- Email: [email protected]
Pure-strategy Nash equilibria in an advertising game with interference
European Journal of Operational Research 216 (2012) 605–612
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Pure-strategy Nash equilibria in an advertising game with interference Bruno Viscolani ⇑ Department of Pure and Applied Mathematics, University of Padua, Via Trieste 63, I-35121 Padua, Italy
a r t i c l e
i n f o
Article history: Received 7 March 2010 Accepted 2 August 2011 Available online 11 August 2011 Keywords: Game theory Marketing Non-cooperative games Advertising Nash equilibrium Nonsmooth optimization
a b s t r a c t Two manufacturers produce substitutable goods for a homogeneous market. The advertising efforts of the two manufacturers determine the demand for the goods and interfere negatively with each other. The demand of each good is a piecewise linear function of the product goodwill, and the latter is a linear function of advertising efforts. In a game with two competing profit-maximizing manufacturers who have access to a set of several advertising media, the pure-strategy Nash equilibria are characterized and their existence is shown. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Two firms compete for demand from the same group of customers, as they provide a homogeneous market with substitutable products (goods or services). The two manufacturers want to maximize their profits and may advertise their brands using two media, possibly different. We represent the system using a static model, instead of a dynamic one, for the sake of simplicity and because it is particularly appropriate for certain situations, for example where fast depreciation of advertising effects on demand is observed (see e.g. [24]). Moreover, we assume that each manufacturer decides in advance his advertising strategy, ignoring the competitor’s choice, so that we can consider the two manufacturers’ decisions as simultaneous. We investigate the system in the natural framework of non-cooperative game theory under complete information (see [19,20,31]), and assume that Nash equilibria provide credible descriptions of the firms’ behaviors. Therefore we want to discuss whether there exists any Nash equilibrium and to determine it explicitly. Dorfman and Steiner [11] were the first to address marketing decisions in a static and monopolistic framework. The unconventional analysis of advertising proposed by Becker and Murphy [6] and the recent approach to advertising decisions in a segmented market [28] provide further examples of static and monopolistic models. Moreover, static games concerning advertising decisions have been widely proposed in marketing contexts (e.g. [13,16,24,27,29,32,33]) and, in particular, Schoonbeek and Kooreman [24] stress the importance of static modelling of advertis⇑ Tel.: +39 049 827 1397; fax: +39 049 827 1479. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.08.002
ing decision making. The view is also reflected in the mostly static analysis of marketing textbooks (see for example [17]). On the other hand, static modelling may prove particularly useful to understand some basic results when introducing model innovations. In the cited literature, with the exception of [28,29], the authors assume that advertising has a direct effect on the product demand; moreover, when different advertising actions are present, their joint effect on demand is sometimes represented as the sum of single action effects (see [16,24,32]), and sometimes as their product (see [13,27,33]). A distinctive feature of our model, consistent with [28,29], is the assumption that the advertising effect on demand is mediated by the goodwill variable. We believe that this idea, proposed by Nerlove and Arrow [22] in a dynamic context (see [15, Section 3.5]), is useful also while considering a static model, in particular wherever one has to represent the effect on the demand of several and simultaneous advertising actions. As for the joint effect of two simultaneous advertising actions on the product goodwill of a manufacturer, we choose an additive representation, with a positive term for the manufacturer’s advertising effort on his goodwill, and a negative term for the competitor’s one. This choice is in line with the static games discussed in [16,24], and with several dynamic models using the framework of Nerlove–Arrow e.g. [14], where both players advertising efforts have a positive effect on goodwill, and [2,12], where the competitor’s effort has a negative effect. The idea that one player’s advertising effort may hamper the competitor’s sales is intuitively appealing, especially when the firms engage in comparative advertising (see [3,5]). Recently, Danaher et al. [9] investigated empirically the advertising interference phenomenon, documenting its importance and extension, whereas, from a theoretical point of view, Nair and Narasimhan
606
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
[21], Amrouche et al. [1,2] and Viscolani and Zaccour [30] considered dynamic game models where the evolution of each player’s goodwill depends positively on his own advertising and negatively on competing brand’s advertising. On the other hand, the same idea is essential in other differential game models which do not refer to the Nerlove–Arrow goodwill dynamics, as those by Leitmann and Schmitendorf [18] or by Lanchester [15, p. 30]. The latter was originally a combat model, used also to describe the evolution of populations (see [8, p. 19]). Finally, in order to treat also the possibility of observing negative goodwill values, an aspect which has been rather overlooked by the literature, we assume that each product demand is a simple, but non-smooth function of goodwill. As a consequence, the players’ payoff functions are neither differentiable, nor concave and this leads to some technical difficulties in the analysis of optimization problems. Analogous non-smooth optimization features are encountered also in [12,30], in dynamic contexts. The paper is structured as follows. In Section 2 we introduce the model and define the goodwill dependence on both players’ advertising efforts, the demand dependence on goodwill for each player, and the players’ payoff (profit) functions. In Section 3 we analyze the canonical decision rules of the two players, which are used in Section 4 to determine the Nash equilibria of the game. In Section 5 we illustrate the results just obtained, focusing on the case where each manufacturer has only one advertising medium available, without any loss of generality. In Section 6 we summarize the results and propose some future research directions. 2. Manufacturers’ interaction model Let us denote by j a manufacturer (or his product), j 2 {1, 2}. Let Gj represent the stock of goodwill of the product j, and let us admit the possibility of negative goodwill values, as well as positive. We assume that the product demand is a piecewise linear function of its goodwill,
Dj ðGj Þ ¼ b maxf0; Gj g ¼ b½Gj þ ;
ð1Þ
with b > 0. As far as Gj > 0, the demand is a linear function, the most elementary model of dependence on goodwill, which is rather common in the marketing literature (see for example [7, p. 172, footnote 6]). The extension of the demand definition also to negative goodwill values is the same as in [12], and it is a constant price special case of the demand function considered in [10, p. 303]. Each goodwill value Gj is the result of the advertising actions taken by both manufacturers. There is a set I of communication media; manufacturer j has access to the media subset Ij, ; – Ij # I, j = 1, 2, and may use one advertising medium. Let us represent manufacturers j’s action by the pair (ij, aj) 2 Ij [0, +1), in which the first component denotes a medium and the second one advertising effort. We assume that the goodwill levels resulting from the actions (i1, a1) and (i2, a2) are
Gj ¼ cjij aj gh chih ah þ G0j ;
j – h 2 f1; 2g;
ð2Þ
where
ej ¼
cjij aj 1 1 ¼ > ; cjij aj þ gj cjij aj 1 þ gj 2
ð3Þ
which is invariant with respect to the medium ij chosen, and we observe that
ej < 1 () gj > 0:
ð4Þ
As ej > 0, j = 1, 2, we have that a negative change in the competitor’s product goodwill may only occur together with a positive change in own product goodwill. If gj > 0, we have that any advertising message by manufacturer j has a comparative focus, because of (4). We do not rule out the possibility that manufacturer j’s advertising has a noncomparative focus, i.e. that ej = 1 (or gj = 0), but in view of the assumption that g1 + g2 > 0, at least one manufacturer’s advertising has a comparative focus. Manufacturer j incurs the cost cji(aj) when using the advertising medium i with advertising effort aj P 0 and we assume that cji(aj) is an increasing, convex, and continuously differentiable function, where cji ð0Þ ¼ 0; c0ji ð0Þ ¼ 0; c00ji ðaj Þ > 0, and limaj !þ1 c0ji ðaj Þ ¼ þ1. This is a standard assumption, in particular with the specification that the cost is a quadratic function (see, for example, [15, p. 103]). Finally, we assume that production costs are linear and that the transfer and market prices are constant: let pj > 0 be the overall marginal profit of the product j, gross of advertising costs. Therefore, the agents’ profits are
Pj ðij ; aj ; ih ; ah Þ ¼ bpj ½Gj þ cjij ðaj Þ;
j – h 2 f1; 2g:
ð5Þ
3. Canonical decision rules In order to investigate the possible behaviors of the players and, in particular, to discuss the existence and features of Nash equilibria, we need to know the canonical decision rule of player j, j = 1, 2, i.e. the set-valued map Cj : Ih [0, +1) ? Ij [0, +1), which associates each strategy of the competitor (i.e. player h) with the set of profit maximizing strategies of the (first) player (see [4, p. 100] and [19]). The involved optimization problems need special care because the players’ payoff functions are not differentiable, nor concave. Here we preliminarily obtain an upper bound on player j’s profit increment from using any specific medium. Lemma 1. Let j, h 2 {1, 2}, j – h, and ij 2 Ij; for every action (ih, ah) 2 Ih [0, +1) of player h, the profit contribution DPjij ¼ Pj ðij ; aj ; ih ; ah Þ Pj ðij ; 0; ih ; ah Þ manufacturer j can obtain from advertising through the medium ij 2 Ij is upper bounded by
^jij cjij ða ^jij Þ > 0; qjij ¼ bpj cjij a
ð6Þ
^jij > 0 is the unique solution of equation where the advertising effort a
c0jij ðaj Þ ¼ bpj cjij :
ð7Þ
G0j
P 0 is the goodwill level of the product j in the case of no advertising effort by either manufacturer, cjij aj represents the effect of using the advertising medium ij by manufacturer j, with effort level aj, and we assume that cjij > 0, gh 2 [0, 1) is the interference factor of firm h advertising on the brand j’s goodwill making, and we further assume that g1 + g2 > 0.
change cjij aj in own product goodwill, and a negative change gj cjij aj in the competitor’s product goodwill, the latter being 0 if and only if gj = 0. In the terms of Shaffer and Zettelmeyer [25], an advertising message of manufacturer j has emphasis
Proof. From the profit definition (5) and the goodwill Eq. (2) we have that
DPjij ¼ bpj
h
cjij aj gh chih ah þ G0j
iþ
h iþ gh chih ah þ G0j
cjij ðaj Þ: The goodwill production Eq. (2) implies that a nontrivial advertising action (ij, aj), with aj > 0, by manufacturer j induces a positive
If gh chih ah þ G0j P 0, then also cjij aj gh chih ah þ G0j P 0 and
607
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
DPjij ¼ wjij ðaj Þ ¼ bpj cjij aj cjij ðaj Þ;
ð8Þ
which is invariant with respect to (ih, ah). We observe that the function wjij ðaj Þ, just defined, is continuously differentiable and strictly concave in [0, +1), and that w0jij ð0Þ > 0 and limaj !þ1 w0jij ðaj Þ ¼ 1, because of the assumptions on the cost function cjij ðÞ. It follows ^jij > 0 and it satisfies that there exists a unique maximum point a the first order condition w0jij ðaj Þ ¼ 0, which is equivalent to Eq. (7). Then
DPjij 6 max wjij ðaj Þ ¼ wjij ða^jij Þ ¼ qjij : aj
Moreover, wjij ð0Þ ¼ 0 so that qjij > 0. Alternatively, if gh chih ah þ G0j < 0, then
Pðij ; aj ; ih ; ah Þ ¼ wjij ðaj Þ þ bpj G0j : This occurs if and only if it maximizes wjij ðaj Þ, and we observe that
^jij Þ ¼ max qji : max wjij ðajij Þ ¼ max wjij ða j ij ;aj
ij
ij
A manufacturer, whose competitor’s advertising has a noncomparative focus, chooses a medium with maximum profitability index and the associate positive optimal effort. 3.2. Disturbed player’s behavior
h iþ DPjij ¼ bpj cjij aj gh chih ah þ G0j cjij ðaj Þ < qjij :
We consider here the more interesting situation of the canonical decision rule of player j, when his goodwill value Gj is affected by the competitor’s advertising action, which has a comparative focus. Now, if player j knew his opponent’s action, he would take account of such information in deciding his own advertising medium and effort.
In fact, either cjij aj gh chih ah þ G0j 6 0 and
DPjij < qjij ; or cjij aj gh chih ah þ G0j > 0 and
DPjij ¼ wjij ðaj Þ þ bpj gh chih ah þ G0j < wjij ðaj Þ 6 qjij : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Proof. For every (ih, ah) 2 Ih [0, +1), any player j’s strategy (ij, aj) 2 Cj(ih, ah) maximizes the profit
Lemma 3. Let j, h 2 {1, 2}, j – h, and let gh > 0; for all ij 2 Ij, the canonical decision rule of manufacturer j, conditional on his choice of the advertising medium ij, is the set-valued map Cj(jij) : Ih [0, +1) ? [0, +1) such that
<0
The quantity qjij , introduced by Lemma 1, is an index of advertising medium profitability, as it is the maximum profit contribution manufacturer j can obtain from advertising through the medium ij.
8 ^ g; ~ih jij ; fa ah < a > < jij ~ih jij ; ^jij g; ah ¼ a C j ðih ; ah jij Þ ¼ f0; a > : ~ih jij ; f0g; ah > a
^jij > 0 is the advertising effort characterized by Eq. (7) in Lemwhere a ma 1 and the ah-threshold is
3.1. Undisturbed player’s behavior
~ih jij ¼ a We analyse here the canonical decision rule of a player, say j, which is not affected by the other player’s advertising action, i.e. the situation with gh = 0, or eh = 1 in terms of emphasis, when the manufacturer h’s advertising has a noncomparative focus. Of course, in this case, player j decides his own advertising medium and effort ignoring player h’s action.
ð13Þ
1
gh chih
G0j þ
qjij bpj
> 0;
ð14Þ
with qjij > 0 as defined by Eq. (6) in Lemma 1.
Lemma 2. Let j, h 2 {1, 2}, j – h, and let gh = 0; for all ij 2 Ij, the canonical decision rule of manufacturer j, conditional on his choice of the advertising medium ij, is the constant set-valued map Cj(jij) : Ih [0, +1) ? [0, +1) such that
Proof. Let manufacturer j know the medium ih and the advertising effort ah of his competitor; let him have chosen the advertising medium ij; then he wants to choose an advertising effort aj so as to maximize his profit u(aj) = Pj(ij, aj, ih, ah), which is a continuous function on the domain [0, +1), but not concave, nor differentiable everywhere. Precisely, u(aj) is continuously differentiable at all points aj P 0 such that
^jij g; C j ðih ; ah jij Þ ¼ fa
aj – a0j ðah Þ ¼
ð9Þ
^jij > 0 is determined by Eq. (7). where the advertising effort a
we obtain the derivative
(
0
Proof. From the definitions (5), (2), and the assumption that gh = 0, we have that manufacturer j’s profit is
Pðij ; aj ; ih ; ah Þ ¼ bpj cjij aj þ G0j cjij ðaj Þ ¼ wjij ðaj Þ þ bpj G0j ;
ð10Þ
where wjij ðaj Þ is the function defined in (8). From the proof of Lemma 1 we have that the profit has the unique maximum point ^jij > 0. h a Corollary 1. Let j, h 2 {1, 2}, j – h, and let gh = 0; the canonical decision rule of manufacturer j is the constant map
C j ðih ; ah Þ ¼
] ^ji] ; ij ; a
ð11Þ
j
where ]
ij 2 arg max qjij ; ij 2Ij
^ji] is defined as in Lemma 1. and a j
gh chih ah G0j ; cjij
ð12Þ
u ðaj Þ ¼
c0jij ðaj Þ;
aj < a0j ðah Þ;
bpj cjij c0jij ðaj Þ; aj > a0j ðah Þ;
and we observe that u0 (aj) < 0, for all sufficiently large aj, because we have assumed that limaj !þ1 c0jij ðaj Þ ¼ þ1. Hence the function u has at least one maximum point in [0, +1). If such a point is not 0, then it belongs to the open interval ða0j ðah Þ; þ1Þ and satisfies the first order condition u0 (aj) = 0, which is equivalent to Eq. (7): then it is the adver^jij > 0 already introduced by Lemma 1. tising effort a Now, we distinguish two situations with respect to the sign of a0j ðah Þ. If a0j ðah Þ 6 0, i.e. ah 6 G0j =gh chih , then u is continuously differentiable and concave in [0, +1), and u0 (0) > 0, hence u has a ^jij . unique maximum point, which is a If a0j ðah Þ > 0, i.e. ah > G0j =gh chih , then u is decreasing in h i 0; a0j ðah Þ , and it is concave in ½a0j ðah Þ; þ1Þ. In this situation, ^jij a0j ðah Þ, and 0 is the unique maximum point of u either a ^jij Þ < 0), or a ^jij > a0j ðah Þ > 0, then a ^jij and 0 are the (moreover uða local maximum points of u.
608
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
Summarizing the above analysis, we have that the set of global maximum points of u is
f0g;
^jij Þ < 0; if uða
^jij g; fa
(i) ah 6
^jij Þ ¼ 0; if uða
^jij g; f0; a
We can distinguish three possibilities for the profit associated with 0 00 the media ij ; ij 2 Ij , with respect to the advertising effort ah P 0:
^jij Þ > 0: if uða
Corollary 2. Let j, h 2 {1, 2}, j – h; let Cj(ih, ah) be the canonical decision rule of player j; then
^jij g; ðij ; aj Þ 2 C j ðih ; ah Þ ) aj 2 C j ðih ; ah jij Þ # f0; a
N N i1 ; aN1 ; i2 ; aN2
ð15Þ
is a (pure-strategy) Nash equilibrium of the manufacif ^jiN g; j ¼ 1; 2. turers’ advertising game, then aNj 2 f0; a j
Proof. We observe that if (ij, aj) 2 Cj(ih, ah), i.e. if (ij, aj) is a best response by player j to the strategy (ih, ah) chosen by player h, then aj is an optimal advertising effort for player j, while using the medium ij, and given player h’s strategy (ih, ah). Therefore, we have that aj 2 Cj(ih, ahjij). Hence the inclusion in (15) follows from (9) and (13). N Let iN 1 ; i2 be the media chosen by the two players in a Nash N N N is a Nash equilibrium if equilibrium; a bistrategy iN 1 ; a1 ; i2 ; a2 and only if it is a fixed point of the set-valued map N N N iN 1 ; a1 ; i2 ; a2 # C 1 i2 ; a2 C 2 i1 ; a1 (see [4, p. 100]). It follows N N N that for a Nash equilibrium iN 1 ; a1 ; i2 ; a2
N N ij ; aNj 2 C j ih ; aNh ;
j – h 2 f1; 2g;
and, in view of (15), we obtain the second statement.
h
By using the conditional canonical decision rule provided by Lemma 3, we can compare the performances of the different media available to manufacturer j even when the competitor’s advertising has a comparative focus, as we did for the alternative case in the previous subsection. Corollary 3. Let j; h 2 f1; 2g; j – h; gh > 0; i0j ; i00j 2 Ij and let qji0j ; qji00j be the medium profitability index values associated with i0j ; i00j as in (6); if qji0j < qji00j , then, for every (ih, ah) 2 Ih [0, +1), the conditional canonical decision rule of player j, using the medium i00j , gives a higher payoff than the canonical decision rule using the medium i0j .
gh chih
G0j þ
qji0j
bpj
:
Pj ði0j ; a^ji0j ; ih ; ah Þ Pj ði00j ; a^ji00j ; ih ; ah Þ ¼ qji0j qji00j < 0;
uða^jij Þ P 0 () ah 6 a~ih jij ;
As a consequence of Lemmas 2 and 3, the images of the conditional canonical decision rules are finite sets. If manufacturer j is given a medium ij, then his advertising effort must either be 0 or ^jij . Moreover, if a ^jij 2 C j ðih ; ah jij Þ, then a positive goodwill for brand a ^jij . Such equation states j is observed, Gj > 0, and Eq. (7) holds at a ^jij equals the that the marginal advertising cost of medium j at a marginal revenue of advertising for player j.
^ji00 2 C j ih ; ah ji00j , because of (13), ^ji0 2 C j ih ; ah ji0j and a then a j j so that
^jij Þ is a monotonically decreasing function of ah We observe that uða and
~ih jij defined as in Eq. (14), so Eq. (13) is proved. Finally, with a uða^jij Þ > 0 at ah = 0, so that a~ih jij > 0. h
1
(ii)
1
gh chih
G0j þ
qji0j
bpj
< ah <
1
gh chih
G0j þ
qji00j
bpj
:
0 ^ji00 2 C j ih ; ah ji00j , because of (13), so then 0 2 C j ih ; ah jij and a j that
h
Pj i0j ; 0; ih ; ah Pj i00j ; a^ji00j ; ih ; ah ¼ bpj gh chih ah þ G0j
iþ
qji00j bpj gh chih ah þ G0j 6 qji00j < 0; qji00
1 (iii) ah P G0j þ j : gh chih bpj
0 00 then 0 2 C j ih ; ah jij ¼ C j ih ; ah jij , because of (13), so that
Pj i0j ; 0; ih ; ah Pj i00j ; 0; ih ; ah ¼ 0:
The result that the advertising media can be ordered with respect to their performance, stated by Corollaries 1 and 3, is important for the features and variety of equilibria. We notice that it depends crucially on the assumption that the advertising effects combine additively, as represented by Eq. (2). Such a comparability result may not hold for different goodwill production models. By applying the same kind of analysis to the case of two media 0 00 ij ; ij 2 Ij with the same profitability index value, qji0j ¼ qji00j , we ob0 00 serve that ij ; ij have the same performance: we may call them equivalent. In the following we assume that, for j 2 {1, 2}, all manufacturer j’s media ij 2 Ij have different profitability index values qjij , so that there is not any pair of equivalent media. 4. Equilibria N N A bistrategy i1 ; aN1 ; i2 ; aN2 is a Nash equilibrium if and only if it is a fixed point of the set-valued map (i1, a1, i2, a2) ´ i1, a1, i2, a2) ´ C1(i2, a2) C2(i1, a1) (see [4, p. 100]), i.e. if and only if
N N N N i1 ; aN1 ; i2 ; aN2 2 C 1 i2 ; aN2 C 2 i1 ; aN1 :
ð16Þ
Such a bistrategy is also qualified as a pure-strategy Nash equilibrium and has an obvious implementation, as it requires that each manufacturer uses a specific medium with a specific effort. It is a representation of the possible behaviors of rational and intelligent players. As all the strategies (ij, 0), ij 2 Ij, amount to manufacturer j’s unique choice of no advertising with any medium, we treat them as a unique strategy, which we denote by (, 0). Here, we obtain preliminarily an exclusive result, concerning the trivial bistrategy at which neither manufacturer advertises his product. Lemma 4. The bistrategy (, 0, , 0) is not a Nash equilibrium.
Proof. Let ih 2 Ih be a fixed advertising medium of manufacturer h, 0 00 and let ij ; ij 2 Ij be two media of manufacturer j, such that qji0j < qji00j . From (14) we obtain that the following inequality holds for the ah-thresholds associated with the two media,
~i ji0 ¼ a h j
qji0
qji00
1 ~i ji00 : G0j þ j < G0j þ j ¼ a h j gh chih bpj gh chih bpj 1
Proof. Let j 2 {1,2}; for every ij 2 Ij, we have from Lemma 3 that ] ] ^jij g. Then, C j ð; 0Þ ¼ ^ji] ij ; a – fð; 0Þg, where ij is C j ð; 0jij Þ ¼ fa j
the medium index defined in (12). h In the limit situation in which no advertising interference is observed, the optimal choices of a player are independent of the other player’s choice, as seen in Lemma 2. This fact continues to be
609
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
observed as far as the advertising interference is weak enough. From Lemma 3 we obtain the following sufficient condition for the existence of a pure-strategy Nash equilibrium, which entails positive advertising efforts by both manufacturers. ^hih < a ~ ; gh > 0; j – h 2 f1; 2g; ij 2 Ij ; ih 2 Ih , a ih jij ] ^1i] ; i]2 ; a ^1i] , involving the media with then the bistrategy E ¼ i]1 ; a
Theorem
1. If
1
Proof. 1, we know that the bistrategy From Theorem ] ^1i] ; i]2 ; a ^2i] may be a (pure-strategy) Nash equilibrium. E] ¼ i1 ; a 1 2 Now let us assume that E] is not an equilibrium; this may occur if and only if, for either j = 1 or j = 2,
] ^ji] R C j i]h ; a ^hi] : ij ; a j
h
2
]
the maximum profitability index values (12), is the unique pure-
Let this be true for j = 1. From Corollary 3, for every i1 – i1 ,
strategy Nash equilibrium.
P1 i1 ; a^1i1 ; i]2 ; a^2i]
2
6 P1 ðE] Þ;
and hence Proof. Let j – h 2 {1, 2}; for every ij 2 Ij and ih 2 Ih, we have that ^hih jij Þ ¼ fa ^jij g, because of the Theorem hypothesis and Lemma C j ðih ; a 3. The profit associated with the bistrategy resulting by the use of ^jij Þ is Pj ð. . .Þ ¼ qji þ bpj gh chi ah þ G0j , which ðij ; a reaches the j h ] ] ^hih Þ ¼ ^ji] maximum value at ij ¼ ij . Therefore, C j ðih ; a ij ; a . j
A different Nash equilibrium should have an advertising effort component equal to 0, say a2 = 0; but then necessarily ^1i1 Þ, for some i1 2 I1, which contradicts the result ði2 ; 0Þ 2 C 2 ði1 ; a just proved. h
^1i1 Þ R C 1 i]2 ; a ^2i] : ði1 ; a 2
Therefore, we must have that
] ^2i] ; ð; 0Þ 2 C 1 i2 ; a 2
so that the bistrategy equilibrium. h
]
^2i] ; 0; i2 ; a
is a (pure-strategy) Nash
2
Player j knows that the game has at least one equilibrium in ] ^ji] and (, 0) are his possible stratepure strategies and that ij ; a j
If the interference factors g1, g2 are sufficiently small, then the ~ih jij assumption of Theorem 1 is verified, because the threshold a ^hih is invaritends to +1 as gj ? 0, whereas the advertising effort a ant with respect to (g1, g2). Therefore, a pure-strategy Nash equilibrium exists in a market where the advertising interference is weak and each manufacturer, according to it, advertises his product using a medium with maximum profitability index qji] . j A further special situation, with respect to interference, is that of unilateral interference, when one player uses an aggressive advertising policy (his advertising message has a comparative focus), whereas the other player does not: then only the second player would condition his decisions on the possible information on the competitor’s choice. Theorem 2. If gj > 0, gh = 0, j, h 2 {1, 2}, then there exists a pure ^ji] , and strategy Nash equilibrium, where player j’s strategy is i]j ; a j ] ^ player h’s one is either ih ; ahi] or (, 0). h
Proof. The theorem is a straightforward consequence of Corollary 1 and Lemma 3. h If only one manufacturer’s advertising interferes with his competitor’s goodwill evolution, then there exists a pure-strategy Nash equilibrium. Accordingly, the first – aggressive – manufacturer advertises his product, whereas the latter – inoffensive – manufacturer may advertise or not, depending on the profitability of the alternative choices. This may be the situation of a market where an entrant firm – player 1 – advertises by comparing its product to that of an established incumbent – player 2: this is the context considered by [5]. For small values of gj, the equilibrium is the bistrategy E], consistently with the weak interference result. Only for relatively large values of gj, player h chooses not to advertise, i.e. strategy (, 0). We notice that any manufacturer, when advertising his product, uses the medium with maximum profitability index. Finally, we admit that the interference factors g1, g2 have any values in (0, 1), so that we observe the situation of bilateral interference: both manufacturers advertising messages have a comparative focus. Theorem 3. There exists at least one pure-strategy Nash equilibrium; ^ji] ; j ¼ 1; 2. player j’s equilibrium strategy is either (, 0), or i]j ; a j
gies for an equilibrium. In practice, as far as the model assumptions ]
hold, each manufacturer can find a suitable advertising medium ij among those available to him, and determine the possibly positive advertising effort for an equilibrium, irrespectively of the media available to his competitor. Therefore, we may focus simply on the analysis of a reduced game, in which each player may choose between two strategies. Consistently with an observation at the end of Section 3, we notice that this simplification depends on the additive model assumption (2) for the joint effect of the players’ advertising efforts on goodwill and may not extend to more general situations. 5. One medium for each manufacturer In view of the final remark in the previous Section, we find it interesting to explore the different cases one may encounter among the advertising games with one medium for each player. In this context, each manufacturer’s decision concerns simply his advertising effort. Therefore we can simplify the notations by dropping the specifications ij, ih, or changing them into j, h, in a natural way, so that, for instance, the profit of manufacturer j will be represented, using (5) and (2), by
h
Pj ða1 ; a2 Þ ¼ bpj cj aj gh ch ah þ G0j
iþ
cj ðaj Þ;
j – h 2 f1; 2g: ð17Þ
Theorem 3 may be restated as follows. Corollary 4. The manufacturers’ advertising game, in which one medium is available to each manufacturer, jI1j = jI2j = 1, has at least one pure-strategy Nash equilibrium. Such equilibria belong to the set ^1 ; 0Þ; ð0; a ^2 Þ; ða ^1 ; a ^2 Þg, and the advertising effort a ^j > 0 is characfða terized by
^j Þ ¼ bpj cj : c0j ða
ð18Þ
As far as the pure-strategy Nash equilibria are concerned, each manufacturer j is interested in 2 strategies only: either strategy 0, no advertising, ^j , as or strategy 1, advertising and using his medium with effort a characterized by (18). Therefore, the essential features of the game may be represented in the (normal) form of a 2 2 game.
610
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
5.1. Examples of one–one media games
Table 1 Manufacturers’ payoffs – Example 1.
Let us focus on the special case in which the available media have quadratic costs, i.e.
cj ðaÞ ¼
1 2
jj a2 ; a P 0;
pj cj ; jj
ð20Þ
whereas the threshold point (14) is
~hj ¼ a
1
(
gch
G0j
)
pj c2j þb : 2jj
ð21Þ
^j , incurs the Manufacturer j, who advertises with the effort level a cost
^ j Þ ¼ b2 c j ða
c2j p2j : 2jj
a2 = 1
5, 5 10, 3
3, 10 8, 8
ð19Þ
where jj > 0, and assume further that the interference parameters are equal, g1 = g2 = g > 0. Then the advertising messages of both players have a comparative focus, as their Shaffer–Zettelmeyer emphasis is e = 1/(1 + g) < 1. Now, the characteristic positive advertising effort of manufacturer j is
^j ¼ b a
a1 = 0 a1 = 1
a2 = 0
ð22Þ
From Lemma 4 we know that the bistrategy (0, 0) cannot be a Nash equilibrium of the advertising game. On the other hand, each one of the three remaining bistrategies may be a Nash equilibrium, as it is shown by the following examples, built on the assumption of quadratic advertising costs. In particular, the games of Examples 1, 2 and 4 have one equilibrium each, the game of Example 3 has two equilibria. Moreover, in Examples 1–3 games with the special feature that only one (identical) advertising medium is available to both manufacturers are considered. Example 1. Let us consider the symmetric game with quadratic costs, where
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:1; j 2 f1; 2g; b = 1 and g = 0.2. The Shaffer–Zettelmeyer emphasis is e ffi 0.83: the advertising messages of both players have a weakly comparative focus. The positive values of the canonical decision rules are ^1 ¼ a ^2 ¼ 10, whereas the advertising effort thresholds are a ~12 ¼ a ~21 ¼ 50. Therefore the bistrategy (10, 10) is the unique a pure-strategy Nash equilibrium, as it is the unique fixed point of the set-valued map (a1, a2) ´ C1(a2) C2(a1). This is a rather interesting situation, as both manufacturers find it profitable to advertise in equilibrium. If we consider the alternative representation of the game as a 2 2 game, its payoff matrix is given by Table 1. Here we observe that a1 = 1 and a2 = 1 are dominant strategies for the two players and that the bistrategy (a1, a2) = (1, 1) is a Nash equilibrium: it corresponds to (a1, a2) = (10, 10) found above. The following example is a variant of Example 1, in which the interference factor and the cost parameters are larger; it provides an example of the prisoner’s dilemma. Example 2. Let us consider the symmetric game with quadratic costs, where
the previous example. The positive values of the canonical decision ^1 ¼ a ^2 ¼ 4, whereas the advertising effort thresholds are rules are a ~12 ¼ a ~21 ffi 13:9. Therefore the bistrategy (4,4) is the unique purea strategy Nash equilibrium. This is an undesirable situation for the players, who find it rational to advertise in equilibrium, but nevertheless see a better way out. In fact, a market regulation to constrain the interference factor g below the threshold 0.5 (the messages emphasis above the threshold 0.66) would avoid this inefficient situation. In the alternative representation, the payoff matrix of the 2 2 game is given by Table 2. As in Example 1, we observe that a1 = 1 and a2 = 1 are dominant strategies for the two players and that the bistrategy (a1, a2) = (1, 1), corresponding to (a1, a2) = (4, 4), is the unique pure-strategy Nash equilibrium. Nevertheless, at the bistrategy (a1, a2) = (0, 0) both manufacturers would be better off than at the Nash equilibrium. By reducing the values of the cost parameters to jj = 0.019, we obtain an example of crossroads game: it exhibits two pure-strategy Nash equilibria in which the manufacturers have symmetric roles. Example 3. Let us consider the symmetric game with quadratic costs, where
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:019; j 2 f1; 2g; b = 1 and g = 0.6; then the positive values of the canonical decision ^1 ¼ a ^2 ffi 52:63, whereas the advertising effort thresholds rules are a ~12 ¼ a ~21 ffi 52:19. Therefore the two symmetric bistrategies are a (52.63, 0) and (0, 52.63) are the pure-strategy Nash equilibria. Again, this is an undesirable situation for the players, who do not have a unique strategy to choose for an equilibrium and cannot be sure to obtain an equilibrium. As in Example 2, the difficulty comes from the high value of the interference factor g (low emphasis value, e = 0.63). In the alternative representation, the payoff matrix of the 2 2 game is given by Table 3. Here we observe that no player has any dominant strategy. The bistrategies (a1, a2) = (0, 1) and (a1, a2) = (1, 0), corresponding to (a1, a2) = (0, 52.63) and (a1, a2) = (52.63, 0), are Nash equilibria. The following is an example where the advertising media have different cost parameters, so that one manufacturer is stronger than the other. Example 4. Let us consider the asymmetric game with quadratic costs, where
b ¼ 1;
and the manufacturers differ in the advertising cost parameters,
j1 ¼ 0:1; j2 ¼ 0:5: Table 2 Manufacturers’ payoffs – Example 2.
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:15; j 2 f1; 2g; b = 1 and g = 0.6. The emphasis is e ffi 0.63: the advertising messages of both players have a stronger comparative focus than in
g ¼ 0:7; cj ¼ 1; G0j ¼ 5; pj ¼ 1; j 2 f1; 2g;
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 8.33, 1
1, 8.33 4.33, 4.33
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612 Table 3 Manufacturers’ payoffs – Example 3.
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 31.32, 0
0, 31.32 0.26, 0.26
Table 4 Manufacturers’ payoffs – Example 4.
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 10, 0
3.6, 6 8.6, 1
611
models are already analyzed by some papers, e.g. [7,23]. The main drawback of the additive goodwill production model is that it cannot represent any positive/negative enhancement effect of combining several advertising messages, as discussed in [26] from the positive (synergy) viewpoint. We have assumed here that each manufacturer has a characteristic interference parameter gj and hence a characteristic advertising message emphasis. Now, it is rather true that the advertising message emphasis is a decision variable of the manufacturer (see [25]). This observation suggests that a different and relevant definition of advertising action of a manufacturer in a competitive context should have three dimensions: medium, effort and interference parameter (or emphasis). Acknowledgements
The messages emphasis is e ffi 0.59: the advertising messages of both players have a strong comparative focus. The positive values ^1 ¼ 10; a ^2 ¼ 2, whereas the of the canonical decision rules are a ~12 ffi 8:6; a ~21 ffi 14:3. Therefore advertising effort thresholds are a the bistrategy (10, 0) is the unique pure-strategy Nash equilibrium. This is a problematic situation, as only one manufacturer, i.e. the one with the cheaper medium, finds it profitable to advertise in equilibrium, whereas the competitor does not. In the alternative representation, the payoff matrix of the 2 2 game is given by Table 4. The advertising medium available to manufacturer 1 is more efficient than manufacturer 2’s one. Here, a1 = 1 is a dominant strategy for player 1, whereas player 2 has not any dominant strategy; the bistrategy (a1, a2) = (1, 0) is the unique pure-strategy Nash equilibrium of the game and it corresponds to (a1, a2) = (10, 0) found above. Of course, the bistrategy (a1, a2) = (0, 1) is the unique Nash equilibrium of the 2 2 game if the second manufacturer has lower advertising costs than the first one, with all other features being equal.
6. Conclusion and directions for future research This paper proposes a model for two competing manufacturers of substitutable goods in a homogeneous market. It considers a static decision framework where each player can choose his advertising effort, which has a double effect: positive on the player’s sales and negative on the competitor’s sales. The effect of the players’ advertising efforts on the two product sales is mediated by the goodwill variables of the products, as in the dynamic model by Nerlove and Arrow. The use of the intermediate goodwill variables is an original contribution to the literature on static game models of advertising competition. Interference among advertising efforts makes it possible to observe negative goodwill values, a circumstance which entails some analytical difficulties. From a mathematical point of view, a two-player continuous game with non-smooth payoff functions is analyzed and is shown to be equivalent to a suitable 2 2 game. We obtain a complete description of feasible Nash equilibria in pure strategies, while accounting for possible negative goodwill. The treatment of the situation where one (or each) player can choose among different advertising media has no analogous counterpart in the dynamic models literature. The solution to the model shows that in equilibrium only one advertising medium is used by each manufacturer. However, such result seems to be strictly dependent on the assumption that the joint effect of the players’ advertising efforts on goodwill is additive. Therefore, an interesting direction for future research would be to explore different and more general models for this joint effect. Then, the obtained result that only one advertising medium is used in equilibrium is not expected to hold. Such non-linear
The paper is dedicated to the memory of Giorgio Pederzoli, a great friend. Financial support from the Italian Ministry of University and Research and the University of Padua is gratefully acknowledged. The author is indebted to some colleagues and to three referees for proper criticism and relevant suggestions. References [1] N. Amrouche, G. Martı´n-Herrán, G. Zaccour, Feedback Stackelberg equilibrium strategies when the private label competes with the national brand, Annals of Operations Research 164 (2008) 79–95. [2] N. Amrouche, G. Martı´n-Herrán, G. Zaccour, Pricing and advertising of private and national brands in a dynamic marketing channel, Journal of Optimization Theory and Applications 137 (3) (2008) 465–483. [3] S.P. Anderson, R. Renault, Comparative advertising: disclosing horizontal match information, RAND Journal of Economics 40 (3) (2009) 558–581. [4] J.-P. Aubin, Optima and Equilibria, Springer, Berlin, 1993. [5] F. Barigozzi, P.G. Garella, M. Peitz, With a little help from my enemy: Comparative advertising as a signal of quality, Journal of Economics and Management Strategy 18 (4) (2009) 1071–1094. [6] G.S. Becker, K.M. Murphy, A simple theory of advertising as a good or bad, Quarterly Journal of Economics 108 (1993) 941–964. [7] P.K. Chintagunta, D. Jain, A dynamic model of channel member strategies for marketing expenditures, Marketing Science 11 (1992) 168–188. [8] B.W. Conolly, Techniques in Operational Research, Models, Search and Randomization, vol. 2, Ellis Horwood, Chichester, 1981. [9] P.J. Danaher, A. Bonfrer, S. Dhar, The effect of competitive advertising interference on sales for packaged goods, Journal of Marketing Research XLV (2008) 211–225. [10] E. Dockner, S. Jørgensen, N. Van Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge Univ. Press, Cambridge, 2000. [11] R. Dorfman, P.O. Steiner, Optimal advertising and optimal quality, American Economic Review XLIV (1954) 826–836. [12] L. Grosset, B. Viscolani, Optimal dynamic advertising with an adverse exogenous effect on brand goodwill, Automatica 45 (2009) 863–870. [13] Z. Huang, S.X. Li, V. Mahajan, An analysis of manufacturer–retailer supply chain coordination in cooperative advertising, Decision Sciences 33 (2002) 469–494. [14] S. Jørgensen, G. Zaccour, Equilibrium pricing and advertising strategies in a marketing channel, Journal of Optimization Theory and Applications 102 (1) (1999) 111–125. [15] S. Jørgensen, G. Zaccour, Differential Games in Marketing, Kluwer Academic Publishers, Boston, 2004. [16] S. Karray, G. Zaccour, Effectiveness of coop advertising programs in competitive distribution channels, International Game Theory Review 9 (2) (2007) 151–167. [17] P. Kotler, Marketing Management, 11th ed., Prentice-Hall, Upper Saddle River, 2003. [18] G. Leitmann, W.E. Schmitendorf, Profit maximization through advertising: a nonzero sum differential game approach, IEEE Transactions on Automatic Control 23 (4) (1978) 645–650. [19] M. Leng, M. Parlar, Game theoretic applications in supply chain management: A review, INFOR 43 (3) (2005) 187–220. [20] K.S. Moorthy, Using Game Theory to model competition, Journal of Marketing Research 22 (1985) 262–282. [21] A. Nair, R. Narasimhan, Dynamics of competing with quality- and advertisingbased goodwill, European Journal of Operational Research 175 (2006) 462– 474. [22] M. Nerlove, K.J. Arrow, Optimal advertising policy under dynamic conditions, Economica 29 (2) (1962) 129–142. [23] A. Prasad, S.P. Sethi, Integrated marketing communications in markets with uncertainty and competition, Automatica 45 (3) (2009) 601–610.
612
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
[24] L. Schoonbeek, P. Kooreman, The impact of advertising in a duopoly game, International Game Theory Review 9 (4) (2007) 565–581. [25] G. Shaffer, F. Zettelmeyer, Comparative advertising and in-store displays, Marketing Science 28 (6) (2009) 1144–1156. [26] A. Sorato, B. Viscolani, Using several advertising media in a homogeneous market, Optimization Letters, doi:10.1007/s11590-010-0220-z. [27] J.G. Szmerekovsky, J. Zhang, Pricing and two-tier advertising with one manufacturer and one retailer, European Journal of Operational Research 192 (2009) 904–917. [28] B. Viscolani, Advertising decisions for a segmented market, Optimization 58 (4) (2009) 469–477. [29] B. Viscolani, Advertising decisions in a vertical distribution channel, International Game Theory Review 11 (3) (2009) 273–284.
[30] B. Viscolani, G. Zaccour, Advertising strategies in a differential game with negative competitor’s interference, Journal of Optimization Theory and Applications 140 (1) (2009) 153–170. [31] Q. Wang, M. Parlar, Static game theory models and their applications in management science, European Journal of Operational Research 42 (1) (1989) 1–21. [32] J. Xie, J.C. Wei, Coordinating advertising and pricing in a manufacturer–retailer channel, European Journal of Operational Research 197 (2009) 785–791. [33] J. Yue, J. Austin, M.-C. Wang, Z. Huang, Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount, European Journal of Operational Research 168 (2006) 65–85.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Stochastics and Statistics
Pure-strategy Nash equilibria in an advertising game with interference Bruno Viscolani ⇑ Department of Pure and Applied Mathematics, University of Padua, Via Trieste 63, I-35121 Padua, Italy
a r t i c l e
i n f o
Article history: Received 7 March 2010 Accepted 2 August 2011 Available online 11 August 2011 Keywords: Game theory Marketing Non-cooperative games Advertising Nash equilibrium Nonsmooth optimization
a b s t r a c t Two manufacturers produce substitutable goods for a homogeneous market. The advertising efforts of the two manufacturers determine the demand for the goods and interfere negatively with each other. The demand of each good is a piecewise linear function of the product goodwill, and the latter is a linear function of advertising efforts. In a game with two competing profit-maximizing manufacturers who have access to a set of several advertising media, the pure-strategy Nash equilibria are characterized and their existence is shown. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Two firms compete for demand from the same group of customers, as they provide a homogeneous market with substitutable products (goods or services). The two manufacturers want to maximize their profits and may advertise their brands using two media, possibly different. We represent the system using a static model, instead of a dynamic one, for the sake of simplicity and because it is particularly appropriate for certain situations, for example where fast depreciation of advertising effects on demand is observed (see e.g. [24]). Moreover, we assume that each manufacturer decides in advance his advertising strategy, ignoring the competitor’s choice, so that we can consider the two manufacturers’ decisions as simultaneous. We investigate the system in the natural framework of non-cooperative game theory under complete information (see [19,20,31]), and assume that Nash equilibria provide credible descriptions of the firms’ behaviors. Therefore we want to discuss whether there exists any Nash equilibrium and to determine it explicitly. Dorfman and Steiner [11] were the first to address marketing decisions in a static and monopolistic framework. The unconventional analysis of advertising proposed by Becker and Murphy [6] and the recent approach to advertising decisions in a segmented market [28] provide further examples of static and monopolistic models. Moreover, static games concerning advertising decisions have been widely proposed in marketing contexts (e.g. [13,16,24,27,29,32,33]) and, in particular, Schoonbeek and Kooreman [24] stress the importance of static modelling of advertis⇑ Tel.: +39 049 827 1397; fax: +39 049 827 1479. E-mail address: [email protected] 0377-2217/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2011.08.002
ing decision making. The view is also reflected in the mostly static analysis of marketing textbooks (see for example [17]). On the other hand, static modelling may prove particularly useful to understand some basic results when introducing model innovations. In the cited literature, with the exception of [28,29], the authors assume that advertising has a direct effect on the product demand; moreover, when different advertising actions are present, their joint effect on demand is sometimes represented as the sum of single action effects (see [16,24,32]), and sometimes as their product (see [13,27,33]). A distinctive feature of our model, consistent with [28,29], is the assumption that the advertising effect on demand is mediated by the goodwill variable. We believe that this idea, proposed by Nerlove and Arrow [22] in a dynamic context (see [15, Section 3.5]), is useful also while considering a static model, in particular wherever one has to represent the effect on the demand of several and simultaneous advertising actions. As for the joint effect of two simultaneous advertising actions on the product goodwill of a manufacturer, we choose an additive representation, with a positive term for the manufacturer’s advertising effort on his goodwill, and a negative term for the competitor’s one. This choice is in line with the static games discussed in [16,24], and with several dynamic models using the framework of Nerlove–Arrow e.g. [14], where both players advertising efforts have a positive effect on goodwill, and [2,12], where the competitor’s effort has a negative effect. The idea that one player’s advertising effort may hamper the competitor’s sales is intuitively appealing, especially when the firms engage in comparative advertising (see [3,5]). Recently, Danaher et al. [9] investigated empirically the advertising interference phenomenon, documenting its importance and extension, whereas, from a theoretical point of view, Nair and Narasimhan
606
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
[21], Amrouche et al. [1,2] and Viscolani and Zaccour [30] considered dynamic game models where the evolution of each player’s goodwill depends positively on his own advertising and negatively on competing brand’s advertising. On the other hand, the same idea is essential in other differential game models which do not refer to the Nerlove–Arrow goodwill dynamics, as those by Leitmann and Schmitendorf [18] or by Lanchester [15, p. 30]. The latter was originally a combat model, used also to describe the evolution of populations (see [8, p. 19]). Finally, in order to treat also the possibility of observing negative goodwill values, an aspect which has been rather overlooked by the literature, we assume that each product demand is a simple, but non-smooth function of goodwill. As a consequence, the players’ payoff functions are neither differentiable, nor concave and this leads to some technical difficulties in the analysis of optimization problems. Analogous non-smooth optimization features are encountered also in [12,30], in dynamic contexts. The paper is structured as follows. In Section 2 we introduce the model and define the goodwill dependence on both players’ advertising efforts, the demand dependence on goodwill for each player, and the players’ payoff (profit) functions. In Section 3 we analyze the canonical decision rules of the two players, which are used in Section 4 to determine the Nash equilibria of the game. In Section 5 we illustrate the results just obtained, focusing on the case where each manufacturer has only one advertising medium available, without any loss of generality. In Section 6 we summarize the results and propose some future research directions. 2. Manufacturers’ interaction model Let us denote by j a manufacturer (or his product), j 2 {1, 2}. Let Gj represent the stock of goodwill of the product j, and let us admit the possibility of negative goodwill values, as well as positive. We assume that the product demand is a piecewise linear function of its goodwill,
Dj ðGj Þ ¼ b maxf0; Gj g ¼ b½Gj þ ;
ð1Þ
with b > 0. As far as Gj > 0, the demand is a linear function, the most elementary model of dependence on goodwill, which is rather common in the marketing literature (see for example [7, p. 172, footnote 6]). The extension of the demand definition also to negative goodwill values is the same as in [12], and it is a constant price special case of the demand function considered in [10, p. 303]. Each goodwill value Gj is the result of the advertising actions taken by both manufacturers. There is a set I of communication media; manufacturer j has access to the media subset Ij, ; – Ij # I, j = 1, 2, and may use one advertising medium. Let us represent manufacturers j’s action by the pair (ij, aj) 2 Ij [0, +1), in which the first component denotes a medium and the second one advertising effort. We assume that the goodwill levels resulting from the actions (i1, a1) and (i2, a2) are
Gj ¼ cjij aj gh chih ah þ G0j ;
j – h 2 f1; 2g;
ð2Þ
where
ej ¼
cjij aj 1 1 ¼ > ; cjij aj þ gj cjij aj 1 þ gj 2
ð3Þ
which is invariant with respect to the medium ij chosen, and we observe that
ej < 1 () gj > 0:
ð4Þ
As ej > 0, j = 1, 2, we have that a negative change in the competitor’s product goodwill may only occur together with a positive change in own product goodwill. If gj > 0, we have that any advertising message by manufacturer j has a comparative focus, because of (4). We do not rule out the possibility that manufacturer j’s advertising has a noncomparative focus, i.e. that ej = 1 (or gj = 0), but in view of the assumption that g1 + g2 > 0, at least one manufacturer’s advertising has a comparative focus. Manufacturer j incurs the cost cji(aj) when using the advertising medium i with advertising effort aj P 0 and we assume that cji(aj) is an increasing, convex, and continuously differentiable function, where cji ð0Þ ¼ 0; c0ji ð0Þ ¼ 0; c00ji ðaj Þ > 0, and limaj !þ1 c0ji ðaj Þ ¼ þ1. This is a standard assumption, in particular with the specification that the cost is a quadratic function (see, for example, [15, p. 103]). Finally, we assume that production costs are linear and that the transfer and market prices are constant: let pj > 0 be the overall marginal profit of the product j, gross of advertising costs. Therefore, the agents’ profits are
Pj ðij ; aj ; ih ; ah Þ ¼ bpj ½Gj þ cjij ðaj Þ;
j – h 2 f1; 2g:
ð5Þ
3. Canonical decision rules In order to investigate the possible behaviors of the players and, in particular, to discuss the existence and features of Nash equilibria, we need to know the canonical decision rule of player j, j = 1, 2, i.e. the set-valued map Cj : Ih [0, +1) ? Ij [0, +1), which associates each strategy of the competitor (i.e. player h) with the set of profit maximizing strategies of the (first) player (see [4, p. 100] and [19]). The involved optimization problems need special care because the players’ payoff functions are not differentiable, nor concave. Here we preliminarily obtain an upper bound on player j’s profit increment from using any specific medium. Lemma 1. Let j, h 2 {1, 2}, j – h, and ij 2 Ij; for every action (ih, ah) 2 Ih [0, +1) of player h, the profit contribution DPjij ¼ Pj ðij ; aj ; ih ; ah Þ Pj ðij ; 0; ih ; ah Þ manufacturer j can obtain from advertising through the medium ij 2 Ij is upper bounded by
^jij cjij ða ^jij Þ > 0; qjij ¼ bpj cjij a
ð6Þ
^jij > 0 is the unique solution of equation where the advertising effort a
c0jij ðaj Þ ¼ bpj cjij :
ð7Þ
G0j
P 0 is the goodwill level of the product j in the case of no advertising effort by either manufacturer, cjij aj represents the effect of using the advertising medium ij by manufacturer j, with effort level aj, and we assume that cjij > 0, gh 2 [0, 1) is the interference factor of firm h advertising on the brand j’s goodwill making, and we further assume that g1 + g2 > 0.
change cjij aj in own product goodwill, and a negative change gj cjij aj in the competitor’s product goodwill, the latter being 0 if and only if gj = 0. In the terms of Shaffer and Zettelmeyer [25], an advertising message of manufacturer j has emphasis
Proof. From the profit definition (5) and the goodwill Eq. (2) we have that
DPjij ¼ bpj
h
cjij aj gh chih ah þ G0j
iþ
h iþ gh chih ah þ G0j
cjij ðaj Þ: The goodwill production Eq. (2) implies that a nontrivial advertising action (ij, aj), with aj > 0, by manufacturer j induces a positive
If gh chih ah þ G0j P 0, then also cjij aj gh chih ah þ G0j P 0 and
607
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
DPjij ¼ wjij ðaj Þ ¼ bpj cjij aj cjij ðaj Þ;
ð8Þ
which is invariant with respect to (ih, ah). We observe that the function wjij ðaj Þ, just defined, is continuously differentiable and strictly concave in [0, +1), and that w0jij ð0Þ > 0 and limaj !þ1 w0jij ðaj Þ ¼ 1, because of the assumptions on the cost function cjij ðÞ. It follows ^jij > 0 and it satisfies that there exists a unique maximum point a the first order condition w0jij ðaj Þ ¼ 0, which is equivalent to Eq. (7). Then
DPjij 6 max wjij ðaj Þ ¼ wjij ða^jij Þ ¼ qjij : aj
Moreover, wjij ð0Þ ¼ 0 so that qjij > 0. Alternatively, if gh chih ah þ G0j < 0, then
Pðij ; aj ; ih ; ah Þ ¼ wjij ðaj Þ þ bpj G0j : This occurs if and only if it maximizes wjij ðaj Þ, and we observe that
^jij Þ ¼ max qji : max wjij ðajij Þ ¼ max wjij ða j ij ;aj
ij
ij
A manufacturer, whose competitor’s advertising has a noncomparative focus, chooses a medium with maximum profitability index and the associate positive optimal effort. 3.2. Disturbed player’s behavior
h iþ DPjij ¼ bpj cjij aj gh chih ah þ G0j cjij ðaj Þ < qjij :
We consider here the more interesting situation of the canonical decision rule of player j, when his goodwill value Gj is affected by the competitor’s advertising action, which has a comparative focus. Now, if player j knew his opponent’s action, he would take account of such information in deciding his own advertising medium and effort.
In fact, either cjij aj gh chih ah þ G0j 6 0 and
DPjij < qjij ; or cjij aj gh chih ah þ G0j > 0 and
DPjij ¼ wjij ðaj Þ þ bpj gh chih ah þ G0j < wjij ðaj Þ 6 qjij : |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
Proof. For every (ih, ah) 2 Ih [0, +1), any player j’s strategy (ij, aj) 2 Cj(ih, ah) maximizes the profit
Lemma 3. Let j, h 2 {1, 2}, j – h, and let gh > 0; for all ij 2 Ij, the canonical decision rule of manufacturer j, conditional on his choice of the advertising medium ij, is the set-valued map Cj(jij) : Ih [0, +1) ? [0, +1) such that
<0
The quantity qjij , introduced by Lemma 1, is an index of advertising medium profitability, as it is the maximum profit contribution manufacturer j can obtain from advertising through the medium ij.
8 ^ g; ~ih jij ; fa ah < a > < jij ~ih jij ; ^jij g; ah ¼ a C j ðih ; ah jij Þ ¼ f0; a > : ~ih jij ; f0g; ah > a
^jij > 0 is the advertising effort characterized by Eq. (7) in Lemwhere a ma 1 and the ah-threshold is
3.1. Undisturbed player’s behavior
~ih jij ¼ a We analyse here the canonical decision rule of a player, say j, which is not affected by the other player’s advertising action, i.e. the situation with gh = 0, or eh = 1 in terms of emphasis, when the manufacturer h’s advertising has a noncomparative focus. Of course, in this case, player j decides his own advertising medium and effort ignoring player h’s action.
ð13Þ
1
gh chih
G0j þ
qjij bpj
> 0;
ð14Þ
with qjij > 0 as defined by Eq. (6) in Lemma 1.
Lemma 2. Let j, h 2 {1, 2}, j – h, and let gh = 0; for all ij 2 Ij, the canonical decision rule of manufacturer j, conditional on his choice of the advertising medium ij, is the constant set-valued map Cj(jij) : Ih [0, +1) ? [0, +1) such that
Proof. Let manufacturer j know the medium ih and the advertising effort ah of his competitor; let him have chosen the advertising medium ij; then he wants to choose an advertising effort aj so as to maximize his profit u(aj) = Pj(ij, aj, ih, ah), which is a continuous function on the domain [0, +1), but not concave, nor differentiable everywhere. Precisely, u(aj) is continuously differentiable at all points aj P 0 such that
^jij g; C j ðih ; ah jij Þ ¼ fa
aj – a0j ðah Þ ¼
ð9Þ
^jij > 0 is determined by Eq. (7). where the advertising effort a
we obtain the derivative
(
0
Proof. From the definitions (5), (2), and the assumption that gh = 0, we have that manufacturer j’s profit is
Pðij ; aj ; ih ; ah Þ ¼ bpj cjij aj þ G0j cjij ðaj Þ ¼ wjij ðaj Þ þ bpj G0j ;
ð10Þ
where wjij ðaj Þ is the function defined in (8). From the proof of Lemma 1 we have that the profit has the unique maximum point ^jij > 0. h a Corollary 1. Let j, h 2 {1, 2}, j – h, and let gh = 0; the canonical decision rule of manufacturer j is the constant map
C j ðih ; ah Þ ¼
] ^ji] ; ij ; a
ð11Þ
j
where ]
ij 2 arg max qjij ; ij 2Ij
^ji] is defined as in Lemma 1. and a j
gh chih ah G0j ; cjij
ð12Þ
u ðaj Þ ¼
c0jij ðaj Þ;
aj < a0j ðah Þ;
bpj cjij c0jij ðaj Þ; aj > a0j ðah Þ;
and we observe that u0 (aj) < 0, for all sufficiently large aj, because we have assumed that limaj !þ1 c0jij ðaj Þ ¼ þ1. Hence the function u has at least one maximum point in [0, +1). If such a point is not 0, then it belongs to the open interval ða0j ðah Þ; þ1Þ and satisfies the first order condition u0 (aj) = 0, which is equivalent to Eq. (7): then it is the adver^jij > 0 already introduced by Lemma 1. tising effort a Now, we distinguish two situations with respect to the sign of a0j ðah Þ. If a0j ðah Þ 6 0, i.e. ah 6 G0j =gh chih , then u is continuously differentiable and concave in [0, +1), and u0 (0) > 0, hence u has a ^jij . unique maximum point, which is a If a0j ðah Þ > 0, i.e. ah > G0j =gh chih , then u is decreasing in h i 0; a0j ðah Þ , and it is concave in ½a0j ðah Þ; þ1Þ. In this situation, ^jij a0j ðah Þ, and 0 is the unique maximum point of u either a ^jij Þ < 0), or a ^jij > a0j ðah Þ > 0, then a ^jij and 0 are the (moreover uða local maximum points of u.
608
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
Summarizing the above analysis, we have that the set of global maximum points of u is
f0g;
^jij Þ < 0; if uða
^jij g; fa
(i) ah 6
^jij Þ ¼ 0; if uða
^jij g; f0; a
We can distinguish three possibilities for the profit associated with 0 00 the media ij ; ij 2 Ij , with respect to the advertising effort ah P 0:
^jij Þ > 0: if uða
Corollary 2. Let j, h 2 {1, 2}, j – h; let Cj(ih, ah) be the canonical decision rule of player j; then
^jij g; ðij ; aj Þ 2 C j ðih ; ah Þ ) aj 2 C j ðih ; ah jij Þ # f0; a
N N i1 ; aN1 ; i2 ; aN2
ð15Þ
is a (pure-strategy) Nash equilibrium of the manufacif ^jiN g; j ¼ 1; 2. turers’ advertising game, then aNj 2 f0; a j
Proof. We observe that if (ij, aj) 2 Cj(ih, ah), i.e. if (ij, aj) is a best response by player j to the strategy (ih, ah) chosen by player h, then aj is an optimal advertising effort for player j, while using the medium ij, and given player h’s strategy (ih, ah). Therefore, we have that aj 2 Cj(ih, ahjij). Hence the inclusion in (15) follows from (9) and (13). N Let iN 1 ; i2 be the media chosen by the two players in a Nash N N N is a Nash equilibrium if equilibrium; a bistrategy iN 1 ; a1 ; i2 ; a2 and only if it is a fixed point of the set-valued map N N N iN 1 ; a1 ; i2 ; a2 # C 1 i2 ; a2 C 2 i1 ; a1 (see [4, p. 100]). It follows N N N that for a Nash equilibrium iN 1 ; a1 ; i2 ; a2
N N ij ; aNj 2 C j ih ; aNh ;
j – h 2 f1; 2g;
and, in view of (15), we obtain the second statement.
h
By using the conditional canonical decision rule provided by Lemma 3, we can compare the performances of the different media available to manufacturer j even when the competitor’s advertising has a comparative focus, as we did for the alternative case in the previous subsection. Corollary 3. Let j; h 2 f1; 2g; j – h; gh > 0; i0j ; i00j 2 Ij and let qji0j ; qji00j be the medium profitability index values associated with i0j ; i00j as in (6); if qji0j < qji00j , then, for every (ih, ah) 2 Ih [0, +1), the conditional canonical decision rule of player j, using the medium i00j , gives a higher payoff than the canonical decision rule using the medium i0j .
gh chih
G0j þ
qji0j
bpj
:
Pj ði0j ; a^ji0j ; ih ; ah Þ Pj ði00j ; a^ji00j ; ih ; ah Þ ¼ qji0j qji00j < 0;
uða^jij Þ P 0 () ah 6 a~ih jij ;
As a consequence of Lemmas 2 and 3, the images of the conditional canonical decision rules are finite sets. If manufacturer j is given a medium ij, then his advertising effort must either be 0 or ^jij . Moreover, if a ^jij 2 C j ðih ; ah jij Þ, then a positive goodwill for brand a ^jij . Such equation states j is observed, Gj > 0, and Eq. (7) holds at a ^jij equals the that the marginal advertising cost of medium j at a marginal revenue of advertising for player j.
^ji00 2 C j ih ; ah ji00j , because of (13), ^ji0 2 C j ih ; ah ji0j and a then a j j so that
^jij Þ is a monotonically decreasing function of ah We observe that uða and
~ih jij defined as in Eq. (14), so Eq. (13) is proved. Finally, with a uða^jij Þ > 0 at ah = 0, so that a~ih jij > 0. h
1
(ii)
1
gh chih
G0j þ
qji0j
bpj
< ah <
1
gh chih
G0j þ
qji00j
bpj
:
0 ^ji00 2 C j ih ; ah ji00j , because of (13), so then 0 2 C j ih ; ah jij and a j that
h
Pj i0j ; 0; ih ; ah Pj i00j ; a^ji00j ; ih ; ah ¼ bpj gh chih ah þ G0j
iþ
qji00j bpj gh chih ah þ G0j 6 qji00j < 0; qji00
1 (iii) ah P G0j þ j : gh chih bpj
0 00 then 0 2 C j ih ; ah jij ¼ C j ih ; ah jij , because of (13), so that
Pj i0j ; 0; ih ; ah Pj i00j ; 0; ih ; ah ¼ 0:
The result that the advertising media can be ordered with respect to their performance, stated by Corollaries 1 and 3, is important for the features and variety of equilibria. We notice that it depends crucially on the assumption that the advertising effects combine additively, as represented by Eq. (2). Such a comparability result may not hold for different goodwill production models. By applying the same kind of analysis to the case of two media 0 00 ij ; ij 2 Ij with the same profitability index value, qji0j ¼ qji00j , we ob0 00 serve that ij ; ij have the same performance: we may call them equivalent. In the following we assume that, for j 2 {1, 2}, all manufacturer j’s media ij 2 Ij have different profitability index values qjij , so that there is not any pair of equivalent media. 4. Equilibria N N A bistrategy i1 ; aN1 ; i2 ; aN2 is a Nash equilibrium if and only if it is a fixed point of the set-valued map (i1, a1, i2, a2) ´ i1, a1, i2, a2) ´ C1(i2, a2) C2(i1, a1) (see [4, p. 100]), i.e. if and only if
N N N N i1 ; aN1 ; i2 ; aN2 2 C 1 i2 ; aN2 C 2 i1 ; aN1 :
ð16Þ
Such a bistrategy is also qualified as a pure-strategy Nash equilibrium and has an obvious implementation, as it requires that each manufacturer uses a specific medium with a specific effort. It is a representation of the possible behaviors of rational and intelligent players. As all the strategies (ij, 0), ij 2 Ij, amount to manufacturer j’s unique choice of no advertising with any medium, we treat them as a unique strategy, which we denote by (, 0). Here, we obtain preliminarily an exclusive result, concerning the trivial bistrategy at which neither manufacturer advertises his product. Lemma 4. The bistrategy (, 0, , 0) is not a Nash equilibrium.
Proof. Let ih 2 Ih be a fixed advertising medium of manufacturer h, 0 00 and let ij ; ij 2 Ij be two media of manufacturer j, such that qji0j < qji00j . From (14) we obtain that the following inequality holds for the ah-thresholds associated with the two media,
~i ji0 ¼ a h j
qji0
qji00
1 ~i ji00 : G0j þ j < G0j þ j ¼ a h j gh chih bpj gh chih bpj 1
Proof. Let j 2 {1,2}; for every ij 2 Ij, we have from Lemma 3 that ] ] ^jij g. Then, C j ð; 0Þ ¼ ^ji] ij ; a – fð; 0Þg, where ij is C j ð; 0jij Þ ¼ fa j
the medium index defined in (12). h In the limit situation in which no advertising interference is observed, the optimal choices of a player are independent of the other player’s choice, as seen in Lemma 2. This fact continues to be
609
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
observed as far as the advertising interference is weak enough. From Lemma 3 we obtain the following sufficient condition for the existence of a pure-strategy Nash equilibrium, which entails positive advertising efforts by both manufacturers. ^hih < a ~ ; gh > 0; j – h 2 f1; 2g; ij 2 Ij ; ih 2 Ih , a ih jij ] ^1i] ; i]2 ; a ^1i] , involving the media with then the bistrategy E ¼ i]1 ; a
Theorem
1. If
1
Proof. 1, we know that the bistrategy From Theorem ] ^1i] ; i]2 ; a ^2i] may be a (pure-strategy) Nash equilibrium. E] ¼ i1 ; a 1 2 Now let us assume that E] is not an equilibrium; this may occur if and only if, for either j = 1 or j = 2,
] ^ji] R C j i]h ; a ^hi] : ij ; a j
h
2
]
the maximum profitability index values (12), is the unique pure-
Let this be true for j = 1. From Corollary 3, for every i1 – i1 ,
strategy Nash equilibrium.
P1 i1 ; a^1i1 ; i]2 ; a^2i]
2
6 P1 ðE] Þ;
and hence Proof. Let j – h 2 {1, 2}; for every ij 2 Ij and ih 2 Ih, we have that ^hih jij Þ ¼ fa ^jij g, because of the Theorem hypothesis and Lemma C j ðih ; a 3. The profit associated with the bistrategy resulting by the use of ^jij Þ is Pj ð. . .Þ ¼ qji þ bpj gh chi ah þ G0j , which ðij ; a reaches the j h ] ] ^hih Þ ¼ ^ji] maximum value at ij ¼ ij . Therefore, C j ðih ; a ij ; a . j
A different Nash equilibrium should have an advertising effort component equal to 0, say a2 = 0; but then necessarily ^1i1 Þ, for some i1 2 I1, which contradicts the result ði2 ; 0Þ 2 C 2 ði1 ; a just proved. h
^1i1 Þ R C 1 i]2 ; a ^2i] : ði1 ; a 2
Therefore, we must have that
] ^2i] ; ð; 0Þ 2 C 1 i2 ; a 2
so that the bistrategy equilibrium. h
]
^2i] ; 0; i2 ; a
is a (pure-strategy) Nash
2
Player j knows that the game has at least one equilibrium in ] ^ji] and (, 0) are his possible stratepure strategies and that ij ; a j
If the interference factors g1, g2 are sufficiently small, then the ~ih jij assumption of Theorem 1 is verified, because the threshold a ^hih is invaritends to +1 as gj ? 0, whereas the advertising effort a ant with respect to (g1, g2). Therefore, a pure-strategy Nash equilibrium exists in a market where the advertising interference is weak and each manufacturer, according to it, advertises his product using a medium with maximum profitability index qji] . j A further special situation, with respect to interference, is that of unilateral interference, when one player uses an aggressive advertising policy (his advertising message has a comparative focus), whereas the other player does not: then only the second player would condition his decisions on the possible information on the competitor’s choice. Theorem 2. If gj > 0, gh = 0, j, h 2 {1, 2}, then there exists a pure ^ji] , and strategy Nash equilibrium, where player j’s strategy is i]j ; a j ] ^ player h’s one is either ih ; ahi] or (, 0). h
Proof. The theorem is a straightforward consequence of Corollary 1 and Lemma 3. h If only one manufacturer’s advertising interferes with his competitor’s goodwill evolution, then there exists a pure-strategy Nash equilibrium. Accordingly, the first – aggressive – manufacturer advertises his product, whereas the latter – inoffensive – manufacturer may advertise or not, depending on the profitability of the alternative choices. This may be the situation of a market where an entrant firm – player 1 – advertises by comparing its product to that of an established incumbent – player 2: this is the context considered by [5]. For small values of gj, the equilibrium is the bistrategy E], consistently with the weak interference result. Only for relatively large values of gj, player h chooses not to advertise, i.e. strategy (, 0). We notice that any manufacturer, when advertising his product, uses the medium with maximum profitability index. Finally, we admit that the interference factors g1, g2 have any values in (0, 1), so that we observe the situation of bilateral interference: both manufacturers advertising messages have a comparative focus. Theorem 3. There exists at least one pure-strategy Nash equilibrium; ^ji] ; j ¼ 1; 2. player j’s equilibrium strategy is either (, 0), or i]j ; a j
gies for an equilibrium. In practice, as far as the model assumptions ]
hold, each manufacturer can find a suitable advertising medium ij among those available to him, and determine the possibly positive advertising effort for an equilibrium, irrespectively of the media available to his competitor. Therefore, we may focus simply on the analysis of a reduced game, in which each player may choose between two strategies. Consistently with an observation at the end of Section 3, we notice that this simplification depends on the additive model assumption (2) for the joint effect of the players’ advertising efforts on goodwill and may not extend to more general situations. 5. One medium for each manufacturer In view of the final remark in the previous Section, we find it interesting to explore the different cases one may encounter among the advertising games with one medium for each player. In this context, each manufacturer’s decision concerns simply his advertising effort. Therefore we can simplify the notations by dropping the specifications ij, ih, or changing them into j, h, in a natural way, so that, for instance, the profit of manufacturer j will be represented, using (5) and (2), by
h
Pj ða1 ; a2 Þ ¼ bpj cj aj gh ch ah þ G0j
iþ
cj ðaj Þ;
j – h 2 f1; 2g: ð17Þ
Theorem 3 may be restated as follows. Corollary 4. The manufacturers’ advertising game, in which one medium is available to each manufacturer, jI1j = jI2j = 1, has at least one pure-strategy Nash equilibrium. Such equilibria belong to the set ^1 ; 0Þ; ð0; a ^2 Þ; ða ^1 ; a ^2 Þg, and the advertising effort a ^j > 0 is characfða terized by
^j Þ ¼ bpj cj : c0j ða
ð18Þ
As far as the pure-strategy Nash equilibria are concerned, each manufacturer j is interested in 2 strategies only: either strategy 0, no advertising, ^j , as or strategy 1, advertising and using his medium with effort a characterized by (18). Therefore, the essential features of the game may be represented in the (normal) form of a 2 2 game.
610
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
5.1. Examples of one–one media games
Table 1 Manufacturers’ payoffs – Example 1.
Let us focus on the special case in which the available media have quadratic costs, i.e.
cj ðaÞ ¼
1 2
jj a2 ; a P 0;
pj cj ; jj
ð20Þ
whereas the threshold point (14) is
~hj ¼ a
1
(
gch
G0j
)
pj c2j þb : 2jj
ð21Þ
^j , incurs the Manufacturer j, who advertises with the effort level a cost
^ j Þ ¼ b2 c j ða
c2j p2j : 2jj
a2 = 1
5, 5 10, 3
3, 10 8, 8
ð19Þ
where jj > 0, and assume further that the interference parameters are equal, g1 = g2 = g > 0. Then the advertising messages of both players have a comparative focus, as their Shaffer–Zettelmeyer emphasis is e = 1/(1 + g) < 1. Now, the characteristic positive advertising effort of manufacturer j is
^j ¼ b a
a1 = 0 a1 = 1
a2 = 0
ð22Þ
From Lemma 4 we know that the bistrategy (0, 0) cannot be a Nash equilibrium of the advertising game. On the other hand, each one of the three remaining bistrategies may be a Nash equilibrium, as it is shown by the following examples, built on the assumption of quadratic advertising costs. In particular, the games of Examples 1, 2 and 4 have one equilibrium each, the game of Example 3 has two equilibria. Moreover, in Examples 1–3 games with the special feature that only one (identical) advertising medium is available to both manufacturers are considered. Example 1. Let us consider the symmetric game with quadratic costs, where
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:1; j 2 f1; 2g; b = 1 and g = 0.2. The Shaffer–Zettelmeyer emphasis is e ffi 0.83: the advertising messages of both players have a weakly comparative focus. The positive values of the canonical decision rules are ^1 ¼ a ^2 ¼ 10, whereas the advertising effort thresholds are a ~12 ¼ a ~21 ¼ 50. Therefore the bistrategy (10, 10) is the unique a pure-strategy Nash equilibrium, as it is the unique fixed point of the set-valued map (a1, a2) ´ C1(a2) C2(a1). This is a rather interesting situation, as both manufacturers find it profitable to advertise in equilibrium. If we consider the alternative representation of the game as a 2 2 game, its payoff matrix is given by Table 1. Here we observe that a1 = 1 and a2 = 1 are dominant strategies for the two players and that the bistrategy (a1, a2) = (1, 1) is a Nash equilibrium: it corresponds to (a1, a2) = (10, 10) found above. The following example is a variant of Example 1, in which the interference factor and the cost parameters are larger; it provides an example of the prisoner’s dilemma. Example 2. Let us consider the symmetric game with quadratic costs, where
the previous example. The positive values of the canonical decision ^1 ¼ a ^2 ¼ 4, whereas the advertising effort thresholds are rules are a ~12 ¼ a ~21 ffi 13:9. Therefore the bistrategy (4,4) is the unique purea strategy Nash equilibrium. This is an undesirable situation for the players, who find it rational to advertise in equilibrium, but nevertheless see a better way out. In fact, a market regulation to constrain the interference factor g below the threshold 0.5 (the messages emphasis above the threshold 0.66) would avoid this inefficient situation. In the alternative representation, the payoff matrix of the 2 2 game is given by Table 2. As in Example 1, we observe that a1 = 1 and a2 = 1 are dominant strategies for the two players and that the bistrategy (a1, a2) = (1, 1), corresponding to (a1, a2) = (4, 4), is the unique pure-strategy Nash equilibrium. Nevertheless, at the bistrategy (a1, a2) = (0, 0) both manufacturers would be better off than at the Nash equilibrium. By reducing the values of the cost parameters to jj = 0.019, we obtain an example of crossroads game: it exhibits two pure-strategy Nash equilibria in which the manufacturers have symmetric roles. Example 3. Let us consider the symmetric game with quadratic costs, where
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:019; j 2 f1; 2g; b = 1 and g = 0.6; then the positive values of the canonical decision ^1 ¼ a ^2 ffi 52:63, whereas the advertising effort thresholds rules are a ~12 ¼ a ~21 ffi 52:19. Therefore the two symmetric bistrategies are a (52.63, 0) and (0, 52.63) are the pure-strategy Nash equilibria. Again, this is an undesirable situation for the players, who do not have a unique strategy to choose for an equilibrium and cannot be sure to obtain an equilibrium. As in Example 2, the difficulty comes from the high value of the interference factor g (low emphasis value, e = 0.63). In the alternative representation, the payoff matrix of the 2 2 game is given by Table 3. Here we observe that no player has any dominant strategy. The bistrategies (a1, a2) = (0, 1) and (a1, a2) = (1, 0), corresponding to (a1, a2) = (0, 52.63) and (a1, a2) = (52.63, 0), are Nash equilibria. The following is an example where the advertising media have different cost parameters, so that one manufacturer is stronger than the other. Example 4. Let us consider the asymmetric game with quadratic costs, where
b ¼ 1;
and the manufacturers differ in the advertising cost parameters,
j1 ¼ 0:1; j2 ¼ 0:5: Table 2 Manufacturers’ payoffs – Example 2.
cj ¼ 1; G0j ¼ 5; pj ¼ 1; jj ¼ 0:15; j 2 f1; 2g; b = 1 and g = 0.6. The emphasis is e ffi 0.63: the advertising messages of both players have a stronger comparative focus than in
g ¼ 0:7; cj ¼ 1; G0j ¼ 5; pj ¼ 1; j 2 f1; 2g;
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 8.33, 1
1, 8.33 4.33, 4.33
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612 Table 3 Manufacturers’ payoffs – Example 3.
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 31.32, 0
0, 31.32 0.26, 0.26
Table 4 Manufacturers’ payoffs – Example 4.
a1 = 0 a1 = 1
a2 = 0
a2 = 1
5, 5 10, 0
3.6, 6 8.6, 1
611
models are already analyzed by some papers, e.g. [7,23]. The main drawback of the additive goodwill production model is that it cannot represent any positive/negative enhancement effect of combining several advertising messages, as discussed in [26] from the positive (synergy) viewpoint. We have assumed here that each manufacturer has a characteristic interference parameter gj and hence a characteristic advertising message emphasis. Now, it is rather true that the advertising message emphasis is a decision variable of the manufacturer (see [25]). This observation suggests that a different and relevant definition of advertising action of a manufacturer in a competitive context should have three dimensions: medium, effort and interference parameter (or emphasis). Acknowledgements
The messages emphasis is e ffi 0.59: the advertising messages of both players have a strong comparative focus. The positive values ^1 ¼ 10; a ^2 ¼ 2, whereas the of the canonical decision rules are a ~12 ffi 8:6; a ~21 ffi 14:3. Therefore advertising effort thresholds are a the bistrategy (10, 0) is the unique pure-strategy Nash equilibrium. This is a problematic situation, as only one manufacturer, i.e. the one with the cheaper medium, finds it profitable to advertise in equilibrium, whereas the competitor does not. In the alternative representation, the payoff matrix of the 2 2 game is given by Table 4. The advertising medium available to manufacturer 1 is more efficient than manufacturer 2’s one. Here, a1 = 1 is a dominant strategy for player 1, whereas player 2 has not any dominant strategy; the bistrategy (a1, a2) = (1, 0) is the unique pure-strategy Nash equilibrium of the game and it corresponds to (a1, a2) = (10, 0) found above. Of course, the bistrategy (a1, a2) = (0, 1) is the unique Nash equilibrium of the 2 2 game if the second manufacturer has lower advertising costs than the first one, with all other features being equal.
6. Conclusion and directions for future research This paper proposes a model for two competing manufacturers of substitutable goods in a homogeneous market. It considers a static decision framework where each player can choose his advertising effort, which has a double effect: positive on the player’s sales and negative on the competitor’s sales. The effect of the players’ advertising efforts on the two product sales is mediated by the goodwill variables of the products, as in the dynamic model by Nerlove and Arrow. The use of the intermediate goodwill variables is an original contribution to the literature on static game models of advertising competition. Interference among advertising efforts makes it possible to observe negative goodwill values, a circumstance which entails some analytical difficulties. From a mathematical point of view, a two-player continuous game with non-smooth payoff functions is analyzed and is shown to be equivalent to a suitable 2 2 game. We obtain a complete description of feasible Nash equilibria in pure strategies, while accounting for possible negative goodwill. The treatment of the situation where one (or each) player can choose among different advertising media has no analogous counterpart in the dynamic models literature. The solution to the model shows that in equilibrium only one advertising medium is used by each manufacturer. However, such result seems to be strictly dependent on the assumption that the joint effect of the players’ advertising efforts on goodwill is additive. Therefore, an interesting direction for future research would be to explore different and more general models for this joint effect. Then, the obtained result that only one advertising medium is used in equilibrium is not expected to hold. Such non-linear
The paper is dedicated to the memory of Giorgio Pederzoli, a great friend. Financial support from the Italian Ministry of University and Research and the University of Padua is gratefully acknowledged. The author is indebted to some colleagues and to three referees for proper criticism and relevant suggestions. References [1] N. Amrouche, G. Martı´n-Herrán, G. Zaccour, Feedback Stackelberg equilibrium strategies when the private label competes with the national brand, Annals of Operations Research 164 (2008) 79–95. [2] N. Amrouche, G. Martı´n-Herrán, G. Zaccour, Pricing and advertising of private and national brands in a dynamic marketing channel, Journal of Optimization Theory and Applications 137 (3) (2008) 465–483. [3] S.P. Anderson, R. Renault, Comparative advertising: disclosing horizontal match information, RAND Journal of Economics 40 (3) (2009) 558–581. [4] J.-P. Aubin, Optima and Equilibria, Springer, Berlin, 1993. [5] F. Barigozzi, P.G. Garella, M. Peitz, With a little help from my enemy: Comparative advertising as a signal of quality, Journal of Economics and Management Strategy 18 (4) (2009) 1071–1094. [6] G.S. Becker, K.M. Murphy, A simple theory of advertising as a good or bad, Quarterly Journal of Economics 108 (1993) 941–964. [7] P.K. Chintagunta, D. Jain, A dynamic model of channel member strategies for marketing expenditures, Marketing Science 11 (1992) 168–188. [8] B.W. Conolly, Techniques in Operational Research, Models, Search and Randomization, vol. 2, Ellis Horwood, Chichester, 1981. [9] P.J. Danaher, A. Bonfrer, S. Dhar, The effect of competitive advertising interference on sales for packaged goods, Journal of Marketing Research XLV (2008) 211–225. [10] E. Dockner, S. Jørgensen, N. Van Long, G. Sorger, Differential Games in Economics and Management Science, Cambridge Univ. Press, Cambridge, 2000. [11] R. Dorfman, P.O. Steiner, Optimal advertising and optimal quality, American Economic Review XLIV (1954) 826–836. [12] L. Grosset, B. Viscolani, Optimal dynamic advertising with an adverse exogenous effect on brand goodwill, Automatica 45 (2009) 863–870. [13] Z. Huang, S.X. Li, V. Mahajan, An analysis of manufacturer–retailer supply chain coordination in cooperative advertising, Decision Sciences 33 (2002) 469–494. [14] S. Jørgensen, G. Zaccour, Equilibrium pricing and advertising strategies in a marketing channel, Journal of Optimization Theory and Applications 102 (1) (1999) 111–125. [15] S. Jørgensen, G. Zaccour, Differential Games in Marketing, Kluwer Academic Publishers, Boston, 2004. [16] S. Karray, G. Zaccour, Effectiveness of coop advertising programs in competitive distribution channels, International Game Theory Review 9 (2) (2007) 151–167. [17] P. Kotler, Marketing Management, 11th ed., Prentice-Hall, Upper Saddle River, 2003. [18] G. Leitmann, W.E. Schmitendorf, Profit maximization through advertising: a nonzero sum differential game approach, IEEE Transactions on Automatic Control 23 (4) (1978) 645–650. [19] M. Leng, M. Parlar, Game theoretic applications in supply chain management: A review, INFOR 43 (3) (2005) 187–220. [20] K.S. Moorthy, Using Game Theory to model competition, Journal of Marketing Research 22 (1985) 262–282. [21] A. Nair, R. Narasimhan, Dynamics of competing with quality- and advertisingbased goodwill, European Journal of Operational Research 175 (2006) 462– 474. [22] M. Nerlove, K.J. Arrow, Optimal advertising policy under dynamic conditions, Economica 29 (2) (1962) 129–142. [23] A. Prasad, S.P. Sethi, Integrated marketing communications in markets with uncertainty and competition, Automatica 45 (3) (2009) 601–610.
612
B. Viscolani / European Journal of Operational Research 216 (2012) 605–612
[24] L. Schoonbeek, P. Kooreman, The impact of advertising in a duopoly game, International Game Theory Review 9 (4) (2007) 565–581. [25] G. Shaffer, F. Zettelmeyer, Comparative advertising and in-store displays, Marketing Science 28 (6) (2009) 1144–1156. [26] A. Sorato, B. Viscolani, Using several advertising media in a homogeneous market, Optimization Letters, doi:10.1007/s11590-010-0220-z. [27] J.G. Szmerekovsky, J. Zhang, Pricing and two-tier advertising with one manufacturer and one retailer, European Journal of Operational Research 192 (2009) 904–917. [28] B. Viscolani, Advertising decisions for a segmented market, Optimization 58 (4) (2009) 469–477. [29] B. Viscolani, Advertising decisions in a vertical distribution channel, International Game Theory Review 11 (3) (2009) 273–284.
[30] B. Viscolani, G. Zaccour, Advertising strategies in a differential game with negative competitor’s interference, Journal of Optimization Theory and Applications 140 (1) (2009) 153–170. [31] Q. Wang, M. Parlar, Static game theory models and their applications in management science, European Journal of Operational Research 42 (1) (1989) 1–21. [32] J. Xie, J.C. Wei, Coordinating advertising and pricing in a manufacturer–retailer channel, European Journal of Operational Research 197 (2009) 785–791. [33] J. Yue, J. Austin, M.-C. Wang, Z. Huang, Coordination of cooperative advertising in a two-level supply chain when manufacturer offers discount, European Journal of Operational Research 168 (2006) 65–85.