Stochastic perturbations for a duopoly Stackelberg model

Journal Pre-proof Stochastic perturbations for a duopoly Stackelberg model Baodan Tian, Yong Zhang, Jiamei Li

PII: DOI: Reference:

S0378-4371(19)32111-9 https://doi.org/10.1016/j.physa.2019.123792 PHYSA 123792

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Physica A

Received date : 14 May 2019 Revised date : 28 October 2019 Please cite this article as: B. Tian, Y. Zhang and J. Li, Stochastic perturbations for a duopoly Stackelberg model, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123792. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

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Highlights for the manuscript ● A Stochastic duopoly Stackelberg model is firstly proposed in the paper.

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● The globally asymptotical stability of the Nash equilibrium is derived. ● The change regulation of the output of two firms in the competition game is

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deduced by stochastic numerical simulations.

*Manuscript Click here to view linked References

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Stochastic Perturbations for a Duopoly Stackelberg Model*

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Baodan Tian1,2 , Yong Zhang1 , Jiamei Li1

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1.School of Science, Southwest University of Science and Technology, Mianyang 621010, China 2. Institute of Modeling and Algorithm, Southwest University of Science and Technology, Mianyang 621010, China

Abstract: In this paper, a duopoly Stackelberg model of competition on output with

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stochastic perturbations is proposed. Sufficient conditions for the existence and the uniqueness of global positive solution of the stochastic system are derived. For the stochastic Stackelberg model (SSM), the authors obtained some sufficient conditions for the globally asymptotical stability of the Nash equilibrium. When ignoring the stochastic perturbations, sufficient criteria for the corresponding deterministic Stackelberg model (DSM) is also obtained. In addition, by applying Milstein’s numerical algorithm, we solve the stochastic

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differential model then plot the time-series diagram, which can help us deduce the change regulation of the output of two firms in the competition game. Finally, some discussions are

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given for the economic significance for the model. Key words: Duopoly Stackelberg model; stochastic perturbations; Nash equilibrium; globally asymptotical stability; numerical simulations.

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1. Introduction

It is well-known that the game of oligopoly is a very important problem in the theory of economics market. There are two classic oligarchic model, i.e., Cournot’s duopoly model and Stackelberg’s duopoly model. They are the earliest research objects in the game theory. * Corresponding author e-mail address: [email protected](B. Tian).

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Cournot model is also called Cournot duopoly model (see [1]). It is the earliest applied version of Nash equilibrium, and usually regarded as the origin for the analysis of the

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oligopolistic game theory. It is a simple idealized model with only two oligarchs, which is also called as duopoly model. And it succeed to explain the production decision-making behavior

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of two competitive firms with cooperation in the same market condition by solving an equilibrium result between a competitive model and a complete monopoly model. This result is also applicable to the three-oligarch game model or the game model with more oligarchs. The premise of the Cournot model is that an industry consists of only two oligarchs, and

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there is no cooperative behavior between them. They know the reaction functions of each other and make an optimal production decision to maximize the profits before the action. Moreover, in the Cournot model, it is assumed that the production costs of the two firms are zero and have the same marginal cost.

The Stackelberg model is a production leadership competition model proposed by the German economist H. Von Stackelberg in 1934 (see [2]), which reflect the competitive be-

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havior of the game enterprises in the oligopolistic market. It is a decision model that is more suitable for the production of duopoly market. And there are two firms and two stages in the game process of the Stackelberg model, that is, the leader firm and follower firm, the

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decision stage and the production stage. On the one hand, the leader firm announces the pre-production according to the rule of thumb firstly, then the follower makes a choice of production by observing the behavior of the leader. On the other hand, the leader firm needs to consider the reaction that the follower will make when choosing a pre-production.

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However, the common goal of the production decision of the leader and the follower firm is to maximize the benefits as much as possible. More details in the economic meanings can be seen in references (see [3, 4, 5, 6, 7]). In many realistic economic markets, the position of the firm is asymmetric, and this will lead to the asymmetry in decision-making order in the game, (see see [8, 9, 10, 11]). In

2

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contrast of above two models, the obvious advantage of Stackelberg’s duopoly model is to consider a better depiction of this asymmetric in the competition game. During the output

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competition between the two firms, the leader firm have to consider how the follower will react and predict the reaction by and large, and it has the priority of selection the output

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and no longer needs its own response function because the output decision is determined by the response function of the follower firm. At the same time, the follower firm will make response and then decide the output after acquiring the output of the leader firm. Based on above research background, the research on the Stackelberg’s duopoly model

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and the Nash equilibrium seems to be a hot issue. Recently, Peng et al. have finished a series of research work on the discrete duopoly Stackelberg model of competition on output, see [12, 13, 14, 15, 16]. During these work, when assuming that the leader firm is bounded rational and the follower firm is adaptable, and the leader firm does not have knowledge of the market completely, so he tries his best to use local information to improve marginal profit (see [16]) . At the moment, the duopoly Stackelberg model with heterogeneous players

(1)

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can be defined by the following two nonlinear equations (see the system (11) in [16]),  n o  q (t + 1) = q (t) 1 + v [(a + 2c Q ) − 2(b + c )q (t) − bq (t)] ,  1 1 1 1 1 1 2  1    b a + 2c2 Q2   q2 (t + 1) = (1 − v2 )q2 (t) + v2 (t) − q1 (t) . 2(b + c2 ) 2(b + c2 )

where qi (t) represent the output of firm i at discrete time t, Qi represent the announced plan quantities, ci are the positive shift parameters to the cost function and ki represent marginal cost, a, b > 0 are both positive constants.

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On the other hand, for a continuous function q(t), if we assume that the time variable t takes discrete integer values in equal time, k = 0, 1, 2, · · · , and denote qk = q(k), then the values of qk is a sequence. When the time variable changes from k to k +1, the corresponding function increment is ∆qk = qk+1 −qk = q(k+1)−q(k). Besides, according to the relationship between the differentiation and difference, the derivative

dq dt

can be replaced by the difference

q(k + 1) − q(k). Furthermore, let us denote variable k with variable t, then model (1) can 3

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be transformed as following differential form, (2)

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Thus,


dqi (t) ≈ qi (t + 1) − qi (t) dt, i = 1, 2.

 h i   dq (t) = v q (t) (a + 2c Q ) − 2(b + c )q (t) − bq (t) , 1 1 1 1 1 1 1 2 

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(3) h a + 2c Q i  b 2 2   dq2 (t) = v2 − q1 (t) − q2 (t) . 2(b + c2 ) 2(b + c2 ) which is a deterministic model (which is called as DSM for short in the following paper). The coefficients of the model is the same as system (1). Where vi is a positive constant which

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is called the speed of adjustment of the i − th firm. However, in the real world, the speed of the adjustment of a firm is always affected by financial crisis, macro-policy adjustment and other stochastic factors. Therefore, it is more reasonable to describe the dynamics of the firms by stochastic models. For this means, the speed of adjustment of the i − th firm becomes,

vi → vi + σi B˙ i (t), i = 1, 2.

(4)

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where Bi (t) is standard Brown motion defined on a complete probability space (Ω, F , P) with a filtration {Ft }t∈R+ satisfying the usual conditions (i.e., it is right continuous and

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increasing while σi (t) is a continuous and bounded function on t ≥ 0 and σi2 (t) represents the intensity of the white noise, i = 1, 2. Thus, in the present paper, we propose a Stackelberg duopoly model with stochastic

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perturbations as follows,     dq1 (t) = q1 (t) [(a + 2c1 Q1 ) − 2(b + c1 )q1 (t) − bq2 (t)] × [v1 dt + σ1 dB1 (t)],    b a + 2c2 Q2   dq2 (t) = − q1 (t) − q2 (t) × [v2 dt + σ2 dB2 (t)]. 2(b + c2 ) 2(b + c2 )

(5)

which is called as SSM for short in the following paper.

Remark. When σ1 = σ2 = 0, the SSM (5) degenerates into the DSM (3). Since system (2) is a discrete form of system DSM (3), then the SSM (5) proposed in this paper is more extensive in economical meanings. 4

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Inspired by the recent studies on population ecology models and epidemic models with random perturbations (see [17, 18, 19]), we can also study the stochastic economic models

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proposed in this paper. The rest of this paper is organized as follows: In Section 2, since only when the output the i − th firm should be positive from economical meanings, so we

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will discuss the existence and uniqueness of the global positive solution of the SSM (5) . In section 3, we try to find controlling conditions for the global stability of Nash equilibrium of the SSM (5) and the DSM (3). In the last section, we give some numerical examples to

2. Global positive solution

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support the theoretical results, then we will give some economical explanations.

In this section, since the output the i-th firm should be non-negative by economical meaning. Thus, we should study the existence and uniqueness of the positive solution of SSM (5) .

Lemma 2.1. There exists a τe > 0, such that there is a unique positive solution (q1 (t), q2 (t))T

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almost surely for the SSM on the interval t ∈ [0, τe ) for any initial conditions q1 (0) = q10 > 0 and q2 (0) = q20 > 0, where τe is also called as the explosion time.

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Proof. For the SSM (5), if we take a transformation, q1 (t) = ex(t) , q2 (t) =

p

y(t),

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Then, by Ito’s formula, we can derive,     x(t) y(t)  dx(t) = v (a + 2c Q ) − 2(b + c )e − be dt  1 1 1 1     2   − 12 σ12 (a + 2c1 Q1 ) − 2(b + c1 )ex(t) − bey(t) dt          +σ1 (a + 2c1 Q1 ) − 2(b + c1 )ex(t) − bey(t) dB1 (t)],     p p a + 2c2 Q2 b x(t) dy(t) = 2v2 y(t) − e − y(t) dt   2(b + c ) 2(b + c )  2 2   2  p  b  x(t) 2 a + 2c2 Q2  +σ2 − e − y(t) dt    2(b +c2 ) 2(b + c2 )    p p  a + 2c2 Q2 b  x(t)  +2σ2 y(t) − e − y(t) dB2 (t)].  2(b + c2 ) 2(b + c2 ) 5

(6)

(7)

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For the stochastic equations (7) with initial values x(0) = ln q10 , y(0) = ln q20 , since all the coefficients for (7) satisfy local Lipschitz condition, then there is a unique local solution

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(x(t), y(t))T on t ∈ [0, τe ). (see [19]) p Therefore, q1 (t) = ex(t) , q2 (t) = y(t) is the unique local positive solution for the SSM

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(5) with any positive initial values q1 (0) = q10 > 0, q2 (0) = q20 > 0.

Lemma 2.1 shows that there is a unique local positive solution for the SSM (5). Further, we should prove the solution is global for economic meanings. To achieve this aim, we only need to prove τe = ∞.

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2 Theorem 2.2. For any given initial value (q10 , q20 ) ∈ R+ = {(q1 , q2 )|qi > 0 (i = 1, 2)}, if

the intensity of the noises in the system satisfy, (H1)

σ12 b2 4

+ σ22 ≤ 2v2

holds, then the SSM (5) has a unique positive solution on [0, +∞) and the solution remain in R2+ with probability one.

Let k0 > 0 be sufficiently large such that q10 , q20 fall in the interval [1/k0 , k0].

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Define a sequence of stopping time for each integer k ≥ k0 by τk = inf{t ∈ [0, τe )|q1 (t) ∈ / (1/k, k) or q2 (t) ∈ / (1/k, k)}.

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Obviously, τk is nondecreasing as k increasing to ∞. Since τk ≤ τe for any k, by the monotonic bounded criterion, the limit of the sequence τk exists. If we denote lim τk = τ∞ , k→∞

then τ∞ ≤ τe almost surely.

In the following, we will prove τ∞ = ∞ almost surely.

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Otherwise, there exists a T > 0 and ε ∈ (0, 1), such that P {τ∞ ≤ T } > ε. Thus, there exists a k1 ≥ k0 , such that P {τk ≤ T } ≥ ε, for all k ≥ k1 . Now, if we consider a Lyapunov function V (q1 , q2 ) = V1 (q1 ) + V2 (q2 ) where V1 (q1 ) =

√

q1 − 1 − 12 ln q1 , V2 (q2 ) = q22 . 6

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Thus, by Ito’s formula, one can calculate, h i σ2 1 1 1 2 L V1 (q1 ) = v1 (q1 − 1) (a + 2c1 Q1 ) − 2(b + c1 )q1 − bq2 + 1 (2 − q12 ) 2 8 h 2 2 2 2 2 × (a + 2c1 Q1 ) + 4(b + c1 ) q1 + b q2 − 4(a + 2c1 Q1 )(b + c1 )q1 i − 2b(a + 2c1 Q1 )q2 + 4b(b + c1 )q1 q2

3 1 v1 1 v1 v1 (a + 2c1 Q1 )q12 − v1 (b + c1 )q12 − bq12 q2 − (a + 2c1 Q1 ) 2 2 2 σ12 σ 2 b2 bv1 (a + 2c1 Q1 ) + v1 (b + c1 )q1 + q2 + (a + 2c1 Q1 )2 + σ12 (b + c1 )2 q12 + 1 q22 2 4 4 2 bσ (a + 2c1 Q1 ) − σ12 (a + 2c1 Q1 )(b + c1 )q1 − 1 q2 + b(b + c1 )σ12 q1 q2 2 2 1 5 3 σ12 σ σ2 − (a + 2c1 Q1 )2 q12 − 1 (b + c1 )2 q12 + 1 (a + 2c1 Q1 )(b + c1 )q12 8 2 2 2 2 1 2 1 3 σ b bσ bσ12 + (a + 2c1 Q1 )q12 q2 − 1 q12 q22 − 1 (b + c1 )q12 q2 4 8 2  a + 2c Q  h (a + 2c Q )2 bq1 b2 q12 2 2 2 2 L V2 (q2 ) =2v2 q2 − − q2 + σ22 + 2(b + c2 ) 2(b + c2 ) 4(b + c2 )2 4(b + c2 )2 i b(a + 2c2 Q2 ) a + 2c2 Q2 b + q22 − q − q + q q 1 2 1 2 2(b + c2 )2 b + c2 b + c2   σ22 b2 b(σ22 − v2 ) 2 q + q1 q2 = σ22 − 2v2 q22 + 4(b + c2 )2 1 b + c2 bσ 2 (a + 2c2 Q2 ) (v2 − σ22 )(a + 2c2 Q2 ) σ22 (a + 2c2 Q2 )2 q + q + . − 2 1 2 2(b + c2 )2 b + c2 4(b + c2 )2 which yields,

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=

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L V (q1 , q2 ) = L V1 (q1 ) + L V2 (q2 ) h i 5 σ2 σ22 b2 i 2 h σ12 = − 1 (b + c1 )2 q12 + σ12 (b + c1 )2 + q + (a + 2c Q ) − v 1 1 1 2 4(b + c2 )2 1 2 h 3 bσ 2 (a + 2c2 Q2 ) i × (b + c1 )q12 + v1 (b + c1 ) − σ12 (a + 2c1 Q1 )(b + c1 ) − 2 q1 2(b + c2 )2 1 σ2 σ 2 (a + 2c2 Q2 )2 v1 + g1 (q2 )q12 + 1 (a + 2c1 Q1 )2 + 2 − (a + 2c1 Q1 ) + f1 (q1 )q2 4 4(b + c2 )2 2  σ 2 b2  h bv (v2 − σ22 )(a + 2c2 Q2 ) bσ12 (a + 2c1 Q1 ) i 1 1 2 2 + + σ2 − 2v2 q2 + + − q2 4 2 b + c2 2

where,

f1 (q1 ) = −

3 bσ12 b(σ22 − v2 ) (b + c1 )q12 + b(b + c1 )σ12 q1 + q1 , 2 b + c2

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g1 (q2 ) = −

bv1 v1 σ2 σ12 b2 2 bσ12 q2 + (a + 2c1 Q1 )q2 − q2 + (a + 2c1 Q1 ) − 1 (a + 2c1 Q1 )2 . 8 4 2 2 8

According to the form of f1 (q1 )and g1 (q2 ), it is easy to prove that there exist two constants

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M1 and M2 , such that f1 (q1 ) < M1 , g1 (q2 ) < M2 , which yields,

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L V (q1 , q2 ) ≤ f (q1 ) + g(q2) where

i h 5 σ12 σ22 b2 i 2 h σ12 (b + c1 )2 q12 + σ12 (b + c1 )2 + q + (a + 2c Q ) − v 1 1 1 2 4(b + c2 )2 1 2 h 3 1 bσ 2 (a + 2c2 Q2 ) i 2 ×(b + c1 )q12 + v1 (b + c1 ) − σ12 (a + 2c1 Q1 )(b + c1 ) − 2 q + M q 1 2 1 2(b + c2 )2

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f (q1 ) = −

 h bv1 (v2 − σ22 )(a + 2c2 Q2 ) bσ12 (a + 2c1 Q1 ) i 2 + − 2v2 q2 + M1 + + − q2 g(q2 ) = 4 2 b + c2 2 σ12 σ22 (a + 2c2 Q2 )2 v1 2 + (a + 2c1 Q1 ) + − (a + 2c1 Q1 ) 4 4(b + c2 )2 2  σ 2 b2 1

σ22

Thus, when condition (H1) holds, i.e.,

σ12 b2 4

+ σ22 − 2v2 < 0, then there exist another two

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constants M3 and M4 , such that f (q1 ) < M3 , g(q2 ) < M4 , which yields, L V (q1 , q2 ) ≤ f (q1 ) + g(q2 ) < M3 + M4 , M.

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Then the function L V (q1 , q2 ) is upper bounded by M. Integrating dV (q1 , q2 ) from 0 to T ∧ τk and taking expectations yields, Z T ∧τk EV (q1 (T ∧ τk ), q2 (T ∧ τk )) ≤ V (q10 , q20 ) + Mdt ≤ V (q10 , q20 ) + MT.

(9)

0

Thus,

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On the other hand, for every ω ∈ Ωk , either q1 (τk , ω) or q2 (τk , ω) equals either k or 1/k. √ ln k p ln k V (q1 (T ∧ τk , ω), q2(T ∧ τk , ω)) ≥ min{ k − 1 − , 1/k − 1 + } + min{k 2 , 1/k 2 } , m. 2 2 which yields, EV (q1 (T ∧ τk , ω), q2(T ∧ τk , ω)) ≥ E[1Ωk V (q1 (T ∧ τk , ω), q2(T ∧ τk , ω))] ≥ mε. 8

(10)

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where 1Ωk is the indicator function on Ωk .

V (q10 , q20 ) + MT ≥ mε.

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Then it follows from (9) that (11)

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Considering k → ∞, we can derive that ∞ > V (q10 , q20 ) + MT = ∞, which is a contradiction. Thus, τ∞ = ∞ a.s. Then τe = ∞ a.s. So we complete the proof of this theorem. Since Nash equilibrium is a very important term in the game of multiple firms, which can provide a very important means of analysis so that game theory research can find more

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meaningful results in a game structure. Thus, we will try to discuss the stability of the Nash equilibrium for the Stackelberg model under the effects of the stochastic noises in the following section.

3. Nash equilibrium and its global stability

linear algebra equations,

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For the corresponding deterministic model (3), there are two equilibriums: boundary   2 Q2 equilibrium E0 0, a+2c and Nash equilibrium E ∗ (q1∗ , q2∗), where q1∗ , q2∗ satisfy following 2(b+c2 )

  bq ∗ + 2(b + c )q ∗ = a + 2c Q . 2 2 2 2 1

 2(b + c2 )(a + 2c1 Q1 ) − b(a + 2c2 Q2 )  ∗  ,  q1 = 4(b + c1 )(b + c2 ) − b2  2(b + c1 )(a + 2c2 Q2 ) − b(a + 2c1 Q1 )   q2∗ = . 4(b + c1 )(b + c2 ) − b2

(12)

(13)

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which yields,

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   2(b + c1 )q1∗ + bq2∗ = a + 2c1 Q1 ,

In [12, 13], it is proved that q1∗ > 0, q2∗ > 0 for economic meanings, and the local stability of the Nash equilibrium E ∗ for the deterministic discrete model (2) is also studied. However, in the present paper, we will try to discuss the global stability of the Nash equilibrium of SSM (5). 9

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For the sake of convenience, firstly we denote following notations, A = 2q1∗ (b + c1 )2 σ12 +

b2 σ 2 − 2(b1 + c1 )v1 , 4(b + c2 )2 2

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of

q1∗ b2 σ12 B= + σ22 − 2v2 , 2 i h (σ 2 − v2 ) − v1 , C = b 2q1∗ σ12 (b + c1 ) + 2 b + c2

∆ = C 2 − 4AB.

Then, we have the global stability theorem for the Nash equilibrium of the SSM (5). Theorem 3.1. If (H1) and following conditions,

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(H2) A < 0, ∆ = C 2 − 4AB < 0,

then the Nash equilibrium E ∗ of SSM (5) is globally asymptotically stable. That is, lim q1 (t) = q1∗ ,

t→∞

lim q2 (t) = q2∗ .

t→∞

(14)

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Proof. By the expressions of the Nash equilibrium, we can transform the SSM (5) as follows,  ∗ ∗    dq1 (t) = −q1 (t) [2(b + c1 )(q1 − q1 ) + b(q2 − q2 )] × [v1 dt + σ1 dB1 (t)],   (15)  b  ∗ ∗  dq2 (t) = − (q1 − q1 ) + (q2 − q2 ) × [v2 dt + σ2 dB2 (t)]. 2(b + c2 )

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If we consider two Lyapunov functions,

V1 (q1 ) = q1 − q1∗ − q1∗ ln

h i L V1 (q1 ) = − v1 (q1 − q1∗ ) 2(b + c1 )(q1 − q1∗ ) + b(q2 − q2∗ )

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Thus,

q1 , V2 (q2 ) = (q2 − q2∗ )2 . q1∗

i2 q1∗ σ12 h 2(b + c1 )(q1 − q1∗ ) + b(q2 − q2∗ ) , 2 h i q ∗ b2 σ 2 = 2q1∗ σ12 (b + c1 )2 − 2v1 (b + c1 ) (q1 − q1∗ )2 + 1 1 (q2 − q2∗ )2 2 h i + 2bq1∗ σ12 (b + c1 ) − bv1 (q1 − q1∗ )(q2 − q2∗ ). +

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L V2 (q2 ) =

b(σ22 − v2 ) b2 σ22 ∗ 2 2 ∗ 2 (q − q ) + (σ − 2v )(q − q ) + (q1 − q1∗ )(q2 − q2∗ ). 1 2 2 1 2 2 2 4(b + c2 ) b + c2

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Denote V (q1 , q2 ) = V1 (q1 ) + V2 (q2 ), then

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L V (q1 , q2 ) =L V1 (q1 ) + L V2 (q2 ) h i b2 σ22 = 2q1∗ σ12 (b + c1 )2 + − 2v (b + c ) (q1 − q1∗ )2 1 1 4(b + c2 )2 h i (σ 2 − v2 ) + b 2q1∗ σ12 (b + c1 ) + 2 − v1 (q1 − q1∗ )(q2 − q2∗ ) b + c2 i h q ∗ b2 σ 2 + 1 1 + σ22 − 2v2 (q2 − q2∗ )2 . 2

(16)

of A, B, C, we can obtain,

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If we denote |Q−Q∗ | = (|q1 −q1∗ |, |q2 −q2∗ |)T , then with the help of the previous expressions L V (q1 , q2 ) ≤A(q1 − q1∗ )2 + C|q1 − q1∗ ||q2 − q2∗ | + B(q2 − q2∗ )2   2A C 1  |Q − Q∗ |. ≤ |Q − Q∗ |T  2 C 2B

(17)

Therefore, when condition(H1) and (H2): A < 0, ∆ = C 2 − 4BC < 0 holds, the matrix  2A C   is negative defined, which leads to L V (q1 , q2 ) < 0 along all the solutions in C 2B 2 R+ except (q1∗ , q2∗ ). By the stability theorem for the stochastic differential equations (see

al



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[19]), the Nash equilibrium E ∗ is globally asymptotically stable. Especially, if there is no stochastic perturbations, i.e., σ1 = σ2 = 0, then we can easily obtain the global stability for corresponding DSM (3) as follows,

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Corollary 3.2. For the DSM (3), if following inequality,  v2 2 (H3) b2 v1 + < 16(b + c1 )v1 v2 b + c2 hold, then the Nash equilibrium E ∗ of the DSM (3) is globally asymptotically stable. 4. Numerical simulations and discussions In the previous sections, we have some discussed sufficient conditions for the existence and uniqueness of the global positive solution and the global asymptotic stability of the Nash 11

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5 q1(t) of DSM q2(t) of DSM q1(t) of SSM q2(t) of SSM

4.5

of

4

3

2.5

2

1.5

0

5

10

15

20

25 t

30

35

40

45

50

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1

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q(t)

3.5

Figure 1: The time series of DSM (3) and SSM (5) with stochastic noises strength σ12 = 0.12 , σ22 = 0.152 under the same initial values q1 (0) = 2.5, q2 (0) = 1.5.

equilibrium from the perspective of theoretical analysis. In order to verify the correctness of above theoretical results and the feasibility of the given conditions, we will select the

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appropriate parameters and solve the DSM (3) and the SSM (5) numerically to further support our conclusions in this section.

0.7

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To achieve this aim, we will utilize Milstein’s idea of numerical solution of stochastic 0.7 Frequency histogram of q1(t) Corresponding shape−preserving

Frequency histogram of q1(t) Corresponding shape−preserving

0.5

0.4

0.3

0.2

0.1

0

0.6

0.5

0.4

0.3

0.2

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0.6

0.1

0

3.72−3.75 3.75−3.78 3.78−3.81 3.81−3.84 3.84−3.87 3.87−3.90 3.90−3.93 3.93−3.96 3.96−3.99 3.99−4.02 4.02−4.05

2.2−2.32.3−2.42.4−2.52.5−2.62.6−2.72.7−2.82.8−2.92.9−3.03.0−3.13.1−3.23.2−3.33.3−3.43.4−3.53.5−3.63.6−3.73.7−3.8

Figure 2: The histograms and corresponding shape-preserving fitting curve of the values of qi (t) for SSM (5) with σ12 = 0.12 , σ22 = 0.152 based on 1000 stochastic simulations.

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differential equations (see [20]), and use following numerical algorithm for the SSM (5):  h i   q = q + v q (a + 2c Q ) − 2(b + c )q − bq 1,k+1 1,k 1 1,k 1 1 1 1,k 2,k ∆t,     h i √     +σ1 q1,k (a + 2c1 Q1 ) − 2(b + c1 )q1,k − bq2,k ξk ∆t     h i2     +0.5σ12 q1,k (a + 2c1 Q1 ) − 2(b + c1 )q1,k − bq2,k (ξk2 − 1)∆t,  (18) h a + 2c Q i b 2 2   q = q + v − q − q ∆t,  2,k+1 2,k 2 1,k 2,k   2(b + c2 ) 2(b + c2 )   h i √  b a + 2c2 Q2    ∆t +σ − q − q η 2 1,k 2,k k   2(b + c2 ) 2(b + c2 )   h a + 2c Q i2  b  2 2   +0.5σ22 − q1,k − q2,k (ηk2 − 1)∆t. 2(b + c2 ) 2(b + c2 )

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where ξk , ηk , k = 1, 2, · · · , N are the Gaussian random variable which follow N(0, 1).

In order to ensure the economic rationality of the selected parameters, we choose a set of parameters similar to those in [12], and only make minor adjustments to a few parameters as follows:

a = 8.5, b = 0.9, c1 = 1, c2 = 2, v1 = 0.08, v2 = 0.1.

(19)

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with step length ∆t = 0.01 and initial conditions q1 (0) = q10 = 2.5, q2 (0) = q20 = 1.5. Note the relationship between Q1 , Q2 and other parameters a, b, c1 , c2 (formula (4) and (5) in [12]). It is easy to calculate Q1 = 4.3809, Q2 = 3.1933 for above coefficients (19).

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Further, if we consider the stochastic noises strength σ12 = 0.12 , σ22 = 0.152 for the SSM (5). On the one hand, one can easily calculate that, (20)

A = −0.0281 < 0, ∆ = C 2 − 4AB = −0.0170 < 0.

(21)

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σ12 b2 + σ22 − 2v2 = −0.1755 < 0. 4

which means condition (H1) and (H2) hold. According to Theorem 2.2 and Theorem 3.1, not only there exists a unique and positive global solution for the SSM (5), but also the Nash equilibrium E ∗ of the system should be global asymptotically stable. On the other hand, we solve the SSM (5) by above algorithm (18) numerically and plot the time series diagram in Figure 1. It is clear to see the output of the two firms 13

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6 q1(t) of DSM q2(t) of DSM q1(t) of SSM q2(t) of SSM

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5

3

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q(t)

4

2

0

0

5

10

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1

15

20

25 t

30

35

40

45

50

Figure 3: The time series of DSM (3) and SSM (5) with stochastic noises strength σ12 = σ22 = 0.352 under the initial values q1 (0) = 2.5, q2 (0) = 1.5.

will asymptotically tends to the Nash equilibrium E ∗ (3.8141, 3.0760) with the stochastic

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perturbations. At the moment, in order to observe the behavior of the solutions of corresponding deter-

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ministic model (3) , we set the intensity of the random disturbance to 0, i.e., σ12 = σ22 = 0, then one can easily calculate,  v2 2 b2 v1 + − 16(b + c1 )v1 v2 = −0.2326 < 0. b + c2

(22)

which means condition (H3) in Corollary 3.2. holds. According to the Corollary, the Nash

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equilibrium E ∗ of the DSM (3) will be globally asymptotically stable. In fact, by solving the DSM (3) numerically and plot the time series in the same diagram (see Figure 1), we can clearly confirm this point. Furthermore, we can see the solutions of SSM (5) will asymptotically tend to those of DSM (3). When the system is subject to external noise, it is clear to see that the output of the two firms has a significant oscillation for a long period of time, and then eventually stabilized 14

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around the Nash equilibrium of the deterministic model from Figure 1. In order to deeply observe the stationary distribution law of the output of the two firms, we repeated above

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numerical experiments for 1000 times and count the output distribution law of the two firms at t = 30 in Figure 2. One can see that, under the controlling conditions (H1) and (H2), the

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output of the two firms will be stabilize around the Nash equilibrium with a large probability after a period of game between two firms.

Furthermore, if the intensity of the random disturbances in the market is very large, such as sudden economic crisis happens, then the conditions of the global asymptotical stability

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of the Nash equilibrium will be destroyed. At the moment, one of the two firms might be bankrupt due to the failure in the game of the competition, while the other firm will be successful in the game. In order to see this phenomenon, we change the stochastic noises strength σ12 = σ22 = 0.352 while other parameters are the same as above, we solve the DSM (3) and SSM (5) numerically and plot the time series of the two firms for this case in Figure 3. To our surprise, we find the leader firm suddenly becomes bankrupt while the follower

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Acknowledgments:

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firm survives in the game of the competition.

This work is supported by Sichuan Science and Technology Program under Grant 2017JY0336 and Hunan Science and Technology Program under Grant 2019JJ50399, Longshan Talent Research Fund of Southwest University of Science and Technology under Grant 17LZX670 and 18LZX622. The authors would like to express their deep gratitude to the editor and the

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anonymous referee for his/her careful reading and valuable comments. References

[1] A A Cournot. Researches into the principles of the theory of wealth. Irwin Paper Back Classics in Economics, 1963.

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[2] Von Stackelberg H. Probleme der unvollkommenen konkurrenz. Weltwirtschaftliches Archiv, 1938, 48: 95-141.

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[3] X Pu, J Ma. Complex dynamics and chaos control in nonlinear four-oligopolist game with different expectations. Int. J. Bifurc. Chaos., 2013, 23(3):1350053. (15 pages). [4] W Li , J Chen. Backward integration strategy in a retailer Stackelberg supply chain. Omega,

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2018, 75:118-30.

[5] J Ding, Q Mei, H Yao. Dynamics and adaptive control of a duopoly advertising model based on heterogeneous expectations. Nonlinear Dynamics, 2012, 67(1): 129-38 .

Econ. Model., 2007, 24:138-48.

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[6] J Zhang, Q Da, Y Wang. Analysis of nonlinear duopoly game with heterogeneous players.

[7] N Angelini, R Dieci, F Nardini. Bifurcation analysis of a dynamic duopoly model with heterogeneous costs and behavioural rules. Math. Comput. Simul., 2009, 79: 3179-3196. [8] T Puu. Chaos in duopoly pricing. Chaos Solitons Fractals, 1991, 1: 573-81. [9] J B Rosser. The development of complex oligopoly dynamics theory. Text book oligopoly dynamics: Models and tools. Springer-Verleg; 2002.

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[10] M T Yassen, H N Agiza. Analysis of a duopoly game with delayed bounded rationality. Appl. Math. Comput. 2003,138: 387-402.

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[11] H N Agiza, AA Elsadany. Chaotic dynamics in nonlinear duopoly game with heterogeneous players. Appl. Math. Comput., 2004,149: 843-860. [12] Y. Peng, Q Lu, Y Xiao. A dynamic Stackelberg duopoly model with different strategies. Chaos, Solitons and Fractals, 2016, 85: 128-134.

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[13] Y. Peng, Q Lu. Complex dynamics analysis for a duopoly stackelberg game model with bounded rationality. Appl. Math. Comput., 2015, 271: 259-268. [14] Y Peng, Q Lu, Y Xiao et al. Complex dynamics analysis for a remanufacturing duopoly model with nonlinear cost. Physica A, 2019, 514: 658-670. [15] X N Yang, Y Peng, Y Xiao et al. Nonlinear dynamics of a duopoly Stackelberg game with marginal costs. Chaos, Solitons and Fractals 2019, 123: 185-191.

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[16] Y Xiao, Y Peng, Q Lu et al. Chaotic dynamics in nonlinear duopoly Stackelberg game with heterogeneous players. Physica A: Statistical Mechanics & Its Applications, 2018, 492: 1980-

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1987. [17] B D Tian, L Yang, S M Zhong. Global stability of a stochastic predator-prey model with Allee effect. International Journal of Biomathematics, 2015, 8: 1550044. (15 pages).

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[18] Y L Cai, J J Jiao, Z J Gui, et al. Environmental variability in a stochastic epidemic model. Applied Mathematics and Computation, 2018, 329: 210-226.

[19] W M Wang, Y L Cai, Z Q Ding, et al. A stochastic differential equation SIS epidemic model incorporating Ornstein-Uhlenbeck process. Physica A, 2018, 509: 921-936.

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[20] X R Mao. Stochastic differential equations and applications(second edition). Chichester: Horwood Publishing Publishing, 2007.

[21] D J Higham. An algorithmic introduction to numerical simulation of stochastic differential

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equations. SIAM Reviews, 2001, 43: 525-46.

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*Declaration of Interest Statement

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CRediT author statement

Baodan Tian: Corresponding author, Conceptualization, Methodology,

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Original draft preparation

Yong Zhang: Software, Validation, English grammar

Disclosure and conflicts of interest:

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Jiamei Li: Reviewing and Editing

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The authors declare that they have no competing interests.