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4Contributions to Game Theory and Management, XI, 207-223
A Dynamic Oligopoly Marketing Model of Advertising
Lihong Shi and Ovanes Petrosian
St. Petersburg State University 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia E-mail: [email protected] E-mail: petrosian. ovanes@yandex. ru
Abstract We consider a dynamic oligopoly advertising model for both non-cooperative and cooperative setting. Feedback Nash equilibrium strategies and cooperative strategies are found to determine the optimal advertising efforts of each firm for both setting respectively. Besides, depending upon the cooperative strategies, imputation is introduced as an optimal allocation of joint payoff and Imputation Distribution Procedure is used to guarantee the time consistency for cooperation.
Keywords: Advertising competition, Optimal control, Dynamic programming, Time consistency.
1. Introduction
267 billions of dollars were spent in advertising by the world's 100 largest advertisers in 2016 according to the statistics from the source: Ad Age Datacenter, it's easy to imagine that a company couldn't survive in the market without advertising. Hence, advertising as a strategy for market share competition has been studied in many marketing models, so as we are doing in this paper, where we are concentrating on the dynamic advertising model. Here each firm's market share depends on its own and its competitors' current and past advertising expenditures. Competition between firms is formulated by the noncooperative differential game as it was done by (Erickson, 2003), (Sorger, 1989), (JFergensen and Zaccour, 2004), (Prasad and Sethi, 2004), (Naik et al., 2008) and (Prasad et al., 2009). In this paper we derived a dynamic oligopoly advertising model from (Prasad et al., 2009), in the paper the Sethi model was extended to model awareness of auto brands with churn term, which is the extension of the decay of market share term in monopoly-models capturing forgetting and noise. Closed-loop Nash equilibrium concept is used to obtain the optimal advertising expenditure for noncooperative game. Similar to this paper's work, we also consider the noncooperative dynamic game and use feedback Nash equilibrium concept to obtain the optimal advertising expenditure, but more than that, the cooperative dynamic game which wasn't considered by the previous work was elicited to this model and mainly studied in our paper.
Noncooperative game theory deals with strategic interactions among multiple decision makers with the objective functions depending on the choices of all the players and suggests solution concepts for a case when players do not cooperate or make any arrangements about their actions. A player cannot simply optimize her own objective function independent from the choices of the other players. In 1950 and 1951 in (Nash, 1950),(Nash, 1951) by John Nash, such a solution concept was introduced, which is now called the Nash equilibrium. For differential game models Nash equilibrium can be defined in an open-loop strategies or in closed-loop strategies. For both types consult (Basar and Olsder, 1999). In this paper we use the
approach with feedback strategies as the most preferable for the game theoretical problems.
The cooperative dynamic game theory offers socially convenient and group efficient solutions to different decision problems involving strategic actions. One of the fundamental questions in the theory of cooperative dynamic games is the formulation of optimal behavior for players. A characteristic function of a coalition is an essential concept in the theory of cooperative games. This function is defined as indicated in (Chander and Tulkens, 1995) as total payoff of players from coalition S in Nash equilibrium in the game with following set of players: coalition S (acting as one player) and players from the set I \ S. A computation of Nash equilibrium fully described in (Basar and Olsder, 1995) is necessary for this approach. A set of imputations or a solution of the game is determined by the characteristic function as the set of individually rational vectors. To guarantee the cooperation of all players will holds during the game, which means the cooperation is time consistent. Notion of time consistency was formalized mathematically by Petrosyan in the paper (Petrosjan, 1977). In the next paper on time consistency (Petrosyan and Danilov, 1979) L. Petrosyan defined the notion of imputation distribution procedure (IDP), which is used to compose a time consistent cooperative solution or single imputation. Later on, L. Petrosyan defined the notion of strong time consistency in the paper (Petrosyan, 1993), it was introduced to guarantee the time-consistency for a set-value cooperative solutions, such as Core, Nucleus etc.. See recent publications on this topic in (Petrosyan and Yeung, 2006), (Jorgensen and Yeung, 1999) and (Jorgensen et al., 2003). The property of time consistency introduced by Petrosyan in (Petrosyan, 1993) is examined for Shap-ley value.
The paper is organized as follows. In the next section, we formulate a dynamic oligopoly marketing model corresponding to the model from (Prasad et al., 2009). In section 3, feedback Nash equilibrium strategies for noncooperative setting are presented. In section 4, cooperative case is considered, Shapley value is used as cooperative solution corresponding to the optimal cooperative strategies and IDP is introduced to guarantee the time consistency property. In section 5, a numerical example is used to illustrate the results.
2. Initial model
Consider a n-firm oligopoly market in a mature product category so that the total sales of the category are relatively stable (Prasad et al., 2009). The advertising efforts as strategies are used by firms to compete on the oligopoly market, each firm tries to increase its market share while the competitors try to minimize it using the advertisement efforts. Denote by xi(t) the market share of firm i e I = {1,n} at time t and n > 1.
Market share dynamics of firm i has the following form:
jei
with xi(0) = zi(0), where zj(0) is a positive constant. Payoff function of firm i e I is defined by its profit:
where r is the discount rate of firm i. We transformed this differential game model to the corresponding discrete time game with finite horizon T using Finite Difference Method (FDM). Introduce the equally distributed grid points (tj )j=0N given by tj = jh where N is an integer and the spacing h is given by h = T/N and xk — x(tk) for all k € {0, ...,N}. As a result, the market share dynamics of firm i for dynamic game model became
xk+1 = h(PiukJ 1 - xk - -1- £ PjukJ 1 - xk) - (hS - 1 )xk + h<5—, (2) ' n — 1 v ' n — 1 z—' J V j —
jei
with x0 = z0, where z0 is a positive constant.
Payoff function of firm i in dynamic game model became:
N
K0(xo,uo)=YJ ™ix| - (u|)2. (3)
l=o
Full list of notations is presented in Table 1.
Table 1. List of variables and parameters
Notation Explanation
€ [0,1] Market share of firm i € I = {1, ...,n} at stage k. > 0 Advertising effort rate by firm i at stage k. pi > 0 Advertising effectiveness parameter of firm i. S > 0 Churn parameter.
mi > 0 Industry sales multiplied by the per unit
profit margin of firm i. C(ui(t)) Cost of advertising of firm i, parameterized as (ui(t))2.
There is also a logical consistency requirement that the sum of market shares should be equal to one on each stage, i.e.
Y, xk = 1, fork e {0,...,N} ,
iei
where this requirement can be checked by summing up right-side of all firms' motion equations in (2).
3. Noncooperative game model
Consider a noncooperative case, where each firm acts individually. According to (Yeung and Petrosyan, 2012) one can determine the noncooperative strategies as a feedback Nash equilibrium for the game defined by (2) and (4).
Definition 1. A feedback Nash equilibrium (Yeung, 1994) is an n-tuple of feedback strategies {ukfor k e {0, ...,N}, if for every possible initial condition xk of player i the following inequality holds
i^fc / k -fc -fc k -fc -k \ ^ Tfk j k -fc -fc -fc -k -k \ w- T
Ki (x ; u1 ; ...; ui-1; Ui ; ui+1; ...; UnJ ^ Ki (X ; U1; ...; ui-1; Ui ; ui+1; •••^Un)^ Vi e 1 .
If there exists a feedback Nash equilibrium solution with the set of strategies {wk, for k e {0,..., N}, i e {1,..., n} } for (2) and (4), denote an equilibrium payoff function for firm i e I over stage k to N by
N
(x) = mx - )2,
l = k
where xk = x.
A frequently used way to characterize and derive a feedback Nash equilibrium of the game is a dynamic programming.
Theorem 1. A set of strategies {uk, fork e{0, ...,N} , i € {1,..., n}} provides a feedback Nash equilibrium to the game defined by (2) and (4) if there exist functions Vik(x), for i e {1,..., n} and k e {0,..., N}, such that the following recursive relations are satisfied:
Vk (x) = max {miXk - (uk)2 + V/+1 (xk, w^, ..., Uk-1, «<k, Uk+1,..., <)}
uk >0 (4)
= mixk — (uk)2 + Vjk+1(xk,uk,..., uk-i,uk, uk+i,..., uk),
VN +1(x) = 0.
This theorem can be proved in the same way as it was done in (Yeung and Petrosyan, 2012).
Proposition 1. The game equilibrium value functions in (5) are
Vk (xk) = £ Akxk + Bk (i), i e {1,..., n} k e {0,..., N}
iei
where Ak, Bk (i) are determined from the relations: Ak = mi - (Gk+1 Zi)2 - Ak+1 (Jh - 1), Ak = -2(Gk+1Zj)2 - Ak+1 (Jh - 1), j = i, j e I \ i,
Bk(i) ^[2(Gk+1Zi)2 + ] + Bk+1(i) - (Gk+1 Zi)2,
ie/ n
with initial conditions AN+1 = BN+1(i) =0 i e {1,...,n} and where Gk+1 =
nAk+1 -Ej6/ Ak+S Zi = ^.
Corresponding feedback Nash equilibrium strategies are
uk = Gk+1 Z^ 1 - xk, k e {0,..., N}, i e {1,..., n} . (5)
Proof see Appendix.
Substituting (6) into the original dynamics (2), we derive trajectory corresponding to the feedback Nash equilibrium, i e {1, ...,n} and k e {0, ...,N},
xk+1 = npk+1(1 - xk) - £[pk+1(1 - xk)] + n, (6)
je/
where P/+ = 2Gk+1(Zi)2 + ^n1- The noncooperative trajectory vectors for any stage are denoted by
x == [x^,..., x^n ], k == 0, NN.
4. Cooperative game
Consider the case when all firms agree to cooperate. Maximizing the playerssT)™ joint payoff guarantees grand coalition optimality in a game where payoffs are transferable. To maximize their joint payoff the following optimization should be performed:
N
Y,K°(x0 ,u0) = YH - (ui)2 ^ max (7)
iei iei i=o
subject to
x1+1 = h(Piukj 1 - x)--Y pjukJ 1 - x))-(hS-1)x) + h5— x0 = z0.
n — 1 v — — 1 z—' J V j —
jei
This is a discrete time optimization problem. Dynamic programming (Bellman, 1957) is used to solve it. Suppose that the Bellman function has the following form:
Wk (x))= max jv v mixi - (U)2! ,k g{0,...,n} . (8)
1 ' ' n K iei i=k )
Theorem 2. Assume that there exists function Wk (x), such that the following recursive relations are satisfied:
Wk (x) )= max J Y mi x) - (u) )2 + Wk+1 (xk )! ,k g{0,...,N } , (9)
ukItti J
where WN +1(xN +i) = 0.
We use the following linear functions, observing that they satisfy the Bellman equations, to solve this recursive relations.
Proposition 2. Bellman function Wk (xk) in (9) can be computed in the form
Wk (x) ) = V C) x) + D), (10)
iei
where C.f, Di G {1,—} and k G {0,N} satisfy:
C) = mi - (Qk+1 Zi)2 - C^^h - 1),
D) = {Qk+1 Zt)2 + ^^ + DÎ+1,
with initial conditions CN+1 = DN+1 =0 i G {1,n} .
The optimal cooperative strategies can be obtained as follows:
cpk = Qk+1Z^ 1 - xk, Vi G I,
where Qk+1 = nCk+1 - j Ck+1.
Substituting optimal cooperative strategies in (2), cooperative trajectory of i e I is computed as
xc'fc+1 = 2nQk+1Zi2(1 - xc'fc) - £ 2Qk+1Z2(1 - xf) - (hJ - 1)xc'fc + hJ,
je/ n
where k e {0,..., N}.
It's worth mentioning that the advertising effort rate of each firm is equal to zero when all the firms are symmetric, because of the whole market acts as a monopoly when all symmetric firms decide to cooperate. Once consumers want to buy goods they have to buy it from one of the firm even if firms do not spend on advertising.
Proposition 3. If all firms are identical: mi = m, pi = p for i e I, then the optimal strategies of players are
= 0,Vi e I, k e{0,...,N}.
Proof see Appendix.
4.1. Characteristic function
For each coalition S C I define the values of characteristic function as it was done in (Chander and Tulkens, 1995):
[Eie/Ki(xc; u), S = I, V(S; xc) = < V(S, xc), S C I, (11)
[0, S = 0,
where V(S, xc) is defined as total payoff of players from coalition S in feedback Nash equilibrium u = (u1,...,un) in the game with the following set of players: coalition S (acting as one player) and players from the set |I \ Si.e. in the game of |I \ S| + 1 players. | S| = 1
Vk ({i} ; xk) = a * Vik(xC'fc)= AkxC'fc + Bk (i) , i e I \ S,
we consider a parameter a in this case which is a given discount of characteristic function of |S | = 1 under the cooperation . Parameter a guaranties the property of
a
0«,< Vk(I;xk)
Eie/ Vik(xC'fc)
S=I
defined as
Vk(I; xk) = Wk(xk). SCI
from coalition S in Nash feedback equilibrium in the game with |I \ S + 1| players.
Corresponding system of Bellman equations for coalition S and players i € I \ S will have the form:
Vk(S,xk) = m«^[m¿xk - (uk)2] + Vk+1(S,xk+1) I ,
uk es lás J (i2)
Vk({i} ,xk) = max {m¿xk - (u?)2 + Vk+1({i} ,xk+1)} , i € I \ S,
uk
with initial conditions VN +1(S, xN +1) = 0, VN +1({i} , xf+1) = 0 i € I \ S.
Proposition 4. Bellman function for coalition S and players i € I \ S can be computed in the form:
V k (S, xk ) = £ x? + Ek,
¿e/
Vk({i} , x?) = £ A?xk + B(i)k, i € I \ S,
ie/
where parameters C^, B(i)k are defined as:
C? = mi - (Rk+1Zi)2 - C?+1(Jh - 1), i e S, Ck = -2<5k+1Rk+1z2 - C?+1(Jh - 1), i e I \ S,
E = £ (Rk+1Zi)2 + £ 2(5k+1Rk+1Z2 + £ ^^ + Ek+1,
ies ie/\s ie/ n
Ak = mi - ((5k+1Zi)2 - Ak+1 (Jh - 1), i e I \ S, Ak = -2(Gk+1Zj)2 - Ak+1(Jh - 1), j e I \ (S u i), a? = -2(5k+1Rk+1Z2 - A^1 (Jh - 1), f e S,
B(i)k = £ 2((5k+1Zi)2 + £ 2(5k+1Rk+1 Z2 + £ ^^ + B(i)fc+1-
ie/\s ies ie/ n
- ((5k+1 Zi)2,i e I \ S,
with initial conditions Cf+1 = 0 +1 = 0 AN+1 = 0 and B(i)N+1 = 0, where
Rk+1 = n(7k+1 - £ C^1, je/
(5k+1 = nAk+1 - V Ak+1. i i j
je/
Optimal strategies used by players from coalition S and players i e I \ S can be computed as follows:
up = (nC^1 - £ Cf1) , i e s,
je/ ( )
uk = M*+1 - £ Af1) , i e I \ s.
je/ 2(n 1)
Value of characteristic function for coalition S C I is calculated as follows:
V k (S; xc) = V k (S,xc'k),
where k G {0,..., N}.
4.2. Imputation set
Main problem in the theory of cooperative games is to construct realizable principle of allocation of total payoff Wk (xk) among players. An optimality principle can be introduced if all players agree to allocate the total cooperative payoff along the cooperative trajectory according to an imputation.
Definition 2. Imputation is a vector £k(xk) = [£k(xc{k), (xnk)] for k = 0,N, which satisfies the conditions
^ (xf) > Vk ({i} ; xk),i G I,
E $ (xC,k ) = v k(i; xk).
iei
4.3. Imputation distribution procedure
Following the continuous-time analysis of (Petrosyan and Yeung, 2006) for cooperative differential games, we formulate a discrete-time version of the Imputation Distribution Procedure so that the agreed upon imputations of definition 2 can be realized. By Bk(xcC'k) denote the payment that the firm i will receive at stage k
i ck xi k
Payment scheme involving Bk(xCk) constitutes an IDP in the sense that the imputation of firm i over the stages from k to N:
N
ik(xci'k) = Y Bjj(xCj). (13)
j=k
4.4. Time consistency
The property of time-consistency was introduced in 1977 by Petrosyan (Petrosjan, 1977). To ensure stability in dynamic cooperation over time a stringent condition is required: the specific agreed-upon optimality principle must be maintained at any instant of time throughout the game along the optimal state trajectory. This condition is known as time consistency.
Definition 3. Solution H(xk ,k) is called time consistent if for any imputation £,k(x^ ) G H(xk,k) exists IDP Bk(x'c' ), k = 0,N such that it satisfies:
N
Y,Bk(xf) G H(xk,k), k N
Y.B'k (xc,k ) = £ (xc,k). k
Theorem 3. If optimality principle satisfies the following conditions:
(i) u1 = f, for i G I and k = 0, N, is the set of group optimal strategies for the this game,
(ii) Bf (xf) = Bf (xf), /0r i G I and k = 07N, wAere
Bf(xf)= £Í (xc'fc) - Cfc+1(xc'fc+1), (14)
then [^i (xf ),..., en (xf )] ¿s ¿s time consistent.
This theorem was proved in (Petrosjan and Yeung, 2012).
4.5. Cooperative solution
In this paper, we consider Shapley value as a cooperative solution Shk (xc'fc) (Shapley, 1953), which is calculated in the following way:
Shi(xc'fc)= £ —-|S|I.(|5,1-— • [V1 (S; xc'fc) - V1 (S \ i; xc'fc)j, (15)
for k = 0, N.
5. Numerical example
We consider a specific three-firms oligopoly case within the stage interval N=8. To illustrate our model, let the parameters for each firms to be p = [0.4,0.5,0.3], h = 0.4, a = 0.7, J = 0.09, m = [0.6,1,1.2] and for the initial conditions z0 = [0.3, 0.5, 0.2].
Fig.l. shows feedback Nash equilibrium strategies for noncooperative game and Fig.2. shows how the market share of each firm changes corresponding to their advertising expenditure.
Fig. 1. Feedback Nasli equilibrium strategies of noncooperative game
-1
-
-
-
012345678 Stage
Fig. 2. Noncooperative trajectory
Noncooperative outcomes are obtained respectively, which is shown in Fig.3. The optimal cooperative strategies and cooperative trajectory are illustrated in Fig.4. and Fig.5.
O o
c2
o Z
— Vi -V2 -V3
3 4 5
Stage
Fig. 3. Noncooperative outcomes
5
4
3
0
0
6
7
8
0.16 0.14 0.12 0.1 P 0.08 0.06 0.04 0.02 0
012345678
Stage
Fig. 4. The optimal strategies of cooperative game
On the Fig.C. Shaploy value calculated using characteristic function (11) is presented.
In order to guarantee time consistency for cooperative game. Fig. 7. shows IDP for Shaploy value.
Fig. 5. Cooperative trajectory
Fig. 6. Sliapley value
0.55
0.5
0.45 0.4
Ph
a 0.35 0.3 0.25 0.2
012345678 Stage
Fig. 7. Imputation Distribution Procedure
6. Conclusion
In this paper, we considered an oligopoly dynamic marketing model for the nonco-operative game as in the previous paper, and obtained a feedback Nash equilibrium strategies for the advertising expenditures of all firms. Then the cooperation for all oligopolistic firms is introduced, the optimal cooperative strategies are obtained to determine the advertising expenditures, which maximize the profit for this cooperation. in order to maintain this cooperation for all firms. An agreement, namely Imputation Distribution Procedure, is obtained to guarantee no firms will collude and form a smaller coalition under this agreement.
A simple numerical example is presented in this paper to illustrate the results for both cooperative and noncooperative game obtained of the oligopoly dynamic marketing model.
Appendix 1. First appendix
The Bellman equation for firm i is given by
—B
Vi(X) = maxim^ - )2 + V/+1 (x^1 )} uk >o
Invoking (2) and Vik(x?) = Eie/ Ax? + Bk(i), for i G ^d k G {0, ...,N}, then the above Bellman equation becomes
E A?x? + Bk(i) = max Jmixk - (u?)2 + E Ak+1x,k+1 + Bk+1(i)|
ie/ Ui -0 I ie/ J
= E Ak+1[h( ^ pi u?^17-? - ¿r E pj uk T17^)- (i6)
Ui - l ie/ je/
- (hJ - 1)x? + hJr] + Bk+1 (i) + mix? - (u?)2. n
Performing the indicated maximization in above yields
-2uk + n^nA?+1 - ^ yr-X? E A?+1 = 0,
je/
for i G ^d k G {0,..., N}. Then the feedback Nash equilibrium of firm i can be obtained in the form
u? = max{0, [nA?+1 - E A?+1 ^T^}.
je/ 2(n T)
Anticipating that the controls will be shown to be positive, substituting it into (16), collecting the terms together, then the parameters A? Mid Bk (i) can be expressed as
A? = mi - (Gk+1 Zi)2 - A?+1 (Jh - 1) ?k+1Zj)2 - A?
A? = -2(Gk+1Zj)2 - j1 (Jh - 1), j = i, j G I \ i,
_ Ak+1hJ
Bk(i) ^[2(Gk+1Zi)2 + ] + Bk+1(i) - (Gk+1 Zi)2,
ie/ n
where G?+1 = nA?+1 - Eje/ A?+\ Zi = ^.
2. Second appendix
Substituting (10) into (9) we receive
Wfc(xfc)= m«J y[miX? - (u?)2] + Wfc+1(xfcU
uf.....ie/ J
>E Ckxk + Dk = max \ EKx? - (u?)2] + E C^x,^1 +
—: uf —; —;
ie/ ' n We/ ie/
(17)
W N +1(xn +1) = 0.
substitute the right-side of (2) into (17). Performing the indicated maximization in (17) yields
-2uf + (nCf+1 - Y j1) 1 xf = 0. i i ^ 3 n_ i jei
The optimal cooperative strategies can be obtained in the form
= max{0, (nCf+ - £ Cf+1)}, jei ( )
for i G ^d k G {0, ...,N} .
Anticipating that the cooperative strategies will be shown to be positive, then substituting them into (17), hence Cf and Df are defined as:
Cf = mi - (Qk+1 Zi)2 - Cf+1(Jh - 1), D = (Qf+1 Zi)2 + + Df+1.
3. Third appendix
The asymmetric optimal cooperative strategies can be obtained in the form
tf = max{0, (nCf+ - £ Cf+1)^^n^}, jei (n )
for i G ^d k G {0, ...,N} .
Suppose that all firms are identical: mi = m, pi = P f°r i G I, then the related parameters in Proposition 2.3 can be rewritten as follows
Cf = m - (Qf+1Z)2 - Cf+1 (Jh - 1),
Df = (Qf+1 Z )2 + + D*+1,
with initial conditions CN+1 = DN+1 = 0 and where Z = I) and Qf+1 =
nCf+1 3eI Cf+1. It shows that the value of Cf is equal for Vi G I, k G {0,..., N}, which means the optimal cooperative strategies equal to 0 at any stage when all firms are identical.
4. Fourth appendix
Substituting the Bellman function of coalition S and individual players into (12) respectively and corresponding to (2). Performing the indicated maximization in (12) yields
-2uk + (nC^1 - E ) = 0, i G S,
G
n — 1
jei
-2uk + Mk+1 - E Af1) V^F = 0, i G I \ S.
j ' n - 1 jei
The optimal strategies can be obtained in the form
ufk = max{0, (nCf1 - £ CJ+1) }, i G S,
je/ 2(n i)
uk = max{0, («/+ - £ ALk+1) ^P^Z^}, i G I \ S.
2(n - 1)
je/ v 7
Anticipating that the optimal strategies will be shown to be positive and substituting them into (12), then collecting the terms together we obtain the parameters
Ck, / ,E^d B(i)k
Ck = mi - (Rk+1 Zi)2 - C?k+1 (¿h - 1), i G S, Ck = -2(5k+1 Rk+1Z2 - Ck+1 (¿h - 1), i G I \ S,
Ek = £ (Rk+1 Zi)2 + £ 2Gk+1Rk+1Zi2 + £ ^^ + Ek+1, ie/\s ie/ n
/ = mi - (Gk+1 Zi)2 - Af1 (¿h - 1), i G I \ S,
Ak = -2(<5k+1 Zj)2 - j (¿h - 1), j G I \ (S u i),
Af = -2(5k+1 R^Z^ - Af1 (¿h - 1), f g S,
B(i)k = ]T 2((5k+1Zi)2 + £2<5k+1 Rk+1Zi2 + £ ^^ + JB(i)k+1-ie/\s ies ie/ n
- (<5k+1 Zi)2, i G I \ S.
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