormalized Probability

An alternative axiomization of the linear normalized market share
model brings out the close mathematical connection between attraction
and probability theory.

Let S = {s,, ..., s } = set of all sellers

S C S "" 3 subset of sellers

a(S) - attraction of a subset of sellers.

A sufficient set of axioms is:

Bl . Attraction is non negative ,

a(s.) > , s. e S •

B2 . The attraction of a subset of sellers is the sum of the

attractions of the sellers in the subset.

a(S) - E a(s.), S C S •
S.e S ^

83 . a(s.) is finite for all s. r 3 and a(s.) > for at least one s.
B4 . If two subsets of sellers have equal attractions, their market
shares are equal,

a(S^) = 3(82) ^ m(S^) = m{S^) , S^^S^C S •

The proof of the market share theorem is much the same as before.
The intermediate result

f^ (a) = f^ (a)

can be obtained as follows. Define

(S2, ..., s^

For a = a.

a(S) = z a. - A-a, by B2.
1=2 ^ '


For a^ = a,

n _

a(S) = E a. - A-a, by B2.

- i=2 ^

Therefore, denoting the market share of S given a = a' by m(S|a = £') ,

m(S|a = a ) = m(sla = a) by B4,
and so

f^(a) = 1 - in(S|a - a) - 1 - ni(S|a = a ) - f ^ (a )
as desired. The argument that f^(i) can be written f^(a,A) and f^ = f-
for all i,j is the same. Sinre by B2 and B4 there is an equivalence
between a single seller and a set of sellers with the same total attrac-
tion, we can extend the notation to f^(a,A) and ^c ^ '•'■j ^ f-j ^o'" ^^^

By definition a(S) = A and m(S) = 1 so that f.(A,A) = f5(A,A) = 1.
Consider a seller, say s, , with zero attraction. Let S = {s^, ..-, s^},

f^(0,A) + f3(A,A) = 1,
and so f, (0,A) = 0. This establishes (4) without assuming A2. The rest
of the proof is the same.

Axioms Bl and B2 are two of the three axioms of finite sample space
probability theory. (See, for example, Parzen [10].) The third probabil-
ity axiom is that the probability of a certain event is 1. 83 states two
properties implied by this, namely, finiteness and at least one positive
value, but stops short of the unity normalization. Thus Bl - B3 create
attraction as an unnormalized probability function. B4 makes the connec-
tion to share. Share itself satisfies all the axioms of probability and
so is a probability function defined on the set of sellers.

The axiomization B1-B4 is very appealing but was not chosen as the


basic approach because it introduces the additivity assumption by means
of the attraction of a set of sellers. The concept of the attraction
of a set seems a little artificial. This is because attraction has been
discussed as a property of an individual seller and, although our final
result implies that the concept can be extended to sets it seems more
natural to have this as a deduction than an assumption. The approach
chosen is to use A4, which expresses additivity in terms of increments
of an individual sellers' attraction so that no concept of collective
attraction is required.


illJIIIIIIIIIIIIIIIIIilllllllllllll 703 -7V

3 9080 00367 1523


lllillllllllillililil] loi-li

3 9080 00367 1556


llilllllllllllllliiiillllill 10^- IH

3 9080 00367 1572


llllllllllliliillllllllill "^ - s-TH

3 9080 00367 1598


llllilllillilllllllllllllilll ^(JH - r^

3 9080 00370 2534


illlllllllllllllllllliiilll 70S-7M

3 9080 00370 2559 (


lllllllllllilllllliilllllllilll 10i, ' 7h

3 9080 00370 2575


Online LibraryDavid E BellA market share theorem → online text (page 1 of 1)