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14

MASS. INST. TECH.

JUN 2 4 74

DCVitY LiSRASY

WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

A MARKET SHARE THEOREM

David E. Bell*

Ralph L. Keeney=^

John D. C. Littlef

704-74

March, 1974

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

Â«A5S. INST. TECH.

JUN 24 74

A MARKET SHARE THEOREM

David E. Bell*

Ralph L. Keeney^

John D. C. Littlef

704-74

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.

.M4\4

no.lD4-74

JUL 1 1974

^rt. 1. T. LiBKArtltS

Abstract

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.

0721174

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(sâ€ž),...,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

J J

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 ^

and

< 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

a.

f.(a) = â€” L

1 â€” n

j = l

^j

3. Formal Development

The assumptions are:

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

n

a > and X! a . > .

i^l ^

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

a. = â€” m(s.) -

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

a . = a . â€” â–º 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

a(s.)

m(s.) = â– , 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

with

and

i = l ^

(1)

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

Consider the set

n

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) ,

n

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

Let aÂ° = 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

â€” k â€” =k =

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

Thus, f 1 (a ) ~ f 1 (Â§_) 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,

1

a . = a , i = k ,

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

By (3)

and by (1 )

so that by (3) and (4)

or

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

n n _

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

k k _

E uÂ® - E fi(a) >

i=l ^ i=l ^

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

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

Now consider a vector a with

and

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

1

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

n

where A > (n-1 )a/k

10.

Now

n

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 ,

and

(n-l)a/k < A .

That is, if

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

or

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.

n

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

reasonable?

12.

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.

13.

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.

14.

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

15.

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 . .

J

Then assuming that Al - A4 hold for each customer group, the market share

of s. within customer group j is

n

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

and so the total market share is

r

i(s.) = X) m(s.|c.)p(c.)

1 j = i 1 J J

16.

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.

17.

REFERENCES

[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.

Al.

Appendix

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 â€¢

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 â€¢

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 â€¢

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 ^ '

A2.

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 = Â£') ,

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 ^ 'â€¢'â– j ^ f-j ^o'" ^^^

i,J,S.

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^},

then

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

A3.

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.

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414

104-

14

MASS. INST. TECH.

JUN 2 4 74

DCVitY LiSRASY

WORKING PAPER

ALFRED P. SLOAN SCHOOL OF MANAGEMENT

A MARKET SHARE THEOREM

David E. Bell*

Ralph L. Keeney=^

John D. C. Littlef

704-74

March, 1974

MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139

Â«A5S. INST. TECH.

JUN 24 74

A MARKET SHARE THEOREM

David E. Bell*

Ralph L. Keeney^

John D. C. Littlef

704-74

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.

.M4\4

no.lD4-74

JUL 1 1974

^rt. 1. T. LiBKArtltS

Abstract

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.

0721174

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(sâ€ž),...,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

J J

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 ^

and

< 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

a.

f.(a) = â€” L

1 â€” n

j = l

^j

3. Formal Development

The assumptions are:

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

n

a > and X! a . > .

i^l ^

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

a. = â€” m(s.) -

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

a . = a . â€” â–º 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

a(s.)

m(s.) = â– , 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

with

and

i = l ^

(1)

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

Consider the set

n

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) ,

n

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

Let aÂ° = 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

â€” k â€” =k =

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

Thus, f 1 (a ) ~ f 1 (Â§_) 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,

1

a . = a , i = k ,

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

By (3)

and by (1 )

so that by (3) and (4)

or

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

n n _

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

k k _

E uÂ® - E fi(a) >

i=l ^ i=l ^

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

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

Now consider a vector a with

and

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

1

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

n

where A > (n-1 )a/k

10.

Now

n

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 ,

and

(n-l)a/k < A .

That is, if

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

or

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.

n

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

reasonable?

12.

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.

13.

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.

14.

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

15.

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 . .

J

Then assuming that Al - A4 hold for each customer group, the market share

of s. within customer group j is

n

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

and so the total market share is

r

i(s.) = X) m(s.|c.)p(c.)

1 j = i 1 J J

16.

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.

17.

REFERENCES

[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.

Al.

Appendix

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 â€¢

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 â€¢

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 â€¢

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 ^ '

A2.

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 = Â£') ,

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 ^ 'â€¢'â– j ^ f-j ^o'" ^^^

i,J,S.

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^},

then

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

A3.

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.

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