Dynamic contest model with bounded rationality

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Dynamic contest model with bounded rationality Ming Zhang a, Guanghui Wang a,b, Jin Xu a,c,∗, Cunquan Qu a,b a b c

School of Mathematics, Shandong University, Jinan 250100, PR China Data Science Institute, Shandong University, Jinan, 250100, PR China School of Management, Shandong University, Jinan, 250100, PR China

a r t i c l e

i n f o

Article history: Received 31 May 2019 Revised 26 September 2019 Accepted 3 November 2019 Available online xxx Keywords: Contest Bounded rationality Stability Flip bifurcation Neimark-Sacker bifurcation

a b s t r a c t This paper is devoted to exploring the complex dynamics of contest model, where two agents compete for some object with asymmetric valuations by simultaneously choosing efforts at each step. We build the nonlinear discrete system to describe the dynamic contest with bounded rationality, and discuss the stability conditions of the Nash equilibrium theoretically. Meanwhile, our numerical simulation experiments also reveal that the model can exhibit very complex dynamical behaviors. In particular, there exist two different routes to chaos for the system: the period-doubling (flip) bifurcation which leads to periodic cycles and chaos, and the Neimark-Sacker bifurcation which derives an attractive invariant closed curve. These two routes are significantly different for economic views. In addition, the stability of Nash equilibrium point is badly affected by the system parameters, such as the adjustment speeds, the values of the object, and so on. Therefore, the parameter adjustment method could be properly applied to make the system return to its stable state. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Contest model is widely used in many different areas of society and economy [1], for instance the promotions on internal labor markets, R&D patent competitions [2], election campaigns, military conflicts, sports [3], rent-seeking, advertising, and so on. There are three distinct branches on contests: Tullock contests [4–6], all-pay contests [7,8], and tournaments [9,10]. These three branches differ in terms of the contest success function, i.e., the criteria for allocating prizes. Here, our model is based on Tullock contest in which the agents simultaneously determine their effort expenditures and the probabilities of receiving the prizes are proportional to efforts. In most existing theoretical and experimental studies, it is assumed that competitors are indifference or symmetric [11–13]. However, the contestants may value the prizes differently, and asymmetric player valuations are natural and considerably common in many applications [14–17]. Here, we explore an asymmetric contest where each competitor independently perform high or low value realizations. The academic community has extensively explored one contest game: simultaneous or sequential [18,19]. Yet, in the real market contest, agents cannot obtain enough complete decision information [20]. Meanwhile, due to the limitation of the objective condition of the decision maker, the player’s decision can not be perfect. So we focus on the repeated contest game that is referred to dynamic contest model in this paper. ∗ Corresponding author at: School of Mathematics, Shandong University, Jinan 250100, PR China; School of Management, Shandong University, Jinan, 250100, PR China. E-mail address: [email protected] (J. Xu).

https://doi.org/10.1016/j.amc.2019.124909 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

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In recent years, a number of authors have posited the question of combining game theory with complexity theory [21– 25], in which highly complex dynamical behaviors is involved by several economic models, such as stability, bifurcation and chaos. These behaviors are firstly discovered in the Cournot duopoly case by Puu [26,27]. Studies on the complex dynamics of the games can be categorized by model type, number of players, or expectation type. From the type of the expectation, the researches can be divided into two classes, homogeneous and heterogeneous. In homogeneous oligopoly games, Agiza et al. [23], Bischi et al. [28,29], Naimza-da et al. [30], Ahmed et al. [31], and Zhang et al. [20] investigate the duopoly Cournot model based on the boundedly rational expectation. Ma et al. [32], Yu et al. [33,34], Agiza et al. [35], and Yao et al. [36] analyze complex dynamics for duopoly Cournot-Bertrand mixed model, Stackelberg model, Bowley’s model, and advertising model with bounded rationality, respectively. Stackelberg, Bowley’s and Bertrand models are generalized for three players by Askar et al. [37], Yu et al. [33] and Sun et al. [38], respectively. Besides the bounded rationality, there is delayed bounded rationality. Ahmed et al. [31], Elsadany [39], and Yassen et al. [40] consider a duopoly Cournot model with delayed bounded rationality and discuss the effects of the delayed decisions. Ma et al. [41] generalize delayed bounded rationality to tripoly Cournot Model. Another branch of literature concerns games with heterogeneous players. Agiza and Elsadany [24,42] firstly explore Cournot game with heterogeneous players, and especially, investigate its nonlinear dynamics. Later, Zhang et al. [43] and Dubiel-Teleszynski [44] apply the same technique on a duopoly Cournot game with nonlinear cost function. Angelini et al. [45] and Tramontana [25] consider a duopoly game with a microfounded nonlinearity on the demand function [26] based on the work of Agiza and Elsadany [24,42]. Instead of Cournot model, Yu et al. [46] and Yue et al. [47] investigate a dynamic Stackelbetg duopoly model with heterogeneous players. The analysis of the triopoly game with heterogeneous players is also involved. Elabbasy et al. [48,49] extend the Cournot game to the case of three heterogeneous firms with nonlinear cost functions. Tramontana et al. [50] remove the nonlinearity from the cost function and introduce a microfounded one on the demand function. In addition, four oligarchs Cournot game with heterogeneous players have been studied by Zhang et al. [51]. Kebriaei et al. [52] have popularized the heterogeneous expectations to n-players Cournot game. It could be seen from the research work mentioned above that the combination of game theory and chaos theory is concentrating on the dynamic models such as Cournot and Starkberg etc. To the best of our knowledge, there is very little work that attempts to capture how complex dynamics the asymmetric contest model performs. Closely related to our paper are the works by Bischi et al. [28,29], Angelini et al. [45], and Tramontana et al. [25,50]. Bischi et al. [28] explore the Cournot model with a unit-elastic demand function. Their main result is that there are two routes to chaos: flip and Neimark-Sacker bifurcations, when the marginal costs are different; Otherwise, the Neimark-Sacker bifurcation disappears. Angelini et al. [45] and Tramontana et al. [25,50] consider a similar setting as Bischi et al. [28] with the difference that agents are in heterogeneous expectations. In contrast to their papers, we introduce a novel model where two agents compete for some object by simultaneously choosing efforts under asymmetric valuations and the probabilities of receiving the object are proportional to efforts. The Cournot model with a unit-elastic demand function, that is the model of Bischi et al., is equivalent to the special case of our model, in which the agents are symmetric. So even if the agents have the same marginal cost, we can still get the interesting result: there are two different routes to chaos: flip and Neimark-Sacker bifurcations, if and only if agents are the symmetry, i.e. our model degenerates into Bischi et al.’s, the Neimark-Sacker bifurcation disappears. In this paper, we aim to investigate a dynamic asymmetrical contest game with bounded rationality, where each agent updates her strategy to maximize her utility according to the marginal utility in each step. The complex dynamic properties of the model above, such as the stability, the flip bifurcation, the Neimark-Sacker bifurcation and chaotic motions etc. will be probed. In the traditional economy, unstable fluctuations are considered unfavorable phenomena [20]. In the long run, chaotic motion is always irregular and unpredictable, and believe it as harmful by decision makers. Some methods of controlling chaos have been employed to the economic models, such as OGY method [53], the feedback control method [54], the delayed feedback control method [55,56], the variable feedback control method [41,57], the straight-line stabilization method [58] and so on. In this paper, chaotic motions which are brought about by the flip and Neimark-Sacker bifurcations in the proposed model is controlled by applying the parameter adjustment method [33,38,46]. The following of this paper is organised as: Section 2 reviews the contest model and builds the nonlinear discrete system with bounded rationality. In Section 3, we firstly research the stability conditions of the Nash equilibrium, and obtain the locally stable region of Nash equilibrium point. Next, we devote to the analysis of the two routes (the period-doubling and NeimarkâSacker bifurcations) to complex dynamics and characteristics of chaos. Meanwhile, numerical simulations are also performed. In Section 4, we apply control scheme of parameter adjustment to inhibit the chaotic behaviors that occur in the system. At last, in Section 5 some remarks are discussed. 2. The model In this section, we review the contest model and build its dynamic version with bounded rationality. 2.1. The contest model In the contest model, two identical agents compete for some object by making irreversible effort outlays. The object may be thought of as a piece of legislation, a contract for services, a telecoms license, and so forth. The valuation of the reward by each agent is independent from each other and usually known prior to making effort outlays. Denote by vi > 0 the value Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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3

Fig. 1. (color online) The best response functions BRi (xj ), i = j and the equilibrium solution E = (x∗1 , x∗2 ) of contest (1) with parameters v1 = 4, v2 = 2.1 and c = 1.

of the object to agent i (i = 1, 2 ) and by xi > 0 the effort of agent i, respectively. The probability to get the object for agent x i is proportional to her effort, i.e., pi = x +ix . Thus, the utility of agent i is given by i

xi Ui = vi − ci xi xi + x j

j

i = 1, 2, i = j,

(1)

in which ci > 0 is the marginal cost of the effort. The main point of this work to explore the influence of the value vi on the complex dynamics. Here we consider the simplified model below, that is, let ci = c j = c. This can be interpreted as a contest where agents compete for prizes that are asymmetrically valuated, the probability of winning is proportional to efforts, and the marginal cost of effort is c. Alternatively, this could be a model of an oligopoly with an asymmetric isoelastic inverse v demand function Pi = x +ix and marginal cost ci = c. i

j

The focus in this paper is the complex dynamic of contest model. Before proceeding, it is useful to offer some justification for this model. First, this model has been widely used to analyze a wide variety of imperfectly discriminating contests and auctions across a number of settings including rent-seeking, R&D races, advertising, elections, lobbying competition, and x defence expenditures [13]. Second, in this model, the contest success function is assumed as pi = x +ix , which has been i

j

given axiomatic foundations by Skaperdas [59] and Clark and Riis [60]. Next, we begin by deriving equilibrium contest outlays. The marginal utility for agent i at the point (x1 , x2 ) is given by

xj ∂ Ui = v −c ∂ xi ( xi + x j )2 i

i = 1, 2, i = j.

(2)

In order to maximize the utility for agent i, let her marginal utility equal to zero. We can obtain the best response functions of xi about xj , i = j. The best response function for each agent is denoted as BR1 (x2 ) and BR2 (x1 ), respectively. And it follows that



BRi (x j ) : xi =

x j vi − xj c

i = 1, 2, i = j.

(3)

Simultaneously, we have x1 v2 = x2 v1 . Substituting this back into Eq. (3) yields the equilibrium vector E = (x∗1 , x∗2 ) for the agents,

x∗i = vi v j

vi

c ( vi + v j )2

i = 1, 2, i = j.

(4)

Fig. 1 illustrates the best response functions BRi (xj ), i = j and the equilibrium effort expenditures E = (x∗1 , x∗2 ) of contest (1) for the case with v1 = 4, v2 = 2.1 and c = 1. It’s interesting that the best response functions BRi (xj ), i = j are asymmetric and non-monotonous. And when contest (1) reaches equilibrium, we verify x∗1 > x∗2 if v1 > v2 . 2.2. The dynamic contest model As a tool of expectation, the bounded rationality mechanism has been described in [30,61,62]. Under the bounded rationality mechanism, agents update their effort outlays by using local estimate of the marginal utility. The updated effort Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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outlays are well-defined by the following map T : (x1,t , x2,t ) → (x1,t+1 , x2,t+1 ),

T : xi,t+1 = xi,t + Ki (xi,t )

∂ Ui,t ∂ xi,t

i = 1, 2, i = j,

(5) ∂U

where the product of adjustment function Ki (xi,t ) and marginal utility ∂ x i,t of agent i represents the effort variation. For i,t simplicity, assume that the adjustment function is linear as Ki (xi,t ) = ki xi,t [33,61]. Using (2) and (5), we build the following 2-dimensional nonlinear discrete dynamic system,

⎧  

x2,t ⎪ ⎪ v −c , ⎨ x1,t+1 = x1,t 1 + k1 2 1   (x1,t + x2,t )

x1,t ⎪ ⎪ v − c , ⎩ x2,t+1 = x2,t 1 + k2 2 (x1,t + x2,t )2

(6)

where ki > 0 (i = 1, 2 ) indicates the adjustment speed of agent i [40,61]. That means, if the marginal utility of agent i is positive/negative, she will increase/decrease her effort outlays in the next period. 3. Equilibrium point and local stability So as to quest the qualitative behavior of solutions of the two-dimensional nonlinear system (6), let xi,t+1 = xi,t (i = 1, 2 ) in Eq. (6). Then, we have the nonlinear algebraic system

⎧  ⎨x1,t (x x+2x,t )2 v1 − c = 0, 1,t 2,t  x1,t ⎩x2,t v − c = 0. 2 2

(7)

(x1,t +x2,t )

Clearly, since xi > 0 (i = 1, 2 ), the algebraic system (7) has only one equilibrium point:

E = (x∗1 , x∗2 ) = (v1 γ , v2 γ ), where γ

v v = c(v 1+v2 )2 . Since 1 2

(8)

vi > 0 (i = 1, 2 ), the equilibrium point E is known as Nash or interior equilibrium point. Obvi-

ously, the Nash equilibrium point E exists provided that c (v1 + v2 )2 = 0. To sum up, the Nash equilibrium point E is existent and unique. So as to quest the local stability of the equilibrium point of the nonlinear system (6), we need to discuss the eigenvalues of the Jacobian matrix. The Jacobian matrix J(x1 , x2 ) at any point (x1 , x2 ) of system (6) takes the following form,

⎛

∂ x1,t+1 | ⎜ ∂ x1,t x1 ,x2 J ( x1 , x2 ) = ⎜ ⎝ ∂ x2,t+1 | ∂ x1,t x1 ,x2

⎞ ⎛ ∂ x1,t+1 x2 ( x2 − x1 ) |x1 ,x2 1 + k1 [v1 − c] ⎟ ⎜ ∂ x2,t ( x1 + x2 )3 ⎟=⎜ ⎠ ⎝ ∂ x2,t+1 x2 ( x2 − x1 ) | k2 v2 ∂ x2,t x1 ,x2 ( x1 + x2 )3

k1 v1

x1 ( x1 − x2 ) ( x1 + x2 )3

⎞

⎟ ⎟. ⎠

x1 ( x1 − x2 ) 1 + k2 [v2 − c] ( x1 + x2 )3

(9)

3.1. Local stability of Nash equilibrium This section dissects the local stability of the Nash equilibrium E = (v1 γ , v2 γ ) of system (6). Proposition 1. The unique Nash equilibrium point E of system (6) is locally stable. The Jacobian matrix at Nash equilibrium E is



J (E ) =

1 + k1 v1 δ k2 v2 v v(v2 −+vv1 ) 1 1 2

 k1 v1 v v(v1 −+vv2 ) 2 1 2 1 + k2 v2 δ

,

(10)

c where δ = v −2 . The characteristic equation of (10) is 1 +v2

(λ ) = λ2 − T r (JE )λ + Det (JE ) = 0,

(11)

in which Tr(JE) and Det(JE) represent the trace and the determinant of J(E), respectively. The math expressions of Tr(JE) and Det(JE) are as follows,

T r (JE ) = 2 + δ (k1 v1 + k2 v2 ), Det (JE ) = 1 + δ (k1 v1 + k2 v2 ) + k1 k2 c2 .

(12)

Nash equilibrium E is in a stable status if and only if the eigenvalues of the Jacobian matrix J(E) are within the unit circle of the complex plane, i.e., the following Jury’s conditions [31,42] hold,

(i ) (ii ) (iii )

(1 ) = 1 − T r (JE ) + Det (JE ) > 0, (−1 ) = 1 + T r (JE ) + Det (JE ) > 0, |(0 )| = |Det (JE )| < 1.

(13)

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Fig. 2. The stable region OABCD of the Nash equilibrium of system (6) in k1 k2 -plane for v1 = 4, v2 = 2.1 and c = 1. The blue and green curves represent   Eqs. (15)(ii ) and (15)(iiia ), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The above inequalities of (i), (ii) and (iii), define a region in which the Nash equilibrium point E is local stable. The violation of any single inequality in (i), (ii) and (iii) leads to: (I) a fold or transcritical bifurcation (a real eigenvalue that passes through 1) when (1 ) = 1 − T r (JE ) + Det (JE ) = 0; (II) a flip bifurcation (a real eigenvalue that passes through −1) when (−1 ) = 1 + T r (JE ) + Det (JE ) = 0; (III) a Neimark-Sacker bifurcation (i.e., the modulus of a complex eigenvalue pair that passes through 1) when (0 ) − 1 = Det (JE ) − 1 = 0 and Tr(JE) < 2 [46,63]. For the partical case of the Jacobian matrix (10), the stability conditions described in (13) yield 

(i )  (ii )  (iiia )  (iiib )

(1 ) = k1 k2 c2 > 0, (−1 ) = 4 + 2δ (k1 v1 + k2 v2 ) + k1 k2 c2 > 0, (0 ) − 1 = δ (k1 v1 + k2 v2 ) + k1 k2 c2 < 0, (0 ) + 1 = 2 + δ (k1 v1 + k2 v2 ) + k1 k2 c2 > 0.

(14)



In fact, we find inequality (14)(i ) is always satisfied, and when (0 ) − 1 = 0, there is Tr(JE) < 2 constant. On the other   hand, inequality (14)(iiib ) can be derived from inequality (14)(ii ) (see Proposition 2). 



Proposition 2. When inequality (14)(ii ) is satisfied, inequality (14)(iiib ) holds. Proof. (See Appendix A.)







So, inequalities (ii ) and (iiia ) in (14) define a region of stability in the plane of adjustment speeds (k1 , k2 ). This region of stability is delimited by the positive portion of hyperbolas, the equations of which are given by the zeroing of the left-hand   side of inequalities (ii ) and (iiia ) in (14), i.e., 

(−1 ) = 4 + 2δ (k1 v1 + k2 v2 ) + k1 k2 c2 = 0, (0 ) − 1 = δ (k1 v1 + k2 v2 ) + k1 k2 c2 = 0.

(ii )  (iiia )

(15)

For the values of (k1 , k2 ) in the region of stability (see Fig. 2), the Nash equilibrium E is in a stable state and loses its   stability via a flip ((15 )(ii )) or Neimark-Sacker ((15 )(iiia )) bifurcation curve. The flip bifurcation curve intersects the axes k2 and k1 in points A and D, respectively, and the flip bifurcation curve intersects the Neimark-Sacker bifurcation curve in points B and C, whose coordinates are given by



A = 0,

ω + 1 c

, B=

 2 2ω 2 2 ω + 1 , ,C= , , D= ,0 , cω c c c cω

(16)

v

where ω = v1 . We find that the stable region in the plane (k1 , k2 ) is determined, once parameters v1 , v2 , and c are fixed. 2 Particularly, if agents are symmetrical about the value of the prize, i.e. ω = 1, B and C merge, so that the stability region becomes a square, as shown in Fig. 3, and the possibility of Neimark-Sacker bifurcations is lost. Here, we mainly discuss the asymmetrical contest. Without loss of generality, assume that v1 > v2 . From the above analysis, it is natural to get the conclusion that the stability of the Nash equilibrium E relies on parameters k1 , k2 , v1 , v2 and c. Firstly, we dissect the impact of k2 on the stability of the Nash equilibrium E. In Fig. 2, let Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Fig. 3. The stable region OABCD of the Nash equilibrium of system (6) in k1 k2 -plane for v1 = 2, v2 = 2 and c = 1. The blue and green curves represent   Eqs. (15)(ii ) and (15)(iiia ), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)





v1 = 4, v2 = 2.1 and c = 1. The blue and green curves represent the Eqs. (15)(ii ) and (15)(iiia ), respectively. We draw the   stability region OABCD by inequalities (14)(ii ) and (iiia ). It’s found the stability region OABCD can be divided into three parts. When k1 is fixed in [0, 1.05) and [1.05, 1.525), an increase in k2 , starting from a parameter which guarantees the local stability of the Nash equilibrium, can take the point (k1 , k2 ) out of the stable region to cross the flip bifurcation and the Neimark-Sacker bifurcation curve, respectively. When k1 is fixed in [1.525, 2.0], the increasing and decreasing of k2 , can take the point (k1 , k2 ) out of the region of stability to cross the Neimark-Sacker bifurcation and the flip bifurcation curve, respectively. Similar arguments can be applied to k1 . The above phenomenon is different from those of the Cournot and Stackelberg dynamic models with the linear demand function[33,37]. This is essentially caused by the difference of the best response functions BRi (xj ), i = j. As shown in Fig. 1, the best response functions of the contest model are asymmetric and non-monotonic: xi first increases and then decreases in xj . Next, we dissect the impact of c and vi on the stability of the Nash equilibrium E. From Eq. (16), we can conclude these parameters could control the size, position and shape of the stable region. Specifically, when parameters v1 and v2 are fixed, the increase of cost parameter c makes the region of stability smaller, which causes a loss of stability of E. Given v2 and c, the increase of value parameter v1 causes the enlargement of k2 ’s stability range, namely point A to move up, D left shift, B to move up and left, and C to be unchanged. Similarly, given v1 and c, the increase of v2 causes the shrinking of k2 ’s stability range. namely point A to move down, D to the right shift, B to move down and right, and C to be unchanged. 3.2. Numerical simulation and complexity analysis In this section, we will employ numerical simulation to describe the dynamic behaviors of system (6) and its characteristics, such as the strange attractors, the sensitive dependence on the initial conditions, the fractal dimension. Without special explanation, we take v1 = 4, v2 = 2.1 and c = 1. First, numerical simulations are performed to demonstrate the stability, period-doubling bifurcation route and PomeauManneville route to chaos for system (6). When we change one of system parameters with others fixed, bifurcations and chaos take place, which is materialized by the maximal Lyapunov exponent. Fig. 4 illustrates the bifurcation diagrams regarding adjustment speed k2 , when adjustment speed k1 is equal to 0.5, 1.05, 1.4 and 1.6, respectively. Fig. 4 confirms our theoretical analysis that E may lose stability in two different ways, which is determined by the value of k1 . When the value of adjustment speed k1 is in [0, 1.05), the Nash equilibrium may loose stability into a flip bifurcation (Fig. 4(a)). When the value of adjustment speed k1 is in [1.05, 1.525), the Nash equilibrium may loose stability via a Neimark-Sacker bifurcation (Fig. 4(c)). When the value of adjustment speed k1 is in [1.525, 2], the Nash equilibrium may throw away stability into a flip bifurcation or a Neimark-Sacker bifurcation(Fig. 4(d)). Fig. 4 visualizes the bifurcation and chaos of the solution of system (6), and also proves that, even if marginal cost is the same in the asymmetrical contest dynamic model, the Neimark-Sacker bifurcation still appears, which extends the conclusion of Bischi et al. [28]. It is worth pointing out that parameters k1 and k2 are symmetric in system (6). Therefore, in case we select k1 as the bifurcation parameter, the similar conclusion can be unearthed (see Fig. 5). Also the maximum Lyapunov exponent LE is sketched in Figs. 4 and 5 to demonstrate bifurcations and chaos. The Lyapunov exponent is one of the parameters Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Fig. 4. The bifurcation diagram of the solutions x1 (blue) and x2 (green) of system (6) with regard to k2 , and down the maximum Lyapunov exponent LE (magenta) with regard to k2 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. The bifurcation diagram of solutions x1 (blue) and x2 (green) of system (6) with regard to k1 , and down the maximum Lyapunov exponent LE (magenta) with regard to k1 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

to characterize the motion state of the system. If the system is stable, then the maximum Lyapunov exponent is always negative and equal to zero only at the bifurcation points. Otherwise, the maximum Lyapunov exponent of the system is positive. On the other hand, let vi or c as the bifurcation parameter. As shown in Fig. 4, we can get four points UK1 = (0.5, 3.1 ), U K2 = (1.4, 2.7 ), U K3 = (1.6, 0.5 ), and U K4 = (1.6, 2.5 ) about the speeds of adjustment, where the equilibrium point E is erratic. Fig. 6 reveals the bifurcation diagram with respect to parameters v1 , v2 and c in system (6) at UK1 (see Figs. 12, 13 and 14 in Appendix B at U K2 − U K4 , respectively). In Fig. 6, when v1 = 4, v2 = 2.1, and c = 1, system (6) is in flip bifurcation, which is consistent with theoretical analysis (see Fig. 2). In this case, from Fig. 6 we see that the system can return to a stable state by increasing v1 , decreasing v2 or decreasing c appropriately, which is also consistent with theoretical analysis above. Next, numerical simulations are performed to demonstrate the main characteristics of chaos. The strange attractor is one of the important characteristics of chaos, which reveals the inherent regularity of complex dynamic in chaotic state. Therefore, when the system is in a state of chaos, the competitor can predict the opponent’s effort outlay in a short term Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Fig. 6. (color online) The bifurcation diagram of system (6) regarding other parameters at UK1 for each case. (a) displays the bifurcation diagram regarding v1 with v2 = 2.1 and c = 1. (b) displays the bifurcation diagram regarding v2 with v1 = 4 and c = 1. (c) displays the bifurcation diagram regarding c with v1 = 4 and v2 = 2.1.

according to the inherent regularity. As a matter of fact, via a Neimark-Sacker bifurcation, the dynamical system converts from stable to quasiperiodic, owing to the occurrence of a closed invariant curve (see Fig. 7). With the increasing of k1 and k2 , the area surrounded by the closed invariant curve is gradually enlarging until chaos occurs. However, via a flip bifurcation, the system turns into firstly periodic and then chaotic when the values of k1 and k2 are large enough (see Fig. 8). Figs. 7 and 8 simulate the process of chaos, and verify the above theoretical analysis that there are two routes to chaos at points outside the stable region. The sensitive dependence on initial conditions is another vital feature of chaos. So as to confirm the sensitiveness to initial conditions of system (6), the relationship between effort and time is depicted in Fig. 9. By calculating two orbits of x1 -coordinate with initial points (0.9030, 0.4741) and (0.9030 + 10−5 , 0.4741 ) at UK4 , we can get the game results revealed in Fig. 9(a). At the beginning, the two orbits of x1 -coordinate are indistinguishable; As time goes on, the difference between them expands rapidly. Similarly, Fig. 9(b) verifies the sensitive dependence on initial effort of agent 2. This means that the small difference between initial values will make a significant impact on results of the contest. It even more confirms, with (k1 , k2 ) outside the stable region, system (6) drops into chaos. When the system is in a chaotic state, the market changes into inconstant and agents are highly difficult to design long-term strategy. The minor adjustment of the initial effort has a great effect on the competing result. Also, the fractal plays a critical role in characteristic of chaos. Fractal dimension is a quantitative description of the complexity of the attractor geometry structure. Fractal means that the chaotic motion state has a multi-leaf, multi-layer structure, and the leaf layer is more and more fine, which is expressed as an infinite level self-similar structure. The Lyapunov dimension is defined in [64–66] as follows:

i = j dL = j +

λ i=1 i , |λ j+1 |

(17)

where λ1 ≥ λ2 ≥ · · · ≥ λn are values of Lyapunov exponents, and j is the largest largest integer such that i= j+1 λi < 0. i=1 In our 2-dimensional map (6), Lyapunov dimension is given by

dL = 1 +

λ1 |λ2 |

λ1 > 0 > λ2 ,

i = j

i=1

λi ≥ 0 and

(18)

where λ1 and λ2 are Lyapunov exponents, which meet the given condition. As the parameters value (k1 , k2 , v1 , v2 , c) are given, system (6) has two Lyapunov exponents λ1 and λ2 . By the definition of the Lyapunov dimension of the strange λ attractor of system (6), map (6) has a fractal dimension dL = 1 + |λ1 | (see Table 1). It can be found system (6) exhibits a 2

fractal structure and its attractor has the fractal dimension. Finally, numerical simulations are performed to demonstrate the Poincare section and power spectrum that can reveal the universal characteristics of chaotic behavior (see Fig. 10). Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Fig. 7. The invariant curves for system (6) with parameters: (a), (d) and (g) k1 = 1, 45 and k2 = 2.45, 2.50 and 2.55, respectively; (b), (e) and (h) k1 = 1.50 and k2 = 2.45, 2.50 and 2.55, respectively; (c), (f) and (i) k1 = 1.55 and k2 = 2.45, 2.50 and 2.55, respectively. Table 1 The Lyapunov exponents λ1 and λ2 , and the fractal dimension dL of system (6) when the parameters value (k1 , k2 , v1 = 4, v2 = 2.1, c = 1 ) are given. The dot · indicates that there is no fractal dimension under this given set of parameters. k1

k2

λ1

λ2

dL

k2

λ1

λ2

dL

k2

λ1

λ2

dL

1.45 1.50 1.55 1.80 1.85 1.90

2.50 2.50 2.50 0.60 1.00 1.65

−0.0032 0.0626 0.0831 0.2694 0.3771 0.3034

−0.0032 −0.1231 −0.3541 −0.5792 −1.6639 −1.1708

· 1.51 1.23 1.47 1.23 1.26

2.55 2.55 2.55 0.65 1.05 1.70

0.0432 0.1032 −0.0135 0.2605 0.2753 0.2721

−0.0883 −0.2288 −0.4059 −0.6476 −1.6073 −0.8438

1.49 1.45 · 1.40 1.17 1.32

2.60 2.60 2.60 0.70 1.10 1.75

0.1346 0.0238 0.1686 0.2484 0.2598 0.2505

−0.2968 −0.2962 −0.3475 −0.7165 −1.6997 −0.6348

1.45 1.08 1.49 1.35 1.15 1.39

4. Chaos control From the analysis of the above section, we draw a conclusion that there may be chaos in the dynamic behavior of system (6). The chaotic motion has critically influenced dialogue on economics and political economy. In the perspective of realistic claim, there is not always expected for the chaotic motion, wherefore we require to avert the emergence of chaos. In our model, it can be evidenced that with an increase of effort adjustment speed ki , system (6) will turn into unstable and eventually get caught in chaos. Thence, the parameter adjustment method is applied to control the affect of parameters ki on system (6). The controlled system is as follows:

 

⎧ x2,t ⎪ ⎪ x = 1 − η x 1 + k v − c + ηx1,t ; ( ) 1 ,t+1 1 ,t 1 1 ⎨ (x + x2,t )2   1,t

⎪ x1,t ⎪ ⎩ x2,t+1 = (1 − η )x2,t 1 + k2 v2 − c + ηx2,t . (x1,t + x2,t )2

(19)

Here η is a control parameter and other parameters are the same as above. At η = 0, controlled system (19) demotes into original system (6). It can be seen from Fig. 11, with control parameter η increasing, the flip bifurcation and Neimark-Sacker bifurcations are gone. The system gets rid of the chaos by degrees and turns into stable when the control parameter η is properly increased. Therefore chaos control is successful. The method discussed here could be employed in majority chaotic dynamic systems. Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Fig. 8. The strange attractors for system (6) with parameters: (a), (d) and (g) k1 = 1, 80 and k2 = 0.60, 0.65 and 0.70, respectively; (b), (e) and (h) k1 = 1.85 and k2 = 1.0, 1.05 and 1.10, respectively; (c), (f) and (i) k1 = 1.90 and v2 = 1.65, 1.70 and 1.75, respectively.

Fig. 9. (color online) Sensitive dependence on initial conditions at UK4 : (a) the two orbits of x1 -coordinate with initial points (0.9030, 0.4741) and (0.9030 + 10−5 , 0.4741 ), respectively; (b) the two orbits of x2 -coordinate with initial points (0.9030, 0.4741) and (0.9030, 0.4741 + 10−5 ), respectively.

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Fig. 10. The Poincare section and power spectrum for system (6) at UK4 . (a) The blue dot diagram represents the phase portrait, and the red star diagram represents the Poincare section; (b) and (c) The power spectrum of the solutions x1 and x2 of system (6), respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 11. (color online) The bifurcation diagram regarding control parameter η at UK1 , UK2 , UK3 and UK4 , respectively.

5. Conclusions This paper features the dynamical behaviors of the asymmetric contest model with bounded rationality. Firstly, by exploring the local stability and bifurcations of the equilibrium point, we give the stability conditions and prove that the asymmetric contest model inherently produces two routes to chaos: the flip and Neimark-Sacker bifurcations, even if the agents have the same marginal cost. Secondly, itâs indicated that the emergency of chaos extremely relies on values of bifurcation parameters. For example, the sufficiently large adjustment speed of the boundedly rational agent will give rise to volatility of the system, via a flip or Neimark-Sacker bifurcation of the Nash equilibrium. Hence, to control flip and NeimarkSacker bifurcation, we apply the parameter adjustment method which could make agentâs effort stable at the Nash equilibrium point. Finally, it should be pointed out there are many variations and extensions of our proposed dynamic model which could be left as future research content, such as heterogeneous expectations, quadratic adjustment functions, n-player games, different contest success functions, and so on. Please cite this article as: M. Zhang, G. Wang and J. Xu et al., Dynamic contest model with bounded rationality, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124909

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Declaration of Competing Interest The authors declare that they have no conflict of interest. Acknowledgements This work was supported in part by National Natural Science Foundation of China under Grant 11631014, Grant 11871311, in part by Innovation Method Fund of China under Grant 2018IM020200, and in part by China Postdoctoral Science Foundation under Grant 2019TQ0188, Grant 2019M662315, and Grant 2019M662380. Appendix A. The proof of Proposition 2 



Proposition 2. When inequality (14) (ii ) is satisfied, inequality (14) (iiib ) holds. Proof. When 0 < k1 ≤

−δv2 c2

=

 δv2 + k1 c2 ≤ 0,

2v2 , c 2 ( v1 +v2 )

we have

−δ k1 v1 − 2 < 0. That can deduce



2δv2 + k1 c2 < 0, −2δ k1 v1 − 4 < 0. 

Therefore, from inequality (14) (ii ), we obtain

k2 <

−2δ k1 v1 − 4 −δ k1 v1 − 2 −δ k1 v1 − 2 = < . 2 2δv2 + k1 c2 δv2 + k1 c2 δv2 + k1 c 2



Hence, we can deduce that inequality (14) (iiib ) holds in this case. When

2v2 c 2 ( v1 +v2 )

=

−δv2 c2

 δv2 + k1 c2 > 0,

−2 < k1 ≤ δv = 1

v1 +v2 cv1 , we have

−δ k1 v1 − 2 ≤ 0. That can deduce

k2 > 0 ≥

−δ k1 v1 − 2 . δv2 + k1 c2 

Hence, we can deduce that inequality (14) (iiib ) holds in this case. −2 When k1 > δv = 1

v1 +v2 −2δv2 4v = c2 (v +2 v ) . So we can get cv1 , we have k1 > c2 1 2

 δv2 + k1 c2 > 0,

−δ k1 v1 − 2 > 0, and



2δv2 + k1 c2 > 0, −2δ k1 v1 − 4 > 0. 

Therefore, from inequality (14) (ii ), we obtain

k2 >

−2δ k1 v1 − 4 −δ k1 v1 − 2 −δ k1 v1 − 2 = > . k1 c 2 2δv2 + k1 c2 δv2 + k1 c2 δv2 + 2



Hence, we can deduce that inequality (14) (ii ) holds in this case.   In conclusion, when inequality (14) (ii ) is satisfied, inequality (14) (ii ) holds.



Appendix B. The bifurcation diagram

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Fig. 12. (color online) The bifurcation diagram of system (6) regarding other parameters at UK2 for each case. (a) displays the bifurcation diagram regarding v1 with v2 = 2.1 and c = 1. (b) displays the bifurcation diagram regarding v2 with v1 = 4 and c = 1. (c) displays the bifurcation diagram regarding c with v1 = 4 and v2 = 2.1.

Fig. 13. (color online) The bifurcation diagram of system (6) regarding other parameters at UK3 for each case. (a) displays the bifurcation diagram regarding v1 with v2 = 2.1 and c = 1. (b) displays the bifurcation diagram regarding v2 with v1 = 4 and c = 1. (c) displays the bifurcation diagram regarding c with v1 = 4 and v2 = 2.1.

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Fig. 14. (color online) The bifurcation diagram of system (6) regarding other parameters at UK4 for each case. (a) displays the bifurcation diagram regarding v1 with v2 = 2.1 and c = 1. (b) displays the bifurcation diagram regarding v2 with v1 = 4 and c = 1. (c) displays the bifurcation diagram regarding c with v1 = 4 and v2 = 2.1.

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