Negative Advertisement Strategies. Changrong Deng

Transcription

1 Negative Advertisement Strategies Changrong Deng Saša Pekeµc The Fuqua School of Business, Duke University Preliminary Draft, July PLEASE DO NOT CIRCULATE Abstract We develop a multi-period model of dynamically evolving market shares. Firms compete by choosing long-run investment strategies that determine customer retention and new customer attraction in every period. We investigate equilibrium behavior and focus on conditions for emergence of negative strategies, i.e., strategies that put more emphasis on hurting the competitor s market share than on increasing one s own market share. We show that negative strategies are more likely to be triggered by rms that lag behind the market share leader, and that negative strategies are unlikely to emerge in the nal period of the competition.

2 Introduction Negative advertising and activities whose primary goal is hurting a competitor are not standard business practices. Yet, businesses occasionally engage in direct attacks on their competitors. This is best observed in consumer goods industry since negative attacks are often prominent in advertising campaigns aimed at general public. Several advertising campaigns in the soft drink industry during "cola wars" between Coca-Cola Company and PepsiCo during 98s and 99s are examples of such practices. More recent examples include Apple and Microsoft ad war with Apples s "Get a Mac" campaign and Microsoft s "I am a PC" response, and Verizon Wireless and AT&T ad war that started with Verizon s attack "There is a map for that" ad campaign followed by AT&T s "Set the record straight" response. (Beard,, documents and discusses numerous examples.) While businesses generally stay away from attacking competitors and tend to focus on directly helping their bottom line, negative advertising and promotional activities aimed at competitors, appear to be widespread in political contests. The topic of negative advertising and negative political campaigns is well-studied in the political science literature (e.g., Skaperdas and Grofman, 995 Lau et al., 999 Che et al., 7 Lau et al., 7 Lau and Rovner, 9). There are many attempts to explain emergence of negative advertising strategies, ranging from emotional decision-making to intensity of competition (Beard, tries to classify motives for several prominent negative campaigns). Given prevalence of negative campaigns in politics, it is natural to point out the distinguishing features of political competition as

3 fundamentals of an environment in which negative advertising is likely to happen. The most prominent such feature is the tournament nature of political contests: the winner does not care about the number of votes (i.e., market share) as long as it has more votes than the opponents (relative market share). In addition, in some political contests there is a winner-takes-all e ect since the election loser is locked out of any political power. (Quelch and Jocz, 7, use this argument to explain discrepancy in use of negative advertising in politics vs. business competitions.) Another prominent feature of political contests is that there is clearly de ned ending time which is the election day (which somewhat compares to selling perishable seasonal goods case in business competitions). If the competition has no prede ned ending, engaging in negative advertising allows for a possibly of the retaliation from competitors which would mitigate incentive to engage in negative advertising in the rst place. With a prede ned ending time of the competition (or if there exist a small number of critical time points in the competition in which most of the utility will be realized), it is plausible that negative advertising strategies could emerge just before that moment so that no time is left for opponents to implement possible retaliatory strategies. Infamous "October surprise" in US Presidential Elections, i.e., expectations of some negativity emerging within last couple of weeks of the campaign, ts within such explanation. In this paper we develop a theoretical model with the goal of understanding conditions for negative advertising strategies to emerge as (a rational) equilibrium behavior, as well as conditions under which a decision to engage in negative advertising would be sub-optimal. Our model tracks changes in market-shares over discrete time periods. Market-shares 3

4 change as a result of activities of all players ( rms, candidates). In each time period, players implement (costly) investment decisions that jointly de ne ow between any pair of market volumes (those currently belonging to any of the players or the volume of potential market that is not yet captured by any of the players). The aggregate result of all investment decisions is observed in market-shares of all players in the next period. If there is a prede ned end period T, the game ends by de ning market-shares in that period. Players are assumed to be utility maximizing and their utility is a time-discounted sum of utilities from each time period. The utility in each time period is a linear function of the player s market share and the opponents market share (if opponents market share linear coe cient is zero, the player cares only about its own market-share in that period) minus the cost of investments made in that period. We study equilibrium behavior in this model. Since players are ex ante symmetric in the model, we focus on a single player activities. In order to facilitate our analysis, we make a simplifying assumption of aggregating the behavior of all opponents into a single combined opponent. Thus, the model we present and analyze has two players competing for market-shares. However, the modeling choice that is of critical importance for any relevance and meaningfulness of our results is the de nition of negative advertising strategies. We de ne player s advertising strategy at a given time period t to be negative if player s decisions are aimed at decreasing opponent s market share than in increasing one s own market share. Our de nition of negative advertising is similar to that of Skaperdas and Grofman (995). (However, our model captures more general behavior, since Skaperdas and Grofman, 995, 4

5 framework is static, does not involve equilibrium analysis, and focuses on player utilities that are relevant for political competition only.) While majority of marketing literature deals with empirical and experimental studies focusing on e ects of negativity on consumers (e.g., Shiv et al., 997), de ning negative advertising strategies is mostly discussed as a special case of comparative advertising "in which a di erentiative technique is employed" (James and Hansel, 99). However, several theoretical models in marketing have similar features to our model. The core idea of evolving market shares through targeted investment of players is similar to that behind targeted advertising strategies (e.g., Iyer et al., 5), while optimization behind decision-making in our dynamic multi-period model has a similarity to dynamic optimal control models in advertising (Heller and Chakrabarti, Feichtinger et al., 994). The formulation of the dynamic game in our model and use of the Lagrange method in numerical computations is somewhat similar to the formulation and analysis of a dynamic game presented in Chow (997, Chapter 6). We nd that in our model negative advertising strategies could emerge as an equilibrium behavior both in political competition and in business competition. There are several parameters relevant for existence of equilibria with negative advertising strategies. For example, it is not surprising to see that negative advertising strategies emerge when (i) player cares about the opponent s market share and (ii) it is much more costly to advertise with the goal of increasing one s market share than to invest in advertising aimed at reducing opponent s market share. (We model costs to be convex.) The parameter that turns out to be relevant is the initial market share endowment of 5

6 the player relative to the size of the market which is captured by the player or the opponent (i.e., potential customers who might be attracted at later stages of the game are not part of this calculation.) We show that players who are lagging behind are more likely to engage in negative advertising than players who are market leaders. This ts within examples mentioned in the opening paragraph (PepsiCo vs. Coca Cola, Apple vs. Microsoft, Verizon vs. AT&T): the company who triggered negative advertising war was lagging behind the market leader. In terms of political competition, this result indicates that the candidate who is lagging behind is more likely to turn to negative campaigning, which is also found to hold in models of Skaperdas and Grofman (995) and Harrington and Hess (996). The dynamic multi-period feature of our model enables to get some insights into timing of the start of negative advertising activities. We nd that it is nearly impossible (except for some extreme unrealistic choices for cost functions) for negative advertising to emerge in the ending period of the game, and in any period of the game in which player s own market share is of signi cant importance to its utility. Under assumption that businesses overwhelmingly focus in present and in immediate future, the latter can be viewed as an argument why negative advertising is not prevalent in business competition. The in nite horizon version of our model (i.e., game without ending) does allow for existence of negative strategies in the equilibrium. Other equilibria that do not involve negative advertising strategies might exist (multiplicity of equilibria is often the case with in nite horizon games) and could Paretodominate equilibria with negative advertising strategies a cautionary note that businesses could be drawn into negative advertising war and that it is not easy to get out of it. 6

7 The paper is organized as follows. In the next section we de ne the basic model. In Section 3 we discuss how the Luce choice model can be reduced to our basic model, de ne negative advertising strategies and discuss equilibrium existence. We then in Section 4 discuss the two-period game model which allows us to present most of our ndings. In Section 5 we generalize the model and extend results for the case of nitely many periods. We adjust the form of cost functions in our model in Section 6 and analyze the model with cost functions that are scaled by current market volume. In Section 7 we discuss the game with an in nite horizon and in Section 8 we brie y discuss the one-shot game in our model and its relevance for understanding no-commitment strategies. We conclude the paper with brief remarks in Section 9. A Dynamic Model of Market-Share Competition There are two players in the market and market structure is observed in nitely many discrete time periods t = : : : T. The players are denoted and. The market shares at time t are given by a non-negative vector V (t) = (V (t) V (t) V (t) ) with V (t) + V (t) + V (t) =. The market share of player i = at time t is denoted by V (t) i the proportion of the market not captured by the players is V (t). The changes in market shares from period t to period t+ (t = ::: T ) are de ned 7

8 by the transition matrix P (t) = () where ij by, and P j p(t) ij = for i =. The market shares at period t + are de ned V (t+) = V (t) P (t) () Player and Player can both potentially invest into market share transitions from i to j. Let the intensity or magnitude of their e ort be ij 8 and m(t) ij, respectively with

9 ij [ m], m(t) ij [ m] and t = ::: T. Then, market share transitions are determined as in Luce s choice axiom (Luce, 959), ij = 8 >< >: for i j f g h h i +m(t) i +h ij +m(t) ij i +m(t) i +h i +m(t) i if P jfg =3 if P jfg ij + m(t) ij 6= ij + m(t) ij = (3) where h () is a non-negative, continuous and strictly increasing function with homogeneous degree k (k > ) with h () =. Remarks: ) The determination of ij can be regarded as a generalization of the Luce s model, the main result of which excludes perfect discrimination. Here, we allow the occurence of zero and one transitions. ) ij + m(t) ij can be considered as the aggregate wealth that the representative voter in party i can get from Player and Player if choosing the transition from i to j. Hence, h () can be understood as the indirect utility function of the representative voter given the consumption price level. Since utility function is unique up to positive a ne transformation, it is feasible to require h () to be non-negative and P (t) ij is well de ned. In addition, ij = =3 if there occurs the zero aggregate investment, i.e., ij = m(t) ij = for some i and all j f g. Continuity and monotonicity are also standard assumptions. In addition, if the utility function is homogeneous degree k (k > ), the indirect utility function is also homogeneous degree k (k > ) in intensities. 9

10 3) h () is de ned on the domain [ ) instead of the range of intensities [ m]. Then, the domain of h () is [ ). The cost functions associated with intensities are C e ij ( ij ) and C e ij ( ij ). We assume the cost functions are continuous, increasing, and convex (although we will show that the existence of the equilibrium does not require the convexity). The convexity indicates the non-decreasing marginal cost. We also assume C e ij () = C e ij () =. Cij e () and C e ij () are also de ned on the domain [ ) instead of the range of intensities [ m]. Player s lifetime utility is TX t= r t [ (t) V (t) (t) V (t) ] XT X t= ijfg r t [ e C ij ( ij )] where r ( ] is the time discount factor, (t) i share at time t (i.e., (t) i V (t) i is the time-varying weight for player i market describes revenues drawn from player i market share at time t), i is the time-invariant weight that player i puts on the impact of market share of its opponent. Analogously, player s lifetime utility is TX t= r t [ (t) V (t) (t) V (t) ] XT X t= ijfg r t [ e C ij ( ij )] Such utility function attempts to capture various possibilities. For example, setting (t) i = for t = ::: T, allows the market share in the nal period only to a ect players utilities. Such situation could occur where market share captured by a player is converted into utility in a predetermined time period. Such is the case in highly seasonal sales and in

11 sales of perishable goods as well as in politics. In political competition, players (candidates) compete for votes cast and the market shares are only relevant on the election day. One important distinction of political competition is that actual market share is secondary to relative market share, i.e., it is important to have more votes than the opponent, regardless of the actual vote count. This can be captured by setting i =. (On the other hand, setting i = models players whose utility is drawn exclusively from their own market shares.) Before we represent our equilibrium concept, we show that zero aggregate intensities is strictly dominated. Lemma. Either of the buyer has incentive to deviate from zero aggregate intensities, i.e., P jfg ij + m(t) ij =, for i f g. Proof. Without loss of generality, we can take player for example. Let the ow from i to j is bene cial to player. If player increases her intensity from to some small positive amount, e", the market share transition p ij increases from =3 to. However, since the cost function is continuos, this deviation can be always pro table. Therefore, player has incentive to deviate from zero aggregate intensities. Similarly, player also has the incentive to deviate from zero aggregate intensities. By Lemma., we will consider the intensity decision with positive aggregate intensities

12 in the rest of this paper. Hence, player s optimization problem is max (t) fm ij : t= ::: T Subject to V (t+) = V (t) P (t) i= j= g P T t= rt [ (t) V (t) (t) V (t) ] P T t= Pijfg rt [ C e ij ( ij )] = V (t) + V (t) + V (t) ij = h ij + m(t) ij for i j f g V (t+) V (t+) V (t+) P jfg ij + m(t) ij are given. = h i + m(t) i + h i + m(t) i + h i + m(t) i ij m ", " is a small positive number, We impose the Nash equilibrium concept and thus it is noticed that ij for i j f g are equilibrium values chosen by player. " can be imagined as the smallest unit of money.

13 Player s problem is similar max (t) fm ij : t= ::: T Subject to V (t+) = V (t) P (t) i= j= g P T t= rt [ (t) V (t) (t) V (t) ] P T t= Pijfg rt [ C e ij ( ij )] = V (t) + V (t) + V (t) ij = h ij for i j f g + m(t) ij V (t+) V (t+) V (t+) P jfg ij + m(t) ij are given. = h i + m(t) i + h i + m(t) i + h i + m(t) i ij m ", " is a small positive number, The following result demonstrate the existence of the equilibrium. Theorem. The game has a (mixed strategy) Nash equilibrium. Proof. We use Glicksberg s Fixed Point Theorem (Glicksberg, 95). Obviously, the strategy space is not empty. Therefore, the optimization problems for player and are feasible. The strategy space of the optimization problem is a compact metric space, since ij m, m(t) ij m, P jfg ij + m(t) ij " where " is a small positive number, and the constraints are continuous functions. The utility functions are continuous by de nition. 3

14 Remark:. Note that the theorem does not require the cost functions to be convex.. The existence of a pure strategy Nash equilibrium requires the utility function to be quasi-concave and the strategy be set to a non-empty convex, and compact set. Therefore, the only requirement that is not automatically satis ed is convexity. Assumption. Player sets,,, and to zero, while Player sets,,, and to zero. This assumption is intuitive, players will not invest in the activities that hurt themselves. Lemma.3 Let the equilibrium intensities be ij and m(t) ij. With Assumption, Player and Player s equilibrium transition probabilities will not be a ected by rescaling the intensities, i.e., ( i ) =k ij and ( i ) =k ij, where = = h = = h and = = h + + h + h, + h + h, + h + h. Moreover, given cost functions associated with intensities, there exists a unique new set of continuous and strictly increasing cost functions, i.e., C ij (), C e ij ( i ) k h () and C ij (), C e ij ( i ) k h () 4

15 n associated with market share transitions, among which player chooses n o player chooses, and = h h + h it is equivalent to model with intensities and with market share transitions., such that o, Proof. With Assumption, market transitions are = = = h h h +h h +h +h h +h +h h +h = = = h h h +h h +h +h h +h +h h +h and = = = h +m(t) h +m(t) +h h +m(t) h +m(t) +h h +h +h h +h +h : Let the equilibrium intensities be ij and m(t) ij. Then we can normalize the denominators by introducing, = h + h + h 5

16 , = h + h + h and, = h + + h + h : Take for example, = = h h h +h h +h = h +h +h : Since h () is homogeneous degree k, and is de ned on [ ), = h ( ) =k. Therefore, if we de ne, ( ) =k, ( ) =k and, ( ) =k the transition is just the function of the intensity, i.e, = h,, and are not necessarily in the range [ m] Similarly,. It is noticed that = h, = h Let the cost function associated with market share transition be C :. To match 6

17 the utility function, we require C = e C = e C ( ) k = e C ( ) k h : h () exists since h () is strictly increasing. Similarly, C = e C ( ) k h and C = e C ( ) k h : Similarly, we get cost functions associated with market share transitions on party. C C C = e C ( ) k h = e C ( ) k h = e C ( ) k h For party, and are similar, C C = e C ( ) k h = e C ( ) k h : However, there is slightly di erence for since both player and player will a ect this 7

18 probability. De ne Then = h = h = h h, ( ) =k, ( ) =k : + h and C C = e C ( ) k h = e C ( ) k h : Since h () is a strictly increasing function, by the construction, there are one-to-one mapping between and respectively. Similarly, there is one-to-one mapping between and respectively. 8

19 As there is also one-to-one mapping between ij() and m(t) function of player with cost function C e ij, which is optimized by n o ij, can be optimized by C ij (), C e ij ( i ) k h (), given ij n function of player with cost function C e ij n o ij, can be optimized by C ij (), C e ij ( i ) k h (), given On the other hand, since every ij() ij() given i, the objective n o ij given o with the adjusted cost function o. Similarly, the objective n o, which is optimized by ij given o with the adjusted cost function o. n ij n n can be replicated by m(t) ij(), the optimization problem with transitions can be achieved by some optimization problem with intensities. Therefore, modeling with intensities is equivalent to modeling with market share transitions, given the adjusted cost function and = h h + h. As a result, we will adopt the modeling where players decide on their costly investments in transitions, ij. With Assumption, Player invests in market share transitions that could positively in uence his market share (p p p ) and/or negatively in uence his opponents market share (p, in addition to p ). Analogously, Player invests in transitions p p p and/or p. (Both players could also have interest in controlling p.) Looking at the changes of market shares in a given time period t that are result of chosen transitions, we can measure whether Player invested more into increasing its own market share V (t) + V (t) (4) 9

20 or into decreasing the opponent s (i.e., Player ) market share V (t) + V (t) : (5) Comparing (4) and (5) and analogous quantities for Player, we have the following de nitions for negative (positive) advertisement strategy. De nition.4 We de ne d (t) i as Positive Advertisement Index at time t for player i: d (t) = V (t) V (t) (6) d (t) = V (t) V (t) : (7) If d (t) i >, then Player i chooses a strictly positive advertisement strategy at time t since it focuses more on increasing its market share than on decreasing the opponent s market share. Conversely, if d (t) i <, Player i chooses a strictly negative advertisement strategy since it focuses more on increasing its market share than on decreasing the opponent s market share. (Note that a direct transition between V and V has both a positive and negative impact, in context of labeling activities as positive or negative advertising, and cancels out in comparison of quantities (4) and (5).)

21 We can rewrite player s optimization problem as P max T (t) fp p(t) p(t) p(t) p(t) :t=:::t g t= rt [ (t) V (t) (t) V (t) ] P T t= rt [C ( ) + C ( ) + C ( ) + C ( ) + C ( )] subject to V (t+) = V (t) P (t) = V (t) = V (t) + V (t) + V (t) = h h V (t+) V (t+) V (t+), are given. 3 3 = h,,,, and are equilibrium values chosen by Player. For player,

22 the optimization problem is similar, P max T (t) fp p(t) p(t) p(t) p(t) :t=:::t g t= rt [ (t) V (t) (t) V (t) ] P T t= rt [C ( ) + C ( ) + C ( ) + C ( ) + C ( )] subject to V (t+) = V (t) P (t) = V (t) = V (t) + V (t) + V (t) h = h V (t+) V (t+) V (t+), V () are given. 3 3 = h,,,, and are equilibrium values chosen by Player. In the following discussions, we will consider a linear function h (). Then = h h + h = +.

23 3 Linear Cost Functions Consider linear cost functions C (p) = c ij p ij. This can be attained by assuming linear cost functions associated with intensities. We rst discuss the case T =. We can express the market shares at t = are V () = p () [p () + p () + (p () + p () )] +p () [p () + p () + p () ] +p () [p () + p () + p () ] and V () = p () [p () + p () + (p () + p () )] +p () [p () + p () + p () ] +p () [p () + p () + p () ]. The market shares at t = are V () = p () + p () + p () and V () = p () + p () + p (). 3

24 We rst make the following cost assumptions. Assumption A c c. Assumption A c c. These assumptions indicate that it is cheaper to directly attract opponent s customers than to turn them away from the opponent. Assumption B c c. Assumption B c c. These assumptions indicate that it is cheaper to attract uncommitted customers than to focus on keeping them uncommitted. By Theorem., since linear cost functions are also convex, there exists a pure strategy Nash Equilibrium in the subgame at t =. The following Proposition 3. shows the suboptimality of negative advertisement in the last period. Proposition 3. With Assumption A, or with Assumption B and there are more uncommitted customers a ected by the player than opponent s customers who will become uncommitted, i.e., p () V () p () p () V (), (8) Player will not engage in strictly negative advertisement in the last period, i.e., d (). Remarks: ) Assumption A alone can induce no strictly negative advertisement in the last period. ) In fact, Assumption A drives p () down to Assumption B and condition (8) drive 4

25 p () up to p () p (). Therefore, condition (8) holds if Assumption A holds. Proof. Proof is by contradiction. If V () =, it is obvious that there is no strictly negative advertisement. If V () 6= and V () 6=, suppose there exists a set of choices in the support such that d () <. Rewriting d () < we get p () V () < p () V (). (9) Note that V () = p () + p () + (p () + p () ) and V () = p () + p () + p () are between and by de nition. Also, p () >. By (9), p () < p () p () if p () V () p () p () V () which also means that the maximal value of p () can reverse the negative advertisement. We try to nd pro table deviations at t = for p () (p () V () =V () p () ] and p () [p () min n p () V () =V () p () p () o ). It is feasible for player to decrease p () by " >, i.e. p ()new = p () ", or increase p () by " >, i.e. p ()new = p () + ". To balance these changes in order to satisfy the feasible constraints, we must have p ()new = p () + ", or p ()new = p () ". These are the choice variables that could be controlled by player. 5

26 Player s revenue is () () +rf [ () p () () p () ] + [ () p () () p () ] + [ () p () () p () ]g +r [ () p () () p () ][p () + p () + (p () + p () ) ] +r [ () p () () p () ][p () + p () + p () ] +r [ () p () () p () ][p () + p () + p () ]. and transition probability is determined by (3). Case : Decreasing p () by " > is pro table if r () [p () + p () + p () ] r[ C p () + " C p () + C p () " C p () ]: Let " goes to zero, then we have r () [p () + p () + p () ] r[c p () C p () ], r () V () c c 6

27 By Assumption A, this obviously holds. Case : Increasing p () by " > is pro table if r () [p () + p () + (p () + p () ) ]" r[ C p () + " C p () C p () " C p () ]: Let " goes to zero, then we have r () [p () + p () + (p () + p () ) ] r[c p () C p () ], r () V () c c : By Assumption B, this obviously holds. If V () =, by the de nition of d (), d () = p () V (). There is no strictly negative advertisement i p () V () =. This can be achieved by pro table deviation from Assumption A or p () V () V () =. Corollary 3. With Assumption A, A, B and B, neither of the Players devotes to maintaining the uncommitted customers in the last period, i.e., p () + p () =, p () =, and p () =. Remarks: Since the equilibriums are not unique and there are at least three degrees of freedom, 7

28 p (), p () and p () can be regarded as undetermined choices. Proof. With Assumption A and Assumption B for Player, p () =, p () = p () p (), p () = with Assumption A and Assumption B for Player, p () =, p () = p () p (), and p () =. Therefore, p () + p () =. Then, the uncommitted customers are completely turned into other parties. Lemma 3.3 With Assumption A, A, B and B, there exists a pure strategy subgame perfect Nash Equilibrium in the two periods game with linear cost functions. Proof. By Theorem., since linear cost functions are also convex functions, there exists a pure strategy Nash Equilibrium in the subgame at t =. By Corollary 3., the equilibrium transitions do not depend on V (), V (), or V (). Hence, the value function at t =, i.e., take Player for example, J V () V () = max () fp p() p() p() h p() +r gf() V () () V () C (p () ) + C (p () ) + C (p () ) + C (p () ) + C (p () () V () () V () g for t T, ) subject to V (T ) = V P and feasibility constraints, i is linear and continuous in V () and V (). Therefore, the subgame at t = is also a continuous game and the objective function is linear in all transitions. By the above argument again, there exists a pure strategy Nash Equilibrium in the two periods game. 8

29 We turn to identifying situations in which Player chooses strictly negative advertisement in the rst period. If =, there is no strictly negative advertisement in the rst period. As a result, the following Proposition 3.4 focuses on 6=. Proposition 3.4 With Assumption A, A, B and B, Player will engage in strictly negative advertisement in the rst period when 6=, i.e. d () <, if and only if p () > p () () and r A () c c () where A () = () + () p () p () () =r: () A su cient condition is p () <, condition () and r A () c c : (3) Remarks: Similar to Proposition 3., condition (3) drives p () down to, while conditions () and () drive p () up to p (). Moreover, condition (3) leads to condition () if p () <. Proof. The su cient part is similar to the proof of Proposition 3.. If 6=, and 6=, we rst prove the su cient condition for d (). Suppose 9

30 d () >, i.e., p () > p (). n o We try to nd pro table deviations at t = for p () [p () min p () = ) and p () (p () = p () ]. Case : p () >. Obviously, more substantially Player lags behind the opponent, i.e., smaller = larger p () has to be. This indicates a larger space of deviation, p ()new = p () ", and such deviation would have to be balanced by p ()new = p () + ". Looking at the utility function for Player, such pro table deviation exists if +, r () " + r [ () p () () p () ] r [ () p () () p () ] [C (p () ) C (p () )] for small ". The above condition is equivalent to r A () c c where A () = () p () () p () () p () + () p () () r : Case : p () < p (). A requirement for the analysis in this case is p () p (). A pro table deviation here involves p ()new = p () + ", which should be balanced by p ()new = p () ". 3

31 Again, looking at the utility function for Player, such pro table deviation exists if r [ () p () () p () () p () + () p () [C (p () ) C (p () )]: () r ] The above condition is equivalent to r A () c c : Since condition (3) drives p () down to, while conditions () and () drive p () up to p (), by the de nition of d (), the su cient condition for strictly negative advertisement is p () > p () and r A () c c or r A () c c, p () <, and r A () c c : To prove the necessary part, we will rst identify the su cient conditions for positive advertisement following the similar argument as above. There is positive advertisement in the rst period if r A () c c 3

32 or p () p () p () and r A () c c : Suppose either condition () or condition () does not hold, i.e., p () p () (4) or r A () < c c : (5) Case : condition (4) holds. Since p () p (), p () p () p () for equilibrium choice of p (). There must be no strictly negative advertisement in the rst period by any feasible p () values. Case : condition (5) holds. This is just one of the su cient conditions for positive advertisement in the rst period. Hence, if either condition () or condition () does not hold, there is no strictly negative advertisement in the rst period. By the proof of Corollary 3., with Assumption A, A, B and B, we must have 3

33 p () + p () =, p () =, and p () =. Then A () = () p () () p () = p () () + () p () = () + () p () p () () p () + () p () () + () () r : () r () r If =, the above conditions are still valid. This completes the proof. The following Corollary 3.5 shows the feasibility of negative advertisement. Corollary 3.5 There always exist equilibrium choices at t = satisfying condition (). Moreover, with Assumption A, A, B and B, there exist equilibrium choices at t = satisfying conditions () or (3) if r () + () () =r c c or r () + () () =r c c respectively. Remarks: ) Consider the simple cases c = c and c = c. There is no strictly negative 33

34 advertisement in the last period. But, if () + () () =r, it is still possible for player to play strictly negative advertisement in the rst period. This condition obviously holds if either () = () = () = r =. ) In the political competition, i.e., >, these conditions are easier to satisfy. 3) If only the nal market shares matter, these conditions are easier to satisfy. Proof. By the proof of Proposition 3.4, the equilibrium choices are not unique at t =. Similarly, the equilibrium choices are not unique at t =, either. At t =, at least p () can be regarded as undetermined. If condition (3) holds, p () is driven down to. Therefore, there exists p () making condition () satis ed. If (3) does not hold, p () can be regarded as undetermined. Hence, there still exist p () and p () making condition () satis ed. At t =, since there are at least three degrees of freedom, p (), p () and p () can be regarded as undetermined equilibrium choices. It is also noticed that the maximum possible value of A () is A () max = () + () () r : There exist equilibrium choices at t = satisfying conditions () or (3) if r () + () () =r c c 34

35 or r () + () () =r c c respectively. Corollary 3.6 gives how initial market shares a ect negative advertisement. It also sheds light on the optimal strategy of a player who is falling behind. Corollary 3.6 With Assumption A, A, B and B, if xed, decreasing the initial market share of Player, i.e., a smaller value of or xed, decreasing the initial market share of uncommitted party, i.e., a smaller value of, there occur more equilibriums with Player s strictly negative advertisement in the rst period and, moreover, Player s advertisement is of more negativity for some particular equilibrium. Proof. In order to study the e ect of initial market shares on equilibriums, we consider the necessary and su cient conditions. With Assumption A, condition () requires that A (). Moreover, with p () = and condition (), there is no strictly negative advertisement. Hence, conditions () and () are equivalent to < + p () = p () and (c c ) = r A () : 35

36 First, it is noticed that < p () = p () is equivalent to p () = p () < : Since the equilibrium choices at t = is not unique, p () and p () can be regarded to be undetermined equilibrium choices. Fixed, a smaller, or xed, a smaller allows greater upper bounds for both p () and p (). Second, by the proof of Proposition 3.4, With Assumption A, A, B and B, A () is rewritten as A () = () + () p () p () () r where p (), p () and p () can be regarded as undetermined equilibrium choices. It is noticed that (c c ) = r A () is equivalent to (c c ) = r + () =r p () p () : = () + () Hence, a smaller or a smaller allows smaller lower bound for p () p (). Since 36

37 p () p () is free within a unit square, there exist more equilibriums choices at t = with strictly negative advertisement in the rst period. Therefore, there exist more equilibriums with strictly negative advertisement for the whole game with a smaller or. To prove the e ect of initial market shares on the degree of negativity, recall the de nition of positive advertisement index d () = p () p () : If conditions () and () hold, there is strictly negative advertisement and p () is driven up to p (). In addition, if condition (3) holds, p () is driven down to if not, p () is regarded as undetermined equilibrium choices. If conditions () or () does not hold, there is positive advertisement. p () is driven down to (or undetermined) or p () is driven up to p () p (). Since xed, a smaller, or xed, a smaller facilitates the necessary and su cient condition of negative advertisement, the negativity of advertisement increases at the edge of jumping from positive to negative advertisement. Within some particular equilibrium, either p () (or p () ) is driven to the corner or undetermined, so the negativity of advertisement increases when xed, decreasing, or xed, decreasing. The following Corollary 3.7 shows the comparative statics of parameters in the utility function. We also expect to see more strictly negative advertisement with political compe- 37

38 tition. Corollary 3.7 With Assumption A, A, B and B, there exist more equilibriums with Player s strictly negative advertisement in the rst period, () ) if in political competition, i.e., >, than those in business competition, i.e., = ) if Player cares more about the opponent s nal market share, i.e., a larger value of 3) if Player cares more about its own nal market share, i.e., a larger value of () 4) if Player cares less about its own market share at t =, i.e., a smaller value of () 5) if Player is more patient, i.e., a larger discount factor r. More the player cares about the opponent s market share in the last period (i.e., larger () ), as in the political competition, the negative advertising strategy in the initial period is more likely. Similarly, if player s utility puts a considerable importance on its market share at t =, i.e. if () is signi cant, a strictly negative advertisement will not be likely to occur. Conversely, if () is small (or even equal to zero), a strictly negative advertisement in the initial period is likely. These correspond to a two-shot advertising strategy in which (not very signi cant) market share at t = is ignored and all of the investment in the initial period is geared towards optimal market structure for potentially large market share gains at t = T = (when the market share matters). If a player stands to gain at t = from large V () through positive advertising e ort p () then it could make sense to focus on enlarging V () which could call for negative advertising e orts p () in the initial period. 38

39 Proof. By the proof of Proposition 3.4, with Assumption A, A, B and B, A () = () + () p () p () () r : Obviously, (), (), (), r and a ect the decision of negative advertisement at t = only through r A (). However, the value of r A () will not a ect the multiple equilibrium choices at t =. By Corollary 3.5, there exist equilibrium choices satisfying condition (). Therefore, in order to study the comparative statics of (), (), (), r and, we focus on the necessary and su cient conditions. Then, condition () require A () A (), c c r : Hence, they require p () p () A () + () =r = () + () : Moreover, since the equilibrium choices at t = are not unique and thus p () p () is free within a unit square, there exist more equilibriums with Player s strictly negative advertisement at t = if A () + () =r = () + () is smaller. Thus we have the conclusions. 39

40 We close this section by providing the simple examples as follows. Example 3.8 Consider the special case () = () = and () = () =. If =, only the nal market shares matter. Suppose there is no discount, i.e., r =, all the marginal costs are equal to c, and = =. By Corollary 3., neither of the players will adopt negative advertisement. The equilibrium choices at t = satisfy p () + p () =, p () =, and p () =. At t =, by Proposition 3.4, the necessary and su cient condition for Player to adopt strict negative advertisement is p () > p () and p () p () = ( + ) : Moreover, condition (3) in Proposition 3.4 is also p () p () = ( + ), which drives p () down to. Therefore, in this case, Player adopts strict negative advertisement if and only if p () < and p () p () = ( + ). Similarly, Player adopts negative advertisement if and only if p () < and p () p () = ( + ). The following gure 4

41 shows the partitions of equilibrium choices at t =. With the increase of p () + = ( + ) and p () + = ( + ), the region for both Player and Player s strictly negative advertisement shrinks and then the region for both positive advertisement appears. Moreover, the increase of will increase the regions of strictly negative advertisement. In order to illustrate the e ect of initial market shares, we give up the equal marginal costs assumption and provide the following example. Example 3.9 Consider the special case () = () = and () = () =. Suppose there is no discount, i.e., r =, and = =. All the marginal costs are equal to c except 4

42 that c = c = c = c = ( ) c, where >. Similar to Example 3.8, neither of the players will adopt negative advertisement. The equilibrium choices at t = satisfy p () + p () =, p () =, and p () =. To focus more on e ects of the initial market shares, we consider the symmetric equilibrium, i.e., p () = p (), and p () = p () < p (). Denote p, p () p () = p () p (). At t =, by Proposition 3.4, the necessary and su cient condition for Player to adopt strictly negative advertisement is p () > p () and p () p () c= r ( + ) : It is noticed that condition (3) will drive p () down to. Similarly, Player adopts strictly negative advertisement if and only if p () > p () and p () p () c= r ( + ) : Therefore, we have the following partitions on initial market shares, where we make 4

43 condition () strict. For some p (), p (), p (), and p () Player adopts strictly negative advertisement if and only if n > max p () = p () c= pr ( + ) o Player adopts strictly negative advertisement if and only if n > max p () = p () c= pr ( + ) o : The following gure shows the partition of initial market shares. 43

44 The decrease of, given increases the region of negative advertisement. Fixed, smaller and larger make negative advertisement more possible. 4 Convex Cost Functions We consider strictly convex cost functions. The Bellman equations for player are J T V (T ) V (T ) = (T ) V (T ) (T ) V (T ) (6) J t V (t) V (t) = max (t) fp p(t) p(t) p(t) p(t) gf(t) V (t) (t) V (t) i h C ( ) + C ( ) + C ( ) + C ( ) + C ( +rj t+ V (t+) V (t+) g for t T, ) (7) where V (t+) = V (t) + V (t) + V (t) (8) V (t+) = V (t) + V (t) + V (t) (9) and = V (t) + V (t) + V (t). 44

45 The feasibility constraints are = () + + = () and + + = : () 4. Optimal Transitions Lemma 4. With strictly convex cost functions, the subgame in the last period, i.e., t = T, has a pure strategy Nash Equilibrium. Remarks: The equilibrium is not unique. since the feasibility constraints are the same for both player and player, there must be at least 3 degrees of freedom. Hence, the equilibrium is not unique. Proof. At t = T, given V V, we take play for example and player s objective function V V h C (p ) + C (p ) + C (p ) + C (p ) + C (p +rj T V (T ) V (T ) ) i n is concave in p p p p p o, since J T V (T ) V (T ) is linear in V (T ) n o and V (T ), which are also linear in p p p p p by (8) and (9). 45

46 Moreover, the strategy space is non-empty, compact and convex by () (), () and the non-negativity constraints. By Kakutani xed point, there exists a pure strategy Nash Equilibrium. Lemma 4. For the subgame game starting at T (take p, p and p as free equilibrium choices), there exists a pure strategy subgame perfect Nash Equilibrium if marginal costs satisfy n min C(p (T ) ) C(p (T ) ) C(p (T ) ) o rj T V V where J T V V = +r r (T ) (T ) (T ) C p =C p C p +C p r (T ) C p +C p : Remarks: ) To have a simple view, let the cost functions be quadratic, i.e., C (p) = cp ij= with equal marginal cost rate c, r =, and (T ) = (T ) =. Then this condition is c ( =). ) If is su ciently large such that J T V V is negative, the condition hold. Proof. Denote Player s value function as J and Player s value function as H. Without loss of generality, take Player for example. J T V (T ) V (T ) is linear in both n V (T ) and V (T ). By (8) and (9), which are linear in o, J T V (T ) V (T ) n o must also be linear in p p p p p. We will show Player s objec- 46

47 tive at t = T n (T ) (T ) (T ) (T ) is concave in the p p p p p (T ) o, (T ) OBJ (T ) = (T ) V (T ) (T ) V h (T ) (T ) (T ) (T ) C (p ) + C (p ) + C (p ) + C (p ) + C (p (T ) (T ) (T ) (T ) (T ) (T ) +rj T (p + V p p + V p p (T ) (T ) (T ) (T ) (T ) (T ) p + V p p + V p p (T ) ) (T ) (T ) i ): Suppose the rst order conditions are valid. Let (t), (t), and (t) be the Lagrangian multipliers on = + + = and + + = respectively. We consider the interior solution. First order conditions are as follows. : C : C : C : C : C + rj t+v V (t) (t) = (3) + rj t+v V (t) (t) = (4) + rj t+v V (t) (t) = (5) (t) = (6) (t) = : (7) 47

48 Envelope theorem indicates J tv = (t) + rj t+v + rj t+v (8) J tv = (t) + rj t+v + rj t+v (9) where J T +V = J T +V =. Let t = T, rst order conditions are valid. Envelope theorem indicates J T V = + r (T ) p p + r (T ) p p and J T V = + r (T ) p p + r (T ) p p : Since the equilibriums are not uniuqe and there are three degrees of freedom, we can regard p, p and p as freely chosen, which are independent of the state variable V, V, and V. 48

49 By the rst order conditions and envelope theorem, C C p p C C p p = r (T ) = r (T ) V V : Hence, C p C p = = r (T ) and (or ) (or ) = r (T ) : Moreover, since p + p = p p + p = p p (or ) (or ) (or ) (or ) (or ) : 49

50 Similarly, for Player, we have = C p = r (T ) = r (T (or ) (or ) = (or ) (or ) : Since the cost functions are strictly convex, we @V = C p r (T ) + = C p r (T ) + (or ) = r (T ) C p 5

51 (or ) = = r (T ) + =@V (or ) p r (T ) r (T ) C C p + C p p + C p C p =C (or ) = = r (T ) p =@V (or ) r (T ) r (T ) C C p + C p p + C p C p =C p : Hence, the Hessian for Player and Player s value functions are J T V V = +r r (T ) (T ) (T ) C p =C p C p +C p r (T ) C p +C p J T V V = J T V V = +r r (T ) (T ) (T ) C p =C p C p +C p +r r (T ) (T ) (T ) C p =C p C p +C p H T V V = J T V V = +r r (T ) (T ) (T ) C p =C p C p +C p r (T ) + C p +C p r (T ) +r r (T ) C p +C + (T ) (T ) C p =C p p C p +C p! 5

52 H T V V = H T V V = +r r (T ) (T ) (T ) C p =C p C p +C p +r r (T ) (T ) (T ) C p =C p C p +C p and H T V V = r (T ) C p +C p +r r (T ) (T ) (T ) C p =C p C p +C : p (T ) Then, since p, p (T ) and n (T ) (T ) p p p (T ) o are separable, we (T ) OBJ = C(p (T p (T ) (T ) OBJ = C (T p (T ) (p (T ) OBJ = C (T ) (p ) + rj T V V V (T (T ) OBJ (T ) p (T ) (T ) p OBJ (T = rj T V V V = rj T V V V (T ) (T ) V (T ) (T ) V (T ) 5

53 @ (T ) OBJ = C (T ) (p ) + rj T V V V (T (T ) OBJ (T ) p (T ) (T ) p OBJ (T = rj T V V V = rj T V V V (T ) (T ) V (T ) (T ) V (T ) (T ) OBJ = C (T ) (p ) + rj T V V V (T (T ) OBJ (T ) p (T ) (T ) p OBJ (T = rj T V V V = rj T V V V (T ) (T ) V (T ) (T ) V : (T ) n (T ) (T ) (T ) (T ) (T ) Therefore, OBJ is concave in p p p p p (T ) o i 6 (T ) OBJ (T (T ) OBJ (T ) (T (T ) OBJ (T ) (T (T ) OBJ (T ) (T (T ) OBJ (T (T ) OBJ (T ) (T (T ) OBJ (T ) (T (T ) OBJ (T ) (T (T ) OBJ (T p is negative semide nite. The requirement is 53

54 rj T rj T rj T V V V V V V V V V (T ) (T ) (T ) C(p C(p C(p (T ) ) (T ) ) (T ) ) rj T V V C(p rj T V V C(p rj T V V C(p (T ) )C (T ) )C (T ) )C (p (p (p (T ) )= (T ) )= (T ) )= V V V (T ) (T ) (T ) C (T ) (p ) + V C (T ) (p ) + V C (T ) (p ) + V (T ) (T ) (T ) C (p C (p C (p (T ) ) (T ) ) (T ) ) and rj T V V C ) (p(t )C ) (p(t )C ) (p(t ) (T ) C V ) (p(t )C ) (T ) C (p(t )+ V ) (p(t )C ) (T ) C : (p(t )+ V ) (p(t )C ) (p(t ) Then a su cient condition is rj T n V V min C(p (T ) ) C(p (T ) ) C(p (T ) ) o : n (T ) (T ) (T ) (T ) Similarly, Player s objective is concave in p p p p p (T ) o if rh T n V V min C(p (T ) ) C(p (T ) ) C(p (T ) ) o 54

55 which obviously holds since H T V V. Therefore, by Kakutani xed point, there exists a pure strategy Nash Equilibrium at t = T as well. 4. Two period case T = At t =, by the rst order conditions, C C p () p () C C p () p () = r () V () = r () V () : Assume quadratic cost function, i.e., C (p) = c ij p ij=. With quadratic cost functions, rst order conditions lead to p () = p () p () p () = c +c c p () p () p () = c +c c p () p () p () = c +c c p () p () = c +c c p () + r () V () r () V () + r () V () r () V () : The following assumptions ensure that the rst order conditions are valid (the interior solutions). Assumption 3A c r () V () 55

56 If c < r () V (), p () =. This indicates no strictly negative advertisement in the last period. Assumption 3B c r () V () If c < r () V (), p () =. However, p () can still be the interior solution. We have the following proposition as the necessary and su cient condition for no negative advertisement in the last period. Proposition 4.3 With quadratic cost functions, Assumption 3A and 3B, Player will not play strictly negative advertisement in the last period if and only if V () c c p () c () p < or r or V () c c p () + p c () r for where = c c p () 4c (c + c ) rv () c p () + r () V () : Moreover, xed V (), smaller Player s market share is, more negative the advertisement is i Player is not too strong, i.e., V () c p () = ( r) : 56

57 Proof. The condition is just a direct result from the equilibrium solution, V () c V () c p () + r () V () p () r () V () =c = (c + c ) > : To understand the proposition 4.3, The following corollary provides a su cient condition of no negative advertisement. Corollary 4.4 With quadratic cost functions and Assumption 3B, Player will not play strictly negative advertisement in the last period if V () s c c = 4 (c + c ) r () or V () c = r () : Remarks: To have a simple view, let c = c = c = c, r = and () =. Then the su cient conditions above are V () c p =8 or V () c. Proof. From the proof of proposition 4.3, with quadratic cost functions, Assumption 3A 57

58 and 3B the positive advertisement strategy index is d () = p () V () p () V () = V () V () V () c p () + r () V () c p () () r =c V () =c r () V () = (c + c ) c r () V () = (c + c ) : A su cient condition of no strictly negative advertisement is V () () r =c V () c r () V () = (c + c ) : The maximal value of the right hand side is achieved when V () = c = r () and thus the maximal value is c = 4 (c + c ) r (). Therefore, this requires V () c c = 4 (c + c ) r () : 58

59 Another su cient condition of no negative advertisement is V () c r () V () = (c + c ) : This requires V () or V () c = r () : These are contradicted with the feasibility constraint and Assumption 3A if V () > c = However, we know if c < r () V (), p () =, there is no strictly negative advertisement. Therefore, V () c = r () is valid for no strictly negative advertisement. The following proposition consider t =. By the rst order conditions and envelope theorem, we have r (). C C p () p () C C p () p () = r A () = r A () where A () is de ned as (similar in the linear cost functions), A () = () p () () p () () p () + () p () () r : 59