Competative Advertsing

journal of optimization theory and applications:

Vol. 123, No. 1, pp. 163–185, October 2004 (© 2004)

Competitive Advertising under Uncertainty: A Stochastic Differential Game Approach1 A. Prasad2 and S. P. Sethi3 Communicated by G. Leitmann

Abstract. We analyze optimal advertising spending in a duopolistic market where each firm’s market share depends on its own and its competitor’s advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented. Key Words. Advertising, dynamic duopoly, competitive strategy, differential games, stochastic differential equations.

1. Introduction The advertising spending decision has been the focus of considerable interest for researchers in marketing as evidenced by the large body of literature devoted to this subject starting with Vidale and Wolfe (Ref. 1) and reported in surveys by Sethi (Ref. 2) and by Feichtinger, Hartl, and Sethi (Ref. 3). The annual expenditure on advertising by firms was 117 billion dollars in 2002 in the US alone (Ref. 4). At the same time, marketers have 1 The

authors thank the PhD seminar participants at the University of Texas at Dallas for helpful comments. 2 Assistant Professor, School of Management, University of Texas at Dallas, Richardson, Texas. 3 Ashbel Smith Professor, School of Management, University of Texas at Dallas, Richardson, Texas.

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noted that firms advertise often in a suboptimal manner. For example, Patti and Blasko (Refs. 5–6) found in surveys that a large percentage of industrial and consumer good firms do advertising budgeting based on the affordable method, the percentage-of-sales method, and the competitiveparity method. These methods are rarely optimal. A number of researchers have concluded also that firms tend to overadvertise (Refs. 7–8). The combination of large amounts of money spent on advertising and potential inefficiencies in the advertising budgeting process motivates the need to better understanding optimal advertising budgeting. However, one must be careful to limit the conclusions of optimality only to those markets for which the model applies. Thus, we define our market context and research question as follows. We examine a duopoly market in a mature product category where two firms compete for market share using advertising as the dominant marketing tool. The firms are strategic in their behavior; that is, they take actions that maximize their objective while considering also the actions of the competitor. Furthermore, they interact dynamically for the foreseeable future. This is due in part to the carryover effect of advertising, which means that advertising spending today will continue to influence sales several months down the line. Each firm’s advertising acts to increase its market share, while the competitor’s advertising acts to reduce its market share. In addition to market share decay caused by competitive advertising, noncompetitive factors described in the next paragraph can cause market share churn. However, marketing and competitive activities alone do not govern market shares in a deterministic manner, because there is inherent randomness in the marketplace and in the choice behavior of customers. The market for cola drinks, dominated by Coke and Pepsi, is an example of a market with such features (Refs. 9–11). Explanations for churn are product obsolescence, forgetting (Ref. 1), lack of market differentiation (Ref. 12), lack of information (Ref. 13), variety seeking (Ref. 14), and brand switching. These factors do not necessarily cause market share to decay because the decay of market share for one firm is a gain in market share for the other. Hence, we use the term “churn” rather than “decay”. Due to churn, the market shares converge to a long-run equilibrium when neither brand is advertised for a very long duration. The proposed model takes into account decay due to competitive advertising as well as churn due to noncompetitive factors. For a competitive market with stochastic disturbances and other features as described above, our objective is to recommend optimal advertising expenditures over time for the two firms. A stochastic differential game model is formulated based on the monopoly model of Sethi (Ref. 15). We obtain explicit closed-loop solutions.

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Our research follows in the operations research tradition in marketing (Refs. 9–11 and Refs. 16–19). Whereas elements of the marketing environment described above, such as dynamics, competition, competitive and noncompetitive decay, and also stochasticity, have been commonly used by individual models, there exist few attempts to study them together. We provide a focused discussion of the literature in Section 2.

2. Background Among the earliest aggregate response models is the Vidale-Wolfe model whose dynamics is given by dx(t)/dt = ρu(t)(1 − x(t)) − δx(t),

x(0) = x0 ,

(1)

where x(t) is the sales rate (expressed as a fraction of the total market) at time t, u(t) is the advertising expenditure rate, ρ is a response constant, and δ is a market share decay constant; ρ determines the effectiveness of advertising, while δ determines the rate at which consumers are lost due to product obsolescence, forgetting, etc. The formulation has several desirable properties; for example, market share has a concave response to advertising, and there is a saturation level (Ref. 20). The optimal advertising path for the Vidale-Wolfe dynamics is provided by Sethi (Ref. 21). Subsequent research extended the basic framework to incorporate competitive advertising (Refs. 22–23). A competitive extension of the Vidale-Wolfe model, based on the Lanchester model of combat (Refs. 20 and 24), is dx(t)/dt = ρ1 u1 (x(t), y(t))(1 − x(t)) −ρ2 u2 (x(t), y(t))x(t), x(0) = x0 ,

(2)

dy(t)/dt = ρ2 u2 (x(t), y(t))x(t) −ρ1 u1 (x(t), y(t))(1 − x(t)),

(3)

y(0) = 1 − x0 ,

where x(t) and y(t) represent the market shares of the two firms, whose parameters and decision variables are indexed 1 and 2 respectively. Note that x(t) + y(t) = 1. In addition to competitive extensions, recent research has examined the problem where the state variable, usually market share, is determined by stochastic disturbances in addition to advertising spending. We start

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with the stochastic, monopoly advertising model of Sethi (Ref. 15). The Sethi model is given by the Itoˆ equation
  dx(t) = ρu(t) 1 − x(t) − δx(t) dt + σ (x)dw(t), x(0) = x0 , (4) where σ (x) represents a variance term and w(t) represents a standard Wiener process. The model has the useful feature in that it has a basic resemblance to the Vidale-Wolfe model and at the same time it permits an explicit solution to the advertising spending decision. We wish to extend this model to incorporate competition. A related extension is due to Sorger (Ref. 19). He uses a special case of the Lanchester model to obtain a duopoly version of the Sethi model. This is √ √ dx/dt = ρ1 u1 (x, y) 1 − x − ρ2 u2 (x, y) x,  √ dy/dt = ρ2 u2 (x, y) 1 − y − ρ1 u1 (x, y) y,

x(0) = x0 ,

(5)

y(0) = 1 − x0 .

(6)

Chintagunta and Jain (Ref. 17) test this specification using data from the pharmaceuticals, soft drinks, beer and detergent industries, and find it to be appropriate. Sorger describes the appealing characteristics of the specification, noting that it is compatible with word-of-mouth and nonlinear effects. However, the decay constant δ is not included here and it is assumed to be replaced totally by competitive effects. On the other hand, we extend the Sethi model to allow for competition. We are able to do so, while retaining the decay constant. Note that a stochastic version of the Sorger model is a special case of ours (specified in the next section) when δ = 0. The decay constant, which goes back to the Vidale-Wolfe formulation, is not solely replaced by competitive advertising effects. Thus, in order to capture effects such as forgetting, the decay parameter has been morphed into the churn parameter. We will discuss how including the term affects the outcome. 3. Model We consider a duopoly market in a mature product category where total sales are distributed between two firms, denoted firm 1 and firm 2, which compete for market share through advertising spending. We denote the market shares of firms 1 and 2 at time t as x(t) and y(t), respectively. Using the subscript i ∈ {1, 2} to reference the two firms, ui (x(t), y(t), t) ≥ 0 is the advertising control, ρi > 0 is the advertising effectiveness parameter, ri > 0 is the discount rate, δ > 0 is the churn parameter, and ci > 0 is a cost

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parameter. The time argument will be suppressed in future where no confusion arises. The model dynamics is given by √ √ dx = [ρ1 u1 (x, y) 1 − x − ρ2 u2 (x, y) x − δ(x − y)]dt +σ (x, y)dw, x(0) = x0 , (7)  √ dy = [ρ2 u2 (x, y) 1 − y − ρ1 u1 (x, y) y − δ(y − x)]dt −σ (x, y)dw, y(0) = 1 − x0 . (8) This specification has the same desirable properties of concave response with saturation as the Vidale-Wolfe model. The market share is nondecreasing with own advertising and nonincreasing with the competitor’s advertising expenditure. Consistent with the literature, churn is proportional to the market share. As discussed, it is caused by influences other than competitive advertising, such as a lack of perceived differentiation between brands, so that the market shares tend to converge in the absence of advertising. Finally, the market shares are subject to a white noise term σ (x, y)dw. Since dx + dy = 0

and x(0) + y(0) = 1,

we have x(t) + y(t) = 1, for all t ≥ 0. Now that y(t) = 1 − x(t), we need use only the market share of firm 1 to completely describe the market dynamics. Thus, ui (x, y), i = 1, 2, and σ (x, y) can be written as ui (x, 1 − x) and σ (x, 1 − x). With a slight abuse of notation, we will use ui (x) and σ (x) in place of ui (x, 1 − x) and σ (x, 1 − x), respectively. Thus, √ √ dx = [ρ1 u1 (x) 1 − x − ρ2 u2 (x) x − δ(2x − 1)]dt+σ (x)dw, x(0) = x0 , (9) with 0 ≤ x0 ≤ 1. As noted in Ref. 15, when choosing a formulation, an important consideration is that the market share should remain bounded within [0, 1], which can be problematic given stochastic disturbances. In our model, it is easy to see that x ∈ [0, 1] almost surely for t > 0, as long as ui (x) and σ (x) are continuous functions that satisfy the Lipschitz conditions on every closed subinterval of (0, 1) and further that ui (x) ≥ 0, x ∈ [0, 1], and σ (x) > 0, x ∈ (0, 1) and

σ (0) = σ (1) = 0.

(10)

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With this, we have a strictly positive drift at x = 0 and a strictly negative drift at x = 1, i.e., √ (11) ρ1 u1 (0) 1 − 0 + δ > 0 and − ρ2 u2 (1) − δ < 0. Then, from Gihman and Skorohod (Ref. 25, Theorem 2, pp. 149 and 157–158), x = 0 and x = 1 are natural boundaries for the solutions of (9) with x0 ∈ [0, 1]; i.e., x ∈ (0, 1) almost surely for t > 0. Let mi denote the industry sales volume multiplied by the per-unit profit margin for firm i. We formulate the optimal control problem faced by the two firms as    ∞ Max V1 (x0 ) = E e−r1 t [m1 x(t) − c1 u1 (t)2 ]dt , (12) u1 ≥0



0



Max V2 (x0 ) = E u2 ≥0

∞

e

−r2 t

 2

[m2 (1 − x(t)) − c2 u2 (t) ]dt ,

(13)

0

√ √ s.t., dx = [ρ1 u1 (x) 1 − x − ρ2 u2 (x) x − δ(2x − 1)]dt + σ (x)dw, x(0) = x0 ∈ [0, 1].

(14) (15)

Each firm seeks to maximize its expected, discounted profit stream subject to the market share dynamics. Note that, when the advertising expenditure enters linearly in the dynamic equation, its cost in the objective function is assumed to be quadratic as in Ref. 15. Equivalently, one can take the square root of the advertising expenditure in the dynamic equation and subtract the advertising expenditure linearly in the objective function (e.g., Ref. 19). See Refs. 26–27 for a discussion. 4. Analysis To find the closed-loop Nash equilibrium strategies, we form the Hamilton-Jacobi-Bellman (HJB) equation for each firm,  √ r1 V1 = max m1 x − c1 u21 + V1 (ρ1 u1 1 − x u1  √ −ρ2 u∗2 x − δ(2x − 1)) + σ (x)2 V1 /2 , (16)  √ r2 V2 = max m2 (1 − x) − c2 u22 + V2 (ρ1 u∗1 1 − x u2  √ −ρ2 u2 x − δ(2x − 1)) + σ (x)2 V2 /2 , (17) where Vi = dVi /dx, Vi = d 2 Vi /dx 2 and where u∗1 and u∗2 denote the competitor’s advertising policies in (16) and (17), respectively. We obtain the

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optimal feedback advertising decisions 
√ u∗1 (x) = max 0, V1 (x)ρ1 1 − x/2c1 ,

169

(18a)



√ u∗2 (x) = max 0, −V2 (x)ρ2 x/2c2 .

(18b)

Since 0 ≤ x ≤ 1 and since it is reasonable to expect V1 ≥ 0 and V2 ≤ 0, we can reduce the advertising decisions (18) to √ (19a) u∗1 (x) = V1 (x)ρ1 1 − x/2c1 , √ u∗2 (x) = −V2 (x)ρ2 x/2c2 , (19b) which hold as we shall see later. Substituting (19) in (16) and (17), we obtain the Hamilton-Jacobi equations r1 V1 = m1 x + V12 ρ12 (1 − x)/4c1 + V1 V2 ρ22 x/2c2 −V1 δ(2x − 1) + σ (x)2 V1 /2,

r2 V2 = m2 (1 − x) + V22 ρ22 x/4c2 + V1 V2 ρ12 −V2 δ(2x − 1) + σ (x)2 V2 /2.

(20) (1 − x)/2c1 (21)

Following Ref. 15, we obtain the following forms for the value functions: V1 = α1 + β1 x,

(22a)

V2 = α2 + β2 (1 − x).

(22b)

These are inserted into (20) and (21) to determine the unknown coefficients α1 , β1 , α2 , β2 . Equating powers of x in Eq. (20) and powers of 1 − x in Eq. (21), the following four equations emerge, which can be solved for the unknown coefficients: r1 α1 = β12 ρ12 /4c1 + β1 δ,

(23)

r1 β1 = m1 − β12 ρ12 /4c1 − β1 β2 ρ22 /2c2 − 2β1 δ,

(24)

r2 α2 = β22 ρ22 /4c2 + β2 δ,

(25)

r2 β2 = m2 − β22 ρ22 /4c2 − β1 β2 ρ12 /2c1 − 2β2 δ.

(26)

Since for firms having different parameter values, the solutions to these equations are complicated, we consider first the case of symmetric firms in Section 4.1. The case of asymmetric firms will be dealt with in Section 4.2.

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4.1. Symmetric Firms.

For the symmetric firm case,

α = α 1 = α2 , β = β1 = β2 , m = m1 = m2 , c = c1 = c2 , ρ = ρ1 = ρ2 , r = r1 = r2 . The four equations in (23)–(26) reduce to the following two: rα = β 2 ρ 2 /4c + βδ, 2 2

rβ = m − 3β ρ /4c − 2βδ.

(27a) (27b)

There are two solutions for β. One is negative, which clearly makes no sense. Thus, the remaining positive solution is the correct one. This gives also the corresponding α. The solution is
  α = (r − δ)(W − W 2 + 12Rm) + 6Rm /18Rr, (28a)
  β= W 2 + 12Rm − W /6R, (28b) where R = ρ 2 /4c,

W = r + 2δ.

We can see now that, with the solution for the value function, the controls specified in Eq. (18) reduce to (19). This validates our choice of (19) in deriving the value function. Table 1 provides the comparative static results for the parameters (proofs with the authors). When ρ increases or c decreases (i.e., there is a marginal increase in the value of advertising or a reduction in its cost), then as one might expect, the amount of advertising increases. However, contrary to what one would expect to see in a monopoly model of advertising, the value function decreases. This occurs because, in this market, all advertising occurs from competitive motivations, since the optimal advertising expenditure would be zero if a single firm were to own both identical products. Advertising does not increase the size of the marketing pie, but affects only its allocation. Thus, the increase in advertising causes a decrease in the value function. The same logic does not apply when m increases

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Table 1. Comparative statics for symmetric firms. Variables

α β u∗ V (x)

Parameters c

ρ

m

δ

r

+ + − +

− − + −

+ + + +

+ − − ?

− − − −

Legend: increase (+), decrease (−), ambiguous (?).

or r decreases. In these cases, it is true that the wasteful advertising is increased, but it is also true that the size of the pie is increased. The churn parameter δ reduces competitive intensity. Hence, it might be expected that an increase in δ should increase the profitability by reducing advertising. In fact, only the constant α part of the value functions increases, and it is ambiguous what happens to the value functions overall. We can derive the exact conditions under which there is an increase or a decrease in the value function of a firm due to an increase in δ. We find that, if the market share of a firm is less than half, the effect on the firm’s value function is always positive. However, if the market share of a firm is greater than half, its value function can decrease, because of an increase in δ, if the following inequality is satisfied: 

 x> (r + 2δ)2 + 12Rm − (r + 2δ) 6r + 1/2. The reason is that, when a firm has a market share advantage over its rival, δ helps the rival unequally by tending to equalize market shares. 4.2. Asymmetric Firms. We return now to the general case of asymmetric firms. For asymmetric firms, we reexpress equations (23)–(26) in terms of a single variable β1 , which is determined by the solution to the equations 
3R12 β14 + 2R1 (W1 + W2 )β13 + 4R2 m2 − 2R1 m1 − W12 + 2W1 W2 β12 +2m1 (W1 − W2 )β1 − m21 = 0,

(29)

α1 = β1 (β1 R1 + δ)/r1 ,

(30)

β2 = (m1 − β12 R1 − β1 W1 )/2β1 R2 ,

(31)

α2 = β2 (β2 R2 + δ)/r2 ,

(32)

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where R1 = ρ12 /4c1 ,

R2 = ρ22 /4c2 ,

W1 = r1 + 2δ,

W2 = r2 + 2δ.

Once we obtain the correct value of β1 out of the four solutions of the quartic equation (29), the other coefficients can be obtained by solving for α1 , β2 , and then α2 . The solution is given in Appendix A (Section 6). We collect the main results of the analysis into Theorem 4.1. Theorem 4.1. For the advertising game described by Eqs. (12)–(15): (i) (ii)

There exists a unique closed-loop Nash equilibrium solution the differential game. See proof in Appendix √ A, Section ∗6. ∗ (x) = β ρ Optimal advertising is u 1 − x/2c1 , u2 (x) 1 1 1  β 2 ρ2 1 − y/2c2 ; in the symmetric firm case, β1 = β2 ( W 2 + 12Rm − W )/6R; in the asymmetric firm case, and β2 are given by (61) and (31).

to = = β1

We observe that the optimal advertising policy is to spend in proportion to the competitor’s market share. Consistent with Ref. 19, the firm that is in a disadvantageous position fights harder than its opponent and should succeed in wrestling market share from the opponent. Spending is decreasing in the own market share; thus, the advertising-to-sales ratio is higher for the lower market share firm. As noted in the introduction, many firms do advertising budgeting based on the affordable method, the percentage-of-sales method, and the competitive-parity method (Refs. 5–6). These methods suggest that the firm with lower market share should spend less on advertising in contrast to the optimal advertising policy derived here. Table 2 provides the comparative static results for α, β, and V1 (x) with respect to the parameters, see proofs in Appendix B, Section 7. Table 2. Comparative statics for asymmetric firms. Variables

αi βi u∗i Vi (x)

Parameters ci , cj

ρi , ρj

mi , mj

δ

ri , rj

?, + ?, + −, + ?, +

?, − ?, − +, − ?, −

+, − +, − +, − +, −

? − − ?

−, + −, + −, + −, +

Legend: increase (+), decrease (−), ambiguous (?).

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A comparison of Tables 1 and 2 shows the following features. First, due to the complexity of the asymmetric case, more effects are ambiguous. Second, a change in the own parameters has the same effect in the asymmetric case as a change in these parameters for the symmetric case. This is to be expected, since the first-order effects likely dominate the second-order effects, thus yielding the same results as in the symmetric case. Third, a beneficial change in the own parameters (ρi , ci , mi , ri ) has a negative effect on the competitor’s profits. Fourth, the optimal advertising policy does not depend on the noisiness of the selling environment. Finally, the results for the amount of advertising u∗1 are unambiguous and follow the same intuition as in the symmetric case. 4.3. Characterization of the Evolution Path. We examine next the market share paths analytically. Inserting the optimal controls into Eqs. (7)–(8), we obtain 
 dx = β1 ρ12 /2c1 + δ − x β1 ρ12 /2c1 + β2 ρ22 /2c2 + 2δ dt +σ (x)dw, x(0) = x0 , 
 dy = β2 ρ22 /2c2 + δ − y β1 ρ12 /2c1 + β2 ρ22 /2c2 + 2δ dt −σ (1 − y)dw,

y(0) = 1 − x0 .

(33) (34)

To keep the intermediate steps simple, we make use of the notation A1 ≡ β1 ρ12 /2c1 + δ,

A2 ≡ β2 ρ22 /2c2 + δ,

and derive the results only for firm 1. Rewriting (33) as the stochastic integral equation  t  t x(t) = x0 + (A1 − x(s)(A1 + A2 ))ds + σ (x)dw, (35) 0

0

it is obvious that the mean evolution path is independent of the nature of the stochastic disturbance. Specifically,  t E[x(t)] = x0 + (A1 − E[x(s)](A1 + A2 ))ds. (36) 0

This can be expressed as an ordinary differential equation in E[x(t)], with the initial condition E[x(0)] = x0 , whose solution is given by E[x(t)] = exp[−(A1 +A2 )t]x0 +(1−exp[−(A1 +A2 )t])A1 /(A1 +A2 ).

(37)

An analogous result is obtained for firm 2. The long run equilibrium market shares. (x, ¯ y) ¯ are obtained by taking the limit as t → ∞ and are given by x¯ = A1 /(A1 + A2 ), y¯ = A2 /(A1 + A2 ). (38)

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Thus, the expected market shares converge to the form resembling the attraction models commonly used in marketing. However, while an attraction model would rate the attractiveness of each firm based on its lower cost, higher productivity of advertising, and higher advertising, it would exclude exogenous market phenomena such as churn. To further characterize the evolution path, we can calculate the variance of the market shares at each point in time. A specification of the disturbance function that satisfies (10) is required for this purpose. We use  σ (x)dw = σ x(1 − x)dw, where σ is a positive constant. An application of the Itoˆ formula to Eq. (33) provides the result, d(x(t)2 ) = [2x(A1 − x(A1 + A2 )) + σ 2 x(1 − x)]dt  +2xσ x(1 − x)dw.

(39)

Rewriting this as a stochastic integral, taking the expected value, and rewriting as a differential equation, we get (d/dt)E[x(t)2 ] = (2A1 + σ 2 )E[x(t)] − (2A1 + 2A2 + σ 2 )E[x(t)2 ].

(40)

Inserting the solution for E[x(t)] from (37), we obtain the following first-order linear differential equation in the second moment E[x(t)2 ]: dE[x 2 ]/dt + (2A1 + 2A2 + σ 2 )E[x 2 ] = A1 (2A1 + σ 2 )/(A1 + A2 )   +exp[−(A1 + A2 )t] (2A1 + σ 2 )x0 − A1 (2A1 + σ 2 )/(A1 + A2 ) ,

(41)

with the initial condition E[x(0)2 ] = x02 . The solution is A1 (A1 + σ 2 /2) (A1 + A2 + σ 2 /2)(A1 + A2 ) −2(A1 +A2 +σ 2 /2)t ] ×[1 − e

E[x(t)2 ] = x02 e−2(A1 +A2 +σ

2 /2)t

+

e−(A1 +A2 )t − e−2(A1 +A2 +σ /2)t A1 + A 2 + σ 2   2A1 (A1 + σ 2 /2) 2 . × 2(A1 + σ /2)x0 − A 1 + A2 2

+

(42)

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We can calculate the convergence of the second moment as the influence of the initial condition disappears. That is, lim E[x(t)2 ] = A1 (A1 + σ 2 /2)/(A1 + A2 + σ 2 /2)(A1 + A2 ).

t→∞

(43)

Written in this form, it becomes clear that, when σ = 0, the expression is just x¯ 2 , so that the variance is appropriately zero in the absence of stochastic effect. More generally, when σ = 0, E[x(t)2 ] = (E[x(t)])2 holds for all t. For σ > 0, the variance is given by the formula E[x(t)2 ] − (E[x(t)])2 , which allows us to find that, in the long run, the variance of the market shares for both firms is given by A1 A2 σ 2 /(2A1 + 2A2 + σ 2 )(A1 + A2 )2 .

(44)

4.4. Illustration. Illustrative market shares may be obtained for different parameter values. We choose the parameter values r = 0.05, δ = 0.01, symmetric margins m1 = m2 = 1, asymmetric firm strengths R1 = 1, R2 = 4, and an initial starting point at x(0) = 0.5. In practice, a decision calculus approach could be followed to obtain the parameter values. Using Mathematica, we find that the only real positive root for the quartic polynomial is β1 = 0.264545 and the corresponding β2 = 0.43069. Finally, we specify  σ (x)dw = σ x(1 − x)dw, with σ 2 = 0.5. The procedure described in Zwillinger (Ref. 28, pp. 702, Eq. 182.3) is used; i.e., for an SDE dx(t) = a(x(t))dt + b(x(t))dw(t), the numerical approximation √ x(t + ) = x(t) + a(x(t)) + b(x(t)) ς (t) can be used, where the {ς(t)} are i.i.d. standard normal random variables. The time step  = 0.01 is used. Figure 1 shows a sample path. One can see that the path hovers around the mean. It never stays on the mean as it is continuously disrupted due to the Brownian motion. We can calculate a confidence interval if the path is normally distributed around the mean. Then, E[x(t)] ± 1.96 E[x(t)2 ] − (E[x(t)])2

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Fig. 1.

Market share trajectories given optimal advertising decisions.

provides the 95% confidence interval for the market share path. While we know that the distribution is not normal, nevertheless, as Fig. 2 shows, the proposed confidence interval does an adequate job of tracking the market shares. Since the normal distribution is not bounded between zero and one, the confidence interval may exceed the minimum or maximum market share as happened in this case, In Section 4.5, we obtain the equilibrium distribution of the market share, which enables us to provide the exact confidence intervals for the equilibrium market shares. This analysis has the value that it provides a diagnostic tool for management to handle market share fluctuations. Whereas minor fluctuations within the confidence bands may call for cursory examination, overstepping the bands indicates the need for a detailed review. It may be indicative of a shift in the underlying market parameters and hence it requires a reevaluation of the advertising spending policies. Also, the market share fluctuations directly cause fluctuations in advertising spending according to Theorem 4.1; hence, one can simulate the advertising budget as well.

4.5. Probability Distribution of Market Shares. In Section 4.4, we mentioned that the probability densities of the market shares are not necessarily normally distributed. This raises the question of whether the density functions can be determined explicitly or at least approximated. We devote this section to examining this issue.

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Fig. 2.

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Market share for firm 1: Normal density 95% confidence interval (dashed lines), and equilibrium market share 95% confidence interval (dotted lines).

An important property of the solution x(t) of an Itoˆ stochastic differential equation, dx(t) = a(x, t)dt + b(x, t)dw(t),

x(s) = z,

is that it is a Markov process. The transition probability of this Markov process has a density p(t, x; s, z) for going from market share z at time s to market share x at time t > s, which satisfies the Fokker-Planck equation, ∂p/∂t + (∂/∂x)(ap) − (1/2)(∂ 2 /∂x 2 )(b2 p) = 0, p(t, x; t, z) = δ(x − z). For our problem, we shall obtain first and then attempt to solve the Fokker-Planck equation. Firm 1’s stochastic differential equation from (33) is  dx = (A1 − (A1 + A2 )x)dt + σ x(1 − x)dw, x(0) = x0 . (45) The corresponding Fokker-Planck equation is given by ∂p/∂t + (∂/∂x)([A1 − (A1 + A2 )x]p) −(1/2)(∂ 2 /∂x 2 )[σ 2 x(1 − x)p] = 0,

(46)

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which simplifies to (∂p/∂t) + [σ 2 x(x − 1)/2]∂ 2 p/∂x 2 +([2σ 2 − (A1 + A2 )]x + A1 − σ 2 )∂p/∂x + [σ 2 − (A1 + A2 )]p = 0.

(47)

This partial differential equation could not be solved explicitly. Nevertheless, we will attempt to find the density of the steady-state market share by lim p(t, x; s, z). Let f (x) denote this density, since it can be shown to t→∞ be independent of s, z, t. To recapitulate, what we started off wanting to know was the density p(t, x; 0, x0 ) of firm 1’s market share at time t, given that it starts at a point x0 at time zero. By looking for the long-run stationary probability density of the market share, essentially we are willing to ignore the initial transient part of the solution. For the density f (x), we can set ∂p/∂t = 0 in (47) and obtain the second-order ordinary differential equation [σ 2 x(x − 1)/2] d 2 f /dx 2 + ([2σ 2 − (A1 + A2 )]x + A1 − σ 2 ) df /dx +[σ 2 − (A1 + A2 )]f = 0.

(48)

This is identifiable as a Gaussian hypergeometric equation with solution (Ref. 29, pp. 234) f (x) = x (2A1 /σ −1) (1 − x)(2A2 /σ −1)    2 2 × C1 + C2 x −2A1 /σ (1 − x)−2A2 /σ dx . 2

2

(49)

To determine the constants of integration, we can employ the following two properties. First, the density should integrate to 1; second, the expected value of the market share should be x, ¯ which we have already calculated is equal to A1 /(A1 + A2 ). The result is given in the following theorem. Theorem 4.2. The density of the stationary distribution of a firm’s market share is given by the beta density. For firm 1, f (x) = where

(2A1 /σ 2 + 2A2 /σ 2 ) 2A1 /σ 2 −1 2 (1 − x)2A2 /σ −1 , x 2 2 (2A1 /σ ) (2A2 /σ ) 

(s) = 0

∞

x s−1 e−x dx,

s > 0,

(50)

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is the gamma function. For firm 2, by symmetry, f (y) is obtained by interchanging x with y and A1 with A2 in (50). Proof. The density integrates to 1 by definition, while the mean of the beta distribution is given by A1 /(A1 + A2 ). Thus, all the required conditions are satisfied. It can be verified that the variance of the beta distribution matches the expression in (44) obtained by using the Itoˆ formula. We now return to the illustrative example of Section 4.4. Inserting the parameter values A1 = 0.539, A2 = 3.46, σ 2 = 0.5, the beta density is f (x) = 292.39x 1.156 (1 − x)12.84 .

(51)

We compute the 95% confidence bounds to be l = 0.02 and m = 0.33. These are sketched in Fig. 2. They provide a more accurate 95% confidence interval for the equilibrium market share of firm 1 than by assuming a normal distribution. The confidence intervals for firm 2 can be obtained in a similar manner. 5. Conclusions We examine a dynamic duopoly with stochastic disturbances and employ closed-loop methods to solve the problem. The model is analyzed using stochastic differential game theory and explicit solutions are obtained. The effects of several different parameters are discussed for symmetric and asymmetric firms. The paper extends the work of Sethi (Ref. 15) to include competitive advertising response and work of Sorger (Ref. 19) by including stochastic analysis and a churn term in the dynamics that is consistent with the original Vidale-Wolfe formulation and which ensures that, in the absence of competitive advertising, the market shares will converge. The effect of churn can be decomposed into two parts: one is to reduce competition by making advertising less effective, hence causing a decrease in equilibrium advertising; the other is to disproportionately reduce share of the higher market share firm. Thus, higher churn benefits a firm with low market share, but has ambiguous effects for a higher share firm.

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A simple rule describes the optimal advertising control, which is that it should be proportional to the square root of the opponent market share (Theorem 4.1). Sections 4.1 and 4.2 are devoted to determining the proportionality constant, particularly its endogenous component βi , and to obtaining an analytical expression for it. While it is given by a simple expression when the firms are symmetric, it is not simple to state the proportionality constant for asymmetric firms. An explicit formula is provided in Appendix A, Section 6. Furthermore, the dependence of βi as well as the amount of advertising is provided by means of comparative statics. In Section 4.3, we characterize the evolution path by using stochastic calculus to provide the mean and variance of the market shares. The former resembles an attraction model. In Section 4.5, we examine the probability distribution for the market shares by solving the Fokker-Planck equation in the limiting case and showing that it is the beta distribution. The fact that these commonly used forms for market share emerge endogenously from the analysis validates additionally our model assumptions. An illustration demonstrated the usability of the analysis in terms of tracking the market shares (Section 4.4). A limitation of the analysis is that it is restricted to duopoly competition, whereas many markets are characterized by more then two competitors (Refs. 30–31). Future research should examine the possibility of extending the present model to oligopoly markets. Extending the model to incorporate decision variables such as price is also relevant. Finally, the comparative statics presented in the paper represent hypotheses for empirical testing which deserves further attention. 6. Appendix A: Proof of Uniqueness of Solution We want to show that there exists a unique solution to the differential game. This implies showing that there exists a unique β1 > 0 that satisfies the quartic equation (29). We express (29) as F (β1 ) ≡ β14 + κ1 β13 + κ2 β12 + κ3 β1 − κ4 = 0,

(52)

where κ1 = 2(W1 + W2 )/3R1 ,
 κ2 = 4m2 R2 − 2m1 R1 − W12 + 2W1 W2 /3R12 ,

(53b)

κ3 = 2m1 (W1 − W2 )/3R12 ,

(53c)

κ4 = m21 /3R12 .

(53d)

(53a)

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Every quartic equation has four roots. Excluding the fortuitous cases, where two or more roots are equal, the following results are easily observed: When β1 → ±∞, F (β1 ) → ∞; when β1 = 0, F (β1 ) < 0. Since F (β1 ) is differentiable, it is continuous. Thus, it must cross the x-axis at least twice ensuring at least one positive real root and one negative real root. If there are only two real roots, one will be positive and one negative. If all four roots are real, they will be either three positive and one negative or three negative and one positive. (A2) In the case of three real positive roots, ordering them from the smallest to the largest, the slope at the second largest root must be negative. To see this, write

(A1)

F (β1 ) = (β1 − β1 (1))(β1 − β1 (2))(β1 − β1 (3))(β1 − β1 (4)), where β1 (1) < 0, β1 (2) > 0, β1 (3) > β1 (2), β1 (4) > β1 (3) are the four roots. Then, the slope at β1 (3), F  (β1 )|β1 =β1 (3) = (β1 (3) − β1 (1))(β1 (3) − β1 (2))(β1 (3) − β1 (4)),

(A3)

is negative. We can calculate the slope F  (β1 ) directly from (52) and evaluate it at any positive real root. The result is  F  (β1 )|β1 =F −1 (0) = 2 3R12 β14 + R1 (W1 + W2 )β13 + m1 W2 β1  +m1 (m1 − W1 β1 ) /β1 > 0.

(54)

It follows from points (A2) and (A3) above that there is only one real positive root and hence a unique solution to the differential game. To obtain an explicit solution, we utilize the

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Mathematica 4.1 software to generate four solutions to (52): β1 (1) = −κ1 /4 − g/2 − 0.5

× 3κ12 /4 − 2κ2 − g 2 + κ13 − 4κ1 κ2 + 8κ3 /4g,

(55)

β1 (2) = −κ1 /4 − g/2 + 0.5

× 3κ12 /4 − 2κ2 − g 2 + κ13 − 4κ1 κ2 + 8κ3 /4g,

(56)

β1 (3) = −κ1 /4 + g/2 − 0.5

× 3κ12 /4 − 2κ2 − g 2 − κ13 − 4κ1 κ2 + 8κ3 /4g,

(57)

β1 (4) = −κ1 /4 + g/2 + 0.5

× 3κ12 /4 − 2κ2 − g 2 − κ13 − 4κ1 κ2 + 8κ3 /4g,

(58)

where the two intermediate terms g and h are defined as

g ≡ κ12 /4 − 2κ2 /3 + 21/3 κ22 − 3κ1 κ3 − 12κ4 /3h + h/321/3 , (59)  h ≡ 2κ23 − 9κ1 κ2 κ3 + 27κ32 − 27κ12 κ4 + 72κ2 κ4

3 + −4 κ22 − 3κ1 κ3 − 12κ4



2 1/3 + 2κ23 − 9κ1 κ2 κ3 + 27κ32 − 27κ12 κ4 + 72κ2 κ4 . (60) We pick β1 as the only real positive solution out of the four roots, i.e., β1 = β1 (i ∗ ), While

i∗

where i ∗ = {i ∈ {1, 2, 3, 4}|β1 (i) > 0}.

(61)

i∗

in every case.

may depend on the data, there will only be one

7. Appendix B: Outline of Proof of Comparative Statics for Table 2 (B1)

To obtain comparative static results for βi , we define G1 (β1 , β2 ) ≡ m1 − β12 ρ12 /4c1 − β1 β2 ρ22 /2c2 − β1 (r1 + 2δ) = 0, (62) G2 (β1 , β2 ) ≡ m2 − β22 ρ22 /4c2 − β1 β2 ρ12 /2c1 − β2 (r2 + 2δ) = 0. (63)

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Then, for any parameter θ, the implicit function theorem may be written as  ∂β  1 ∂θ = −(1/) ∂β2 ∂θ





− × 

β2 ρ22 β1 ρ12 2c2 + 2c1 +r2 +2δ

β2 ρ12 2c1

 ×





β1 ρ22 2c2



−

β1 ρ12 β2 ρ22 2c1 + 2c2 +r1 +2δ

  



∂G1 ∂θ ∂G2 ∂θ

(64)

,

where  > 0. The comparative static results for changes in the parameters c2 , m1 , m2 , r1 , r2 , ρ2 , δ follow in a straightforward manner. However, the cases for c1 and ρ1 could not be signed. (B2)

For comparative statics for u∗1 , we insert √ u∗1 = β1 ρ1 1 − x/2c1 , √ u∗2 = β2 ρ2 x/2c1 into (62)–(63) to obtain G1 (u∗1 , u∗2 ) and G2 (u∗1 , u∗2 ). Then, the implicit function theorem can be written as 

∂u∗1 /∂θ ∂u∗2 /∂θ

 = −(1/)

   2c2 u2 2c2 (r2 +2δ) 2c2 u1 ρ1 + √ √ √ + − x ρ2 x ρ2 1 − x x ×  2c2 u2 ρ1 √ √ ρ2 1−x x
 ∂G × ∂G1 /∂θ , /∂θ



2c1 u1 ρ2 √ √ ρ1 1−x x



−

2c u ρ2 2c (r +2δ) + √1 2 √ + 1 √1 1 − x ρ1 1−x x ρ1 1−x 2c1 u1

  

2

(65) where it can be shown that  > 0. The calculations for the results reported in Table 2 follow in a straightforward manner. (B3)

For α1 , we note from Eq. (30) that, in many cases, α1 will have the same comparative statics as β1 . These relationships are as follows: ∂α1 /∂c2 > 0, ∂α1 /∂m1 > 0, ∂α1 /∂m2 < 0, ∂α1 /∂r1 < 0, ∂α1 /∂r2 > 0, ∂α1 /∂ρ2 < 0.

(66)

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the cases for c1 , ρ1 , δ are unclear. (B4)

The unambiguous results for V1 occur when comparative statics for α1 and β1 are in the same direction, which is true for all parameters except c1 , ρ1 , δ.

References 1. Vidale, M. L., and Wolfe, H. B., An Operations Research Study of Sales Response to Advertising, Operations Research, Vol. 5, pp. 370–381, 1957. 2. Sethi, S. P., Dynamic Optimal Control Models in Advertising: A Survey, SIAM Review, Vol. 19, pp. 685–725, 1977. 3. Feichtinger, G., Hartl, R. F., and Sethi, S. P., Dynamic Optimal Control Models in Advertising: Recent Developments, Management Science, Vol. 40, pp. 195–226, 1994. 4. CMR/TNS Media Intelligence, US Advertising Market Shows Healthy Growth: Spending Up 4.2% in 2002; see http://www.tnsmi-cmr.com/news/2003/031003.html (March 10, 2003). 5. Patti, C. H., and Blasko, V. J., Budgeting Practices of Big Advertisers, Journal of Advertising Research, Vol. 21, pp. 23–29, 1981. 6. Blasko, V. J., and Patti, C. H., The Advertising Budgeting Practices of Industrial Marketers, Journal of Marketing, Vol. 48, pp. 104–110, 1984. 7. Aaker, D., and Carman, J. M., Are You Overadvertising? Journal of Advertising Research, Vol. 22, pp. 57–70, 1982. 8. Lodish, L.M., Abraham, M., Kalmenson, S., Livelsberger, J., Lubetkin, B., Richardson, B., and Stevens, M. E., How TV Advertising Works: A MetaAnalysis of 389 Real World Split Cable TV Advertising Experiments, Journal of Marketing Research, Vol. 32, pp. 125–139, 1995. 9. Chintagunta, P. K., and Vilcassim, N., An Empirical Investigation of Advertising Strategies in a Dynamic Duopoly, Management Science, Vol. 38, pp. 1230–1224, 1992. 10. Erickson, G. M., Empirical Analysis of Closed-Loop Duopoly Advertising Strategies, Management Science, Vol. 38, pp. 1732–1749, 1992. 11. Fruchter, G., and Kalish, S., Closed-Loop Advertising Strategies in a Duopoly, Management Science, Vol. 43, pp. 54–63, 1997. 12. Bain, J. S., Barriers to New Competition, Harvard University Press, Cambridge, Massachusetts, 1956. 13. Nelson, P., Advertising as Information, Journal of Political Economy, Vol. 82, pp. 729–754, 1974. 14. McAlister, L., and Pessemier, E., Variety Seeking Behavior: An Interdisciplinary Review, Journal of Consumer Research, Vol. 9, pp. 311–323, 1982. 15. Sethi, S. P., Deterministic and Stochastic Optimization of a Dynamic Advertising Model, Optimal Control Applications and Methods, Vol. 4, pp. 179–184, 1983.

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16. Deal, K., Sethi, S. P., and Thompson, G. L., A Bilinear-Quadratic Differential Game in Advertising, Control Theory in Mathematical Economics, Edited by P. T. Lui and J. G. Sutinen, Manuel Dekker, New York, NY, pp. 91–109, 1979. 17. Chintagunta, P. K., and Jain, D. C., Empirical Analysis of a Dynamic Duopoly Model of Competition, Journal of Economics and Management Strategy, Vol. 4, pp. 109–131, 1995. 18. Horsky, D., and Mate, K., Dynamic Advertising Strategies of Competing Durable Good Producers, Marketing Science, Vol. 7, pp. 356–367, 1988. 19. Sorger, G., Competitive Dynamic Advertising: A Modification of the Case Game, Journal of Economics Dynamics and Control, Vol. 13, pp. 55–80, 1989. 20. Little, J. D. C., Aggregate Advertising Models: The State of the Art, Operations Research, Vol. 27, pp. 629–667, 1979. 21. Sethi, S. P., Optimal Control of the Vidale-Wolfe Advertising Model, Operations Research, Vol. 21, pp. 998–1013, 1973. 22. Erickson, G. M., Dynamic Models of Advertising Competition, Kluwer, Norwell, Massachusetts, 2003. 23. Dockner, E. J., Jorgensen, S., Long, N. V., and Sorger, G., Differential Games in Economics and Management Science, Cambridge University Press, Cambridge, UK, 2000. 24. Kimball, G. E., Some Industrial Application of Military Operations Research Methods, Operations Research, Vol. 5, pp. 201–204, 1957. 25. Gihman, I. I., and Skorohod, A. V., Stochastic Differential Equations, Springer-Verlag, New York, NY, 1972. 26. Gould, J. P., Diffusion Processes and Optimal Advertising Policy, Microeconomic Foundations of Employment and Inflation Theory, Edited by E. S. Phelps et al., W. W. Norton, New York, NY, pp. 338–368, 1970. 27. Sethi, S. P., and Thompson, G. L., Optimal Control Theory: Applications to Management Science and Economics, Kluwer, Norwell, Massachusetts, 2000. 28. Zwillinger, D., Handbook of Differential Equations, Academic Press, San Diego, California, 1998. 29. Polyanin, A. D., and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall, CRC, Boca Raton, Florida, 2003. 30. Fruchter, G., Oligopoly Advertising Strategies with Market Expansion, Optimal Control Applications and Methods, Vol. 20, pp. 199–211, 1999. 31. Erickson, G. M., Advertising Strategies in a Dynamic Oligopoly, Journal of Marketing Research, Vol. 32, pp. 233–237, 1995.