International Game Theory Review, Vol. 3, No.4 (2001) 301-314
@ World Scientific Publishing Company
ADVERTISING
IN A COMPETITIVE
PRODUCT
LINE
GILA E. FRUCHTER
Gmduate School of Business Administmtion,
Bar-nan University, Ramat-Gan 52900, Ismel
/ruchtg@mail.biu.ac.il
In many markets, competing but differentiated firms offer a product line in order to
target each product to a different type of customer. Under competition, firms, while
trying to target different products to different segments, are also interested in "stealing"
customers from the competitors and, in particular, customers from the most profitable
segments. How should a profit-maximising firm allocate the advertising budget along
a competitive product line? Developing a closed-loop time-variant Nash equilibrium
strategy for a multi-player multi-product advertising game, we find the determinants
of the optimal advertising budget allocation in a product line in an oligopoly growing market. Moreover, for a symmetric competition in a fixed market, we derive an
analytic feedback time-variant Nash equilibrium strategy. Marketing implications regarding the firm's strategic orientation in product preference advertising spending, as
well as comparison to rivals' spending, are discussed. Finally, we show that using a
single-product advertising game instead of a multi-product advertising game, leads to
over-advertising.
1. Introduction
In many markets, competing but differentiated firms offer a product line in order
to target each product to a different type of customer. Under competition, firms,
while trying to target different products to different segments,are also interested
in "stealing" customers from the competitors and, in particular, customers from
the most profitable segments.How should a profit-maximising firm allocate the
advertising budget along a competitive product line? The presentpaper deals with
this major question. Assuminga dynamicand cqmpetitive environmentin a growing
market, we consider the case where the strategic partners focus their marketing
activities (such as advertising) on attracting salesfrom eachother and from outside
competition.
An appropriate framework for analysing dynamic marketing expenditure decisions in a competitive setting is that of differential games. A good number
of differential game models in advertising appear in the literature, see for e.g.,
Eliashberg and Chatterjee (1985), Rao (1990), and Erickson (1991). The special
issues of the Journal of Marketing Research (1985) on "Competition in Marketing" and of Marketing Science(1988) on "Competitive Marketing Strategy"
are instants of the interest in incorporating competitive effectsin modeling the
301
302
G. E. FJoouchter
response of firms' sales to such marketing variables as advertising. Still most of
the research has been restricted to open-loop solutions, which are not always
appropriate.
In this paper, we focus on deterring closed-loop solutions. To model the dynamics of the competition, resulting from the firms' marketing efforts to attract
sales from competitors, we chose the combat Lanchester model. Many researchers
in different contexts have used this model. Kimball (1957) first applied it to a
combat problem, and then Vidale and Wolfe (1957) used it as a sales-advertising
responsemodel. Others, among them, Isaacs (1965), Horsky (1977), Little (1979),
Case (1979), Deal et al. (1979), Deal (1979), Sorger (1989), Sethi (1983), Erickson
(1985, 1991, 1992and 1995), Fruchter and Kalish (1997 and 1998) and Fruchter
(1999) applied it to find optimal advertising strategies in a dynamic, competitive
market. Erickson (1992) provides empirical evidencethat a closed-loopsolution fits
the data better than an open-loop solution.
As a review by Erickson (1995) indicates, much of the differential gameanalysis
of advertising competition has been limited to duopoly and ftied market situations,
and very little study of more general oligopolies has been conducted. The extension
to oligopolies, even in a fixed market, is especially difficult as regards closed-loop
strategies, becausethere are more than one-state variables. In a growing market,
even in duopoly, there are more than one-state variables.
Fruchter and Kalish (1997) provide a closed-loopNash equilibrium solution for
a duopoly with fixed market, extending the analytical solution of Case (1979) to
the case of nonzero discount rate. Fruchter and Kalish (1998) analyse the case
of oligopoly with a fixed market and a single product using multi-instruments to
influence the salesdynamics of its product. Further analysesto a single product in
oligopoly markets are provided in Fruchter (1999a and 1999b).
The presentstudy extends the methodology of finding closed-loop strategiesto
an oligopoly growingmarket with multi-products using advertising (one instrument)
to influence the salesdynamics of the product line. We summarise the contributions
of this paper as follows:
(a) We developa suitable differential gamethat extends the single product advertising analysisof Fruchter (1999b) to the problem of advertising in a competitive
product line.
(b) For this game, we find Nash equilibrium closed-loop time-variant strategies.
(c) For a symmetric competition, we go further on and find an analytic feedback
(subgameperfect) time-variant Nash equilibrium strategy.
Moreover, the analysis of the model and solution lead to several marketing
implications.
Specifically we find that:
(1) The market potential of the competitive product line depends on advertising
efforts of all players and all products.
Advertising in a Competitive Product Line
303
(2) Determinants of optimal advertising budget allocation of a product line are:
advertising effectiveness,grossprofit rate, potential salesof the specific product
and the total number of the products in the line.
(3) The optimal advertising expenditure,allocated to a product in the line, is monotonically increasing with the potential sales of that product, and monotonically
decreasingwith the number of the products in the line.
(4) If the products differ in potential sales, advertising effectivenessor gross profit
rates, then imitation of the advertising budget allocated to one product to
another leads to over- or under-advertising.
(5) If the competitors differ in potential sales, advertising effectivenessor gross
profit rates, then imitation of the advertising budget allocated to the same
product of one competitor to another leads to over- or under-advertising.
(6) Using a single-product advertisinggame instead of a multi-product advertising
game leads to over-advertising.
Next we present the model.
2. Model Description
We consider an industry in a growing market where n firms compete with a set
of products& using advertising (as their major marketing instrument) to attract
customersfrom each other as well as new customers, in their efforts to maximise
their discounted profits over the planning horizon. Assuming an infinite planning
horizon, the problem of oligopolist k, k E {I,..., n}, is to develop an advertising
strategy that will maximise the presentvalue of the firm's profit stream, as given by
roo
llk(Ul,...,Un)=}o
0
m
LQkiSki(t)-Ak(t))e-rtdt,
i=l
k=I,...,n.
(1)
In formula (1), Ski(t), i = 1,..., m, represents the sales rate, at time t, of product
i of firm k, and m is the maximal number of products in the competitive set (if
for exampleproduct i of competitor k is not in competition, this means, Ski = 0);
qki, i = 1, . . . , m, representsthegrossprofit rate of product i of firm k; the amount
Ak(t) represents the advertising expenditure of firm k, at time t, for the whole
product line; and r is the discountrate. For simplicity, we assumethat all the firms
havethe same discount rate.
There is a dynamic relation betweenthe rate of changein salesand the competitors' simultaneous advertising efforts to attract and generate sales. A well-known
relation that captures the abovedynamics in the form of a differential equation is
the combat Lanchester model. The studies quoted above have used this model for
a single product. We hereby relax this assumption.
8The literature uses different terms in defining a competitive set of products. Lehmann and Winer
(1997) distinguish between product form and product category competition depending on whether
the products have the same or similar features. A competition that focuses on substitutable
product categories is termed generic competition.
304
G. E. Frnchter
Let the term PkiUki(t) denote the effectivenessof the advertising efforts of firm
k, k E {I, 2,. . ., n}, targeted to product i, i = 1,.. ., m, in attracting potential sales
at time t. The constant Pki is related to media of advertising, and to consumers'
brand perceptions and preferences.We identify the advertising effort of competitor
k, allocated to product i, Uki, with the squareroot of the correspondingadvertising
expenditure, i.e. the total advertising expenditure of firm k will be,
m
= LU~i(t).
Ak(t)
(2)
i=l
Let Ni be a parameter that represents the satumtion level of product i or, in
other words, the limit of the entire industry sales that can be generated from
product i as a result of marketing activities. Let ci(t) be the market potential of
product i at time t, i.e. the difference betweenthe saturation level of the product i's
market, Ni, and product i's total present industry salesat time t, EZ=l Ski(t). Then
n
Ni
=L
Ski(t)+ ci(t).
(3)
k=l
Using the above notation,
we generalize the Lanchester model in the follow-
ing way,
Ski(t)
= PkiUki(t)(Ni -
Ski(O)
= S~i'
n
8ki(t)) - 8ki(t)
L
PjiUji(t)
,n,
;=l,;#k
i = 1,...,m
,n,
i = 1, ,m
k = 1,
or
Ski(t)
= PkiUki(t)Ni
(4)
k = 1,
Ski(0) = S2i'
2.1. Problem statement
Considering (1)-(4) the oligopolist k's problem, k
.
~~1
00
= 1,
n, is:
m
L[qkiSki(t)
0 i=l
- u~i(t)]e-rtdt
. (5)
n
Ski(t) = PkiUki(t)Ni
,
-
Ski(t)
L
PjiUji(t),
Ski(O)= S2i' i = 1,. .., m
J'-1
For this differential game, we want to find Nash equilibrium closed-loopstrategies.
Such strategies, particularly depend on the current level of the sale rates. Therefore, having the ability to capture current changesin the market, they are realistic
strategies. However, while the concept of closed-loopstrategy is more appealing it
is more difficult to compute. Case (1979) found a closed-form,closed-loopsolution
Advertising in a Competitive Product Line
305
for the duopoly advertising game for a fixed and single-product market but only for
a zero discount rate. He mentioned that the oligopoly advertising game is unsolvable, even for a zero discount rate. Finding a closed-loopsolution in the caseof an
oligopoly game is especially difficult becausethere is more than one state variable.
Here, we solve this problem for an oligopoly growing and multi-product market.
2.2. The market potential
of the line
According to (4), changesin the sale rates of product i of competitor k are a result
of the advertising efforts on potential sales of firm k, Ni - Ski, and simultaneous
oppositeefforts of the rivals to generatebrand switching. Note that this modeldoes
not include customer retention activities.
Considering Eqs. (3) and (4), we have
n
=
i:i(t)
-
L Ski(t) =
n
L PkiUki(t),
-E:i(t)
k=l
ci(O) = Ni
-
k=l
n
L Ski(O),
,~_\
= 1,. . ., m.
i
k=l
Solving the differential
Eq. (5a), we obtain
,
n
Ci(t)
= ci(O)Exp -L
t
[
.
pkiUki(T)dT
k=110
.
Therefore (5b) demonstrates that:
Remark 2.1. The market potential of each product in the competitive line is a
function of advertising efforts of all players. Furthermore, it decreasesexponentially
with the effect of the aggregateadvertising of the competitors' efforts. Alternatively,
the total industry of salesof product i of competitor k grow exponentially with the
aggregateadvertising effectivenessof the competitors' efforts.
Moreover, Eq. (5b) implies that
m
m
L ci(t) = L
i=l
n
ci(O)Exp
I
-
k=l
i=l
t
L 1 PkiUki(r)dr
0
,
Let
m
.IV
=
LNi
(6)
i=l
be the saturation level of the line. Since saturation level is related to the demand,
formula (6) expressesthe interdependenceof demandof one product with others in
the line.
Considering (3), (5c) and (6), the difference between the saturation level of the
line and the total presentline salesat time t,
E~l ci(t). Therefore(5c) demonstratesthat:
.IV
- L:::::l E~=l Ski(t), is givenby
306 G. E. Fnschter
Remark 2.2. The market potential of the line in our model is a function of advertising efforts of all players and all products that composethe competitive line.
Remark 2.3. The particular caseof
c,(t) =0,
i=l,...;m
(7)
correspondsto the casewhen the market of product i is saturated and mature and
the total industry salesof the product is fixed.
3. Main
Results
3.1. The Nash equilibrium closed-loop stmtegies
Extending the approachof Fruchter (1999b) to the multi-product problem, we arrive
at the following result. For details seeAppendix A.
Theorem 3.1. Consider the differential game associatedwith (5) and let <I>(t)satisfy the following two-point boundaryvalue problem (TPBVP),
"*
Ski
--
J.
2
cJliie'
Ski(O) = S~i'
k = 1,
S;i)qji<I>~i(t )ert
-e
,n, i = 1,
rt
(8)
m
~ii(OO) = 0,
i = 1,
m
Then
Uki
= ~PkiqkiCPii(t)ert(Ni-
Ski),
k = 1,..., n, i = 1,. . ., m
(9)
are global Nash equilibrium closed-loopstmtegies of the abovedifferential game.
Proof.
SeeAppendix B.
0
Remark 3.1. If m = 1, the results are consistent with Fruchter (1999b).
Considering (9) and (6), we find that the factors that affect the advertising
budget allocation for a product line are: (i) advertising effectiveness,(ii) grossprofit
rate, (iii) potential salesof the specific product, and (iv) the total number of the
products in the line.
Moreover, considering (6), since the total saturation level, lV, is fixed, we find
that Ni is monotonically decreasingwith m. Considering (9), we conclude with the
following result regarding the optimal allocation of the advertising expenditure of
each competitor in a product line, (uki)2, k = 1,... ,n, i = 1,... ,m.
Proposition
allocated
3.1. In a competitive market, the optimal advertising expenditure
(Uki)2, k = 1,. . . , n, i = 1, . . . , m is monotonically increasing
to a product,
Advertising in a Competitive Product Line
with the potential sales of that product, Ni
-
307
Ski, and monotonically decreasingwith
the number of the products in the line, m.
Remark 3.2. The time-variant coefficient of each advertising strategy uk is the
same for each firm. At any moment, the advertising strategy of each firm differs
only by its advertising effectiveness,grossprofit rate and actual salesrate.
3.2. The Nash equilibrium feedback strategy
We consider now a c~
where the market is fixed, i.e.
n
LSki
i=
= Ni,
,m,
k=l
and the competitive effects are symmetrically created, i.
Pki = Pi,
qki = qi,
k = 1,
,n,
i = 1,
,m
This is the case when competitors, having similar products, cannot be differentiated with respect to their advertising effectivenessand gross profits rates, and in
addition, the market is saturated. This leadsto the following result.
Theorem 3.2. If the competition is symmetric and the market is fixed, then the
closed-looptime-variant Nash equilibrium stmtegiesin Theorem 3.1 becomefeedbacktime-variant Nash equilibrium stmtegies.Furthermore, they have thefollowing
analytical functional form, where
Uki=~Piqi'iI!i(t)(Ni-Ski)'
k=l,...,n,
i=l,...,m,
(12)
where
1
Ilfi(t) = ~ + -' .and
bi
= [r2 + 2p~qi(n-
1)Ni]1/2
and
Proof.
See Appendix C.
0
Remark 3.3. The equilibrium strategy (12) has the property of being subgame
perfect, i.e. it is optimal not only for the original game as specified by its initial
conditions but also for every subgameevolving from it.
308 G. E. Fruchter
3.3. A duopoly fixed market
If n = 2 and the market of each product
Sli = Si and
S2i
is fixed, then 81i + 82i = Nio Let
Ni - Si,
i = 1,
m
and let
cI>ii(t)ert = tp(Sio)
(14)
Considering the formula
= cp'(Si* )iJi*
ci>iiert + rcp( Si*)
,
and substitution in the secondequation of (8) yields
1
.. 2 -
2
2 [Pliqli(Ni - s.)
,.2,'.
2
..
P2iQ21iS.]cp(s' )cp(s.
2
..
21 [Pliqli(Ni
- s' ) +
..
2
2
.. -
P2iQ2is' ]cp (s~
lim cp(si' (t) )e-rt = 0
t-+oo
i
, ..
1 + rcp(s' ),
= i,...,m
Remark 3.4. Solving (16) backward we are able to find t.p(si.(0)) and therefore in
a duopoly fixed market, the TPBVP (8) can be transformed into an initial value
problem (which is a much easierproblem).
Remark 3.5. The equilibrium strategy in Theorem 3.1 now becomes
uli*
1
= :2PliQli'P
(Si* )(l'Vi
AT
-
i)
S
an
d
*
u2i
1
(Si* )S,i.
= :2P2iQ2i'P
~
= 1,..., m. (17)
Remark 3.6. If m = 1,takingNl = 1, the resultshereareconsistentwith Fruchter
and Kalish (1997).
4. Marketing Implications
The results obtained in Sec. 2, shed light on the general orientation of a firm's
advertising strategies, as measuredby the ratio of advertising budget allocation to
different products of a firm. In this context, Theorem 3.1 leads to the following
result.
Corollary 4.1. Consider an oligopoly growing market, and let i and j # i, i,
j E {1,.. ., m}, be two different products in the competitive product line of firm k,
k = 1,. .., n. Let Aki be the advertising expenditure allocatedto product i and Akj
to product j, then
Aki
Ak'i
=
, Ni - Ski
N.
.
1
Skj
)2
clJii(t) Pki ~
--<Pjj(t} Pkj qkj
)2
'v'k = I,...,n,
V i,j = 1,
,m
(18)
)
)
Advertising in a Competitive Product Line
309
Another useful orientation of a firm's advertising strategies can be obtained by
measuring the ratio of advertising budget allocated to the same product by different
competitors. Consideringagain Theorem 3.1, we conclude with the following.
(a) Advertising expendituresof two products into a product line of an oligopolist
are equal if and only if the products have the samepotential sales,advertising
effectivenessand grossprofit rates.
(b) For the same product, advertising expenditures of two competitors are equal
if and only if the competitors have the same potential sales, advertising effectiveness and grossprofit rates.
This leads to the following result:
Proposition
4.1. In a competitive product line:
(1) If the products differ in potential sales, advertising effectivenessor grossprofit
rates, then imitation of the advertising budget allocated to one product to another leads to over- or under-advertising.
(2) If the competitors differ in potential sales, advertising effectiveness or gross
profit rates, then imitation of the advertising budgetallocated to the sameproduct of one competitorto another leads to over- or under-advertising.
5. The Multi-Product
versus the Single-Product
Model
In the following, we want to assessthe advertising expenditures of an oligopolist
in a growing market with a multi-product competitive line, using the model of this
study over a single-product model.
If the competitor of the multi-product game is using the single-product game,
then in Theorem 3.1, m = 1 and according to Proposition 3.1, we conclude with
the following result.
Proposition 5.1. In a competitive product line, the use of a single-product advertising game instead of a multi-product advertising game leads to over-advertising.
6. Conclusions
Developing a differential game to solve the problem of advertising budget allocation among different products of a competitive product line of a firm in a growing
market, we find a closed-looptime-variant Nash equilibrium advertising strategy
310
G. E. l'ruchter
for each product in the line. The value of closed-loopstrategies is that they allow
a manager to adjust to changing market conditions in an optimal way. This enhancesthe ability to deal effectively with dynamic competition in the development
of advertising strategies. The resulting strategy providesus with the determinants
of the optimal advertising budget of a product line; We also show that in a duopoly
fixed market, the mathematical computations becomemuch simpler. In a caseof
symmetric competition, with fixed market, we are able to find an analytic feedback
time-variant (subgame perfect) Nash equilibrium strategy. Marketing implications
regarding the firm's strategic orientation in product preferenceadvertising spending, as well as comparison to rivals' spending, are discussed.Finally, we showthat
using a single-product advertising game, instead of a multi-product advertising
game, leads to over-advertising.
Acknow ledgments
This researchwas supported by Technion V.P.R. fund.
Appendix
A. Derivation
of the Closed-Loop Strategies
Considering the differential game (5), using variational calculus, cr. Kamien and
Schwartz (1991), and using a similar relationship betweenthe co-state and function
cIIii we obtain the TPBVP (8). Let u~~be the open-loopstrategies. Then
u~~=~Pkiqkicllii(t)ert(Ni-Ski)'
k=l,...,n,
i=I,...,m,
(A.l)
where ski and cIIii(t) are solutions of the TPBVP (8). FI-om(A.l), we construct the
closed-loopstrategies defined in (9).
Appendix
B. Proof of Theorem 3.1
To prove this theorem we use vectorial and matrix notation. The vectors qk and
Sk are of length m and consist from the elementsqkl,. . . , qkm and SkI,..., Skm,
respectively. The matrix function ~(t) is diagonal and consists of the elements
~ii(t), i = 1,. . . , m. Consider the zero sum
0 = -qk ~(t)skl': +
100
~(qk~(t)sk)dt.
(B.l)
From (B.l), we obtain
0 = q'{;cI>(O)Sk(O)+ 100, (qk ci>(t)Sk+ qf cI>(
t )Sk (t) )dt
Using Eq. (4) we obtain
n
0 = q[cI>(O)Sk(O)+ 100
(qkT' cI>(t)Sk
+
T cI>(t)(BkNuk(t) - ~
qk
L-.JBjUj(t)Sk)dt
j=1
(B.3)
Advertising in a Competitive Product Line
311
In (B.3), Bk, Uj and N are diagonal matrices consistingof Pkl,..., Pkm,Ujl,.
and N1,... , Nm, respectively.Considering (8) for <P, we obtain
0=
- I e-rt
q'[cI>(O)Sk(O)
+ 100 q'[
Sk
n
+ q'[cI>(t) BkNuk(t) -
L Bj Uj (t)Sk
dt
where OJ is a diagonal matrix consisting of the element8j1' .
Considering (B.l) and (9), Eq. (BA) becomes
0=
q'[clI(O)Sk(O)+
100
*
,Sjm
2(Uk - Ukl)TUk + qkBkCP(t)Nertukl
x e-rtdt
Adding the zero sum (B.5) to Ilk
llk(Ul,
= Ilk(ul,
, un) = q[ clI(O)Sk(O)
+ [00
,Un),in (1), we obtain
-Iluk - ukl12+ IIukl12- 2u~lTUk
n
+ 2 L(Uj - ujl)T BjBl;lUZ
I
e-rtdt
;=1
;#k
where II0 112
represents the Euclidean norm of the corresponding vector. Particularly,
IIk(ui.
,u~) =
q[ cI>(O)Sk(O)
+ 100
IIUkl12
-
2u~ITUk + q'f Bk<I>(t)N ertu~l
312
G. E. Fruchter
Considering(B.6) and (B.1), we have
*
Uk-I'
*
*'
Uk, Uk+I
,Un
-Iluk
n
q[cI>(t)ert
NL
Bj(uj
- ukll2
+ IIUkll2n
-
ujl) + 2L(uj
;=1
;..k
= llk(ui,... ,u~)
2UklTUk + q[ Bkcl>{t)Nertukl
-
ujl)T BjB;lUk
I
e-rtdt
;=1
;..k
Iluk - ukI12e-rtdt::;llk(ui,
-100
,u~),
for every Uk. This completes the proof of our theorem.
(Bo8)
0
Appendix C. Proof of Theorem 3.2
Considering(10) and (11), the secondequationin (8) becomes,
ci>ii(t)
= ~p~qi(n- l)Nicll~i(t)ert-e
rt
~ii(OO} = 0,
i = 1, ,m
(c.!)
Let
Wi(t) = ~ii(t)ert
(C.2)
Then (C.!) becomes the following Riccati-type equation,
.
lI!i(t)
1 2
= -1 + rlI!i(t) + 2PiQi(n
-
lim ll1i(t)e-rt=o,
t-too
2
I)NilI!i (t),
i=l,...,m.
(C.3)
Equation (C.3) depends only on time and not on initial conditions. As a result, the
time variant closed-loop Nash equilibrium strategy in (9) (i.e. an equilibrium that
is a function of time, state and initial conditions) becomesa time-variant feedback
Nash equilibrium strategy (i.e. an equilibrium that is a function only of time and
state). Furthermore, below we show that (C.3) can be solved analytically which
completesthe proof of our theorem.
Through the substitution
ll1i(t)
ai
=
-r
- bi
p1qi(n - l)Ni
ai
+ l/~(t),
and b, = [r2+ 2p~q,(n
- 1)N,]1/2
(C.4)
Advertisingin a CompetitiveProduct Line 313
the Riccati Eq. (C.3) can be transformed into the first order linear differential
equation
.
Vi(t)
-
bi Vi(t)
1 2
= -2Pi
qi(n -
l)Ni.
(CJ
Solving(C.5), and consideringthe boundaryconditiongivenin (C.3), we obtain
V(t) = eb.t+ Ci,
(C.!
where
Ci = p~qi(n - 1)Ni/2
bi
0
References
Case, J. H. (1979). Economics and the Competitive Process,New York: New York University Press.
Deal, K. (1979). "Optimizing Advertising Expenditures in a Dynamic Duopoly". Opemtions Resea~h, Vol. 27, No. 4, 682~92.
Deal, K., S. P. Sethi and G. L. Thompson (1979). "A Bilinear-Quadratic Differential
Game in Advertising". In Liu and Satinen (eds.), Control Theory in Mathematical
Economics, New York: Marcel Dekker, Inc., 91-109.
Eliashberg, J. and R. Chatterjee (1985). "Analytical Models of Competition with Implications for Marketing: Issues, Findings, and Outlook". Journal of Marketing Resea~h,
Vol. 22, 237-261.
Erickson, G. M. (1985). "A Model of Advertising Competition". Journal of Marketing
Resea~h, Vol. 22, 297-304.
Erickson, G. M. (1991). Dynamic Models of Advertising Competition: Open- and ClosedLoop Extensions. Boston, MA: Kluwer Academic Publishers.
Erickson, G. M. (1992). "Empirical Analysis of Closed-Loop Duopoly Advertising Strategies". Management Science, Vol. 38, 1732-1749.
Erickson, G. M. (1995). "Advertising Strategies in a Dynamic Oligopoly". Journal of
Marketing Resea~h, 233-237.
Fruchter, E. G. and S. Kalish (1997). "Closed-Loop Advertising Strategies in a Duopoly".
Management Science, Vol. 43, 54~3.
Fruchter, E. G. and S. Kalish (1998). "Dynamic Promotional Budgeting and Media Allocation". European Journal of Opemtional Resea~h, Vol. 111, 15-27.
Fruchter, E. G. (1999a). "The Many-Player Advertising Game". Management Science,
Vol. 45, 1609-1611.
Fruchter, E. G. (1999b). "Oligopoly Advertising Strategies with Market Expansion". Optimal Control Applications and Methods,Vol. 20, 199-211.
Horsky, D. (1977). "An Empirical Analysis of the Optimal Advertising Policy". Management Science, Vol. 23, 1037-1049.
Isaacs, R. (1965). Differential Games. New York: John Wiley.
Kamien, M. I. and N. Schwartz (1991). Dynamic Optimization: The Calculus of Variations
and Optimal Control in Economics and Management.Amsterdam: North-Holland.
Kimball, G. E. (1957). "Some Industrial Applications of Military Operations Research
Methods". Operotions Resea~h, Vol. 5, 201-204.
Lehmann, D. L. and R. S. Winer (1997). Analysis for Marketing Planning. Irwin.
Little, J. D. C. (1979). "Aggregate Advertising Models: The State of the Art". Operotions
Resea~h, Vol. 27, 62~67.
314
G. E. Fruchter
Roo, R. C. (1990). "Impact of Competition on Strategic Marketing Decisions". In G. Day,
B. Weitz and R. Wensley (Eds.), Interface of Marketing and Strategy. Greenwich,
CT: JAI Press.
Sethi, S. P. (1983). "Deterministic and Stochastic Optimization of a Dynamic Advertising
Model", Optimal Control Applications and Methods, Vol. 4, 179-184.
Sorger,G. (1989). "Competitive Dynamic Advertising: A Modification of the Case Game".
Journal of Economic Dynamics and Control, Vol. 13, 55-80.
Vidale, M. L. and H. B. Wolfe (1957). "An Operations ResearchStudy of Sales Response
to Advertising". Operations Resean:h,Vol. 5, 370-381.