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Tripoly Stackelberg game model: One leader versus two followers
Applied Mathematics and Computation 328 (2018) 301–311
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Tripoly Stackelberg game model: One leader versus two followers S.S. Askar a b
Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
Keywords: Bounded rationality Stackelberg game Stability Bifurcation Chaos Control
a b s t r a c t This paper is devoted to introduce and study a Stackelberg game consisting of three competed firms. The three firms are classified as a leader which is the first firm and the other two firms are called the followers. A linear inverse demand function is used. In addition a quadratic cost based on an actual and announced quantities is adopted. Based on bounded rationality, a three-dimensional discrete dynamical system is constructed. For the system, the backward induction is used to solve the system and to get Nash equilibrium. The obtained results are shown that Nash equilibrium is unique and its stability is affected by the system’s parameters by which the system behaves chaotically due to bifurcation and chaos appeared. Some numerical experiments are performed to portrays such chaotic behavior. A control scheme is used to return the system back to its stability state and is supported by some simulations. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Dynamic games carried out in economic market are important because of the complex dynamic characteristics appeared in such games. Two factors in those games should be highlighted, the behaviors on which consumers use and the reactions performed by the competitors. Recent studies have investigated the dynamic behavior of those games such as monopolistic [1], duopolistic [2–9], and tripolistic one [10,11]. Literature has shown that only few studies are dealing with the tripolistic models. The current paper introduces and studies a tripolistic model based on Stackelberg assumptions. In 1934, the German economist Heinrich Freiherr von Stackelberg introduced the so-called Market Structure and Equilibrium that in turns described the Stackelberg model. Stackelberg game is a strategic game in economics in which one firm of the competed firms is called the leader and moves first and then the other firms which they are called the follower firms move sequentially. In such games, it must be a firm that has some sort of advantage making it leads the market and takes the first move. In other words, the leader must have power of commitment. To be the firm that has the first move you should be the incumbent monopolistic firm of the commodity and your competitors (followers) are new entrants. In repeated Stackelberg game if the follower feels that he/she might be punished he/she may adopt a punishment strategy as well to hurt the leader unless the leader chooses a non-optimal strategy in the current period. One has to mention here that both Cournot and Stackelberg games are similar as both use the same decisional variables that are the quantities. However, in the Stackelberg, there is a crucial advantage for the leader as he/she starts the first move in the market. Furthermore,
E-mail addresses: [email protected], [email protected] https://doi.org/10.1016/j.amc.2018.01.041 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
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the perfect information is also important in such game because the follower must watch the quantity chosen by the leader otherwise Stackelberg game will be reduced to Cournot. In formulating economic models one should be aware that expectation is an important role while describing such models. There are several techniques that may be used by the firms to adjust and update their productions. For instance, Puu’s strategy [12] and bounded rationality mechanism [13]. The current paper adopts the bounded rationality approach to introduce a triopoly Stackelberg game and then investigate the complex dynamic characteristics of the game. The proposed game consists of three firms that are competed. The first firm is chosen to lead the game (the leader) and the other two firms (the followers) follow the first move carried out by the leader. The three firms update their productions so as to maximize their profits in the subsequent stages. Depending on the expected marginal profits the firms adjust their decisional variables and of course they use local information about their outputs. In other words, the stackelberg firms maximize their profits based on the local information of their decisional variables. Recent trends in tackling economic competition have adopted networks of networks or multilayer networks. They are aptly used to describe such competition or social systems. In literature, multilayer networks have been devoted to evolutionary games and in particular to the evolution of cooperation. For instance, in [29] a colloquium has been introduced to highlight some important aspects of cooperation under adverse conditions, as well as the cooperative interaction among groups in evolutionary game theory. In [30], a concise and informative review on coevolutionary games has discussed. Other interesting articles are available in literature [31–34]. Outline of the current paper is as follows. In Section 2, a triopoly game is introduced based on Stackelberg assumptions and bounded rationality mechanism. All the equilibrium positions of the system described the game are obtained. Analysis of the local stability of the system’s equilibrium points are illustrated. Some numerical simulations to confirm and verify the obtained theoretical results are performed in Section 2. In Sections 3, we apply control scheme on the proposed system to suppress the chaotic behavior appeared in the system. Finally, some concluding remarks are shown. 2. Model We suppose that there are three firms labelled by i, i = 1, 2, 3. Those firms produce the same commodities so as to sale them in the market. Assuming that firm 1 leads the competition (Stackelberg leader) among the firms and firm 2 and firm 3 are two followers. Decisional variables of the three firms are production quantities and are denoted by qi,t , i = 1, 2, 3 which are updated according to discrete time steps, t ∈ Z+ . The price is determined by the total supply Q = 3i=1 qi,t at time period t, and is given by,
pt = a − b
3
qi,t
(1)
i=1
where, a and b are positive constants. Indeed, qi,t should be positive because negative quantities do not have any valuable a meaning in economy and therefore positive price is guaranteed when Q < . From an economic prospective, the announced b products are different than the actual products for firm i at time t. For the leader the announced products may be less than the actual products but for the followers the converse may be true. Using different strategy of productions, we assume that the firms use the following cost function.
Ci (qi ) = ci (qi,t − θi )2 , i = 1, 2, 3
(2)
and ci , i = 1, 2, 3 is a positive parameter. As in [14], θ i , i = 1, 2, 3 refers to the announced plan products of firm i, i = 1, 2, 3 respectively. Now, the profit of each firm can be written as,
πi,t (q1 , q2 , q3 ) = qi,t
a−b
3
qi,t
− ci (qi,t − θi )2 , i = 1, 2, 3
(3)
i=1
The target of each firm is to maximize its profit and this requires to estimate the marginal profit as follows, 3 ∂πi,t = a + 2 c i θi − 2 ( b + c i ) q i − b qi , i = 1, 2, 3; i = j ∂ qi j=i
So, to maximize the profit we solve the system of algebraic equations,
(4)
∂πi,t = 0, i = 1, 2, 3 then we get the following, ∂ qi
2c1 θ1 [b(3b + 4c2 ) + 4c3 (b + c2 )] − 2b[c2 θ2 (b + 2c3 ) + c3 θ3 (b + 2c2 )] + a(b + 2c2 )(b + 2c3 ) , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )] 2c2 θ2 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2b[c1 θ1 (b + 2c3 ) + c3 θ3 (b + 2c1 )] + a(b + 2c1 )(b + 2c3 ) = , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )] 2c3 θ3 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2b[c1 θ1 (b + 2c2 ) + c2 θ2 (b + 2c1 )] + a(b + 2c1 )(b + 2c2 ) = , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )]
q1,t = q2,t q3,t
(5)
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As one can see from (5), the quantities that maximize the profits depend on the planning products θ i , i = 1, 2, 3. Let us now summarize our Stackelberg game as follows: Phase i: in this phase a subgame perfect Nash equilibrium is solved by the backward induction as follows. • First, the leader (which is the first firm) takes its first move by announcing its plan production θ 1 . • Due to this action by the leader, both followers (firm 2 and firm 3) will choose the plan outputs θ 2 and θ 3 respectively, in order to get the highest profits. Substituting (5) in π 2,t and π 3,t then solve
∂π2,t ∂π3,t = 0 and = 0 we get, ∂θ2 ∂θ3
θ2 = 2 [a(b + 2c1 )(b + 2c3 ) − 2bc1 θ1 (b + 2c3 ) − 2bc3 θ3 (b + 2c1 )], θ3 = 3 [a(b + 2c1 )(b + 2c2 ) − 2bc1 θ1 (b + 2c2 ) − 2bc2 θ2 (b + 2c1 )], ( b + c 2 ) [b( 3 b + 4 c 3 ) + 4 c 1 ( b + c 3 ) ] , 2 = 2b c2 (b + 2c1 ) 8c32 (b + c1 ) + 2bc2 c3 (5b + 6c1 ) + c2 b2 (3b + 4c1 ) + bc3 (3b + 4c1 )[c3 (3b + 4c1 ) + 2b(2b + 3c1 )] + b3 (3b + 3c1 )2 3 =
( b + c 3 ) [b( 3 b + 4 c 1 ) + 4 c 2 ( b + c 1 ) ]
2b c3 (b + 2c1 ) 8c22 (b + c1 ) + 2bc2 c3 (5b + 6c1 ) + c3 b2 (3b + 4c1 ) + bc2 (3b + 4c1 )[c2 (3b + 4c1 ) + 2b(2b + 3c1 )] + b3 (3b + 3c1 )2
(6) Phase ii: Sunstituting (5) and (6) in
θ1 =
π 1,t then solve
∂π1,t = 0 gives the plan products as functions of the game’s parameters only as follows. ∂θ1
a − b(B2 + D2 ) − 2(bA1 − c1 )H2 , −2c1 + b(B1 + D1 ) − 2(bA1 − c1 )H1
θ2 = F1 − F2 θ1 , θ3 = L1 + L2 θ1
(7)
where, A1 = L1 =
2c1 [b(3b + 4c2 ) + 4c3 (b + c2 )] + 2bc2 F2 (b + 2c3 ) − 2bc3 L2 (b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
a3 (b + 2c1 )[(b + 2c2 ) − 2bc2 2 (b + 2c1 )(b + 2c3 )] , 1 − 4b2 c2 c3 2 3 (b + 2c1 )2
L2 =
2bc1 3 2bc2 2 (b + 2c1 )(b + 2c3 ) − (b + 2c2 )2 1 − 4b2 c2 c3 2 3 (b + 2c1 )2
,
F1 = 2 (b + 2c1 )[a(b + 2c3 ) − 2bc3 L1 ], F2 = 2b2 [c1 (b + 2c3 ) + c3 L2 (b + 2c1 )], B1 = B2 = D1 = D2 = H1 = H2 =
−2c2 F2 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2bc1 (b + 2c3 ) − 2bc3 L2 (b + 2c1 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
2c2 F1 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2bc3 L1 (b + 2c1 ) + a(b + 2c1 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
2c3 L2 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2bc1 (b + 2c2 ) + 2bc2 F2 (b + 2c1 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
2c3 L1 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2bc2 F1 (b + 2c1 ) + a(b + 2c1 )(b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
2c1 [b(3b + 4c2 ) + 4c3 (b + c2 )] − 2bc3 L2 (b + 2c2 ) + 2bc2 F2 (b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
,
,
−2bc2 F1 (b + 2c3 ) − 2bc3 L1 (b + 2c2 ) + a(b + 2c2 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
Now the equilibrium point becomes, q∗1 = q∗2 = q∗3 =
[2c1 (b(3b + 4c2 ) + 4c3 (b + c2 )) + 2bc2 F2 (b + 2c3 ) − 2bc3 L2 (b + 2c2 )]θ1 − 2bc2 F1 (b + 2c3 ) − 2bc3 L1 (b + 2c2 ) + a (b + 2c2 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
[−2c2 F2 (b(3b + 4c1 ) + 4c3 (b + c1 )) − 2bc1 (b + 2c3 ) − 2bc3 L2 (b + 2c1 )]θ1 + 2c2 F1 (b(3b + 4c1 ) + 4c3 (b + c1 )) − 2bc3 L1 (b + 2c1 ) + a (b + 2c1 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
[2c3 L2 (b(3b + 4c1 ) + 4c2 (b + c1 )) − 2bc1 (b + 2c2 ) + 2bc2 F2 (b + 2c1 )]θ1 + 2c3 L1 (b(3b + 4c1 ) + 4c2 (b + c1 )) − 2bc2 F1 (b + 2c1 ) + a (b + 2c1 )(b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
(8) which depends only on the game’s parameters.
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2.1. Tripoly Stackelberg dynamic model The bounded rationality mechanism as a tool of expectation is described elsewhere (see, for example [13,15]). The bounded rational players update their quantities by using local estimate of the marginal profit. Based on an adjustment function, k(qi,t ) with such estimation of the marginal profits the updated quantities become well-defined by the following map.
T : qi,t+1 = qi,t +k(qi,t )
∂πi,t , i = 1, 2, 3 ∂ qi,t
(9)
The adjustment function k(qi,t ) represents the quantity variation produced by the firm i according to its computed marginal profit
∂πi,t . We assume that the adjustment function is linear as k(qi,t ) = kqi,t . Using (4) and (9) the following 3∂ qi,t
dimensional discrete dynamical system is built.
q1,t+1 = q1,t + kq1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] q2,t+1 = q2,t + kq2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )] q3,t+1 = q3,t + kq3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )]
(10)
The equilibrium position of (10) can be obtained by solving algebraically the following system,
q1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] = 0, q2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )] = 0, q3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )] = 0 that gives the following equilibria,
E0 = ( 0, 0, 0 ), E1 =
E4 =
E5 =
E6 =
a + 2 c 1 θ1 , 0, 0 , E2 = 2 ( b + c1 )
(11)
a + 2 c 2 θ2 0, , 0 , E3 = 2 ( b + c2 )
a + 2 c 3 θ3 0, 0, , 2 ( b + c3 )
4(b + c2 )c1 θ1 − 2(bθ2 − a )c2 + ab 4(b + c1 )c2 θ2 − 2(bθ1 − a )c1 + ab , ,0 , 4 ( b + c 2 ) c 1 + b( 3 b + 4 c 2 ) 4 ( b + c 1 ) c 2 + b( 3 b + 4 c 1 )
4(b + c3 )c1 θ1 − 2(bθ3 − a )c3 + ab 4(b + c1 )c3 θ3 − 2(bθ1 − a )c1 + ab , 0, , 4 ( b + c 3 ) c 1 + b( 3 b + 4 c 3 ) 4 ( b + c 1 ) c 3 + b( 3 b + 4 c 1 )
4(b + c3 )c2 θ2 − 2(bθ3 − a )c3 + ab 4(b + c2 )c3 θ3 − 2(bθ2 − a )c2 + ab 0, , , 4 ( b + c 3 ) c 2 + b( 3 b + 4 c 3 ) 4 ( b + c 2 ) c 3 + b( 3 b + 4 c 2 )
E7 = (q∗1 , q∗2 , q∗3 ),
(12)
Proposition 1. The boundary equilibrium points E0 , E1 , E2 , E3 , E4 , E5 and E6 are unstable. The proof is straightforward using the Jacobian matrix as follows.
1 + k ( a + 2 c 1 θ1 ) 0 J (E 0 ) = 0
0 1 + k ( a + 2 c 2 θ2 ) 0
0 0 1 + k ( a + 2 c 3 θ3 )
for which the eigenvalues are,
λ1 = 1 + k(a + 2c1 θ1 ), λ2 = 1 + k(a + 2c2 θ2 ), λ3 = 1 + k(a + 2c3 θ3 ) For simplicity we take a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1 which give θ1 = −1.4053, θ2 = 0.9055 and θ3 = 1.0113. This means that both λ2 and λ3 are greater than 1 and hence E0 is unstable. The Jacobain at E1 is given by,
⎡ ⎢ ⎢
J (E 1 ) =⎢ ⎢
⎣
1 − k ( a + 2 c 1 θ1 ) 0
−kb(a + 2c1 θ1 ) 2 ( b + c1 ) 2(b + c1 )(a + 2c2 θ2 ) − b(a + 2c1 θ1 ) 1+k 2 ( b + c1 )
0
whose eigenvalues take the following form,
0
−kb(a + 2c1 θ1 ) 2 ( b + c1 )
⎤
⎥ ⎥ ⎥ ⎥ ⎦ 2(b + c1 )(a + 2c3 θ3 ) − b(a + 2c1 θ1 ) 1+k 2 ( b + c1 ) 0
S.S. Askar / Applied Mathematics and Computation 328 (2018) 301–311
305
λ1 = 1 − k(a + 2c1 θ1 ), a(b + 2c1 ) + 4(b + c1 )c3 θ3 − 2bc1 θ1 λ2 = 1 + k , 2 ( b + c1 ) a(b + 2c1 ) + 4(b + c1 )c2 θ2 − 2bc1 θ1 λ3 = 1 + k 2 ( b + c1 ) For the same values of the parameters, we get λ2 and λ3 are greater than 1 and hence E1 is unstable. Similarly, we can prove that E2 , E3 , E4 , E5 and E6 are unstable. Proposition 2. The Nash equilibrium point E7 is locally stable. At a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055 and θ3 = 1.0113, NE becomes (0.03404114469, 0.6316829651, 0.7244982873) and the Jacobian at this point is as follows,
1 + 0.00555828757k −0.3158414826k J (E 7 ) = −0.3158414826k
−0.01702057234k 1 − 0.6105426220k −0.3158414826k
−0.01702057234k −0.3158414826k 1 − 0.6466411708k
For the above Jacobian, the characteristic equation takes,
λ3 +A1 λ2 +A2 λ + A3 = 0,
A1 = −3 + 1.567197600k, A2 =, 0.5075467970(k − 1.184198510 )(k − 4.991380214 ), A3 = 0.01382795552(k − 0.886382313543441 )(k − 2.44467571604969 )(k − 33.3733409877390 )
(13)
Using (13) then Routh–Hurwiz conditions [28] becomes in the following form,
1 := 1 + A1 +A2 +A3 = 0.01382795552k(k + 0.0 0 02689187061 )(k − 0.0 0 02689187061 ), 2 := 1 − A1 +A2 −A3 = 0.01382795552(k − 1.77276462645378 )(k − 4.88935143393347 )(66.7466819742770 − k ), 3 := 1 − A23 = 0.0 0 01912123539k(3.40336095813135 − k )(33.3010327110959 − k )× 2 × (33.4449953351344 − k )(k −3.25940903460231k + 4.32455028832382 ), 2 2 2 4 := (1 − A3 ) −(A2 −A1 A3 ) = 3.656216428 × 10−8 k2 (k − 1.78953702861959 )(k − 3.14465615802274 )(k − 3.33105839834034 )× × (k − 6.01989470153874 )(k − 28.5082345692734 )(k − 33.9463477209331 )× × (k − 34.2598611601244 )(k − 35.8180063662506 )× 2 −9 × (k −2.288191274 × 10−9 k + 1.224571002 × 10 ) (14) For the NE to be local stable the above conditions should be greater than zero (i > 0, i = 1, 2, 3, 4 ). It is clear to see that the above conditions depend on the parameter k and hence the stability region is identified by it. Simulation has shown that i > 0 when k < 1.7895. 2.2. Numerical simulation In such games, numerical simulation is required to detect more characteristics of the behavior of system (10) and at the same time to confirm the results obtained above. It is really important to illustrate numerically the stability of NE of system (10). To do that we start our simulation by taking the following value parameters: a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1. Substituting these parameters in (7) gives θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113. It is easy to check that q∗1 > θ1 , q∗2 < θ2 and q∗3 < θ3 which means that the equilibrium quantity of firm 1 is more than its announced quantity but for the followers this is not achieved. With these parameters, Fig. 1a shows the stability behavior of system (10). As one can see that the NE is locally stable for certain interval of the parameter k. It seems that the first firm (the game’s leader) gets unstable first due to using these vales of the parameters. The reason for that may be the leader’s announced quantity is less than the equilibrium quantity or it may be the cost adopted by the leader has an impact on the behavior of the leader. In addition, simulation has shown that when we use equal and low costs (c1 = 0.01, c2 = .01, c3 = 0.01) for the three firms the situation of the leader becomes better and the stability of the NE lasts when increasing the parameter k. Its stability becomes more than the stability of the followers however its equilibrium quantity is still much more its announced one. Furthermore, we have given the behavior of the firms against the whole system’s parameters. Fig. 1b shows that the three firms are unstable with respect to the cost c1 . Figs. 2–5 confirm the instability of the system against those parameters. Fig. 6a and 6b show the strange attractor of the system at the same parameters and for different values of the parameter k. 3. Chaos control Chaos appeared in economic systems is not expected and has harmful influences on the system. The analysis investigated in the previous section shows that the system’s parameters have been considered as the main reason that lead to chaos.
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Fig. 1. (a) The bifurcation diagram for the solution of system (10) versus the system’s parameter k and a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113. (b) The bifurcation diagram of the quantity q1 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1.
Fig. 2. (a) The bifurcation diagram of the quantity q2 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1. (b) The bifurcation diagram of the quantity q3 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1.
It is shown that the costs parameters affect the system stabilization and then chaotic behavior appears. It is important to suppress such chaotic behavior and control the system. Avoiding such behavior has helped to introduce a wide variety of control approaches in order to control on some level chaotic behavior of such systems. In [16], the OGY or (Ott–Grebogi– Yorke) approach has been presented to return a system from chaotic to stable state. It has been applied to Kopel duopolistic game model in [17]. This approach depends on making change on the systems’s parameters so as to force it to return back to the stable state. Modification of the straight-line stabilization approach has been applied to a duopolistic model in [18]. In [19], a self-adaptive proportional control method in economic chaotic system is discussed. A time-delayed feedback control approach to control chaos in an economic model has been studied in [20]. For more control approaches readers are advised to see literature [21–27]. In this section, we assume that the system may be controlled depending on a control function ut that may be incorporated to the system as follows.
qt+1 = f (qt , ut )
(15)
Here, to stabilize our model we introduce the following control function,
ui,t = (1 − η )(qi,t+1 − qi,t ), i = 1, 2, 3
(16)
S.S. Askar / Applied Mathematics and Computation 328 (2018) 301–311
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Fig. 3. (a) The bifurcation diagram of the quantity q1 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1. (b) The bifurcation diagram of the quanitity q2 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1.
Fig. 4. (a) The bifurcation diagram of the quantity q3 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1. (b) The bifurcation diagram of the quantity q1 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02.
on which η represents the control parameter. Using (16) in (10), we get the following controllable system.
q1,t+1 = q1,t + q2,t+1 = q2,t + q3,t+1 = q3,t +
k
η k
η k
η
q1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] q2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )]
(17)
q3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )]
Now, we need to suppress chaos appeared in the system (10) and force the system to return back to its stabilization. To do so let us assume k ≈ 1.783 and a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113 , then the Jacobian of system (17) is given by,
⎡
1−
0.07283443268
⎢ η ⎢ −0.5631453634 η ⎣
J ( E7 ) = ⎢ ⎢
−0.6458902231
η
−0.03034768049
η
1−
1.171342355
η
−0.6458902231
η
−0.03034768049
⎤
⎥ ⎥ ⎥ ⎥ η 1.550136535 ⎦ η
−0.5631453634 1−
η
(18)
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Fig. 5. (a) The bifurcation diagram of the quantity q2 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02. (b) The bifurcation diagram of the quantity q3 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02.
Fig. 6. Strange attractor diagram for the system (10) at the parameters: a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.1.0113: (a) k = 0.1, (b) k = 2.3.
For the above Jacobian, the characteristic equation takes,
λ3 +B1 λ2 +B2 λ + B3 = 0, B1 = −3+ B2 = 3−
2.794313323
η
5.588626645
B3 = −1.+
η
,
+
2.794313323
η
1.613536440
−
η
2
1.613536440
η2
(19)
, +
0.07838121695
η3
To study the stability of E7 , we recall Routh–Hurwiz conditions [28] with (19) one gets the following conditions.
1 := 1 + A1 +A2 +A3 =
1 × 10−9
+
0.07838121695
,
η η3 8(η − 0.026712938 )(η − 0.364670044 )(η − 1.005773679 ) 2 := 1 − A1 +A2 −A3 = , η3 3 := 1 − A23
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309
Fig. 7. Bifurcation diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113 and k ≈ 1.783.
Fig. 8. Strange attractor diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113, μ = 1.01 and k ≈ 1.783.
=
5.588626646(η − 0.053311415 )(η − 0.053541882 )(η − 0.523893888 )(η2 − 1.343845239η + 0.735125917 )
1
η6
,
× 1 × 10−8 × (η − 0.0049778896 )(η − 0.052047935 )(η − 0.052519989 ) η12 ×(η − 0.062543416 )(η − 0.296184556 )(η − 0.535265714 )(η − 0.566993233 ) ×(η − 0.996347093 )(η2 − 7.388319168η + 4.951908979 × 109 ) (20)
4 := (1 − A23 )2 − (A2 − A1 A3 )2 =
These conditions guarantee the local stability of E7 if i > 0, i = 1, 2, 3, 4 and this is happened when η > 1.0058 as shown numerically in Fig. 7. This means that when the control parameter goes large the system becomes out of the chaotic region
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Fig. 9. Strange attractor diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113, μ = 0.99 and k ≈ 1.783.
and hence stable which means that the proposed control scheme is successful. Fig. 8 and 9 show the attractor of the three quantities for different values of the control parameter η. 4. Conclusion The current paper has introduced and studied the complex dynamic characteristics of a tripolistic Stackelberg game. The myopic mechanism (bounded rationality) has used to study the dynamic of the game by introducing a three-dimensional discrete system. The obtained results have confirmed that Nash equilibrium point of the game is unique and its stability is affected by bifurcation and chaos. In addition, the control scheme has returned the game’s system back to it stability state when appropriate values of the parameters were used. Numerical simulations have been used and confirmed the obtained results. Acknowledgment The author would like to extend their sincere appreciation to the Deanship of Scientific Research at king Saud University for its funding this Research group NO (RG-1435-054). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Tripoly Stackelberg game model: One leader versus two followers S.S. Askar a b
Department of Statistics and Operations Researches, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
a r t i c l e
i n f o
Keywords: Bounded rationality Stackelberg game Stability Bifurcation Chaos Control
a b s t r a c t This paper is devoted to introduce and study a Stackelberg game consisting of three competed firms. The three firms are classified as a leader which is the first firm and the other two firms are called the followers. A linear inverse demand function is used. In addition a quadratic cost based on an actual and announced quantities is adopted. Based on bounded rationality, a three-dimensional discrete dynamical system is constructed. For the system, the backward induction is used to solve the system and to get Nash equilibrium. The obtained results are shown that Nash equilibrium is unique and its stability is affected by the system’s parameters by which the system behaves chaotically due to bifurcation and chaos appeared. Some numerical experiments are performed to portrays such chaotic behavior. A control scheme is used to return the system back to its stability state and is supported by some simulations. © 2018 Elsevier Inc. All rights reserved.
1. Introduction Dynamic games carried out in economic market are important because of the complex dynamic characteristics appeared in such games. Two factors in those games should be highlighted, the behaviors on which consumers use and the reactions performed by the competitors. Recent studies have investigated the dynamic behavior of those games such as monopolistic [1], duopolistic [2–9], and tripolistic one [10,11]. Literature has shown that only few studies are dealing with the tripolistic models. The current paper introduces and studies a tripolistic model based on Stackelberg assumptions. In 1934, the German economist Heinrich Freiherr von Stackelberg introduced the so-called Market Structure and Equilibrium that in turns described the Stackelberg model. Stackelberg game is a strategic game in economics in which one firm of the competed firms is called the leader and moves first and then the other firms which they are called the follower firms move sequentially. In such games, it must be a firm that has some sort of advantage making it leads the market and takes the first move. In other words, the leader must have power of commitment. To be the firm that has the first move you should be the incumbent monopolistic firm of the commodity and your competitors (followers) are new entrants. In repeated Stackelberg game if the follower feels that he/she might be punished he/she may adopt a punishment strategy as well to hurt the leader unless the leader chooses a non-optimal strategy in the current period. One has to mention here that both Cournot and Stackelberg games are similar as both use the same decisional variables that are the quantities. However, in the Stackelberg, there is a crucial advantage for the leader as he/she starts the first move in the market. Furthermore,
E-mail addresses: [email protected], [email protected] https://doi.org/10.1016/j.amc.2018.01.041 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.
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the perfect information is also important in such game because the follower must watch the quantity chosen by the leader otherwise Stackelberg game will be reduced to Cournot. In formulating economic models one should be aware that expectation is an important role while describing such models. There are several techniques that may be used by the firms to adjust and update their productions. For instance, Puu’s strategy [12] and bounded rationality mechanism [13]. The current paper adopts the bounded rationality approach to introduce a triopoly Stackelberg game and then investigate the complex dynamic characteristics of the game. The proposed game consists of three firms that are competed. The first firm is chosen to lead the game (the leader) and the other two firms (the followers) follow the first move carried out by the leader. The three firms update their productions so as to maximize their profits in the subsequent stages. Depending on the expected marginal profits the firms adjust their decisional variables and of course they use local information about their outputs. In other words, the stackelberg firms maximize their profits based on the local information of their decisional variables. Recent trends in tackling economic competition have adopted networks of networks or multilayer networks. They are aptly used to describe such competition or social systems. In literature, multilayer networks have been devoted to evolutionary games and in particular to the evolution of cooperation. For instance, in [29] a colloquium has been introduced to highlight some important aspects of cooperation under adverse conditions, as well as the cooperative interaction among groups in evolutionary game theory. In [30], a concise and informative review on coevolutionary games has discussed. Other interesting articles are available in literature [31–34]. Outline of the current paper is as follows. In Section 2, a triopoly game is introduced based on Stackelberg assumptions and bounded rationality mechanism. All the equilibrium positions of the system described the game are obtained. Analysis of the local stability of the system’s equilibrium points are illustrated. Some numerical simulations to confirm and verify the obtained theoretical results are performed in Section 2. In Sections 3, we apply control scheme on the proposed system to suppress the chaotic behavior appeared in the system. Finally, some concluding remarks are shown. 2. Model We suppose that there are three firms labelled by i, i = 1, 2, 3. Those firms produce the same commodities so as to sale them in the market. Assuming that firm 1 leads the competition (Stackelberg leader) among the firms and firm 2 and firm 3 are two followers. Decisional variables of the three firms are production quantities and are denoted by qi,t , i = 1, 2, 3 which are updated according to discrete time steps, t ∈ Z+ . The price is determined by the total supply Q = 3i=1 qi,t at time period t, and is given by,
pt = a − b
3
qi,t
(1)
i=1
where, a and b are positive constants. Indeed, qi,t should be positive because negative quantities do not have any valuable a meaning in economy and therefore positive price is guaranteed when Q < . From an economic prospective, the announced b products are different than the actual products for firm i at time t. For the leader the announced products may be less than the actual products but for the followers the converse may be true. Using different strategy of productions, we assume that the firms use the following cost function.
Ci (qi ) = ci (qi,t − θi )2 , i = 1, 2, 3
(2)
and ci , i = 1, 2, 3 is a positive parameter. As in [14], θ i , i = 1, 2, 3 refers to the announced plan products of firm i, i = 1, 2, 3 respectively. Now, the profit of each firm can be written as,
πi,t (q1 , q2 , q3 ) = qi,t
a−b
3
qi,t
− ci (qi,t − θi )2 , i = 1, 2, 3
(3)
i=1
The target of each firm is to maximize its profit and this requires to estimate the marginal profit as follows, 3 ∂πi,t = a + 2 c i θi − 2 ( b + c i ) q i − b qi , i = 1, 2, 3; i = j ∂ qi j=i
So, to maximize the profit we solve the system of algebraic equations,
(4)
∂πi,t = 0, i = 1, 2, 3 then we get the following, ∂ qi
2c1 θ1 [b(3b + 4c2 ) + 4c3 (b + c2 )] − 2b[c2 θ2 (b + 2c3 ) + c3 θ3 (b + 2c2 )] + a(b + 2c2 )(b + 2c3 ) , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )] 2c2 θ2 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2b[c1 θ1 (b + 2c3 ) + c3 θ3 (b + 2c1 )] + a(b + 2c1 )(b + 2c3 ) = , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )] 2c3 θ3 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2b[c1 θ1 (b + 2c2 ) + c2 θ2 (b + 2c1 )] + a(b + 2c1 )(b + 2c2 ) = , 2[b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )]
q1,t = q2,t q3,t
(5)
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As one can see from (5), the quantities that maximize the profits depend on the planning products θ i , i = 1, 2, 3. Let us now summarize our Stackelberg game as follows: Phase i: in this phase a subgame perfect Nash equilibrium is solved by the backward induction as follows. • First, the leader (which is the first firm) takes its first move by announcing its plan production θ 1 . • Due to this action by the leader, both followers (firm 2 and firm 3) will choose the plan outputs θ 2 and θ 3 respectively, in order to get the highest profits. Substituting (5) in π 2,t and π 3,t then solve
∂π2,t ∂π3,t = 0 and = 0 we get, ∂θ2 ∂θ3
θ2 = 2 [a(b + 2c1 )(b + 2c3 ) − 2bc1 θ1 (b + 2c3 ) − 2bc3 θ3 (b + 2c1 )], θ3 = 3 [a(b + 2c1 )(b + 2c2 ) − 2bc1 θ1 (b + 2c2 ) − 2bc2 θ2 (b + 2c1 )], ( b + c 2 ) [b( 3 b + 4 c 3 ) + 4 c 1 ( b + c 3 ) ] , 2 = 2b c2 (b + 2c1 ) 8c32 (b + c1 ) + 2bc2 c3 (5b + 6c1 ) + c2 b2 (3b + 4c1 ) + bc3 (3b + 4c1 )[c3 (3b + 4c1 ) + 2b(2b + 3c1 )] + b3 (3b + 3c1 )2 3 =
( b + c 3 ) [b( 3 b + 4 c 1 ) + 4 c 2 ( b + c 1 ) ]
2b c3 (b + 2c1 ) 8c22 (b + c1 ) + 2bc2 c3 (5b + 6c1 ) + c3 b2 (3b + 4c1 ) + bc2 (3b + 4c1 )[c2 (3b + 4c1 ) + 2b(2b + 3c1 )] + b3 (3b + 3c1 )2
(6) Phase ii: Sunstituting (5) and (6) in
θ1 =
π 1,t then solve
∂π1,t = 0 gives the plan products as functions of the game’s parameters only as follows. ∂θ1
a − b(B2 + D2 ) − 2(bA1 − c1 )H2 , −2c1 + b(B1 + D1 ) − 2(bA1 − c1 )H1
θ2 = F1 − F2 θ1 , θ3 = L1 + L2 θ1
(7)
where, A1 = L1 =
2c1 [b(3b + 4c2 ) + 4c3 (b + c2 )] + 2bc2 F2 (b + 2c3 ) − 2bc3 L2 (b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
a3 (b + 2c1 )[(b + 2c2 ) − 2bc2 2 (b + 2c1 )(b + 2c3 )] , 1 − 4b2 c2 c3 2 3 (b + 2c1 )2
L2 =
2bc1 3 2bc2 2 (b + 2c1 )(b + 2c3 ) − (b + 2c2 )2 1 − 4b2 c2 c3 2 3 (b + 2c1 )2
,
F1 = 2 (b + 2c1 )[a(b + 2c3 ) − 2bc3 L1 ], F2 = 2b2 [c1 (b + 2c3 ) + c3 L2 (b + 2c1 )], B1 = B2 = D1 = D2 = H1 = H2 =
−2c2 F2 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2bc1 (b + 2c3 ) − 2bc3 L2 (b + 2c1 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
2c2 F1 [b(3b + 4c1 ) + 4c3 (b + c1 )] − 2bc3 L1 (b + 2c1 ) + a(b + 2c1 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
2c3 L2 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2bc1 (b + 2c2 ) + 2bc2 F2 (b + 2c1 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
2c3 L1 [b(3b + 4c1 ) + 4c2 (b + c1 )] − 2bc2 F1 (b + 2c1 ) + a(b + 2c1 )(b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
2c1 [b(3b + 4c2 ) + 4c3 (b + c2 )] − 2bc3 L2 (b + 2c2 ) + 2bc2 F2 (b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
,
,
,
−2bc2 F1 (b + 2c3 ) − 2bc3 L1 (b + 2c2 ) + a(b + 2c2 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
Now the equilibrium point becomes, q∗1 = q∗2 = q∗3 =
[2c1 (b(3b + 4c2 ) + 4c3 (b + c2 )) + 2bc2 F2 (b + 2c3 ) − 2bc3 L2 (b + 2c2 )]θ1 − 2bc2 F1 (b + 2c3 ) − 2bc3 L1 (b + 2c2 ) + a (b + 2c2 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
[−2c2 F2 (b(3b + 4c1 ) + 4c3 (b + c1 )) − 2bc1 (b + 2c3 ) − 2bc3 L2 (b + 2c1 )]θ1 + 2c2 F1 (b(3b + 4c1 ) + 4c3 (b + c1 )) − 2bc3 L1 (b + 2c1 ) + a (b + 2c1 )(b + 2c3 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
[2c3 L2 (b(3b + 4c1 ) + 4c2 (b + c1 )) − 2bc1 (b + 2c2 ) + 2bc2 F2 (b + 2c1 )]θ1 + 2c3 L1 (b(3b + 4c1 ) + 4c2 (b + c1 )) − 2bc2 F1 (b + 2c1 ) + a (b + 2c1 )(b + 2c2 )
2 b2 [2b + 3(c1 + c2 + c3 )] + 4bc3 (c1 + c2 ) + 4c1 c2 (b + c3 )
(8) which depends only on the game’s parameters.
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2.1. Tripoly Stackelberg dynamic model The bounded rationality mechanism as a tool of expectation is described elsewhere (see, for example [13,15]). The bounded rational players update their quantities by using local estimate of the marginal profit. Based on an adjustment function, k(qi,t ) with such estimation of the marginal profits the updated quantities become well-defined by the following map.
T : qi,t+1 = qi,t +k(qi,t )
∂πi,t , i = 1, 2, 3 ∂ qi,t
(9)
The adjustment function k(qi,t ) represents the quantity variation produced by the firm i according to its computed marginal profit
∂πi,t . We assume that the adjustment function is linear as k(qi,t ) = kqi,t . Using (4) and (9) the following 3∂ qi,t
dimensional discrete dynamical system is built.
q1,t+1 = q1,t + kq1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] q2,t+1 = q2,t + kq2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )] q3,t+1 = q3,t + kq3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )]
(10)
The equilibrium position of (10) can be obtained by solving algebraically the following system,
q1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] = 0, q2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )] = 0, q3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )] = 0 that gives the following equilibria,
E0 = ( 0, 0, 0 ), E1 =
E4 =
E5 =
E6 =
a + 2 c 1 θ1 , 0, 0 , E2 = 2 ( b + c1 )
(11)
a + 2 c 2 θ2 0, , 0 , E3 = 2 ( b + c2 )
a + 2 c 3 θ3 0, 0, , 2 ( b + c3 )
4(b + c2 )c1 θ1 − 2(bθ2 − a )c2 + ab 4(b + c1 )c2 θ2 − 2(bθ1 − a )c1 + ab , ,0 , 4 ( b + c 2 ) c 1 + b( 3 b + 4 c 2 ) 4 ( b + c 1 ) c 2 + b( 3 b + 4 c 1 )
4(b + c3 )c1 θ1 − 2(bθ3 − a )c3 + ab 4(b + c1 )c3 θ3 − 2(bθ1 − a )c1 + ab , 0, , 4 ( b + c 3 ) c 1 + b( 3 b + 4 c 3 ) 4 ( b + c 1 ) c 3 + b( 3 b + 4 c 1 )
4(b + c3 )c2 θ2 − 2(bθ3 − a )c3 + ab 4(b + c2 )c3 θ3 − 2(bθ2 − a )c2 + ab 0, , , 4 ( b + c 3 ) c 2 + b( 3 b + 4 c 3 ) 4 ( b + c 2 ) c 3 + b( 3 b + 4 c 2 )
E7 = (q∗1 , q∗2 , q∗3 ),
(12)
Proposition 1. The boundary equilibrium points E0 , E1 , E2 , E3 , E4 , E5 and E6 are unstable. The proof is straightforward using the Jacobian matrix as follows.
1 + k ( a + 2 c 1 θ1 ) 0 J (E 0 ) = 0
0 1 + k ( a + 2 c 2 θ2 ) 0
0 0 1 + k ( a + 2 c 3 θ3 )
for which the eigenvalues are,
λ1 = 1 + k(a + 2c1 θ1 ), λ2 = 1 + k(a + 2c2 θ2 ), λ3 = 1 + k(a + 2c3 θ3 ) For simplicity we take a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1 which give θ1 = −1.4053, θ2 = 0.9055 and θ3 = 1.0113. This means that both λ2 and λ3 are greater than 1 and hence E0 is unstable. The Jacobain at E1 is given by,
⎡ ⎢ ⎢
J (E 1 ) =⎢ ⎢
⎣
1 − k ( a + 2 c 1 θ1 ) 0
−kb(a + 2c1 θ1 ) 2 ( b + c1 ) 2(b + c1 )(a + 2c2 θ2 ) − b(a + 2c1 θ1 ) 1+k 2 ( b + c1 )
0
whose eigenvalues take the following form,
0
−kb(a + 2c1 θ1 ) 2 ( b + c1 )
⎤
⎥ ⎥ ⎥ ⎥ ⎦ 2(b + c1 )(a + 2c3 θ3 ) − b(a + 2c1 θ1 ) 1+k 2 ( b + c1 ) 0
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λ1 = 1 − k(a + 2c1 θ1 ), a(b + 2c1 ) + 4(b + c1 )c3 θ3 − 2bc1 θ1 λ2 = 1 + k , 2 ( b + c1 ) a(b + 2c1 ) + 4(b + c1 )c2 θ2 − 2bc1 θ1 λ3 = 1 + k 2 ( b + c1 ) For the same values of the parameters, we get λ2 and λ3 are greater than 1 and hence E1 is unstable. Similarly, we can prove that E2 , E3 , E4 , E5 and E6 are unstable. Proposition 2. The Nash equilibrium point E7 is locally stable. At a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055 and θ3 = 1.0113, NE becomes (0.03404114469, 0.6316829651, 0.7244982873) and the Jacobian at this point is as follows,
1 + 0.00555828757k −0.3158414826k J (E 7 ) = −0.3158414826k
−0.01702057234k 1 − 0.6105426220k −0.3158414826k
−0.01702057234k −0.3158414826k 1 − 0.6466411708k
For the above Jacobian, the characteristic equation takes,
λ3 +A1 λ2 +A2 λ + A3 = 0,
A1 = −3 + 1.567197600k, A2 =, 0.5075467970(k − 1.184198510 )(k − 4.991380214 ), A3 = 0.01382795552(k − 0.886382313543441 )(k − 2.44467571604969 )(k − 33.3733409877390 )
(13)
Using (13) then Routh–Hurwiz conditions [28] becomes in the following form,
1 := 1 + A1 +A2 +A3 = 0.01382795552k(k + 0.0 0 02689187061 )(k − 0.0 0 02689187061 ), 2 := 1 − A1 +A2 −A3 = 0.01382795552(k − 1.77276462645378 )(k − 4.88935143393347 )(66.7466819742770 − k ), 3 := 1 − A23 = 0.0 0 01912123539k(3.40336095813135 − k )(33.3010327110959 − k )× 2 × (33.4449953351344 − k )(k −3.25940903460231k + 4.32455028832382 ), 2 2 2 4 := (1 − A3 ) −(A2 −A1 A3 ) = 3.656216428 × 10−8 k2 (k − 1.78953702861959 )(k − 3.14465615802274 )(k − 3.33105839834034 )× × (k − 6.01989470153874 )(k − 28.5082345692734 )(k − 33.9463477209331 )× × (k − 34.2598611601244 )(k − 35.8180063662506 )× 2 −9 × (k −2.288191274 × 10−9 k + 1.224571002 × 10 ) (14) For the NE to be local stable the above conditions should be greater than zero (i > 0, i = 1, 2, 3, 4 ). It is clear to see that the above conditions depend on the parameter k and hence the stability region is identified by it. Simulation has shown that i > 0 when k < 1.7895. 2.2. Numerical simulation In such games, numerical simulation is required to detect more characteristics of the behavior of system (10) and at the same time to confirm the results obtained above. It is really important to illustrate numerically the stability of NE of system (10). To do that we start our simulation by taking the following value parameters: a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1. Substituting these parameters in (7) gives θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113. It is easy to check that q∗1 > θ1 , q∗2 < θ2 and q∗3 < θ3 which means that the equilibrium quantity of firm 1 is more than its announced quantity but for the followers this is not achieved. With these parameters, Fig. 1a shows the stability behavior of system (10). As one can see that the NE is locally stable for certain interval of the parameter k. It seems that the first firm (the game’s leader) gets unstable first due to using these vales of the parameters. The reason for that may be the leader’s announced quantity is less than the equilibrium quantity or it may be the cost adopted by the leader has an impact on the behavior of the leader. In addition, simulation has shown that when we use equal and low costs (c1 = 0.01, c2 = .01, c3 = 0.01) for the three firms the situation of the leader becomes better and the stability of the NE lasts when increasing the parameter k. Its stability becomes more than the stability of the followers however its equilibrium quantity is still much more its announced one. Furthermore, we have given the behavior of the firms against the whole system’s parameters. Fig. 1b shows that the three firms are unstable with respect to the cost c1 . Figs. 2–5 confirm the instability of the system against those parameters. Fig. 6a and 6b show the strange attractor of the system at the same parameters and for different values of the parameter k. 3. Chaos control Chaos appeared in economic systems is not expected and has harmful influences on the system. The analysis investigated in the previous section shows that the system’s parameters have been considered as the main reason that lead to chaos.
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Fig. 1. (a) The bifurcation diagram for the solution of system (10) versus the system’s parameter k and a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113. (b) The bifurcation diagram of the quantity q1 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1.
Fig. 2. (a) The bifurcation diagram of the quantity q2 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1. (b) The bifurcation diagram of the quantity q3 versus the system’s parameter c1 and a = 1, b = 0.5, k = 2.3, c2 = .02, c3 = 0.1.
It is shown that the costs parameters affect the system stabilization and then chaotic behavior appears. It is important to suppress such chaotic behavior and control the system. Avoiding such behavior has helped to introduce a wide variety of control approaches in order to control on some level chaotic behavior of such systems. In [16], the OGY or (Ott–Grebogi– Yorke) approach has been presented to return a system from chaotic to stable state. It has been applied to Kopel duopolistic game model in [17]. This approach depends on making change on the systems’s parameters so as to force it to return back to the stable state. Modification of the straight-line stabilization approach has been applied to a duopolistic model in [18]. In [19], a self-adaptive proportional control method in economic chaotic system is discussed. A time-delayed feedback control approach to control chaos in an economic model has been studied in [20]. For more control approaches readers are advised to see literature [21–27]. In this section, we assume that the system may be controlled depending on a control function ut that may be incorporated to the system as follows.
qt+1 = f (qt , ut )
(15)
Here, to stabilize our model we introduce the following control function,
ui,t = (1 − η )(qi,t+1 − qi,t ), i = 1, 2, 3
(16)
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Fig. 3. (a) The bifurcation diagram of the quantity q1 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1. (b) The bifurcation diagram of the quanitity q2 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1.
Fig. 4. (a) The bifurcation diagram of the quantity q3 versus the system’s parameter c2 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c3 = 0.1. (b) The bifurcation diagram of the quantity q1 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02.
on which η represents the control parameter. Using (16) in (10), we get the following controllable system.
q1,t+1 = q1,t + q2,t+1 = q2,t + q3,t+1 = q3,t +
k
η k
η k
η
q1,t [a + 2c1 θ1 − 2(b + c1 )q1,t − b(q2,t + q3,t )] q2,t [a + 2c2 θ2 − 2(b + c2 )q2,t − b(q1,t + q3,t )]
(17)
q3,t [a + 2c3 θ3 − 2(b + c3 )q3,t − b(q1,t + q2,t )]
Now, we need to suppress chaos appeared in the system (10) and force the system to return back to its stabilization. To do so let us assume k ≈ 1.783 and a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113 , then the Jacobian of system (17) is given by,
⎡
1−
0.07283443268
⎢ η ⎢ −0.5631453634 η ⎣
J ( E7 ) = ⎢ ⎢
−0.6458902231
η
−0.03034768049
η
1−
1.171342355
η
−0.6458902231
η
−0.03034768049
⎤
⎥ ⎥ ⎥ ⎥ η 1.550136535 ⎦ η
−0.5631453634 1−
η
(18)
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Fig. 5. (a) The bifurcation diagram of the quantity q2 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02. (b) The bifurcation diagram of the quantity q3 versus the system’s parameter c3 and a = 1, b = 0.5, k = 2.3, c1 = 0.1, c2 = 0.02.
Fig. 6. Strange attractor diagram for the system (10) at the parameters: a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.1.0113: (a) k = 0.1, (b) k = 2.3.
For the above Jacobian, the characteristic equation takes,
λ3 +B1 λ2 +B2 λ + B3 = 0, B1 = −3+ B2 = 3−
2.794313323
η
5.588626645
B3 = −1.+
η
,
+
2.794313323
η
1.613536440
−
η
2
1.613536440
η2
(19)
, +
0.07838121695
η3
To study the stability of E7 , we recall Routh–Hurwiz conditions [28] with (19) one gets the following conditions.
1 := 1 + A1 +A2 +A3 =
1 × 10−9
+
0.07838121695
,
η η3 8(η − 0.026712938 )(η − 0.364670044 )(η − 1.005773679 ) 2 := 1 − A1 +A2 −A3 = , η3 3 := 1 − A23
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Fig. 7. Bifurcation diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113 and k ≈ 1.783.
Fig. 8. Strange attractor diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113, μ = 1.01 and k ≈ 1.783.
=
5.588626646(η − 0.053311415 )(η − 0.053541882 )(η − 0.523893888 )(η2 − 1.343845239η + 0.735125917 )
1
η6
,
× 1 × 10−8 × (η − 0.0049778896 )(η − 0.052047935 )(η − 0.052519989 ) η12 ×(η − 0.062543416 )(η − 0.296184556 )(η − 0.535265714 )(η − 0.566993233 ) ×(η − 0.996347093 )(η2 − 7.388319168η + 4.951908979 × 109 ) (20)
4 := (1 − A23 )2 − (A2 − A1 A3 )2 =
These conditions guarantee the local stability of E7 if i > 0, i = 1, 2, 3, 4 and this is happened when η > 1.0058 as shown numerically in Fig. 7. This means that when the control parameter goes large the system becomes out of the chaotic region
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Fig. 9. Strange attractor diagram with respect to the control parameter η at a = 1, b = 0.5, c1 = 0.1, c2 = .02, c3 = 0.1, θ1 = −1.4053, θ2 = 0.9055, θ3 = 1.0113, μ = 0.99 and k ≈ 1.783.
and hence stable which means that the proposed control scheme is successful. Fig. 8 and 9 show the attractor of the three quantities for different values of the control parameter η. 4. Conclusion The current paper has introduced and studied the complex dynamic characteristics of a tripolistic Stackelberg game. The myopic mechanism (bounded rationality) has used to study the dynamic of the game by introducing a three-dimensional discrete system. The obtained results have confirmed that Nash equilibrium point of the game is unique and its stability is affected by bifurcation and chaos. In addition, the control scheme has returned the game’s system back to it stability state when appropriate values of the parameters were used. Numerical simulations have been used and confirmed the obtained results. Acknowledgment The author would like to extend their sincere appreciation to the Deanship of Scientific Research at king Saud University for its funding this Research group NO (RG-1435-054). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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