**Dynamic Targeted Promotions**

**A Customer Retention and Acquisition Perspective**

**Gila E. Fruchter**

*Bar-Ilan University*

**Z. John Zhang**

*University of Pennsylvania*

*This research analyzes the strategic use of targeted *
*promo-tions for customer retention and acquisition in a dynamic*
*and competitive environment. The normative analysis*
*shows that a firm’s optimal targeting strategies, both *
*of-fensive and deof-fensive, depend on its actual market share,*
*the relevant redemption rate of its targeted promotions,*
*customer profitability, and the effectiveness of its targeted*
*promotions. These strategies have the attractive feature of*
*being an adaptive control rule. A firm can operationalize*
*these strategies by adjusting its planned promotional *
*in-centives on the basis of the observed differences between*
*actual and planned market shares and between actual and*
*planned redemption rates. In the long run, a focus on*
*customer retention is not an optimal strategy for all firms.*

**Keywords:**targeted promotions; competitive strategy;

*dif-ferential game*

Marketing information technology is rapidly trans-forming the way in which firms conduct their marketing in service industries. Because of their unprecedented ability to gather, store, and process consumer information on a large scale at a low cost, firms are obtaining ever sharper pictures of individual consumers who have been hitherto largely anonymous and, consequently, gain much flexibil-ity in targeting desired customers with tailored promo-tional incentives to achieve strategic marketing objectives.

For instance, a service firm can offer promotional incen-tives only to its current customers in an effort to defend its market share from competitive encroachment or accu-rately target its rival’s customers with promotional induce-ments to acquire new customers. In this article, we study the strategic use of targeted promotions and characterize the optimal targeting strategies a firm can adopt in a dynamic context.

The intense rivalry between AT&T and MCI in the
mar-ket for consumer long-distance telephone service
pro-vided a good example of how targeted promotions aided
by information technology could help a firm to realize its
strategic objectives. With a market share close to 13%,
MCI launched its promotional program Friends & Family
in the spring of 1990. The program was designed to attract
AT&T’s customers by offering substantial discounts for
family-and-friends calling circles.1 _{Aided by database}
marketing, MCI succeeded in switching at least 5 million
AT&T callers by the end of 1993 and increased its market
share to 19%. In response, AT&T introduced its own
ver-sion of database marketing in 1994, signing more than 28
million callers to its True USA and True World saving
pro-grams to shore up its customer base, and regained close to
1 million subscribers from its competitors. In the
mean-time, AT&T and MCI also engaged in an expensive
pro-motional tug-of-war by targeting each other’s customers

Journal of Service Research, Volume 7, No. 1, August 2004 3-19 DOI: 10.1177/1094670504266130

© 2004 Sage Publications

1. For details on the promotional rivalry between AT&T and MCI since 1990, see Rapp (1995) and Urry (1995).

with wback checks to induce switching. In 1994, for in-stance, the two firms jointly spent an estimated $1.3 billion on such checks, generating tremendous customer churn-ing. It is estimated that consumers switched 27 million times in 1994 alone, notwithstanding the fact that both firms also offered frequency and loyalty programs. Accu-rate targeting of own and rival’s customers with promo-tional incentives has clearly initiated in the industry a dynamic game of strategic promotions for customer retention and acquisition.

Using targeted promotions to retain current customers
and to switch a competitor’s customers is not limited to the
consumer long-distance telephone service industry;
in-deed, it is likely to become a widespread practice in many
other service industries such as banking and hospitality
in-dustries, where customer contact and recognition are
pre-requisite for service delivery.2 _{The advent of targeted}
promotions thus raises many competitive and strategic
is-sues for service managers that have not been raised before.
Most of these issues are concerned not with how much to
spend but with where to spend on promotions (“Briefings”
2003). Specifically, how should a firm use targeted
promo-tions? How should a firm respond to its rival’s targeted
promotions? What are the optimal targeting strategies for a
firm to retain its own customers and to attract a rival’s
cus-tomers, given that the rival will adjust its targeting
strate-gies over time? How do competing firms’ market shares
evolve over time under targeted promotions? We offer
some normative answers to these important managerial
questions by developing a differential game of targeted
promotions.

**Previous Research**

Studies on targeted promotions have not addressed
these questions in a dynamic context. Shaffer and Zhang
(1995), along with others, use a static game-theoretical
model to explore competitive implications of targeted
pro-motions and optimal targeting strategies. Although their
static model generates many valuable insights about how
targeting has changed the nature of market competition, it
cannot address issues related to the dynamics of
competition, especially market share implications of targeted
promotions.3_{Rossi,McCulloch,andAllenby(1995)approach}
targeted promotions from an empirical angle by developing
econometric methods for a firm to use in implementing
tar-geted promotions.

Studies on dynamic competition in marketing are nu-merous but focus mostly on advertising based on the Lanchester model. Early studies in this tradition include those of Kimball (1957), Vidale and Wolfe (1957), Isaacs (1965), Horsky (1977), Little (1979), Case (1979), Deal (1979), and Deal, Sethi, and Thompson (1979). These studies examine how advertising expenditure affects sales in a dynamic setting. In recent years, much progress has been made in modeling dynamic competition in advertis-ing, despite the fact that persistent technical difficulties re-main. Chintagunta and Vilcassim (1992) and Erickson (1992) contribute valuable insights about dynamic adver-tising competition by applying a differential game solu-tion to the case of the cola war and find that the closed-loop solution provides a better empirical fit than an open-loop solution. Erickson (1993) further extends his analysis to allow both offensive and defensive advertising expendi-tures. Because of inherent technical difficulties in solving a differential game, these previous studies have to resort to restrictive assumptions on model parameters and cannot be used directly to address questions of dynamic targeted promotions when promotional incentives are specific to groups of recipients and contingent on sales.

A related stream of literature is on customer retention
as a profitable business strategy.4_{Hart, James, and Sasser}
(1990) and Reichheld and Sasser (1990), for instance,
forcefully advocate a focus on customer retention through
investing in relationship-based assets. Although
influen-tial, such a strategic prescription obviously cannot be
opti-mal for all firms in a competitive context because customer
retention is a meaningful strategy only if customer
switch-ing by a competitor is a real threat. Indeed, McGahan and
Ghemawat (1994), using a two-stage game model, show
that a larger firm can profitably focus on customer
reten-tion, whereas a smaller firm is better off focusing on
cus-tomer switching. We extend the analysis on this important
strategy issue to the context of targeted promotions by
using a more general dynamic model, and we generate
some new insights about firms’ strategic orientations in
equilibrium.

**Study Approach and Main Results**

We develop a differential game suitable for modeling dynamic targeted promotions and provide analytical solu-tions for both closed-loop and open-loop strategies by modifying a technique developed by Fruchter and Kalish (1997) in the context of advertising competition. We con-clude that a firm’s optimal targeting strategies, both offen-sive and defenoffen-sive, in a dynamic setting depend on its actual market share, the relevant redemption rate of its

tar-2. For more examples in service industries, see Feinberg, Krishna, and Zhang (2002). There are many similar examples in other industries. For examples in the consumer goods industry, see Shaffer and Zhang (1995).

3. There are many static models on targeted pricing. See Shaffer and Zhang (2002) for a review of this literature. Most of these models conclude that equilibrium market shares for competing firms are not

lit-geted promotions, customer profitability, and the effec-tiveness of its targeted promotions. In the long run, a focus on customer retention cannot be an optimal strategy for all firms. A firm with a sufficiently large market share should stress customer retention, whereas a firm with a small mar-ket share should stress customer acquisition. When marmar-ket shares are more evenly divided in a market, firms should all focus on customer acquisition. We illustrate through a numerical example the trajectories over time of a firm’s market share, promotional expenditures, and profits as competing firms use targeted promotions optimally.

In what follows, we first set up our model. After provid-ing analytical solutions to the model, we discuss the mana-gerial implications of the solutions. We then conclude with suggestions for future research.

**MODEL**

Consider an industry where two competing firms have the capability of using targeted promotions both offen-sively for customer acquisition and defenoffen-sively for cus-tomer retention. We restrict our analysis to a market of fixed size to focus on competition for market shares. Therefore, customer acquisition entails customer switch-ing in our model. Without any loss of generality, we nor-malize the number of consumers in the market to one and thus denote a firm’s demand by its market share. To intro-duce targeted promotions, we assume that at any point in time, both firms have adequate information to identify their own and their competitor’s customers and hence can implement a different promotional program respectively for customer retention and acquisition.

Targeted promotions, unlike advertising, offer different
incentives to different types of consumers, depending on a
firm’s marketing objectives. To retain their current
cus-tomers, both AT&T and MCI, for instance, offered savings
programs whereby a customer could accumulate credit
points based on usage and redeem them for cash, free
long-distance minutes, or frequent flier miles. They also offered
win-back checks in the amount of $50 on average to bid for
their competitor’s customers. Therefore, to capture the
es-sence of targeted promotions, we introduce heterogeneity
in customer composition and allow a firm to tailor its
pro-motional incentives to each identifiable group of
consum-ers. Let*xk*(t) denote the market share for firm*k*at time*t.*

The*xk*(t) consists of two types of purchases: those from

re-peat customers, denoted by*x tk*1( ), and those from

switch-ing customers, denoted by *x tk*2( ). Therefore, at any time*t,*

we have

*x t x t x tk*( )= *k*1( )+ *k*2( ). (1)

To model a firm’s practice of using different promo-tional incentives to generate repeat and switching sales, let

δ*kd*( ) and*t* δ*ko*( ) denote firm*t* *k’s promotional incentives *

of-fered, respectively, to repeat and switching purchases at
time*t. These incentives can be coupons, premiums, *
spe-cial savings, or others. As these promotional incentives are
typically contingent on consumer purchase, the firm’s
pro-motional expenditures will depend on how many people
actually redeem its promotional offers. Therefore, firm*k’s*
defensive and offensive promotional expenditures are
given byδ*kd*( ) ( )*t x tk*1 andδ*k*

*o*
*k*

*t x t*

( ) ( )_{2} ,5_{respectively.}
We assume that each firm makes independent
promo-tional decisions at any time*t*to maximize its discounted
profit stream over the planning horizon. When this
plan-ning horizon is finite, firm *k’s optimization problem is*
characterized by
*Max* *q x T e*
*q* *t x t* *q*
*k*
*d*
*k*
*o* *k* *k*
*T*
*k* *rT*
*k* *kd* *k* *k* *k*
δ δ
δ δ
, ( )
( ( )) ( ) (
Π =
+ − + −
−
1

### [

*o*

### ]

*k*

*rt*

*T*

*t x t e dt*( )) ( )2 . 0 −

### ∫

(2)The constant*qk*represents the gross profit margin,6and the

constant*qkT* represents the value of firm*k’s market share at*

the terminal time*T.*

Given any promotional incentives, a firm’s payoffs
at time*t*depend on how effective its targeted
promo-tions are in generating sales. Letρ δ*kd*( ( ) ( ))*kd* *t x tk*1

1_{2}

and

ρ δ*ko*( ( ) ( ))*ko* *t x tk*2
1_{2}

be the respective effectiveness measures
of firm *k’s defensive and offensive promotional efforts.*
Here the effectiveness measure is taken as a square root
function of a specific promotional expenditure, as is
com-monly done in the literature for tractability, and it implies a
decreasing return to any promotional expenditure. The
constantsρ*kd* andρ*ko*are firm-specific effectiveness

coeffi-cients that capture the factors that affect the potency of a
firm’s targeted promotions such as database quality, the
accuracy of data analysis and targeted delivery, customer
loyalty, a firm’s product characteristics, and so on. To
sim-plify our notation, let*f t f tkjod*( )( *jkod*( )) denote the difference

between the effectiveness of firm*k’s (firmj’s) offensive*
promotions and that of firm*j’s (firmk’s) defensive *
promo-tions at time*t. The variable* *f t f tkjod*( )( *jkod*( )) thus measures

the competitive strength of firm*k’s (firmj’s) offensive *
tar-geted promotions in attracting the customers that firm*j*

5. Here we ignore any fixed cost in conducting targeted promotions as our focus is on how, rather than whether, a firm should conduct targeted promotions.

6. The implication is that a firm’s regular price and marginal cost of production are constant over time. That assumption is commonly made in studies of dynamic promotional decisions to maintain tractability. How-ever, in many mature industries, such as the consumer goods industry, such an assumption is not unduly restrictive.

(firm*k) tries to retain through its defensive targeted *
pro-motions. Then, we have

*f tkjod*( )=ρ δ*ko*( ( ) ( ))*ko* *t x tk*2 −ρ δ*dj*( ( ) ( ))*dj* *t x tj*
1_{2} 1_{2}
1 , (3)
*f tkjod*( )=ρ δ*oj*( ( ) ( ))*oj* *t x tj*2 −ρ δ*kd*( ( ) ( ))*dk* *t x tk*
1_{2}
1
1_{2}
. (4)

A firm’s market share and its composition change over time because of competitive targeted promotions, which in turn affect its promotional decisions. To capture these dy-namic linkages, we specify a dydy-namic system as follows:

( ) ( )( ( ) ( )) ( ) ( )
*x tk*1 = *f t x t x tkjod* *k* − *k*1 −*f t x tjkod* *k*1 ,
*xk*1 0 *xk*1
0
( )= , (5)
( ) ( )( ( ) ( )) ( ) ( )
*x tk*2 = *f t x t x tkjod* *j* − *k*2 −*f t x tjkod* *k*2 ,
*xk*2 0 *xk*2
0
( )= , (6)

where*j*≠and*k*= 1, 2. Equation (5) captures the effect of
targeted promotions on a firm’s repeat sales. Firm*k’s *
re-peat sales will increase if the firm can effectively convert
switching customers to repeat purchasers through its
of-fensive targeted promotions that overcome the rival’s
de-fensive promotions (the first term). It also increases if the
firm can effectively fend off the rival’s offensive targeted
promotions to retain its repeat purchasers (the second
term). Equation (6) indicates that firm*k*can acquire more
new customers from competition if it can effectively tap
the rival’s customers through its offensive targeted
promo-tions against the rival’s defensive promopromo-tions (the first
term) and/or effectively fend off the rival’s offensive
tar-geted promotions aimed at switching customers (the
sec-ond term). As*x t x t x tk*( )= *k*1( )+ *k*2( ), we have

( ) ( )( ( )) ( ) ( )
*x tk* =*f tkjod* 1−*x tk* −*f t x tjkod* *k* ,

*xk*( )0 =*xk*0.

(7)

Equation (7) describes the market share dynamics, and it is the well-known Lanchester dynamics modified to incor-porate firms’knowledge about consumers and their ability to offer targeted promotions. This dynamic process im-plies that a firm’s overall market share will increase if it can successfully acquire the rival’s customers through its overpowering offensive targeted promotions against the rival’s defensive promotions to retain them (the first term) and/or successfully fend off the rival’s offensive targeted promotions and retain its current customers (the second

term). This specification of dynamics is more general than Erickson’s (1993).7

For the differential game associated with equations (2),
(5), and (6), we want to find closed-loop Nash equilibrium
strategies ( , ,_{δ δ δ δ}* * *, *)
*k*
*d*
*k*
*o*
*j*
*d*
*j*
*o* _{that satisfy}
Π*k* *kd* *ko* *dj* *oj* Π*k* *kd* *ko* *dj* *oj*
*k*
*d*
( ,_{δ δ δ δ}* *, *, *) ( , ,_{δ δ δ δ}*, *),
δ
≥
∀ , ,δ*ko* *k*=1 2, , *j k*≠ .
(8)

In mathematical terms, a closed-loop strategy can be ex-pressed as

δ*kd* =δ*kd*( ,*t x x x xk*1, *k*2, *k*1, *k*2)

0 0 _{and}

δ*ok*=δ*ko*( ,*t x x x xk*1, *k*2, *k*1, *k*2),*k*= ,

0 0 _{1 2.} (9)

The particular case when the strategy does not depend on the initial condition is known in the literature as feedback strategy. Both feedback and closed-loop strategies are contingent on the observed state of the system (market shares in our model) and therefore best capture the dy-namic nature of competitive promotions. However, it is technically very challenging to derive an equilibrium of feedback or closed-loop strategies. An open-loop strategy is only time dependent. Such a strategy prescribes at the beginning of the game how a firm should promote at each point in time and is never revised over the course of the game. This strategy captures some realistic aspects of stra-tegic planning in promotions; however, its wide applica-tion is due mainly to the relative ease with which one can solve differential games by using standard optimal control methods. Indeed, Chintagunta and Vilcassim (1992) and Erickson (1992) provide empirical evidence that feedback solutions fit the actual data better than open-loop solu-tions. Fruchter and Kalish (1997) provide similar evidence for closed-loop strategies.

As*T*→ ∞, we can also extend the previous differential
game to the case of infinite planning horizon as follows:

*Max* *q* *t x t* *q* *t x t*
*k*
*d*
*k*
*o* *k*
*k* *kd* *k* *k* *ko* *k*
δ δ
δ δ
,
[( ( )) ( ) ( ( )) ( )]
Π = − 1 + − 2 *e dt*
*s t*
*x t* *f t x t x t* *f*
*rt*
*k* *kjod* *k* *k* *jkod*
−
∞

### ∫

= − − . . ( ) ( )( ( ) ( )) ( 0 1 1*t x t x*

*x*

*x t*

*f t x t x t*

*k*

*k*

*k*

*k*

*kjod*

*j*

*k*) ( ), ( ) ( ) ( )( ( ) ( )) 1 1 01 2 2 0 = = − − =

*f t x t xjkod*( ) ( ),

*k*2

*k*2( )0

*xk*02 (10) The control functionsδ

*kd*( ),

*t*δ

*ko*( ),

*t*

*k*= 1, 2, are

admissi-ble controls if they are bounded with the values of *f tkjod*( )

and *f tjkod*( ) being positive. For such values, equations (5),

(6), and (7) reveal 0≤*xk*_{1} ≤ ≤*xk* 1and 0≤*xk*2 ≤ ≤*xj* 1, and

.

7. Our specification, in contrast to Erickson’s (1993), allows more freedom in the choice of parameters and leads to more general conclu-sions.

the integrals associated with the above differential games exist.

In steady state, we have

*x* *x* *f*
*f* *f*
*k* *ss* _{k ss}*kjod*
*kjod* *jkok*
*ss*
1 2
2
= =
+
(11)
and

### [

### ]

*x*

*x x*

*f f*

*f*

*f*

*k*

*ss*

*k*

_{j ss}*kjod*

*jkod*

*kjod*

*jkod*

*2 = = 2 + , (12)*

_{ss}with*j*≠*k,k*= 1, 2. It is straightforward to show that for
*f tkjod*( )>0and*f tjkod*( )>0, the steady-state solutions in

equa-tions (10) and (11) are asymptotically stable.8

**OPTIMAL (EQUILIBRIUM)**
**TARGETING STRATEGIES**

As we show in appendix, we can derive explicitly Nash equilibrium strategies, both open loop and closed loop, for the two games specified in the preceding section. The closed-loop equilibrium-targeted expenditures, respec-tively for the finite- and infinite-time horizon, are given by

δ*kd* ρ λ*k*
*d*
*k k*
*k*
*x*
*x*
*_{=}
2
1
2 2
4 andδ
ρ λ
*k*
*o* *ko* *k* *k*
*k*
*x*
*x* *k*
*_{=} ( − ) , ,_{=}
2
2
2 _{1} 2
4 1 2, (13)
δ*kd* ρ*k*
*d*
*k* *rt* *k*
*k*
*q* *t e x*
*x*
*_{=} ( )
2
1
2 2 2 2
4
Φ _{and}
δ*ko* ρ*k*
*o*
*k* *rt* *k*
*k*
*q* *t e* *x*
*x* *k*
*_{=} ( ) ( − ) , ,_{=}
2
2
2 2 2 _{1} 2
4 1 2
Φ _{.}
(14)

The open-loop equivalent expenditures are given by

_{( )} ( ) ( )
_{( )}
δ*kd* ρ λ*k*
*d*
*k* *k*
*k*
*t* *t x t*
*x t*
=
2
1
2 2
4 and
_{( )} ( )( ( ))
( ) , ,
δ*ko* ρ λ*k*
*o*
*k* *k*
*k*
*t* *t* *x t*
*x t* *k*
= − =
2
2
2 _{1} 2
4 1 2,
(15)
_{( )} ( ) ( )
( )
δ*kd* ρ*k*
*d*
*k* *rt* *k*
*k*
*t* *q* *t e x t*
*x t*
=
2
1
2 2 2 2
4
Φ _{and}
_{( )} ( ) ( ( ))
, ,
δ*ko* ρ*k*
*o*
*k* *rt* *k*
*k*
*t* *q* *t e* *x t*
*x* *k*
= − =
2
2
2 2 2 _{1} 2
4 1 2
Φ (16)

The equilibrium strategies contained in equations (13) through (16) can all be relevant optimal strategies for firms practicing targeted promotions. If a firm has a short plan-ning horizon, strategies in equations (13) and (15) are more relevant. However, if a firm has a long planning hori-zon, strategies in equations (14) and (16) are more perti-nent. In addition, the latter case often offers a good approximation to the former with much reduced complex-ity of computation. Open-loop strategies in equations (15) and (16) are better suited to capture the planning stages of promotions, whereas closed-loop strategies in equations (13) and (14) convey a better sense of dynamic competi-tion. Interestingly, all these different strategies yield more or less similar managerial insights (various relationships between optimal promotional strategies are listed in Table 1), which we discuss next.

**TABLE 1**

**Various Relationships Between**
**Optimal Promotional Strategies**

*Type of*

*Comparison* *Relationship* *Finite Time* *Infinite Time*

δ
δ
*k*
*d*
*k*
*o*
*
*
ρ
ρ
*k*
*d*
*k k*
*k*
*o*
*k* *k*
*x x*
*x x*
2
2
2
1
2
2
1
( − )
ρ
ρ
*k*
*d*
*k k*
*k*
*o*
*k* *k*
*x x*
*x x*
2
2
2
1
2
2
1
( − )
δ
δ
*k*
*d*
*j*
*o*
*
*
ρ λ
ρ λ
*k*
*d*
*k j*
*j*
*o*
*j k*
*x*
*x*
2
2
2
1
2
2
ρ
ρ
*k*
*d*
*k j*
*j*
*o*
*j* *k*
*q x*
*q x*
2
2
2
1
2
2
δ
δ
*k*
*d*
*j*
*d*
*
*
ρ λ
ρ λ
*k*
*d*
*k k j*
*j*
*d*
*j* *k* *k*
*x x*
*x x*
2
2
1
2 2
1
2_{(}_{1}_{−} _{)}2
ρ
ρ
*k*
*d*
*k k j*
*j*
*d*
*j* *k* *k*
*q x x*
*q* *x x*
2
2
2 2
1
2 2
1
1
( − )
δ
δ
*k*
*o*
*j*
*o*
*
*
ρ λ
ρ λ
*k*
*o*
*k* *k* *j*
*j*
*o*
*j k k*
*x x*
*x x*
2
2
2
2
2 2
2 2
1
( − ) ρ
ρ
*k*
*o*
*k* *k* *j*
*j*
*o*
*j* *k k*
*q* *x x*
*q x x*
2
2
2
2
2 2
2 2
1
( − )
δ
δ
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
*
* 1
2
ρ
ρ
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
2
2
2
2
1
( − )
ρ
ρ
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
2
2
2
2
1
( − )
δ
δ
*k*
*d*
*k*
*j*
*o*
*j*
*x*
*x*
*
*
1
2
ρ λ
ρ λ
*k*
*d*
*k*
*j*
*o*
*j*
2
2
2
2
ρ
ρ
*k*
*d*
*k*
*j*
*o*
*j*
*q*
*q*
2
2
2
2
δ
δ
*k*
*d*
*k*
*j*
*d*
*j*
*x*
*x*
*
* 1
1
ρ λ
ρ λ
*k*
*d*
*k k*
*j*
*d*
*j* *k*
*x*
*x*
2
2
2 2
2_{(}_{1}_{−} _{)}2
ρ
ρ
*k*
*d*
*k k*
*j*
*d*
*j* *k*
*q x*
*q* *x*
2
2
2 2
2_{(}_{1}_{−} _{)}2
δ
δ
*k*
*d*
*k*
*j*
*o*
*j*
*x*
*x*
*
* 1
2
ρ λ
ρ λ
*k*
*o*
*k* *k*
*j*
*o*
*j k*
*x*
*x*
2
2
2 2
2 2
1
( − ) ρ
ρ
*k*
*o*
*k* *k*
*j*
*o*
*j* *k*
*q* *x*
*q x*
2
2
2 2
2 2
1
( − )
Targeted
promotional
incentives
Total targeted
promotional
expenditures

8. For instance, in the case of equation (7), the first derivative of the
right-hand side with respect to*xk*is negative.

**ANALYSIS OF RESULTS AND**
**MANAGERIAL INSIGHTS**

In a dynamic setting where both competing firms can use targeted promotions and can react to each other based on the history of their interactions, the allocation of a firm’s promotional expenditures over time, both for the defensive and offensive purposes, is an important manage-rial decision that can be quite intractable to make. For in-stance, how should a firm adjust its targeted expenditures over time? Should a firm do what its rival does in terms of its strategic posture, focusing on customer acquisition or retention whenever its rival does? Or should a firm always focus on customer retention regardless of what competi-tion does, as some experts apparently advocate (Reichheld and Sasser 1990)? The answers to all these questions can be quite elusive without any normative modeling.

Our analysis provides an answer to each of these ques-tions. Although our analysis of the differential game itself is complex and even somewhat serendipitous, our answers to all these questions turn out to be surprisingly simple and intuitive, which should enhance the practical applicability of our conclusions.

**Determinants of Optimal Targeting Strategies**
By analyzing the closed-loop strategies in equations
(13) and (14), we can characterize how a firm can
opti-mally engage its promotional expenditure for customer
re-tention and acquisition.

*Proposition 1:* A firm’s optimal defensive (offensive)
promotional incentives increase (decrease) with its
market share*xk*, with defensive (offensive) targeting

effectivenessρ ρ*kd*( ), and with the value of market*ko*

share (λ*k*( ) or*t* *qk*Φ( ) in the infinite-time horizon*t ert*

case), but they decrease with the actual redemption rate of defensive (offensive) targeted promotions

θ*kd* =*xk*1 / (*xk* θ*ok*=*xk*2 / (1−*xk*)).

Proposition 1 is intuitive but important. It essentially
con-firms the optimality of some instinctive managerial
prac-tices. When a firm has a larger (smaller) market share,
ceteris paribus, its offensive (defensive) promotions
gen-erate less (more) a bang on sales, and hence the firm
should rationally provide more incentives for customer
re-tention (customer switching). Thus, one may observe at
any point in time that a firm engages more resources in
customer retention as its market share increases. The fact
that any promotional incentives, defensive and offensive
alike, should increase with respective targeting
effective-ness (ρ*kd* or ρ*ko*) perhaps explains why the Internet has

spawned a rapid growth of corporate expenditures on tar-geted promotions (Booker and Krol 2002). As the value of

market share in our model is essentially customer profit-ability, our analysis thus warns against a widespread prac-tice of implementing customer retention or acquisition programs “with little regard to customer profitability” (“Briefings” 2003). Finally, any targeted promotional in-centives should decrease with the actual redemption rate of specific targeted promotions, as a higher redemption rate reduces the unit profit margin at any given level of sales.

Open-loop strategies in equations (15) and (16) suggest
similar managerial insights for promotional planning. The
key difference between these two types of strategies is that
a firm’s planned promotional incentives are based on
planned market shares, *x tk*( ), and expected redemption
rates, defined analogously as_{} θ*dk*( )*t x t x t*=*k*1( ) / *k*( ) and

( ) ( ) / ( ( ))

θ*ko* *t x t*= *k*2 1−*x tk* . Indeed, these two types of

strate-gies are related. In both finite- and infinite-time cases, we have from equations (13) and (14) or (15) and (16):

δ δ θ
θ
*k*
*d*
*k*
*d* *k kd*
*k kd*
*x*
*x*
*
= andδ δ θ
θ
*k*
*o*
*k*
*o* *k* *ko*
*k* *ko*
*x*
*x*
* ( )
( _{)}
= −
−
1
1 .

This analysis gives us the following proposition:
*Proposition 2:* A firm can implement its promotional

strategies optimally by adjusting its planned promo-tional incentives to reflect the observed difference between actual and planned market shares, as well as between actual and planned redemption rates. If a firm’s actual market share is larger (smaller) than planned and if actual redemption rates are lower than expected, the firm should adjust upward its planned incentives for its defensive (offensive) tar-geted promotions.

Proposition 2 goes one step further than proposition 1 in suggesting that it is an advisable practice to adjust planned promotional incentives for customer acquisition and re-tention in accordance with how the planned market share and redemption rate are fulfilled. This two-stage adjust-mentprocessismoreactionable,demandinglessinformation to implement.

**Strategic Orientation**

We can also shed light on the general orientation of a firm’s promotional strategies, as measured by the ratio of the firm’s total defensive and offensive promotional ex-penditures. From equations (13) and (14), we have

δ
δ
ρ
ρ
*k*
*d*
*k*
*k*
*o*
*k*
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
*x*
*x*
*
* 1 _{(} _{)}
2 1
2
=
−
.

Then, we have the following proposition:

*Proposition 3:* A firm should be defense oriented
(of-fense oriented), spending more on customer
reten-tion (switching) if its overall market share is larger
than its rival’s, weighted by its targeting
effective-ness in respective markets. Furthermore, a firm
should become more defense oriented (offense
ori-ented) as its market share increases (decreases).
Thus, a smaller (larger) share firm may optimally
choose to focus on customer retention (acquisition),
instead of customer acquisition (retention).
Intuitively, the size of a firm’s market share, not just the
ef-fectiveness of the firm’s defensive or offensive targeting—
the commonly recognized factor—should be an important
factor for determining the firm’s strategic orientation. As a
firm’s market share increases, it should increase its
expen-diture on defensive targeting both because customer
ac-quisition is less effective in generating sales and because
customer retention helps to fend off more aggressive
cus-tomer switching from the rival. In the case of a decrease in
market share, a firm should increase its expenditure on
of-fensive targeting to take advantage of the rival’s
vulnera-bility and to build up its market share, as is often done in
practice. Thus, Proposition 3 suggests that the competing
firms are engaging in a tug-of-war in which more offensive
expenditure by one firm triggers more defensive
expendi-ture by the other.

Proposition 3 also suggests that a firm’s strategic orien-tation should indeed depend critically on the relative effec-tiveness of targeted promotions. Everything else being equal, if a firm is more effective in using offensive (defen-sive) targeted promotions to increase its market share, it should spend more on customer switching (retention). This means that a firm should spend relatively more on the type of promotion that increases the firm’s market share the most.

Interestingly, as a firm’s short-term strategic orienta-tion depends on its market share as well as its relative tar-geting effectiveness, the sweeping strategic prescription that all firms or all larger share firms should focus on cus-tomer retention can be misleading. Proposition 3 points out that a smaller (larger) market share firm can optimally choose to focus on customer retention (acquisition) if tar-geting own (rival’s) customers is significantly more effective or less costly for the firm.

Similarly, we can take advantage of our analytical solu-tions and examine many other equilibrium relasolu-tionships. In general, the relative targeted expenditure for a specific group of consumers depends only on the relative targeting effectiveness and on the relative value of market share in-crease for the competing firms. If we compare firms’ tar-geted expenditures across consumer groups, the relative

expenditure also depends on the size of each consumer group.

**Long-Run Market Share**
**and Strategic Orientation**

We can carry out the analysis of the long-run equilib-rium by using equations (11) and (12) (relationships for optimal promotional strategies in steady state are listed in Table 2).

Note first that the asymptotic stability of the
equilib-rium requires
*t* *q*
*q* *t*
*j*
*o*
*k*
*d*
*k*
*j*
*j*
*d*
*k*
*o*
1
2 2
2
=
_{ρ}ρ _{} > >_{}ρ_{ρ} _{} = . (17)
Let*y q*
*qkj*

= denote the relative unit profit margin, and
de-fine
ξ ρ ρ
ρ ρ
( )*y* *y*
*y*
*k*
*o*
*j*
*d*
*j*
*o*
*k*
*d*
= −
−
2 2
2 2 .
**TABLE 2**

**Relationships Between Optimal**
**Promotional Strategies in Steady State**

*Type of*

*Comparison* *Relationship* *Steady-State Value*

δ
δ*k*
*d*
*k*
*o*
*
* ρ
ρ
ρ ρ
ρ ρ
*k*
*d*
*k*
*o*
*k*
*o*
*k* *jd* *j*
*j*
*o*
*j* *kd* *k*
*q* *q*
*q* *q*
2
2
2 2
2 2
1_{2}
−
−
_{}
δ
δ
*k*
*d*
*j*
*o*
*
* ρ
ρ
ρ ρ
ρ ρ
*k*
*d*
*k*
*j*
*o*
*j*
*j*
*o*
*j* *kd* *k*
*k*
*o*
*k* *jd* *j*
*q*
*q*
*q* *q*
*q* *q*
2
2
2 2
2 2
1_{2}
2
2
−
−
_{}
δ
δ
*k*
*d*
*j*
*d*
*
*
ρ
ρ
*k*
*d*
*k*
*j*
*d*
*j*
*q*
*q*
2
2
2
2
δ
δ*k*
*o*
*j*
*o*
*
*
ρ
ρ
ρ ρ
ρ ρ
*k*
*o*
*k*
*j*
*o*
*j*
*j*
*o*
*j* *kd* *k*
*k*
*o*
*k* *jd* *j*
*q*
*q*
*q* *q*
*q* *q*
2
2
2 2
2 2
2
2
−
−
_{}
δ
δ
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
*
* 1
2
ρ
ρ
ρ ρ
ρ ρ
*k*
*d*
*k*
*o*
*k*
*o*
*k* *jd* *j*
*j*
*o*
*j* *kd* *k*
*q* *q*
*q* *q*
2
2
2 2
2 2
−
−
_{}
δ
δ
*k*
*d*
*k*
*j*
*o*
*j*
*x*
*x*
*
* 1
2
ρ
ρ
*k*
*d*
*k*
*j*
*o*
*j*
*q*
*q*
2
2
2
2
δ
δ
*k*
*d*
*k*
*j*
*d*
*j*
*x*
*x*
*
*
1
1
ρ
ρ
ρ ρ
ρ ρ
*k*
*d*
*k*
*j*
*d*
*j*
*k*
*o*
*k* *jd* *j*
*j*
*o*
*j* *kd* *k*
*q*
*q*
*q* *q*
*q* *q*
2
2
2 2
2 2
2
2
−
−
_{}
δ
δ
*k*
*o*
*k*
*j*
*o*
*j*
*x*
*x*
*
* 2
2
ρ
ρ
ρ ρ
ρ ρ
*k*
*o*
*k*
*j*
*o*
*j*
*j*
*o*
*j* *kd* *k*
*k*
*o*
*k* *jd* *j*
*q*
*q*
*q* *q*
*q* *q*
2
2
2 2
2 2
2
2
−
−
_{}
Targeted
promotional
incentives
Total
promotional
expenditures

Condition (17) impliesξ′(y) > 0. In steady state, a firm’s
market share is given by*x y* *y*

*y*
*k*( )=_{1}_{+}ξ( )_{ξ}_{( )}. We can verify
that lim ( ) ,_{y t}_{→} *x yk* =
1 1 *xk*(t2) = 0, and*x yk*
′_{( ) 0 for}_{>} _{y}_{∈}_{(t}
2,*t*1).
This means a firm’s long-run market share increases with
its relative unit profit margin. The reason is that a higher
unit profit margin due either to cost advantage (or perhaps
consumer loyalty) will motivate a firm to spend more on
targeted promotions and thus gain a higher market share in
the long run.

It is also straightforward to show, by taking derivatives of a firm’s steady-state market share with respect to a spe-cific effectiveness measure, that a firm’s long-run market share increases with its targeting effectiveness, both de-fensive and ofde-fensive. Thus, our model suggests that to gain the long-run market share advantage when targeted promotions are feasible, a firm should strive to boost its unit profit margin either by lowering cost or building con-sumer loyalty and to improve its targeting effectiveness.

Most important, we can shed some light on the relation-ship between a firm’s long-run market share and its strate-gic orientation in targeted promotions, as measured by the firm’s relative expenditure on defensive and offensive tar-geting. In steady state, we have

δ
δ
ρ
ρ
ρ ρ
ρ ρ
*k*
*d*
*k*
*k*
*o*
*k*
*k*
*d*
*k*
*o*
*k*
*o*
*k* *dj* *j*
*j*
*o*
*j* *kd* *k*
*x*
*x*
*q* *q*
*q* *q*
*
*
1
2
2
2
2 2
2 2
= −
−
.

Let us define two more structural variables,

*t* *ko* *oj* *kd* *dj*
*k*
*d*
*k*
*o*
3
2 2 2 2
2 2
2
=ρ ρ +ρ ρ
ρ ρ and*t*
*j*
*d*
*j*
*o*
*k*
*o*
*j*
*o*
*k*
*d*
*j*
*d*
4
2 2 2
2 2 2 2
=
+
ρ ρ
ρ ρ ρ ρ ,

where*t*1>*t*3>*t*4>*t*2by condition (17). Firm*k*will be

de-fense oriented (spending more on defensive targeting)—
that is,δ
δ
*k*
*d*
*k*
*k*
*o*
*k*
*x*
*x*
*
*
1
2
1
> , if *q*
*qkj* *t*

> 3—whereas firm*j*will be

de-fense oriented if *q*
*qkj* *t*

< 4. Thus, the strategic orientations for the two firms in the long-run equilibrium are as illus-trated in Figure 1. The following proposition comes di-rectly from Figure 1.

*Proposition 4:*A focus on customer retention is not an
optimal long-run strategy for all firms in a
competi-tive context. A firm with a sufficiently large market
share should focus on customer retention, whereas a
firm with a sufficiently small market share should
stress customer acquisition. This is the case
regard-less of whether or not the firm is more effective in
targeting its current customers.9 _{When market}
shares are more evenly divided, the optimal strategy
for a firm is to focus more on customer acquisition.
Thus, customer acquisition can be a long-term
win-ning strategy for all firms in an industry.

Intuitively, a focus on defensive targeting is a sensible competitive strategy for a firm only if it faces a rival’s in-tense offensive targeting, which is the case when the mar-ket shares are unevenly distributed. When the marmar-ket shares are more or less evenly distributed, the rival exerts little offensive pressure, and a firm is better off reducing its defensive targeting. Consequently, the equilibrium cannot be sustained when both firms focus on customer retention through defensive targeting. These conclusions extend the main result in McGahan and Ghemawat (1994) that only one firm (i.e., the smaller market share firm) can profitably focus on customer acquisition, and they contrast with Erickson’s (1993) conclusion that defensive and offensive promotions should be balanced in steady state. In our model, these two types of promotions are balanced for a firm only if its market share takes on a specific value.

To the extent that offensive targeting is the root cause
for customer churning, Proposition 4 thus predicts that
among industries where targeted promotions are feasible,
**Firm j Defensive**
**Firm k Offensive**
**Both**
**Offensive**
**Firm j Offensive**
**Firm k Defensive**
* t2 (xk =0) t4 t3 t1 (xk* *1) q*
*qkj*
→
**FIGURE 1**

**Long-Run Relationship Between Market Share and Strategic Orientation**

9. If firm*k*, for instance, is more effective in defensive targeted
pro-motions(ρ*kd* ρ*ko* )

2 2

> , condition (24) impliesρ*jo* ρ*jd*

2 2

> . There exists no sta-ble long-run equilibrium in which both firms are more effective in defensive targeting.

customer churning will be more severe in the industries in which competing firms are more equally matched, all else being equal.

**Numerical Example**

The trajectories of a firm’s promotional strategy,
mar-ket share, and profits over time can be illustrated through a
numerical example. For this purpose, we setρ*kd* =ρ*dj* =

ρ*d* _{=}_{0 009}_{.} _{,}_{ρ} _{ρ} _{ρ}
*k*
*o*
*j*
*o* *o*
= = =0 0119. ,*qk*=*qj*= 500,*x*11(0) = 0.4,
*x*12(0) = 0.2,*x*21(0) = 0.1, and*x*22(0) = 0.3.10The results of
our simulation are shown in Figures 2, 3, and 4. Firm 2,
with a smaller initial market share, aggressively pursues
new customers through offensive targeted promotions at
the start of the game. Over time, Firm 2 decreases its
offen-sive expenditure and increases its defenoffen-sive expenditure as

its overall market share increase is due mostly to the
in-crease in repeat purchases, which are below the
steady-state level at the outset. Firm 1, in contrast, initially spends
more on defensive targeting than on offensive targeting in
an effort to maintain its large market share but becomes
in-creasingly offense oriented as its market share (mostly
re-peat purchasers) is chipped away by the rival. In the long
run, both firms spend more on customer switching11_{and}
share the market equally in the dynamic long-run
equilib-rium.12_{In this dynamic process of gravitating toward the}
long-run market shares, Firm 1’s profits decrease while
Firm 2’s increase, as shown in Figure 4.

**CONCLUSIONS**

Targeted promotions are gaining popularity as a strate-gic marketing tool. Many firms across different service in-dustries are embracing new information technologies and gearing up to build their own sophisticated consumer data-bases to improve their targeting accuracy. Our analysis thus addresses a timely managerial question of how a firm should use targeted promotions to achieve the strategic ob-jectives of customer retention and acquisition. Our norma-tive analysis enables us to identify the determinants of optimal targeting strategies and to characterize the optimal overall strategic posture for a firm’s targeting programs:

• A firm’s promotional incentives for customer

reten-tion (acquisireten-tion) should be related positively to its own (its rival’s) market share, its targeting

effective-0
10
20
30
40
50
60
70
80
90
100
1 11 21 31
**Time**
**D**
**ef**
**ens**
**iv**
**e a**
**nd O**
**ffe**
**ns**
**iv**
**e E**
**xp**
**end**
**itur**
**es**
** o**
**f**
** F**
**irm**
**s 1 **
**an**
**d 2**
Defensive-1
Offensive-1
Defensive-2
Offensive-2
**FIGURE 2**

**Dynamics of Promotional Expenditure**

0
0.1
0.2
0.3
0.4
0.5
0.6
1 11 21 31
**Time**
**M**
**ar**
**ke**
**t S**
**ha**
**re**
**s of**
** Fir**
**m**
**s 1**
** a**
**nd 2**
x1
x2
x11
x12
x21
x22
**FIGURE 3**

**Dynamics of Market Shares**

0
50
100
150
200
250
1 11 21 31
**Time**
**Pr**
**of**
**its**
Firm 1
Firm 2
**FIGURE 4**
**Dynamics of Profitability**

10. We choseρ*o*_{= .0119, as in the Erickson (1992) and Fruchter and}

Kalish (1997)studies, and choseρ*d*_{to satisfy the stability condition(24).}

11. This is because we have*t* *q*

*qkj* *t*

4< < 3in this example.

12. The fact that the market is equally divided in steady state and each firm has one quarter of repeat and switching sales is due to the symmetry in this example.

ness, and customer profitability but negatively to the redemption rate of its promotions.

• A firm can promote optimally by adjusting its planned

promotional incentives to reflect the observed dif-ference between actual and planned market shares, as well as between actual and planned redemption rates.

• A firm should spend more on customer retention if

its market share is increasing but spend more on cus-tomer acquisition if its market share is decreasing.

• A focus on either customer retention or acquisition

can be an optimal long-run strategy for a firm. A firm with a large market share should focus more on customer retention. Otherwise, it should focus on customer acquisition. Indeed, a focus on customer acquisition can be a winning business strategy for all competing firms if their market shares are similar, whereas a focus on customer retention is not.

• To build a long-run market share advantage in the

age of information-intensive marketing, a firm should strive to improve its targeting effectiveness

and increase its unit profit margin through, for in-stance, building customer loyalty or improving product quality.

Future research can extend our analysis in two major directions. First, the results from any differential game al-ways hinge on the specification of the dynamics for state variables. Future research can explore the robustness of our conclusions by using different but equally plausible dynamics for state variables. The robustness can also be checked by extending our model to include more firms and allow for a variable market size. Such extension is espe-cially important for addressing targeting issues related to growth industries. Second, our solutions are analytical, and they are well suited to empirical estimation if proper data are available. Such an empirical exercise can help to pin down the parameters in the model and provide more specific managerial guidance for practitioners in different industries.

**APPENDIX**

In this appendix, we derive closed-loop Nash equilibrium strategies for both games specified in the second section. For these games, we proceed first to construct closed-loop strategies from the first-order necessary conditions for the open-loop Nash equilibrium. Then we prove that the closed-loop strategies we have proposed satisfy the Nash equilibrium conditions contained in equation (8). This ap-proach avoids technical difficulties associated with the standard methods of deriving state-dependent Nash equilibrium strategies. The use of standard methods for the problems is intractable and a tricky task indeed. The procedure for determining the open- and closed-loop solutions for both finite- and infinite-time problems is described after we state our solutions formally in the following two theorems.

*Theorem 1*(closed-loop strategies for the finite-time problem): Consider the differential game associated with equations (2), (5), and
(6), where*xk*(*t*) is as in equation (7). Let

δ*kd* ρ λ*k*
*d*
*k k*
*k*
*x*
*x*
*_{=}
2
1
2 2
4 andδ
ρ λ
*k*
*o* *ko* *k* *k*
*k*
*x*
*x*
*_{=} ( − )
2
2
2 _{1} 2
4 , k = 1, 2,
(18)

whereλ*k*satisfies the following two-point boundary value problem (TPBVP):
_{( )} _{( )} _{( ) [} _{}_{( )} _{(} _{}_{( ))]}
λ*k* *t* =*r*λ*k* *t q*− *k*−_{2}1λ*k* *t* ρ*id* *x t* −ρ*oi* 1−*x t* λ
2 2
*i* *k* *kT*
*i t*
*k* *k* *dj* *j*
*t* *T* *q*
*x t* *t* *t*
( ), ( )
( ) [( ( ) ( ))
λ
ρ λκο ρ λ
=
= −
=

### ∑

2 1 2 2 2 (1 ( )) (2 2 ( ) 2 ( ))2( )], ( )0 −*x tk*− ρ λ

*oj*

*j*

*t*−ρ λ

*kd*

*k*

*t x tk*

*xk*=

*xk*0 , (19)

with*j*≠*k*,*k*= 1, 2, and*xj*= −1 *xk*. Then( ,δ δ δ δ*d*1* *o*1*, *d*2*, 2*o**)forms a global Nash equilibrium closed-loop strategy of the above differential

game, that is, it satisfies condition (8).

*Proof*. See Appendix C below.

The*open-loop strategies*are given by (see Appendix A below)

_{( )} ( ) ( )
( )
δ*kd* ρ λ*k*
*d*
*k* *k*
*k*
*t* *t x t*
*x t*
=
2
1
2 2
4 and ( )
( )( _{( ))}
( ) , , ,
δ*ko* ρ λ*k*
*o*
*k* *k*
*k*
*t* *t* *x t*
*x t* *k*
= − =
2
2
2 _{1} 2
4 1 2

( ) ( ( ) ( ))( ( ))( ( )
*x tk*1 1_{2} *ko* *k* *t* *dj* *j* *t* 1 *x t x tk* *k*
2 2
= ρ λ −ρ λ − −*x tk*1 *oj* *j* *t* *kd* *k* *t x t x tk* *k*1 *xk*1
2 2
0
( )) (− ρ λ ( )−ρ λ ( )) ( ) ( )], ( )=*xk*01. (20)

Then,*xk*2is determined through*xk*2=*xk*−*xk*1.

We can now extend Theorem 1 to the case of an infinite-time horizon.

*Theorem 2*(closed-loop strategies for the infinite-time problem): Consider the differential game associated with equation (10). Let

Ψ(*x*)satisfy the following differential equation:

′ − + −
Ψ (*x*) (Ψ *x*)[(ρ*dk* *qk* ρ*oj* *q x tj*)*k*( ) (ρ*ko* *qk* ρ*dj* *qj*)
2 2 _{2} 2 2
(1 ( )) ]2 2() [ ( ) (1 ( ))]
1
2 _{2} _{2}
− + − −
=
*x tk* *x* *id* *x t* *io* *x t qi*
*i*
Ψ

### ∑

ρ ρ = − = →∞ − 2*r x*

_{Ψ}

_{(}

_{}

_{) , lim (}2

_{t}_{Ψ}

*x t e*

_{}

_{( ))}

*rt*0

_{,}(21) where

*x*

*x if k*

*x if k*

*k*= = − = 1 1 2. (22) Let δ

*kd*ρ

*k*

*d*

*k*

*rt*

*k*

*k*

*q*

*t e x*

*x**

_{=}( ) 2

_{2}

_{2}

_{2}

_{2}1 4 Φ

_{and}

_{δ}ρ

*k*

*o*

*ko*

*k*

*rt*

*k*

*k*

*q*

*t e*

*x*

*x*

*k**

_{=}( ) ( − ) , ,

_{=}2

_{2}

_{2}

_{2}

_{2}2 1 4 1 2 Φ

_{,}(23)

whereΦ(*t*) satisfies the following initial value problem

_{( )} _{( )} _{[} _{(} _{)] , ( )}
Φ*t* *e* *rt* Φ *t ert* *x* *x q* Φ Ψ
*i*
*d*
*i* *io* *i* *i*
= − − −1 − − =
2 2 1 0
2 2
ρ ρ (( ))
_{( )} _{[(} _{)(} _{( ))}
*x*
*x t* *q* *q* *x t*
*i*
*k* *ko* *k* *dj* *j* *k*
0
1
2 1
1
2
2
2 2
=

### ∑

= ρ −ρ − −(ρ*oj*

*qj*ρ

*kd*

*q x tk*)

*k*

_{( )] ( ) ,}

*t ert*

*xk*

_{( )}

*xk*2 2

_{2}

_{0}0 − = Φ , (24)

with*j*≠*k*,*k*= 1, 2,*xj*= −1 *xk*, andΦis as in equation (21). Then( ,δ δ δ δ1*d** *o*1*, *d*2*, 2*o**)forms a global Nash equilibrium closed-loop strategy of

the differential game (10); that is, it satisfies the condition (8) for the infinite-time problem.

*Proof*. See Appendix D below.

The corresponding*open-loop strategies*are given by the following (see Appendix B below):

_{( )} ( ) ( )
( )
δ*kd* ρ*k*
*d*
*k* *rt* *k*
*k*
*t* *q* *t e x t*
*x t*
=
2
1
2 2 2 2
4
Φ _{and}_{} _{( )} ( ) ( ( ))
, ,
δ*ko* ρ*k*
*o*
*k* *rt* *k*
*k*
*t* *q* *t e* *x t*
*x* *k*
= − =
2
2
2 2 2 _{1} 2
4 1 2
Φ _{,}
(25)

where

_{{}

Φ,*xk*

_{}}

are as in (24),*xk*1is given by

( ) [( )( ( ))( ( ) (
*x tk*1 1 *ko* *qk* *dj* *qj* *x t x t xk* *k* *k*1
2 1
2 2
= ρ −ρ − − *t*)) (− ρ*oj* *qj* −ρ*dk* *q x t x tk*)*k*( )*k* ( )] ( ) ,*t ert* *xk* ( )=*x*
2 2
1 Φ 1 0 *k*01, (26)

and*xk*2is determined through*xk*2=*xk*−*xk*1.

We first describe the procedure for constructing the open- and closed-loop strategies for the differential games stated in Theorems 1 and 2. Then we prove that the closed-loop strategies we have constructed are Nash equilibrium solutions as stated in Theorems 1 and 2.

**A. THE FINITE-TIME CASE**

**Step 1: Derivation of Open-Loop Strategies**
Define the control variables*ukd*and*uko*,*k*= 1, 2, as

δ*kdxk*1 *ukd*
2
= andδ*koxk*2 *u kko*
2
1 2
= , = , . (A1)

In terms of*ukd*and*uko*, the differential game associated with equations (2), (5), and (6) becomes

*Max* *q x T e* *q x t u t u t*
*u u* *k* *k*
*T*
*k* *rt* *k k* *kd* *ko*
*k*
*d*
*k*
*o*
, Π = ( ) + ( ( )− ( )− ( )
− 2 2
)
( ) ( ( ) ( ))( ( )) (
*e dt*
*x t* *u t* *u t* *x t*
*rt*
*T*
*k* *ko* *ko* *dj* *dj* *k*
−

### ∫

= − − − 0 1 ρ ρ ρ*oju toj*( )−ρ

*dku t x t xkd*( )) ( ),

*k*

*k*( )0 =

*xk*0. (A2)

The current-value Hamiltonian for player*k*for game (A2) is given by

*Hk*=*H x u u u uk*( , , , , , )*k* *kd* *ko* *dj* *jo* λ*k* =*q x uk k* − *kd* −*uko* +λ*k kx*

2 2

, (A3)

whereλ*k*is the current-value multiplier associated with the state equation. The necessary optimality conditions for player*k*’s open-loop

strategies are given by the maximization condition (cf. Kamien and Schwartz 1991),

∂*Hk*/∂*ukd* = −2*ukd* +ρ*kdxk*λ*k*=0 and∂*Hk*/∂*ukd* = −2*uko*+ρ*ko*(1−*xk*)λ*k*=0, (A4)

and the adjoint equations with the terminal conditions

_{(} _{)}
λ*k* λ*k* *H* *k* λ*k* *k* λ*k* ρ ρ
*i* *i*
*d*
*i*
*d*
*i*
*o*
*io*
*r* *k* *x* *r* *q* *u* *u*
= − ∂ ∂ = − − −
=

### ∑

2_{1}, λ

*k*( )

*T*=

*qkT*. (A5)

From (A4), we have

*ukd* =1_{2}ρ*kdxk*λ*k*and*uko*=1_{2}ρ*ko*(1−*xk*) ,λ*k* *k*=1 2, . (A6)

Plugging (A6) into (A5) and the state equation, we obtain the following two-point boundary value problem (TPBVP):

_{( )} _{( )} _{( ) [} _{}_{( )} _{(} _{}_{( ))]}
λ*k* *t* =*r*λ*k* *t q*− *k*−_{2}1λ*k* *t* ρ*id* *x t* −ρ*oi* 1−*x t* λ
2 2
*i* *k* *kT*
*i*
*k* *ko* *k* *dj* *j*
*t* *T* *q*
*x t* *t* *t*
( ), ( )
( ) [( ( ) ( )
λ
ρ λ ρ λ
=
= −
=

### ∑

2_{1}1 2 2 2 )(1 ( )) (2 2 ( ) 2 ( ))2( )], ( )0 −

*x tk*− ρ λ

*oj*

*j*

*t*−ρ λ

*kd*

*k*

*t x tk*

*xk*=

*xk*0 , (A7)

where*j*≠*k*,*k*= 1, 2. Plugging (A6) into the state equation (5), we obtain

( ) [( ( ) ( ))( ( ))( ( )
*x tk*1 1_{2} *k* *t* *dj* *j* *t* 1 *x t x tk* *k*
2
= ρ λκ −ρ λ − −
ο2
( )) ( ( ) ( )) ( ) ( )], (
*x tk*1 *oj* *j* *t* *dk* *k* *t x t x tk* *k*1 *xk*1
2 2
− ρ λ −ρ λ 0)=*x*0*k*1. (A8)

Considering (A7), we find the*open-loop*solutions to be

_{( )} _{( ) ( )}

We have in terms of the original variables
_{( )} ( ) ( )
( )
δ*kd* ρ λ*k*
*d*
*k* *k*
*k*
*t* *t x t*
*x t*
=
2 _{2} _{2}
1
4 and ( )
( )( ( ))
( ) , ,
δ*ko* ρ λ*k*
*o*
*k* *k*
*k*
*t* *t* *x t*
*x t* *k*
= − =
2 _{2} _{2}
2
1
4 1 2,
(A10)

whereλ*k*(*t*),*x tk*( )are as in (A7);*x tk*1( )is as in (A8), and

*xk*2=*xk*−*xk*1. (A11)

From (A7), we obtain

_{,} _{, ,}

λ*k* λ*k*=0= − <*qk* 0 *k*=1 2 (A12)

which, together with the terminal condition in (A7), implies

λ*k*( )*t* >0 (A13)

for all*t*∈[0,*T*].

**Step 2: Constructing the Closed-Loop Strategies**

Considering (A10) and substituting the values of*x x xk*, *k*1, *k*2by the actual states (market shares) as given in equations (1), (5), and (6),

we obtain
δ*kd* ρ λ*k*
*d*
*k* *k*
*k*
*t* *t x t*
*x t*
*_{( )} ( ) ( )
( )
=
2 _{2} _{2}
1
4 andδ
ρ λ
*k*
*o* *ko* *k* *k*
*k*
*t* *t* *x t*
*x t* *k*
*_{( )} ( )( ( ))
( ) , ,
= − =
2 _{2} _{2}
2
1
4 1 2.
(A14)

In Appendix C, we prove that the closed-loop strategies defined in (A14) are Nash equilibrium strategies; that is, they satisfy condi-tion (8).

**B. THE INFINITE-TIME CASE**
Let

λ*k*( )*t q*= Φ*k* ( )*t ert*. (B1)

Then, as*T*→ ∞, the two equations in (A7) forλ*k*can be transformed into only one equation,

_{( )} _{( )} _{[} ( ) ( (
Φ*t* *e* *rt* Φ *t ert* *x t* *x*
*i* *i*
*d*
*i* *io* *i*
= − − − − −
=

### ∑

1 2 2 1 1 2_{2}

_{2}ρ ρ

*t q*))] ,

*i*Φ( )∞ =0. (B2)

Equation (B2) follows immediately by substituting (B1) and its derivative with respect to*t*into the equations forλ*k*in (A7). From (B2), as

before, we obtain Φ(*t*) > 0 for all*t*∈[0,∞), which is a simple consequence of the terminal condition in (B2) and the property

_{( )}
Φ*t* Φ= *e* *rt*
−
= − <
0 0 (B3)
derived from (B2).

_{( )} _{( )} _{[} _{( )} _{(} _{( ))]}
Φ*t* *e* *rt* Φ *t ert* *x t* *x t q*
*i*
*d*
*i* *io* *i*
= − − −1 − −
2 1
2 2 2
ρ ρ , ( )
( ) [( )( ( ))
Φ ∞ =
= − − −
=

### ∑

0 1 2 1 1 2 2 2 2*i*

*k*

*ko*

*k*

*dj*

*j*

*k*

*x t*ρ

*q*ρ

*q*

*x t*(ρ

*oj*

*qj*ρ

*kd*

*q x tk*)

*k*( )] ( ) ,

*t ert*

*xk*( )

*xk*2 2

_{2}

_{0}0 − = Φ , (B4)

where*j*≠*k*,*k*= 1, 2. Note that reducing a TPBVP from two end points to one end point reduces significantly the complexity of
computa-tion.

Now we want to take another step in reducing the complexity of computation. Let

*x* *x if k*
*x if k*
*k* =
=
− =
1
1 2
(B5)
and suppose
Φ( )*t ert* _{=}Ψ(*x*)_{.} _{(B6)}

ThenΨ(*x*)satisfies the following differential equation:

′ − + −
Ψ (*x*) (Ψ *x*)[(ρ*dk* *qk* ρ*oj* *q x tj*)*k*( ) (ρ*ko* *qk* ρ*dj* *qj*)
2 2 _{2} 2 2
(1 ( )) ]2 2() [ ( ) (1 ( ))]
1
2 2
− + − −
=
*x tk* *x* *id* *x ti* *io* *x t qi*
*i*
Ψ 2 ρ ρ
2 2 0

### ∑

= − = →∞ −*r x*

_{Ψ}

_{(}

_{}

_{) , lim (}

_{t}_{Ψ}

*x t e*

_{}

_{( ))}

*rt*

_{.}(B7)

Equation (B7) follows from (B2) and the relationship

_{( )} _{(})

Φ*t ert*_{+}*r*Ψ_{′}*x x*_{.} _{(B8)}

Making use of (B7) and (B6), we can transform the TPBVP (B4) into the following initial value problem:

_{( )} _{( )} _{[} _{( )} _{(} _{( ))] ,}
Φ*t* *e* *rt* Φ *t ert* *x t* *x t q*
*i*
*d*
*i*
*o*
*i*
= − − −1 − −
2 1
2 2 2
ρ ρ Φ( ) Ψ(_{)( ))}
( ) [( )(
0 0
1
2 1
1
2
2 2
=
= − −
=

### ∑

*x*

*x t*

*q*

*q*

*x*

*i*

*k*ρ

*ko*

*k*ρ

*dj*

*j*

*k*( )) (

*t*2

*oj*

*qj*

*dk*

*q x tk*)

*k*2( )] ( ) ,

*t ert*

*xk*( )

*xk*0 2 2 0 − − = ρ ρ Φ , (B9)

where*j*≠*k*,*k*= 1, 2, andΨ(*x*( ))0 is obtained by solving the backward equation (B7). Plugging (B1) into (A8), we also have

( ) [( )( ( ))( ( ) (
*x tk*1 1_{2} *ko* *qk* *dj* *qj* 1 *x t x t x tk* *k* *k*1
2 2
= ρ −ρ − − )) (− ρ*oj* *qj* −ρ*kd* *q x t x tk*)*k*( )*k* ( )] ( ) ,*t ert* *xk* ( )=*xk*
2 2
1 Φ 1 0 01. (B10)
Considering (B1) and (A10), we have the open-loop strategies for this case as

_{( )} ( ) ( )
( )
δ*kd* ρ*k*
*d*
*k* *rt* *k*
*k*
*t* *q* *t e x t*
*x t*
=
2
1
2 2 2 2
4
Φ _{and}_{} _{( )} ( ) ( ( ))
( ) , ,
δ*ko* ρ*k*
*o*
*k* *rt* *k*
*k*
*t* *q* *t e* *x t*
*x t* *k*
= − =
2
2
2 2 2 _{1} 2
4 1 2
Φ _{,} (B11)

whereΦ(*t*),*x tk*( )are as in (B9);*x tk*1( )is as in (B10); and*xk*2is as in (A11).

Considering (B11) and substituting the values of*x x xk*, *k*1, *k*2by the actual states (market shares) as given in equation (10), we construct

δ*kd* ρ*k*
*d*
*k* *rt* *k*
*k*
*t* *q* *t e x t*
*x t*
*_{( )} ( ) ( )
( )
=
2
1
2 2 2 2
4
Φ _{and}_{δ} ρ
*k*
*o* *ko* *k* *rt* *k*
*k*
*t* *q* *t e* *x t*
*x t* *k*
*_{( )} ( ) ( ( ))
( ) , ,
= − =
2
2
2 2 2 _{1} 2
4 1 2
Φ _{.} (B12)

In Appendix D, we prove that the closed-loop strategies defined in (B12) are Nash equilibrium strategies for the infinite-time horizon problem; that is, they satisfy condition (8).

**C. PROOF OF THEOREM 1**
Consider the zero-sum

0
0 _{0}
= − − + −

### ∫

λ*k*

*rt*

*k*λ

*T*

*k*

*rt*

*k*

*T*

*e x*

*d*

*dt*(

*e x dt*) , (C1)

whereλ*k*is as in equation (19) or (A7), and*xk*is as in equation (7). Considering (C1), we obtain

0_{= −}* _{q e x T}*−

_{+}0

*0*

_{x}_{+}

_{x}_{+}

_{−}

_{r}*−*

_{x e dt}*kT* *rT* *k*( ) λ*k*( ) *k* [λ*k k* (λ*k* λ*k*) ]*k* *rt*

*o*
*T*

### ∫

. (C2)Considering equation (7), we obtain

0 0 0 2
1_{2}
= −* _{q e x T}*− +

*+*

_{x}*x*−

*kT*

*rT*

*k*

*k*

*k*

*k*

*ko*

*ko*

*k*

*dj*

*dj*( ) λ ( ) λ ρ δ[( ( ) ρ δ(

*x*

*x*

*x*

*x*

*x*

*r*

*j*

*k*

*j*

*o*

*j*

*o*

*j*

*kd*

*k*

*k*

*k*1 1

_{2}2 1

_{2}1 1

_{2}1 ) )( ) ( ( ) ) ) ] ( − − ρ δ −ρ + λ − λ

*k*

*k*

*rt*

*T*

*x*

*e dt*) −

### ∫

_{0}. (C3)

Now, substituting inλfrom equation (19), we obtain

0 0 0 2
1_{2}
= − + +
−
−
*q e x T* *x*
*x*
*kT* *rT* *k* *k* *k*
*k* *ko* *ko* *k* *dj* *dj*
( ) ( )
[( ( ) (
λ
λ ρ δ ρ δ *x* *x* *x*
*x* *x* *q*
*j* *k* *oj* *oj* *j*
*k*
*d*
*k*
*d*
*k* *k*
1
1_{2}
2
1_{2}
1
1_{2}
1
) )( ) ( ( )
( ) ) ] (
− −
− + −
ρ δ
ρ δ *k* *k*
*i* *i*
*d*
*i* *io* *i* *i* *k*
*t* *x t* *x t* *t x*
− − −
=

### ∑

1 2 1 1 2_{2}

_{2}λ ( ) [ρ ( ) ρ ( ( ))] ( ))λ

### ∫

−*o*

*T*

*rt*

*e dt.*(C4)

Using the notation in (A9), in terms*ukd*and*uko*, we rewrite equation (C4) as

0 0 0 2
1_{2}
= − + +
−
−
*q e x TkT* *rT* *k* *k* *xk* *x*
*k* *ko* *ko* *k* *dj* *dj*
( ) λ ( ) [( ( ) (
λ ρ δ ρ δ *x* *x* *x*
*x* *x* *q*
*j* *k* *oj* *oj* *j*
*k*
*d*
*k*
*d*
*k* *k*
1
1_{2}
2
1_{2}
1
1_{2}
1
) )( ) ( ( )
( ) ) ] [
− −
− + −
ρ δ
ρ δ *k* *k*
*i* *i*
*d*
*id* *io* *io* *k*
*T*
*rt*
*u* *u x* *e dt*
− −
=
−

### ∑

### ∫

_{λ}

_{ρ}

_{ρ}1 2 0 ( )] . (C5)

Considering (18) or (A14), after rearrangement of equation (C5), we have

0 0
2
0
1
1_{2}
= − + +
− +
−
*q e x T* *x*
*q x* *x* *x*
*kT* *rT* *k* *k* *k*
*k k* *kd* *k* *kd* *k*
( ) ( )
[ ( * ) (
λ
δ δ _{1} 1_{2}
2
1_{2}
2
1_{2}
1
1_{2}
2
2 2
) ( ) ( )
( ) (
*
*
+
− −
δ δ
δ δ
*k*
*o*
*k* *ko* *k*
*k*
*d*
*k* *kd* *ko*
*x* *x*
*x* *u* *
*
) ( )
( )[( )
*x* *u* *u* *u*
*x*
*k* *ko* *ko* *ko* *dj* *jd* *k*
*j*
*d*
*j*
*k*
*j*
2
1_{2}
1
1
2
+ −
+
ρ ρ λ
δ
λ
λ 2 _{1}
1_{2}
1
1_{2}
1
1_{2}
(* _{u}* (

*) ) ( **

_{x}*) (*

_{x}*(*

_{u}*) )]*

_{x}*j*

*d*

*j*

*d*

*j*

*oj*

*j*

*jo*

*oj*

*j*− + − δ δ δ

### ∫

− 0*T*

*rt*

*e dt. (C6)*

Adding the zero-sum (C6) toΠ*k*=Π*k*( , , , )δ δ δ δ*d*1 *o*1 2*d* *o*2 as in equation (2) and rearranging, we obtain
Π*k* *kd* *ko* *dj* *oj* *k* *k*
*k*
*d*
*k* *ko* *k* *kd*
*x*
*x* *x*
( , , , ) ( )
[ (
δ δ δ δ λ
δ δ δ
= +
− − +
0
2
0
1 2
* *
*
) ( ) ( ) ( )
(
*x* *x* *x* *x*
*x*
*k* *kd* *k* *ko* *k* *ko* *k*
*k*
*d*
*k*
1 1 2 2
1_{2} 1_{2} 1_{2} 1_{2}
1
2
2
δ δ δ
δ
+
− ) ( ) ( )
(
*
1_{2} 1_{2}
2
2
2
*ukd* *ok* *xk* *uko* *kouko* *djujd* *k*
*k*
− + −
+
δ ρ ρ λ
λ λ*j*)[(δ*dj***xj*1) (*ujd* (δ*djxj*1) ) (δ*oj***xj*1) (*uoj* (
1_{2} 1_{2} 1_{2}
− + − δ*oj* *j*
*T*
*rt*
*x*
*e dt*
1
0 _{1}
2
) )]

### ∫

−_{.}(C7) Or, Π

*k*

*kd*

*ko*

*dj*

*oj*

*k*

*k*

*k*

*d*

*k*

*kd*

*k*

*x*

*x*

*x*( , , , ) ( ) [( * ) ( δ δ δ δ λ δ δ = + − − 0 0 1 1 1

_{2}) ] [(1

_{2}

_{2}* )1

_{2}( ) ]1

_{2}* *

_{1 2}1 2 2 1 2 − − + + − δ

*ko*

*xk*δ

*koxk*δ

*kd*

*xk*δ

*ko*

*xk*2 1 2 2 1

_{2}1

_{2}(

_{δ}* ) (

_{δ}* ) (

_{ρ}

_{ρ}

*k*

*d*

*k*

*kd*

*ko*

*k*

*ko*

*ko*

*ko*

*dj*

*dj*

*x*

*u*−

*x*

*u*+

*u*−

*u*) ( )[( * ) ( ( ) ) ( * ) λ λ λ δ δ δ

*k*

*k*

*j*

*dj*

*xj*

*ujd*

*djxj*

*oj*

*xj*+2 1 − 1 + 2 1

_{2}1

_{2}1

_{2}1

_{2}1 0 (

*u*(

*x*) )]

*e dt*

*j*

*o*

*j*

*o*

*j*

*T*

*rt*−

### ∫

− δ . (C8) In particular, Π*k*

*kd*

*ko*

*dj*

*oj*

*k*

*k*

*k*

*d*

*k*

*ko*

*k*

*x*

*x*

*x*( ,* *, *, *) ( ) [ * * δ δ δ δ =λ 0 0+ δ 1+δ 2 1 2 1

_{2}1

_{2}2 1 2 2 − (

_{δ}* ) − (

_{δ}* ) +(

_{ρ}−

_{ρ}

*k*

*d*

*k*

*kd*

*ko*

*k*

*ko*

*ko*

*ko*

*dj*

*x*

*u*

*x*

*u*

*u*

*ujd*

*k*

*k*

*j*

*dj*

*xj*

*ujd*

*djxj*

*oj*

*xj*) ( )[( * ) ( ( ) ) ( * λ λ λ δ δ δ +2 1 − + 1

_{2}1 1

_{2}1 1 0 ) (12

*u*(

*x*) )]12

*e dt*

*j*

*o*

*j*

*o*

*j*

*T*

*rt*−

### ∫

− δ . (C9)Considering (C8) and (C9), we have

Π*k* *kd* *ko* *dj* *oj* *k* *k*
*k*
*d*
*k* *kd*
*x*
*x* *x*
( , , , ) ( )
[( ) (
* *
*
δ δ δ δ λ
δ δ
= +
− −
0 0
1
1_{2}
*k*1 2 *ok* *xk*1 *koxk*2 2 *kd* *xk*1 *ko* *xk*2
1_{2} 1_{2} 1_{2} _{1}
) ] [(_{−} _{δ} * ) _{−}(_{δ} ) ] _{+}_{δ} * _{+}_{δ} * 2
1_{2} 1_{2}
2 1 2 2
− (_{δ} * ) − _{(}_{δ} * _{)} +_{(}_{ρ} −_{ρ}
*k*
*d*
*k* *kd* *ko* *k* *ko* *ko* *ko* *dj*
*x* *u* *x* *u* *u* *ujd* *k*
*k* *j* *dj* *xj* *ujd* *dj* *xj* *oj* *x*
)
( )[( * ) ( ( * ) ) ( *
λ
λ λ δ δ δ
+2 1 − 1 +
1_{2} 1_{2}
*j* *jo* *oj* *j*
*T*
*rt*
*u* *x*
*e dt*
1 1
0 1_{2} 1_{2}
) ( _{−}( * ) )]

### ∫

− δ . (C10) Therefore, Π*k*

*kd*

*ko*

*dj*

*oj*Π

*k*

*kd*

*ko*

*dj*

*oj*

*k*

*d*( , ,

_{δ δ δ δ}*, *)

_{=}( ,

_{δ δ δ δ}* *, *, *)

_{−}

### [

(δ### ]

### [

### ]

* * ) ( ) ( ) ( )*x*

*x*

*x*

*x*

*k*

*kd*

*k*

*k*

*o*

*k*

*ko*

*k*1 1 2 1 1 2 2 1 1 2 2 1 2 2 − + − δ δ δ ≤ −

### ∫

*e dtrt*

*T*

*k*

*kd*

*ko*

*dj*

*oj*0 Π ( ,

_{δ δ δ δ}* *, *, *), (C11)

for everyδ δ*kd*, . This completes the proof of our theorem.*ko*

**D. PROOF OF THEOREM 2**

To prove this theorem, we proceed exactly as in Theorem 1, with the exception that, instead of the sum (C1), we consider the
zero-sum
0
0 0
= − +
∞ _{∞}

### ∫

*q*

*t x*

*d*

*dt*

*q*

*t x dt*

*k*Φ( )

*k*(

*k*Φ( ) ) ,

*k*(D1)