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.3Rossi,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.4Hart, 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. Letxk(t) denote the market share for firmkat timet.
Thexk(t) consists of two types of purchases: those from
re-peat customers, denoted byx tk1( ), and those from
switch-ing customers, denoted by x tk2( ). Therefore, at any timet,
we have
x t x t x tk( )= k1( )+ k2( ). (1)
To model a firm’s practice of using different promo-tional incentives to generate repeat and switching sales, let
δkd( ) andt δko( ) denote firmt k’s promotional incentives
of-fered, respectively, to repeat and switching purchases at timet. 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, firmk’s defensive and offensive promotional expenditures are given byδkd( ) ( )t x tk1 andδk
o k
t x t
( ) ( )2 ,5respectively. We assume that each firm makes independent promo-tional decisions at any timetto 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 constantqkrepresents the gross profit margin,6and the
constantqkT represents the value of firmk’s market share at
the terminal timeT.
Given any promotional incentives, a firm’s payoffs at timetdepend on how effective its targeted promo-tions are in generating sales. Letρ δkd( ( ) ( ))kd t x tk1
12
and
ρ δko( ( ) ( ))ko t x tk2 12
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 constantsρkd andρkoare 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, letf t f tkjod( )( jkod( )) denote the difference
between the effectiveness of firmk’s (firmj’s) offensive promotions and that of firmj’s (firmk’s) defensive promo-tions at timet. The variable f t f tkjod( )( jkod( )) thus measures
the competitive strength of firmk’s (firmj’s) offensive tar-geted promotions in attracting the customers that firmj
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.
(firmk) tries to retain through its defensive targeted pro-motions. Then, we have
f tkjod( )=ρ δko( ( ) ( ))ko t x tk2 −ρ δdj( ( ) ( ))dj t x tj 12 12 1 , (3) f tkjod( )=ρ δoj( ( ) ( ))oj t x tj2 −ρ δkd( ( ) ( ))dk t x tk 12 1 12 . (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 tk1 = f t x t x tkjod k − k1 −f t x tjkod k1 , xk1 0 xk1 0 ( )= , (5) ( ) ( )( ( ) ( )) ( ) ( ) x tk2 = f t x t x tkjod j − k2 −f t x tjkod k2 , xk2 0 xk2 0 ( )= , (6)
wherej≠andk= 1, 2. Equation (5) captures the effect of targeted promotions on a firm’s repeat sales. Firmk’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 firmkcan 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). Asx t x t x tk( )= k1( )+ k2( ), we have
( ) ( )( ( )) ( ) ( ) x tk =f tkjod 1−x tk −f t x tjkod k ,
xk( )0 =xk0.
(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 Πk kd ko dj oj Πk kd ko dj oj k d ( ,δ δ δ δ* *, *, *) ( , ,δ δ δ δ*, *), δ ≥ ∀ , ,δko k=1 2, , j k≠ . (8)
In mathematical terms, a closed-loop strategy can be ex-pressed as
δkd =δkd( ,t x x x xk1, k2, k1, k2)
0 0 and
δok=δko( ,t x x x xk1, k2, k1, k2),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.
AsT→ ∞, 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( ) ( ),k2 k2( )0 xk02 (10) The control functionsδkd( ),t δko( ),t k= 1, 2, areadmissi-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≤xk1 ≤ ≤xk 1and 0≤xk2 ≤ ≤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 ss 2 = = 2 + , (12)withj≠k,k= 1, 2. It is straightforward to show that for f tkjod( )>0andf 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
δkd ρ λk 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) δkd ρk d k rt k k q t e x x *= ( ) 2 1 2 2 2 2 4 Φ and δko ρk 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
( ) ( ) ( ) ( ) δkd ρ λk d k k k t t x t x t = 2 1 2 2 4 and ( ) ( )( ( )) ( ) , , δko ρ λk o k k k t t x t x t k = − = 2 2 2 1 2 4 1 2, (15) ( ) ( ) ( ) ( ) δkd ρk d k rt k k t q t e x t x t = 2 1 2 2 2 2 4 Φ and ( ) ( ) ( ( )) , , δko ρk 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 toxkis 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 sharexk, with defensive (offensive) targeting
effectivenessρ ρkd( ), and with the value of marketko
share (λk( ) ort qkΦ( ) in the infinite-time horizont ert
case), but they decrease with the actual redemption rate of defensive (offensive) targeted promotions
θkd =xk1 / (xk θok=xk2 / (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 (ρkd or ρko) 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 θdk( )t x t x t=k1( ) / k( ) and
( ) ( ) / ( ( ))
θko t x t= k2 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 = ρρ > >ρρ = . (17) Lety 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 δk 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 12 − − 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 12 2 2 − − δ δ k d j d * * ρ ρ k d k j d j q q 2 2 2 2 δ δk 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 byx y y
y k( )=1+ξ( )ξ( ). We can verify that lim ( ) ,y t→ x yk = 1 1 xk(t2) = 0, andx yk ′( ) 0 for> y∈(t 2,t1). 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 andt 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 ,
wheret1>t3>t4>t2by condition (17). Firmkwill 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 firmjwill 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 firmk, for instance, is more effective in defensive targeted pro-motions(ρkd ρko )
2 2
> , condition (24) impliesρjo ρjd
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 setρkd =ρdj =
ρd =0 009. ,ρ ρ ρ k o j o o = = =0 0119. ,qk=qj= 500,x11(0) = 0.4, x12(0) = 0.2,x21(0) = 0.1, andx22(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 switching11and share the market equally in the dynamic long-run equilib-rium.12In 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 choseρo= .0119, as in the Erickson (1992) and Fruchter and
Kalish (1997)studies, and choseρdto satisfy the stability condition(24).
11. This is because we havet 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), wherexk(t) is as in equation (7). Let
δkd ρ λk 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)
whereλksatisfies the following two-point boundary value problem (TPBVP): ( ) ( ) ( ) [ ( ) ( ( ))] λk t =rλk t q− k−21λk t ρid x t −ρoi 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 λoj j t −ρ λkd k t x tk xk =xk0 , (19)withj≠k,k= 1, 2, andxj= −1 xk. Then( ,δ δ δ δd1* o1*, d2*, 2o*)forms a global Nash equilibrium closed-loop strategy of the above differential
game, that is, it satisfies condition (8).
Proof. See Appendix C below.
Theopen-loop strategiesare given by (see Appendix A below)
( ) ( ) ( ) ( ) δkd ρ λk d k k k t t x t x t = 2 1 2 2 4 and ( ) ( )( ( )) ( ) , , , δko ρ λk o k k k t t x t x t k = − = 2 2 2 1 2 4 1 2
( ) ( ( ) ( ))( ( ))( ( ) x tk1 12 ko k t dj j t 1 x t x tk k 2 2 = ρ λ −ρ λ − −x tk1 oj j t kd k t x t x tk k1 xk1 2 2 0 ( )) (− ρ λ ( )−ρ λ ( )) ( ) ( )], ( )=xk01. (20)
Then,xk2is determined throughxk2=xk−xk1.
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)[(ρdk qk ρoj q x tj)k( ) (ρko qk ρdj 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 = − = →∞ − 2r xΨ() , lim (2 t Ψ x t e( )) rt 0, (21) where x x if k x if k k = = − = 1 1 2. (22) Let δkd ρk d k rt k k q t e x x *= ( ) 2 2 2 2 2 1 4 Φ andδ 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
( ) ( ) [ ( )] , ( ) Φt 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 − −(ρoj qj ρkd q x tk)k( )] ( ) ,t ert xk( ) xk 2 2 2 0 0 − = Φ , (24)withj≠k,k= 1, 2,xj= −1 xk, andΦis as in equation (21). Then( ,δ δ δ δ1d* o1*, d2*, 2o*)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 correspondingopen-loop strategiesare given by the following (see Appendix B below):
( ) ( ) ( ) ( ) δkd ρk d k rt k k t q t e x t x t = 2 1 2 2 2 2 4 Φ and ( ) ( ) ( ( )) , , δko ρk 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),xk1is given by( ) [( )( ( ))( ( ) ( x tk1 1 ko qk dj qj x t x t xk k k1 2 1 2 2 = ρ −ρ − − t)) (− ρoj qj −ρdk q x t x tk)k( )k ( )] ( ) ,t ert xk ( )=x 2 2 1 Φ 1 0 k01, (26)
andxk2is determined throughxk2=xk−xk1.
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 variablesukdanduko,k= 1, 2, as
δkdxk1 ukd 2 = andδkoxk2 u kko 2 1 2 = , = , . (A1)
In terms ofukdanduko, 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 ρoju toj( )−ρdku t x t xkd( )) ( ),k k( )0 =xk0. (A2)The current-value Hamiltonian for playerkfor game (A2) is given by
Hk=H x u u u uk( , , , , , )k kd ko dj jo λk =q x uk k − kd −uko +λk kx
2 2
, (A3)
whereλkis the current-value multiplier associated with the state equation. The necessary optimality conditions for playerk’s open-loop
strategies are given by the maximization condition (cf. Kamien and Schwartz 1991),
∂Hk/∂ukd = −2ukd +ρkdxkλk=0 and∂Hk/∂ukd = −2uko+ρko(1−xk)λk=0, (A4)
and the adjoint equations with the terminal conditions
( ) λk λk H k λk k λk ρ ρ i i d i d i o io r k x r q u u = − ∂ ∂ = − − − =
∑
21 , λk( )T =qkT. (A5)From (A4), we have
ukd =12ρkdxkλkanduko=12ρko(1−xk) ,λk k=1 2, . (A6)
Plugging (A6) into (A5) and the state equation, we obtain the following two-point boundary value problem (TPBVP):
( ) ( ) ( ) [ ( ) ( ( ))] λk t =rλk t q− k−21λk t ρid x t −ρoi 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 = = − =
∑
21 1 2 2 2 )(1 ( )) (2 2 ( ) 2 ( ))2( )], ( )0 −x tk − ρ λoj j t −ρ λkd k t x tk xk =xk0 , (A7)wherej≠k,k= 1, 2. Plugging (A6) into the state equation (5), we obtain
( ) [( ( ) ( ))( ( ))( ( ) x tk1 12 k t dj j t 1 x t x tk k 2 = ρ λκ −ρ λ − − ο2 ( )) ( ( ) ( )) ( ) ( )], ( x tk1 oj j t dk k t x t x tk k1 xk1 2 2 − ρ λ −ρ λ 0)=x0k1. (A8)
Considering (A7), we find theopen-loopsolutions to be
( ) ( ) ( )
We have in terms of the original variables ( ) ( ) ( ) ( ) δkd ρ λk d k k k t t x t x t = 2 2 2 1 4 and ( ) ( )( ( )) ( ) , , δko ρ λk o k k k t t x t x t k = − = 2 2 2 2 1 4 1 2, (A10)
whereλk(t),x tk( )are as in (A7);x tk1( )is as in (A8), and
xk2=xk−xk1. (A11)
From (A7), we obtain
, , ,
λk λk=0= − <qk 0 k=1 2 (A12)
which, together with the terminal condition in (A7), implies
λk( )t >0 (A13)
for allt∈[0,T].
Step 2: Constructing the Closed-Loop Strategies
Considering (A10) and substituting the values ofx x xk, k1, k2by the actual states (market shares) as given in equations (1), (5), and (6),
we obtain δkd ρ λk 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
λk( )t q= Φk ( )t ert. (B1)
Then, asT→ ∞, the two equations in (A7) forλkcan be transformed into only one equation,
( ) ( ) [ ( ) ( ( Φt 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 totinto the equations forλkin (A7). From (B2), as
before, we obtain Φ(t) > 0 for allt∈[0,∞), which is a simple consequence of the terminal condition in (B2) and the property
( ) Φt Φ= e rt − = − < 0 0 (B3) derived from (B2).
( ) ( ) [ ( ) ( ( ))] Φt 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 (ρoj qj ρkd q x tk)k( )] ( ) ,t ert xk( ) xk 2 2 2 0 0 − = Φ , (B4)wherej≠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)[(ρdk qk ρoj q x tj)k( ) (ρko qk ρdj 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 xΨ() , lim (t Ψ x t e( )) rt . (B7)Equation (B7) follows from (B2) and the relationship
( ) ()
Φt ert+rΨ′x x. (B8)
Making use of (B7) and (B6), we can transform the TPBVP (B4) into the following initial value problem:
( ) ( ) [ ( ) ( ( ))] , Φt 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 ρko k ρdj j k( )) (t 2 oj qj dk q x tk)k2( )] ( ) ,t ert xk( ) xk0 2 2 0 − − = ρ ρ Φ , (B9)wherej≠k,k= 1, 2, andΨ(x( ))0 is obtained by solving the backward equation (B7). Plugging (B1) into (A8), we also have
( ) [( )( ( ))( ( ) ( x tk1 12 ko qk dj qj 1 x t x t x tk k k1 2 2 = ρ −ρ − − )) (− ρoj qj −ρkd 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
( ) ( ) ( ) ( ) δkd ρk d k rt k k t q t e x t x t = 2 1 2 2 2 2 4 Φ and ( ) ( ) ( ( )) ( ) , , δko ρk 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 tk1( )is as in (B10); andxk2is as in (A11).
Considering (B11) and substituting the values ofx x xk, k1, k2by the actual states (market shares) as given in equation (10), we construct
δkd ρk d k rt k k t q t e x t x t *( ) ( ) ( ) ( ) = 2 1 2 2 2 2 4 Φ andδ 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 = − − + −
∫
λk rt k λ T k rt k T e x d dt( e x dt) , (C1)whereλkis as in equation (19) or (A7), andxkis as in equation (7). Considering (C1), we obtain
0= −q e x T− + 0x0+ x + −r x e dt−
kT rT k( ) λk( ) k [λk k (λk λk) ]k rt
o T
∫
. (C2)Considering equation (7), we obtain
0 0 0 2 12 = −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 12 2 12 1 12 1 ) )( ) ( ( ) ) ) ] ( − − ρ δ −ρ + λ − λk k rt T x e dt ) −
∫
0 . (C3)Now, substituting inλfrom equation (19), we obtain
0 0 0 2 12 = − + + − − 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 12 2 12 1 12 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 termsukdanduko, we rewrite equation (C4) as
0 0 0 2 12 = − + + − − 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 12 2 12 1 12 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 12 = − + + − + − q e x T x q x x x kT rT k k k k k kd k kd k ( ) ( ) [ ( * ) ( λ δ δ 1 12 2 12 2 12 1 12 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 12 1 1 2 + − + ρ ρ λ δ λ λ 2 1 12 1 12 1 12 (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) toΠk=Πk( , , , )δ δ δ δd1 o1 2d o2 as in equation (2) and rearranging, we obtain Πk 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 12 12 12 12 1 2 2 δ δ δ δ + − ) ( ) ( ) ( * 12 12 2 2 2 ukd ok xk uko kouko djujd k k − + − + δ ρ ρ λ λ λj)[(δdj*xj1) (ujd (δdjxj1) ) (δoj*xj1) (uoj ( 12 12 12 − + − δoj j T rt x e dt 1 0 1 2 ) )]
∫
− . (C7) Or, Πk 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 12 ) ] [(12 2 * )12 ( ) ]12 * * 1 2 1 2 2 1 2 − − + + − δko xk δkoxk δkd xk δko xk 2 1 2 2 12 12 (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 12 12 12 12 1 0 (u ( x ) )] e dt j o j o j T rt − ∫
− δ . (C8) In particular, Πk 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 12 12 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 − + 12 1 12 1 1 0 ) (12 u ( x ) )]12 e dt j o j o j T rt − ∫
− δ . (C9)Considering (C8) and (C9), we have
Πk kd ko dj oj k k k d k kd x x x ( , , , ) ( ) [( ) ( * * * δ δ δ δ λ δ δ = + − − 0 0 1 12 k1 2 ok xk1 koxk2 2 kd xk1 ko xk2 12 12 12 1 ) ] [(− δ * ) −(δ ) ] +δ * +δ * 2 12 12 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 + 12 12 j jo oj j T rt u x e dt 1 1 0 12 12 ) ( −( * ) )]
∫
− δ . (C10) Therefore, Πk kd ko dj oj Πk 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 δkd, . 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 = − + ∞ ∞
