In this section, we will study a competitive extension of the Sethi ad- vertising model discussed in Sect.12.3. This will give us a stochastic differential game, for which we aim to obtain a feedback Nash equilib- rium by using a dynamic programming approach developed in Sect.12.1.
We should note that this approach can also be used to obtain feedback Nash equilibria in deterministic differential games as an alternative to the maximum principle approach developed in Sect.13.2.2.
Specifically, we consider a duopoly market in a mature product cate- gory where total sales are distributed between two firms, labeled as Firm 1 and Firm 2, which compete for market share through advertising ex- penditures. We let Xt denote the market share of Firm 1 at time t, so that the market share of Firm 2 is (1−Xt).Let U1t and U2t denote the advertising effort rates of Firms 1 and 2, respectively, at time t. Using the subscript i ∈ {1,2} to reference the two firms, let ri > 0 denote the advertising effectiveness parameter, πi >0 denote the sales margin, ρi > 0 denote the discount rate, and ci > 0 denote the cost parame- ter so that the cost of advertising effort u by Firm i is ciu2. Further, let δ >0 be the churn parameter, Zt be the standard one-dimensional Wiener process, and σ(x) be the diffusion coefficient function as defined in Sect.12.3. Then, in view of the competition between the firms, Prasad and Sethi (2004) extend the Sethi model dynamics in (12.42) as the Itˆo
13.3. A Feedback Nash Stochastic Differential Game in Advertising393 stochastic differential equation
dXt = [r1U1t#
1−Xt−δXt−r2U2t#
Xt+δ(1−Xt)]dt+σ(Xt)dZt, X(0) =x0∈[0,1]. (13.30) We formulate the optimal control problem faced by the two firms as
maxU1≥0
V1(x0) =E ∞
0 e−ρ1t4
π1Xt−c1U12t$ dt
, (13.31)
maxU2≥0
V2(x0) =E ∞
0 e−ρ2t4
π2(1−Xt)−c2U22t$ dt
, (13.32) subject to (13.30). Thus, each firm seeks to maximize its expected, discounted profit stream subject to the market share dynamics.
To find the feedback Nash equilibrium solution, we form the Hamilton-Jacobi-Bellman (HJB) equations for the value functionsV1(x) and V2(x) :
ρ1V1 = max
U1≥0{H1(x, U1, U2, Vx1) + (σ(x))2Vxx1 /2}
= max
U1≥0{π1x−c1U12+Vx1[r1U1√
1−x−r2U2√
x−δ(2x−1)]
+(σ(x))2Vxx1 /2}, (13.33)
ρ2V2 = max
U2≥0{H2(x, U1, U2, Vx2) + (σ(x))2Vxx2 /2}
= max
U2≥0{π2(1−x)−c2U22 +Vx2[r1U1√
1−x−r2U2√
x−δ(2x−1)]
+(σ(x))2Vxx2 /2}, (13.34)
where the Hamiltonians are as defined in (13.14). We use the first-order conditions for Hamiltonian maximization to obtain the optimal feedback advertising decisions
U1∗(x) =Vx1(x)r1√
1−x/2c1 and U2∗(x) =−Vx2(x)r2√
x/2c2. (13.35) Since it is reasonable to expect that Vx1 ≥0 and Vx2 ≤0,these controls will turn out to be nonnegative as we will see later.
394 13. Differential Games Substituting (13.35) in (13.33) and (13.34), we obtain the Hamilton- Jacobi equations
ρ1V1 = π1x+ (Vx1)2r12(1−x)/4c1+Vx1Vx2r22x/2c2
−Vx1δ(2x−1) + (σ(x))2Vxx1/2, (13.36) ρ2V2 = π2(1−x) + (Vx2)2r22x/4c2+Vx1Vx2r21(1−x)/2c1
−Vx2δ(2x−1) + (σ(x))2Vxx2 /2. (13.37) As in Sect.12.3, we look for the following forms for the value functions
V1 =α1+β1x and V2 =α2+β2(1−x). (13.38) These are inserted into (13.36) and (13.37) to determine the unknown coefficients α1, β1, α2, and β2. Equating the coefficients of x and the constants on both sides of (13.36) and the coefficients of (1−x) and the constants on both sides of (13.37), the following four equations emerge, which can be solved for the unknownsα1, β1, α2,and β2:
ρ1α1 = β21r12/4c1+β1δ, (13.39) ρ1β1 = π1−β21r12/4c1−β1β2r22/2c2−2β1δ, (13.40) ρ2α2 = β22r22/4c2+β2δ, (13.41) ρ2β2 = π2−β22r22/4c2−β1β2r12/2c1−2β2δ. (13.42) Let us first consider the special case of symmetric firms, i.e., when π =π1 =π2, c=c1 =c2, r =r1 = r2, and ρ =ρ1 =ρ2, and therefore α = α1 = α2, β = β1 = β2. The four equations in ((13.39)–(13.42)) reduce to the following two:
ρα=β2r2/4c+βδ and ρβ=π−3β2r2/4c−2βδ. (13.43) There are two solutions for β.One is negative, which clearly makes no sense. Thus, the remaining positive solution is the correct one. This also allows us to obtain the correspondingα. The solution is
α = [(ρ−δ)(W −#
W2+ 12Rπ) + 6Rπ]/18Rρ, (13.44) β = (#
W2+ 12Rπ−W)/6R, (13.45)
where R = r2/4c and W = ρ+ 2δ. With this the value functions in (13.38) are defined, and the controls in (13.35) for the case of symmetric firms can be written as
u∗1(x) = β1r1√ 1−x
2c1 = βr√ 1−x
2c andu∗2(x) = β2r2√ x
2c2 = βr√ x 2c ,
13.4. A Stackelberg Differential Game of Cooperative Advertising 395 which are clearly nonnegative as required.
We return now to the general case of asymmetric firms. For this, we re-express equations ((13.39)–(13.42)) in terms of a single variable β1, which is determined by solving the quartic equation
3R12β41+ 2R1(W1+W2)β31 + (4R2π2−2R1π1−W12+ 2W1W2)β21 + 2π1(W1−W2)β1−π21 = 0. (13.46) This equation can be solved explicitly to give four roots. We will find that only one of these is positive, and select it as our value of β1.With that, other coefficients can be obtained by solving for α1 and β2 and then, in turn, α2,as follows:
α1 = β1(β1R1+δ)/ρ1, (13.47) β2 = (π1−β21R1−β1W1)/2β1R2, (13.48) α2 = β2(β2R2+δ)/ρ2, (13.49) where R1 =r12/4c1, R2=r22/4c2, W1 =ρ1+ 2δ, andW2 =ρ2+ 2δ.
It is worthwhile to mention that firmi’s advertising effectiveness pa- rameterriand advertising cost parametercimanifest themselves through Ri=ri2/4ci.This would suggest thatRi is a measure of firm i’s adver- tising power. This can be seen more clearly in Exercise 13.6 involving two firms that are identical in all other aspects except that R2 > R1. Specifically in that exercise, you are asked to use Mathematica or an- other suitable software program to solve (13.46) to obtain β1 and then obtain the coefficients α1, α2, and β2 by using (13.47)–(13.49), when ρ1 = ρ2 = 0.05, π1 = π2 = 1, δ = 0.01, R1 = 1, R2 = 4, x0 = 0.5, and σ(x) = #
0.5x(1−x).Figure 13.1 represents a sample path of the market share of the two firms with this data.
It is noteworthy to see that both firms are identical except in their advertising powers R1 and R2. With R2 > R1,firm 2 is more powerful and we see that this results in its capture of an increasing share of the market average over time beginning with exactly one half of the market at time 0.