On the stability of quadratic dynamics in discrete ... - Semantic Scholar
Automatica 48 (2012) 1182–1189
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On the stability of quadratic dynamics in discrete time n-player Cournot games✩ Hamed Kebriaei 1 , Ashkan Rahimi-Kian Smart Network Lab/Control and Intelligent Processing Center of Excellence, School of ECE, University of Tehran, P.O. Box 14395-515, Tehran, Iran
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Article history: Received 19 December 2010 Received in revised form 18 September 2011 Accepted 31 October 2011 Available online 10 April 2012 Keywords: Cournot game Lyapunov function Stability analysis Nash equilibrium point
abstract Motivated by analyzing the dynamic adjustment process of players in an n-player Cournot game, in this paper a discrete-time quadratic dynamical system is proposed and the stability of its equilibrium is analyzed. Several output adjustment mechanisms (e.g. best reply, adaptive adjustment and myopic) and expectations (e.g. naïve and adaptive) in a Cournot game with linear price function and quadratic costs, which form a quadratic dynamical system, are the special cases of the proposed model. The stability of the proposed quadratic dynamical system is analyzed with (1) time invariant parameters, (2) time invariant parameters in the presence of disturbance and (3) bounded time varying parameters in the presence of disturbance. In each case, the sufficient condition to find the region that belongs to the basin of attraction is derived using some discrete-time converse Lyapunov theorems. In addition, the proposed model and theorems are utilized to analyze the stability of the boundary and Nash equilibrium points of a Cournot game with three heterogeneous players. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The first dynamic model of oligopoly games was introduced by Cournot (1838). In his game model, players make their decisions assuming the rivals would offer their last output quantity in the next step of the game (naive expectation). The stability of strict Nash equilibrium point in a linear n-player Cournot game with naïve expectations was discussed by Theocharis (1960). He showed that the above mentioned game is stable for n < 3, neutrally stable for n = 3 and unstable for n > 3. Fisher (1961) showed that the Cournot game could be generally stable for more than three players in a more general case, where the players use adaptive adjustment and the quadratic cost function. The stability analysis of the Nash equilibrium point was also generalized in linear oligopolistic models where players use adaptive expectations instead of naive expectations (see, e.g., Bischi & Kopel, 2001, Okuguchi, 1970, Okuguchi, 1976, Okuguchi & Szidarovszky, 1999, Szidarovszky & Okuguchi, 1988, Szidarovszky & Yen, 1991). The stability of continuous-time Cournot games was studied by several researchers. Hahn (1962) analyzed the stability of Cournot oligopoly solutions under general conditions. However, the actual output of each player was adjusted in proportion to the difference
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Editor Berç Rüstem. E-mail addresses: [email protected] (H. Kebriaei), [email protected] (A. Rahimi-Kian). 1 Tel.: +98 21 88027756; fax: +98 21 88778690.
0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.03.021
between its profit maximizing output and the actual one. Okuguchi (1964) generalized Hahn’s results, assuming that the rate of change of each player’s actual output was a sign-preserving and monotonic function with respect to the difference between profit maximizing output and the actual one; then derived the sufficient stability conditions. However, Al-Nowaihi and Levine (1985) presented a counter-example to the Hahn–Okuguchi stability analysis results. They showed that the game equilibrium is globally stable only if the number of firms is less than or equal to five. Okuguchi and Yamazaki (2008) studied a general Cournot game under a gradient dynamics output adjustment. It was shown that the unique equilibrium is globally stable if the game satisfies a set of assumptions and the number of players is less than or equal to three. Complex dynamics of the Cournot game was studied in several research works (Agiza, 1998; Agiza, Hegazi, & Elsadany, 2002; Bischi & Naimzada, 1999; Bischi, Stefanini, & Gardini, 1998; Kopel, 1996; Puu, 1991, 1996, 1998). For example, the dynamics of a three-player game was discussed by Puu (1996). Agiza (1998) studied the dynamics of three and four player Cournot games with identical and linear cost functions for the players. The result was extended to an n-player game by Ahmed and Agiza (1998). The basins of attraction for each (of multiple) equilibrium point were discussed in Agiza, Bischi, and Kopel (1999), where the players used naive expectations. A review of studies on complex dynamics of oligopoly games could be found in Rosser (2002). Many research works have considered the homogeneous expectation of rivals which means that the players assume that the rivals use identical rules for behavior adjustment. However, they could have different adjustment mechanisms.
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
In real dynamic repeated games, it is assumed that none of the players has complete information about the game parameters and the rivals’ future decisions; thus the players should act adaptively in the game. In an adjustment process the players make their decisions based on local estimation of their marginal payoffs (Ahmed, Agiza, & Hassan, 2000; Bischi & Naimzada, 1999). This adjustment mechanism has been called ‘‘myopic’’ by some authors (Dixit, 1986; Flam, 1993). The case of having incomplete information about market price function was considered by Bischi, Sbragia, and Szidarovszky (2008) and Szidarovszky (2003). In those research works an adjustment mechanism was designed to learn the slope or intercept of the market price function. The stability conditions of the adjustment process were given for n-player. However, it was assumed that players had complete information about their rivals’ cost functions. In most of the previous research works with nonlinear dynamic adjustment model, only the local stability of the equilibrium was analyzed. In some other research works, the global stability of Cournot game was analyzed in some special cases; where the players used specific adjustment mechanism (e.g. Best reply, adaptive, myopic) or expectations (e.g. naïve, adaptive). However, the global stability of discrete-time Cournot games with quadratic nonlinearities, bounded time varying parameters and disturbances have not been studied in the literature. In this paper a quadratic dynamical system is proposed for output adjustment of the players in a stochastic dynamic Cournot game which allows for the heterogeneity in players’ adjustment mechanism. The Cournot game with linear price function, quadratic cost and some well known adjustment mechanisms (e.g. best reply, adaptive adjustment and myopic) and expectations (e.g. naïve and adaptive), which are explained in detail in the next section, are all special cases of the proposed model. In addition, the stability analysis of the model is provided in both cases of time invariant and bounded time varying parameters, considering the effect of disturbance. Moreover, the regions that belong to the basin of attraction2 are estimated for all cases. The organization of this paper is as follows; Section 2 provides the preliminaries of this paper. In Section 3, the stability theorems are given for three cases including: time invariant, time invariant in presence of disturbance and time variant in presence of disturbance. In Section 4, a Cournot game example is provided to illustrate the stability analysis results. Finally, the paper is concluded in Section 5. 2. Preliminaries Consider a market consisting of n sellers. The market price function could be expressed as follows:
λ(t ) = λ0 −
n
αi qi (t )
(1)
i =1
where qi is the output quantity (decision) and αi > 0 represents the market power of the player-i. The scalar λ0 is the cap price of the market which is usually set by the market regulator. Eq. (1) implies that increasing the market supply quantities would result in decreasing the market price. The payoff of player-i is calculated as follows:
πi (t ) = λ(t )qi (t ) − Ci (qi (t )) i = 1, . . . , n (2) where Ci (·) is the supply cost function of player-i with the following quadratic form: Ci (qi (t )) = ai q2i (t ) + bi qi (t ) + ci
(3)
where ai , bi , ci > 0.
2 The basin of attraction is a set of points in the space of system variables such that initial conditions chosen in this set dynamically evolve to a particular attractor.
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As it is clear from Eqs. (1)–(2), each player’s payoff is affected by other players’ output quantities through the market price function. The goal of each player is to maximize her payoff function by taking into account the decision of her rivals; therefore, she needs to estimate other players’ decisions. In oligopolistic games, the players may use different adjustment processes (for updating their output quantity) and different expectations/estimations of the rivals’ outputs. The best reply adjustment process could be used together with naïve expectation (Theocharis, 1960) or with adaptive expectation (Bischi & Kopel, 2001). Adaptive adjustment (Huang, 2001) could be used together with naïve expectation (Fisher, 1961) or with adaptive expectation (Okuguchi, 1976). Players may use adjustment processes based on profit gradient that require less information and consequently lower degree of rationality. Bischi and Naimzada (1999) adopted the adjustment process based on profit gradient (myopic mechanism) with naïve expectation. Moreover, the players may use different adjustment processes (heterogeneous expectations) or similar adjustment processes (homogeneous expectations). Consider a Cournot oligopoly market model, where each player is able to predict its rivals’ decisions one step forward. The optimal output quantities of the players could be determined by solving the following set of n equations: qi (t + 1)
= arg max πi (q1i (t + 1), q2i (t + 1), . . . , q(i−1)i (t + 1) qi (t +1)
× qi (t + 1), q(i+1)i (t + 1), . . . , qni (t + 1)) i = 1, . . . , n (4) where qji (j = 1, . . . , n, j ̸= i) is the predicted output of player-j by player-i. If a unique optimal solution for Eq. (4) exists, it could be described as follows: qi (t + 1) = fi (q1i (t + 1), q2i (t + 1), . . . , q(i−1)i (t + 1)
× q(i+1)i (t + 1), . . . , qni (t + 1))
(5)
where, fi (·) is the reaction function of player-i. Cournot assumed that qji (t + 1) = qj (t ), which implies that each player assumes her rivals will repeat their last step decisions in the next time-step. This is called the naive expectation. Therefore, the Cournot reaction function has the following dynamic discrete-time form: qi (t + 1) = fi (q1 (t ), q2 (t ), . . . , qi−1 (t ), qi+1 (t ), . . . , qn (t )).
(6)
Eq. (6) describes the best reply dynamics in an oligopoly game with naive expectations of the players. In a myopic adjustment mechanism based on profit gradient, if the marginal payoff is positive (negative), the player will increase (decrease) its supply quantity for the next time-step. The myopic adjustment equation together with naive expectation (of rivals) and linear speed of adjustment could be written as follows (Bischi & Naimzada, 1999): qi (t + 1) = qi (t ) + γ qi (t )
∂πi (t ) , ∂ qi (t )
(7)
where γ is a positive scalar that shows the relative behavior adjustment speed. The minimum value of qi (t ) should be bounded by zero. Note that Nash equilibria are fixed points of Eq. (7), but not vice versa.3 Another strategic behavior is adaptive adjustment (Huang, 2001). Through this strategy a player calculates its desired supply quantity for the next time-step using the weighted sum of its reaction function and its output quantity in the previous time-step.
3 The boundary fixed points exist with some zero coordinates, which are not located at the intersections of reaction functions, so they are not Nash equilibria.
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The dynamic equation for the adaptive adjustment for player-i is shown as follows: qi (t + 1) = (1 − v)qi (t ) + v fi (q1 (t ), q2 (t ), . . . , qi−1 (t )
× qi+1 (t ), . . . , qn (t )) (8) where v ∈ [0, 1] is the adjustment factor of player-i. Note that for the special case of v = 1, Eq. (8) reduces to Eq. (3), which represents the best reply dynamics with naive expectations. Another method of dynamic decision making in Cournot games is that each player predicts the output quantities of her rivals for the next period, and then makes a profit maximizing decision based on the predicted outputs (Okuguchi, 1976). A general formation of expectations that is usually used in economy is adaptive expectations. The player-i adapts her expectation from player-j’s output as follows: qji (t + 1) = qji (t ) + βi (qj (t ) − qji (t )) 0 ≤ βi ≤ 1.
(9)
It is easy to see that under this formulation, the estimated output is the weighted average of the past actual outputs, where the weights decrease as time proceeds (Okuguchi, 1976). The generalization of the proposed model by Okuguchi (1976) to multi-product firms was considered by Okuguchi and Szidarovszky (1999). Moreover, it was shown by Bischi, Chiarella, Kopel, and Szidarovszky (2010) that the local stability of the best reply dynamics (4) with adaptive expectation (9) is equivalent to adaptive adjustment process with naive expectation in (8). In the following, an n-dimensional quadratic dynamic model is developed. The n-player Cournot game (Eqs. (1)–(3)) together with the above mentioned adjustment mechanisms and expectations which form a quadratic dynamical system are special cases of the proposed model. In addition, the bounded time varying parameters and disturbances are also considered in the proposed model, which had not been studied in the literature of the dynamic Cournot games. The stability analysis of the proposed stochastic dynamic model is presented for different cases in the next section. 3. Problem formulations 3.1. Time invariant case without disturbance Consider an n-player Cournot game with quadratic dynamic output adjustment. The players’ decision model may be represented by the following dynamical system: y(t + 1) = f (y(t )) = C · y(t ) + Y (t ) · B · y(t ) + e
(10)
where, y(t ) = [q1 (t ), q2 (t ), . . . , qn (t )]T , Y = diag(q1 (t ), q2 (t ), . . . , qn (t )), e is a n × 1 vector, C and B are n × n matrices. If y∗ is an equilibrium vector of the system defined by Eq. (10), then by defining: y˜ = y − y∗ , it could be rewritten as follows: y˜ (t + 1) = [Y ∗ · B + C ] · y˜ (t ) + Y˜ (t ) · B · y˜ (t ) + Y˜ · B · y∗
(11)
where, Y ∗ = diag(y∗ ) and Y˜ (t ) = diag(˜y). In addition we have: Y˜ · B · y∗ = diag(B · y∗ ) · y˜ .
(12)
Therefore, Eq. (10) could be represented as following: x(t + 1) = Ax(t ) + X (t ) · B · x(t )
(13)
where: x(t ) = y˜ (t ),
X (t ) = Y˜ (t )
A = Y ∗ · B + C + diag(B · y∗ ).
∥x(t )∥ =
n
(14)
i=1
(16)
Here λmax (·) is the maximal eigenvalue of the corresponding matrix. Remark 1. The proposed matrix norm is a p-norm (p = 2) (Meyer, 2000) and holds the following property:
∥NM ∥ ≤ ∥N ∥ ∥M ∥ where, N and M are n × n matrices. Theorem 1. Consider the dynamical system of Eq. (13), if there exists two positive definite matrices G and H such that AT HA − H = −G, then the origin, equilibrium point of dynamical system (13), is asymptotically stable within the region Λ defined as:
Λ = x ∈ ℜn | xT Hx ≤ r , r ∈ ℜ+
(17)
where Λ is embedded in Γ , defined as:
Γ = x ∈ ℜn | λmin (G) ≥ ∥B∥2 ∥H ∥ |xi |2 + 2 ∥B∥ ∥HA∥ |xi | for i = 1, 2, . . . , n .
(18)
Proof. Consider the following Lyapunov function for the system of Eq. (13): V (x) = xT Hx > 0.
(19)
Therefore, we have:
1V (x(t )) = V (x(t + 1)) − V (x(t )) = (Ax(t ) + X (t )Bx(t ))T H (A · x(t ) + X (t ) · B · x(t )) − x(t )T Hx(t ) = x(t )T (AT HA − H )x(t ) + x(t )T BT X (t )HAx(t ) + x(t )T BT X (t )HX (t )Bx(t ) + x(t )T AT HX (t )Bx(t ) = x(t )T (AT HA − H )x(t ) + 2x(t )T BT X (t )HAx(t ) + x(t )T BT X (t )HX (t )Bx(t ). Now, using AT HA − H = −G we have: V (x(t + 1)) − V (x(t )) = −x(t )T Gx(t ) + 2x(t )T BT X (t )HAx(t )
+ x(t )T BT X (t )HX (t )Bx(t ) = −x(t )T G − 2BT X (t )HA − BT X (t )HX (t )B x(t ) ≤ −x(t )T G − 2 BT ∥X (t )∥ ∥HA∥ − BT ∥X (t )∥ ∥H ∥ ∥X (t )∥ ∥B∥ x(t ). Since: λmin (G)x(t )T x(t ) ≤ x(t )T Gx(t ) V (x(t + 1)) − V (x(t ))
≤ −x(t )T λmin (G) − 2 BT ∥X (t )∥ ∥HA∥ − BT ∥X (t )∥ ∥H ∥ ∥X (t )∥ ∥B∥ x(t ). λmax (X T (t )X (t )) = maxi Furthermore, we have ∥X (t )∥ = |xi (t )| = xmax (t ); therefore: ≤ −x(t )T λmin (G) − 2 BT ∥HA∥ xmax (t ) − ∥B∥2 ∥H ∥ x2 (t ) x(t ). max
1/2 x2 ( t )
1/2 ∥A∥ = λmax AT A .
V (x(t + 1)) − V (x(t ))
Definitions. Consider the following norm definitions for the n × 1 vector x(t ) and n × n matrix A:
and
(15)
Using Eq. (18) we have:
1V (x(t )) = V (x(t + 1)) − V (x(t )) < 0.
(20)
Eqs. (20) guarantee the stability of system (13), if parameter r is selected small enough, such that the domain described by Eq. (17) is embedded within the domain described by Eq. (18).
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
Remark 2. If all eigenvalues of matrix A lay inside the unit circle, for an arbitrary positive-definite symmetric matrix G there exists a positive-definite symmetric matrix H, which is the unique solution of the so-called Lyapunov equation: AT HA−H = −G (Elaydi, 2005). The obtained matrix H could be used to estimate the stability domain. Remark 3. If there exists the positive-definite symmetric matrix G, according to Eq. (18), then 1V (x(t )) will be negative in a hypersphere neighborhood, Uδ = {x ∈ ℜn | ∥x∥ < δ} ⊂ Γ , of the origin if δ is sufficiently small. Since Uδ exists and is not empty therefore, Γ also exists and is not empty. Remark 4. The parameter δ should be determined first in order to estimate Γ (by Uδ ), and then one may set r = δ 2 λmin (H ) to determine Λ. 3.2. Time invariant case with disturbance In this section a bounded disturbance is considered as an arbitrary input to the system of Eq. (13) as follows: x(t + 1) = A · x(t ) + X (t ) · B · x(t ) + d(t )
(21)
where ∥d(t )∥ < d0 (d0 ∈ ℜ+ ). The sufficient condition for a region to belong to the basin of attraction for the system of Eq. (21), is given by the proposed lemmas and theorem in this section.
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Then we have,
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) = ∥A · x(k0 ) + X (k0 )B · x(k0 ) + d(k0 )∥ − ∥x(k0 )∥ ≤ (∥A + X (k0 )B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ ≤ (∥A∥ + ∥X (k0 )∥ ∥B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ . Since x(k0 ) ∈ Ω , we have ∥A∥ + ∥X (k0 )∥ ∥B∥ − 1 < −k. Therefore, 1V (x(k0 )) ≤ −k ∥x(k0 )∥ + ∥d(k0 )∥. V (x(k0 + 1))
= ∥x(k0 + 1)∥ < V (x(k0 )) − k ∥x(k0 )∥ + ∥d(k0 )∥ = (1 − k) ∥x(k0 )∥ + ∥d(k0 )∥ . Since ∥x(k0 )∥
kN and consequently, x(k) ∈ Λ − Ω . Else if x(k0 ) ∈ Ω − Γ then using Lemma 2 we have: x(k0 + 1) ∈ Ω − Γ . Therefore, by induction it could be concluded that x(k) ∈ Ω − Γ for all k > kN .
1−k−∥A∥ Lemma 1. Let Γ be x ∈ ℜn | 1k d0 ≤ ∥x∥ < ∥B∥ < r > 0 , if Γ is not empty and x(k0 ) ∈ Γ then ∥x(k0 + 1)∥ < ∥x(k0 )∥ for some 0 < k < 1 − ∥A∥.
Remark 5. According to Γ in Lemma 1, a sufficient condition for existence of a basin of attraction (Ω ) is: ∥A(t )∥ < 1 − k, which is more restrictive than the case of no disturbance.
Proof. Consider the following Lyapunov function for the system of (21):
Remark 6. Theorem 2 shows that the invariant attracting set of system (21) is conditioned to the bound of disturbance d(t ) (the larger bound of disturbance, the wider bound of convergence).
V (x(k0 )) = ∥x(k0 )∥ .
(22)
3.3. Time variant case with disturbance
Then we have:
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) = ∥A · x(k0 ) + X (k0 )B · x(k0 ) + d(k0 )∥ − ∥x(k0 )∥ ≤ (∥A + X (k0 )B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ .
Consider the time varying version of system (10) as follows: x(t + 1) = f (t , x(t ), d(t ))
= A(t ) · x(t ) + X (t ) · B(t ) · x(t ) + e(t ) + d(t )
Since ∥d(k0 )∥ ≤ d0 ≤ k ∥x(k0 )∥, we have:
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) ≤ (∥A + X (k0 )B∥ − 1 + k) ∥x(k0 )∥ ≤ (∥A∥ + ∥X (k0 )∥ ∥B∥ − 1 + k) ∥x(k0 )∥ = (∥A∥ + xmax (k0 ) ∥B∥ − 1 + k) ∥x(k0 )∥ . Moreover we have: xmax (k0 ) ≤ ∥x(k0 )∥, then:
1V (x(k0 )) ≤ (∥A∥ + ∥x(k0 )∥ ∥B∥ − 1 + k) ∥x(k0 )∥ . Thus, 1V (x(k0 )) < 0 in Γ . Therefore, V (x(k0 + 1)) = ∥x(k0 + 1)∥ < V (x(k0 )) = ∥x(k0 )∥, and the proof is completed. Lemma 2. If x(k0 ) ∈ Ω − Γ then, x(k0 + 1) ∈ Ω − Γ and thus Ω − Γ is an invariant set for the system of Eq. (21); where:
1 − ∥A∥ − k Ω = x ∈ ℜn | ∥x∥ < 0 . ∥B∥
(23)
Proof. Consider the following Lyapunov function for the system of Eq. (21): V (x(k0 )) = ∥x(k0 )∥ .
(24)
(25)
where, A(t ) and B(t ) are the n × n matrices with bounded elements that deviate around their nominal values; e(t ) is a time varying n × 1 vector with ∥e(t )∥ < e0 and d(t ) is a bounded n × 1 disturbance vector with ∥d(t )∥ < d0 . Since the solutions of x(t ) = f (t , x(t ), d(t )) are time dependent in general, the system of Eq. (25) has no equilibrium points. Nevertheless, since the elements of the above mentioned system matrices and vectors are bounded, there might exist a nominal equilibrium point x∗ and a sufficiently large constant l such that each solution of the quadratic equation: x(t ) = f (t , x(t ), d(t )) lays within a compact set like K = {x| ∥x − x∗ ∥ ≤ l}. In this section, the stability of system (25) with respect to K is analyzed. The basic converse Lyapunov theorems for discrete time varying systems with disturbance, was introduced in Jiang and Wang (2002). Definitions. Let K be a nonempty closed subset of Rn . The distance of x with respect to K is defined as follows (Jiang & Wang, 2002):
∥x∥K = d(x, K ) = inf ∥x − η∥ . η∈ K
(26)
4 Invariant attracting set is a set of points in the phase space, invariant under the dynamics, toward which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution.
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Definitions. For the defined vector norm in Eq. (15), consider the following norm definitions for the n × n matrix M (x, t ):
∥M (x, t )∥ = min {b : ∥M (x, t )ς∥ ≤ b ∥ς ∥ for all ς ̸= 0} .
(27)
−ε ∥x(t )∥ + ∥e(t )∥ + ∥d(t )∥ ≤ −ε ∥x(t )∥ + ε
Clearly, ∥M (x, t )x∥ ≤ ∥M (x, t )∥ ∥x∥. Lemma 3. The defined matrix norm in Eq. (27) is a sub-multiplicative norm. Proof. Suppose that c1 , c2 and c are the norms of matrices M , N and MN, respectively; then we have:
∥MN ς ∥ = ∥M (N ς )∥ ≤ c1 ∥N ς ∥ ≤ c1 c2 ∥ς ∥
for all ς ̸= 0.
According to the norm definition for matrix MN:
∥MN ∥ = c = min {b : ∥MN ς ∥ ≤ b ∥ς ∥ for all ς ̸= 0} . Therefore c ≤ c1 c2 and the proof is completed.
Theorem 3. Consider the nonempty set Ω defined as:
Ω =
x ∈ ℜ | ∥x ∥ < n
1 − ∥A(t )∥ − ε
∥B(t )∥
for some 0 < ε < 1 − ∥A(t )∥
(28)
and the set K : (29)
Then, the system of Eq. (25) is uniformly asymptotically stable with respect to K in Ω . Proof. Consider the following function: V (x(t )) = ∥x(t )∥K
(e0 + d0 ) ε (e0 + d0 ) ∥x(t )∥ ≤ . ε ∥x(t )∥ >
1
Therefore:
(e0 + d0 ) 1V (x(t )) < −ε ∥x(t )∥ − ε
If ∥x(t )∥ ≤
(e0 +d0 ) ε
then ∥x(t )∥K = 0 and we have:
1V (x(t )) = V (x(t + 1)) = ∥A(t )x(t ) + X (t )B(t )x(t ) + e(t ) + d(t )∥ (e0 + d0 ) − ε ≤ (∥A(t )∥ + ∥X (t )∥ ∥B(t )∥) ∥x(t )∥ e0 + d0 + ∥e(t ) + d(t )∥ − ε e0 + d0 (e0 + d0 ) ≤ (1 − ε) + ∥e(t ) + d(t )∥ − ε ε ≤ −e0 − d0 + ∥e(t ) + d(t )∥ ≤ −e0 − d0 + ∥e(t )∥ + ∥d(t )∥ ≤ 0 = −ε ∥x∥K . In addition, if ∥x(t + 1)∥ ≥ 0 ε 0 , then V (x(t + 1)) = 0 and 1V (x(t )) = −V (x(t )) ≤ − ∥x∥K . Therefore V (x(t )) is a continuous Lyapunov function with respect to K in Ω for the system of Eq. (25). As a result, according to Lemma 2.7 in Jiang and Wang (2002), there also exists a smooth Lyapunov function with respect to K in Ω . Now using Theorem 1 in Jiang and Wang (2002), it is concluded that the system is uniformly asymptotically stable with respect to K in Ω and the proof is completed.
In the following, a three-player dynamic Cournot game with disturbance (including three myopic, adaptive and naïve players) is considered. The quadratic cost functions of the players are the same as given by Eq. (3) with bi = 0. The market price function is also defined as in Eq. (2).
Thus based on Definition 2.6 in Jiang and Wang (2002), the first necessary condition for Eq. (30) to be a continuous Lyapunov function with respect to K for the system of Eq. (25) is satisfied. ( e +d ) First assume that ∥x(t + 1)∥ < 0 ε 0 .
1V (x(t )) = = ≤ ≤
≤ −ε ∥x(t )∥K .
4. A Cournot game with three heterogeneous players
∥x∥K ≤ V (x) ≤ 2 ∥x∥K .
If ∥x(t )∥ >
(30)
Then we have: 2
(e0 + d0 ) . ε
(e +d )
(e0 + d0 ) . K = x| ∥x∥ ≤ ε
(e0 + d0 ) ∥x(t )∥ − ε = 0
Since ∥x∥ ≥ xmax and ∥A(t )∥ , ∥B(t )∥ are bounded, according to Eq. (28) there exists a constant 0 < γ < 1 such that 1V (x(t )) < −ε ∥x(t )∥ + ∥e(t )∥ + ∥d(t )∥. Moreover, according to Eq. (28) we have:
(e0 +d0 ) : ε
V (x(t + 1)) − V (x(t ))
∥A(t )x(t ) + X (t )B(t )x(t ) + e(t ) + d(t )∥ − ∥x(t )∥ (∥A(t ) + X (t )B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ (∥A(t )∥ + ∥X (t )∥ ∥B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ .
In addition:
∥X ∥ = min {b : ∥X ς ∥ ≤ b ∥ς ∥ for all ς ̸= 0} = min {b : ∥(x1 ς1 , x2 ς2 , . . . , xn ςn )∥ ≤ b ∥ς ∥ for all ς ̸= 0} = max |xi | = xmax .
q1 (t + 1) = q1 (t ) 1 + β
1V (x(t )) ≤ (∥A(t )∥ + xmax ∥B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ .
= (1 + βλ0 )q1 (t ) − β(2α + 2a1 )q21 (t ) − βα q2 (t )q1 (t ) − βα q3 (t )q1 (t ) + d1 cos(t )
(31)
and q2 (t + 1) = (1 − v)q2 (t ) + v
λ0 − α · (q1 (t ) + q3 (t )) 2(α + a2 )
+ d2 sin(t )
(32)
λ0 − α.(q1 (t ) + q2 (t )) + d3 cos(t ) 2(α + a3 )
(33)
and q3 (t + 1) =
where, d1 cos(t ), d2 sin(t ) and d3 cos(t ) are the game disturbances. Considering di = 0 i = 1, 2, 3, the game has a Nash–Cournot equilibrium and one boundary equilibrium as follows:
i
Therefore:
∂π1 (t ) ∂ q1 (t )
λ0 (α3 − α) λ0 (α2 − α) , , α2 α3 − α 2 α2 α3 − α 2 ∗ ∗ ∗ = q1 , q2 , q3
E0 = ENash
0,
(34)
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
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Table 1 The Cournot game parameters. Parameter
a1
a2
a3
α
β
λ0
v
d1
d2
d3
δλ
Value
0.1
0.17
0.27
0.05
0.01
117.83
0.8
10
750.5
15
50
where, αk = 2(α + ak ) k = 1, 2, 3 are positive parameters and: q∗1 = q∗3 =
λ0 (α3 − α)(α2 − α) fα
,
q∗2 =
λ0 (α3 − α)(α1 − α) fα
λ0 (α1 − α)(α2 − α)
,
fα
fα = 2α − α (α1 + α2 + α3 ) + α1 α2 α3 . 3
2
It is easy to show that the obtained boundary and Nash equilibrium points are positive. First, let us analyze the stability of the boundary equilibrium point. As was discussed in Section 3.1 the system dynamic equations could be converted to Eq. (14) ∥X (t )·B·x(t )∥ = 0 in (14) (i.e. and it is easy to verify that lim∥x∥→0 ∥x(t )∥ ∥X (t ) · B · x(t )∥ is O(x(t )) as ∥x(t )∥ → 0). Therefore from Elaydi (2005), if the linear part of system dynamic equations is unstable, then the zero solution of system is unstable too. According to Eqs. (14), matrix A for boundary equilibrium point is defined as in Box I. One could find out that an eigenvalue of A is outside the unit circle as follows:
Fig. 1. The phase plane of the players’ output quantities in the time invariant case in presence of disturbance.
λ0 (α2 − α) λ0 (α3 − α) − βα 2 α2 α3 − α α2 α3 − α 2 βλ0 =1+ (α2 − α)(α3 − α) > 1. α2 α3 − α 2
1 + βλ0 − βα
Therefore, the boundary equilibrium point of the system is unstable even in the time invariant case without disturbance. It is obvious that the boundary equilibrium point would also be unstable in the cases of time varying parameters and in presence of disturbance. Now the stability of Nash equilibrium point of the above game is analyzed. Using Eqs. (14) we have matrices A and B as given in Box II. Moreover, we have: ∥d(t )∥ ≤
d21 + d22 + d23 = l. If l
0} is a basin of attraction for the system. For example, for the dynamic game parameters given in Table 1, we have ∥A∥ = 0.2597, ∥B∥ = 0.0031. The boundary and Nash equilibrium points of the game are: (0, 259.08, 164.64) and (333.57, 213.82, 141.34), respectively. A∥ 1−k−∥A∥ If k is set to 0.2, then 20 < k ∥B∥ = 35.05 and 1−∥k−∥ B∥ = 175.28. Therefore, in the basin of attraction we have: ∥(q1 , q2 , q3 ) − EN −C ∥ < 175.28. The phase plane of the dynamical system (dynamic Cournot game) is depicted in Fig. 1. It can be seen that in the basin of attraction, the trajectories of the players’ output quantities converge. As shown in Fig. 2, if d(t ) is set to zero, then the trajectories of the players’ output quantities converge to the exact Nash equilibrium point. Now to examine the stability of the time-varying case study, it ¯ 0 (t )) belongs is assumed that the intercept of the price function (λ to the interval: [λ0 − δλ , λ0 + δλ ]. Moreover, we have: 1−k−∥A∥
1−k−∥A∥
∥e(t ) + d(t )∥ ≤ (λ0 + δλ ) v 2 /α22 + 1/α32 + l = e0 + d0 . If ε is set to 0.05, then K = {x| ∥x∥ ≤ 2010} , ∥A(t )∥ = 0.45, ∥B∥ = 0.003 and 1−∥A∥(Bt∥)∥−ε = 230.95. Therefore, in the region
Fig. 2. The phase plane of the players’ output quantities in the time invariant case without disturbance.
specified by ∥(q1 , q2 , q3 )∥ < 230.95, the time varying system is ¯ 0 (t ) is set to stable with respect to K . The time varying parameter λ λ0 +δλ sin(t ) in the simulation. The phase plane of this time varying dynamical system is depicted in Fig. 3. It can be seen that within the region specified by ∥(q1 , q2 , q3 )∥ < 230.95, the trajectories of the players’ outputs converge. The resulted convergent bound in the time varying case is bigger than that of the time invariant case and has different shape around the Nash equilibrium point, which is the expected result. 5. Conclusions In a dynamic Cournot game there might be multiple equilibrium points based on the players’ expectations and adjustment mechanisms. In this paper, a quadratic dynamical system was proposed
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1 + βλ0 − βα
A=
λ0 (α2 − α) λ0 (α3 − α) − βα α2 α3 − α 2 α2 α3 − α 2 −α v α2 −α α2
0 1−v
−α α2
0
−α v . α2 0
Box I.
1 + βλ0 −
2βα1 λ0 (α3 − α)(α2 − α) + βαλ0 (α3 − α)(α1 − α) + βαλ0 (α2 − α)(α1 − α) 2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
A= B=
−
βαλ0 (α3 − α)(α1 − α) 2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
−α α2 −α α2
1−v
v
−βα1
−βα
0
0
0
0
−βα
−α α2
−
βαλ0 (α2 − α)(α1 − α)
2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
v
−α α2 0
0 . 0
Box II.
the stochastic dynamic Cournot game, the estimated domain of stability might be restrictive. More research works could be focused on better estimation of the domain of stability of stochastic dynamic game equilibria. References
Fig. 3. The phase plane of the players’ output quantities in the time varying case in presence of disturbance.
to cover the quadratic dynamic output adjustment of the players in the Cournot game with linear price function and quadratic costs. Moreover, time varying parameters and disturbances were also considered in the proposed model. The stability of the quadratic dynamic adjustment was analyzed and the regions that belonged to the basins of attraction were determined using some converse Lyapunov theorems. In the time invariant case without disturbance, the sufficient condition for existence of a basin of attraction was: ‘‘the norm of matrix A (i.e. absolute value of all eigenvalues of matrix A) should be less than unity’’. It was proved that in presence of additive disturbance (as an input to the system), the norm of matrix A should be less than 1-k for an arbitrary positive constant k. In this case the parameter k limited the domain of stability. In addition, the basin of attraction existed only for some limited norm disturbances. The same condition was obtained for the time varying case with disturbance, where the norm of matrix A(t ) should be less than 1-ε for some positive constant ε . It should be noted that, since the proposed lemmas and theorems offered the sufficient conditions for the stability of
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H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189 Jiang, Z. P., & Wang, Y. (2002). A converse Lyapunov theorem for discrete-time systems with disturbances. Systems & Control Letters, 45, 49–58. Kopel, M. (1996). Simple and complex adjustment dynamics in Cournot duopoly models. Chaos, Solitons & Fractals, 12, 2031–2048. Meyer, C. D. (2000). Matrix analysis and applied linear algebra. SIAM. Okuguchi, K. (1964). The stability of the Cournot oligopoly solution: a further generalization. The Review of Economic Studies, 31(2), 143–146. Okuguchi, K. (1970). Adaptive expectations in an oligopoly model. The Review of Economic Studies, 37(2), 233–237. Okuguchi, K. (1976). Expectations and stability in oligopoly models. Berlin: SpringerVerlag. Okuguchi, K., & Szidarovszky, F. (1999). The theory of oligopoly with multi-product firms (2nd ed.). Berlin, Heidelberg, New York: Springer-Verlag. Okuguchi, K., & Yamazaki, T. (2008). Global stability of unique Nash equilibrium in Cournot oligopoly and rent-seeking game. Journal of Economic Dynamics & Control, 32, 1204–1211. Puu, T. (1991). Chaos in duopoly pricing. Chaos, Solitons & Fractals, 1, 573–581. Puu, T. (1996). Complex dynamics with three oligopolists. Chaos, Solitons & Fractals, 7, 2075–2081. Puu, T. (1998). The chaotic duopolists revisited. Journal of Economic Behavior and Organization, 37, 385–394. Rosser, J. B. (2002). The development of complex oligopoly dynamics theory. In Text book oligopoly dynamics: models and tools. Berlin: Springer. Szidarovszky, F. (2003). Global stability analysis of a special learning process in dynamic oligopolies. In Conference on complex oligopolies. Odense, Denmark. 18–21 May. Szidarovszky, F., & Okuguchi, K. (1988). A linear oligopoly model with adaptive expectations: stability reconsidered. Journal of Economics, 48, 79–82. Szidarovszky, F., & Yen, J. (1991). Stability of a special oligopoly market. Pure Mathematics and Applications, 2(2–3), 93–100. Theocharis, R. D. (1960). On the stability of the Cournot solution on the oligopoly problem. Review of Economic Studies, 27, 133–134.
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Hamed Kebriaei was born in Tehran, Iran in 1983. He received his B.Sc. degree in Electrical Engineering from the school of ECE, University of Tehran, Iran, in 2005. He also received his M.Sc. degree in EE at Tarbiat Modares University, Tehran, Iran, in 2007 and received his Ph.D degree in electrical engineering, control systems from the School of ECE, University of Tehran, Iran in 2011. His research interests are game theory and applications to energy and economic systems, estimation theory, reinforcement learning, power system optimization and intelligent control.
Ashkan Rahimi-Kian (SM’08) received the B.Sc. degree in Electrical Engineering from the University of Tehran, Tehran, Iran, in 1992 and the M.S. and Ph.D. degrees in Electrical Engineering from Ohio State University, Columbus, in 1998 and 2001, respectively. He was the Vice President of Engineering and Development with Genscape, Inc., Louisville, KY, from September 2001 to October 2002 and a Research Associate with the School of Electrical and Computer Engineering (ECE), Cornell University, Ithaca, NY, from November 2002 to December 2003. He is currently an Associate Professor of Electrical Engineering (Control and Intelligent Processing Center of Excellence) with the School of ECE, College of Engineering, University of Tehran. His research interests include bidding strategies in dynamic energy markets, game theory and learning, intelligent transportation systems, decision making in multiagent stochastic systems, stochastic optimal control, dynamic stock market modeling and decision making using game theory, smart grid design, operation and control, estimation theory and applications in energy and financial systems, risk modeling, and management in energy and financial systems.
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Brief paper
On the stability of quadratic dynamics in discrete time n-player Cournot games✩ Hamed Kebriaei 1 , Ashkan Rahimi-Kian Smart Network Lab/Control and Intelligent Processing Center of Excellence, School of ECE, University of Tehran, P.O. Box 14395-515, Tehran, Iran
article
info
Article history: Received 19 December 2010 Received in revised form 18 September 2011 Accepted 31 October 2011 Available online 10 April 2012 Keywords: Cournot game Lyapunov function Stability analysis Nash equilibrium point
abstract Motivated by analyzing the dynamic adjustment process of players in an n-player Cournot game, in this paper a discrete-time quadratic dynamical system is proposed and the stability of its equilibrium is analyzed. Several output adjustment mechanisms (e.g. best reply, adaptive adjustment and myopic) and expectations (e.g. naïve and adaptive) in a Cournot game with linear price function and quadratic costs, which form a quadratic dynamical system, are the special cases of the proposed model. The stability of the proposed quadratic dynamical system is analyzed with (1) time invariant parameters, (2) time invariant parameters in the presence of disturbance and (3) bounded time varying parameters in the presence of disturbance. In each case, the sufficient condition to find the region that belongs to the basin of attraction is derived using some discrete-time converse Lyapunov theorems. In addition, the proposed model and theorems are utilized to analyze the stability of the boundary and Nash equilibrium points of a Cournot game with three heterogeneous players. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction The first dynamic model of oligopoly games was introduced by Cournot (1838). In his game model, players make their decisions assuming the rivals would offer their last output quantity in the next step of the game (naive expectation). The stability of strict Nash equilibrium point in a linear n-player Cournot game with naïve expectations was discussed by Theocharis (1960). He showed that the above mentioned game is stable for n < 3, neutrally stable for n = 3 and unstable for n > 3. Fisher (1961) showed that the Cournot game could be generally stable for more than three players in a more general case, where the players use adaptive adjustment and the quadratic cost function. The stability analysis of the Nash equilibrium point was also generalized in linear oligopolistic models where players use adaptive expectations instead of naive expectations (see, e.g., Bischi & Kopel, 2001, Okuguchi, 1970, Okuguchi, 1976, Okuguchi & Szidarovszky, 1999, Szidarovszky & Okuguchi, 1988, Szidarovszky & Yen, 1991). The stability of continuous-time Cournot games was studied by several researchers. Hahn (1962) analyzed the stability of Cournot oligopoly solutions under general conditions. However, the actual output of each player was adjusted in proportion to the difference
✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Editor Berç Rüstem. E-mail addresses: [email protected] (H. Kebriaei), [email protected] (A. Rahimi-Kian). 1 Tel.: +98 21 88027756; fax: +98 21 88778690.
0005-1098/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2012.03.021
between its profit maximizing output and the actual one. Okuguchi (1964) generalized Hahn’s results, assuming that the rate of change of each player’s actual output was a sign-preserving and monotonic function with respect to the difference between profit maximizing output and the actual one; then derived the sufficient stability conditions. However, Al-Nowaihi and Levine (1985) presented a counter-example to the Hahn–Okuguchi stability analysis results. They showed that the game equilibrium is globally stable only if the number of firms is less than or equal to five. Okuguchi and Yamazaki (2008) studied a general Cournot game under a gradient dynamics output adjustment. It was shown that the unique equilibrium is globally stable if the game satisfies a set of assumptions and the number of players is less than or equal to three. Complex dynamics of the Cournot game was studied in several research works (Agiza, 1998; Agiza, Hegazi, & Elsadany, 2002; Bischi & Naimzada, 1999; Bischi, Stefanini, & Gardini, 1998; Kopel, 1996; Puu, 1991, 1996, 1998). For example, the dynamics of a three-player game was discussed by Puu (1996). Agiza (1998) studied the dynamics of three and four player Cournot games with identical and linear cost functions for the players. The result was extended to an n-player game by Ahmed and Agiza (1998). The basins of attraction for each (of multiple) equilibrium point were discussed in Agiza, Bischi, and Kopel (1999), where the players used naive expectations. A review of studies on complex dynamics of oligopoly games could be found in Rosser (2002). Many research works have considered the homogeneous expectation of rivals which means that the players assume that the rivals use identical rules for behavior adjustment. However, they could have different adjustment mechanisms.
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
In real dynamic repeated games, it is assumed that none of the players has complete information about the game parameters and the rivals’ future decisions; thus the players should act adaptively in the game. In an adjustment process the players make their decisions based on local estimation of their marginal payoffs (Ahmed, Agiza, & Hassan, 2000; Bischi & Naimzada, 1999). This adjustment mechanism has been called ‘‘myopic’’ by some authors (Dixit, 1986; Flam, 1993). The case of having incomplete information about market price function was considered by Bischi, Sbragia, and Szidarovszky (2008) and Szidarovszky (2003). In those research works an adjustment mechanism was designed to learn the slope or intercept of the market price function. The stability conditions of the adjustment process were given for n-player. However, it was assumed that players had complete information about their rivals’ cost functions. In most of the previous research works with nonlinear dynamic adjustment model, only the local stability of the equilibrium was analyzed. In some other research works, the global stability of Cournot game was analyzed in some special cases; where the players used specific adjustment mechanism (e.g. Best reply, adaptive, myopic) or expectations (e.g. naïve, adaptive). However, the global stability of discrete-time Cournot games with quadratic nonlinearities, bounded time varying parameters and disturbances have not been studied in the literature. In this paper a quadratic dynamical system is proposed for output adjustment of the players in a stochastic dynamic Cournot game which allows for the heterogeneity in players’ adjustment mechanism. The Cournot game with linear price function, quadratic cost and some well known adjustment mechanisms (e.g. best reply, adaptive adjustment and myopic) and expectations (e.g. naïve and adaptive), which are explained in detail in the next section, are all special cases of the proposed model. In addition, the stability analysis of the model is provided in both cases of time invariant and bounded time varying parameters, considering the effect of disturbance. Moreover, the regions that belong to the basin of attraction2 are estimated for all cases. The organization of this paper is as follows; Section 2 provides the preliminaries of this paper. In Section 3, the stability theorems are given for three cases including: time invariant, time invariant in presence of disturbance and time variant in presence of disturbance. In Section 4, a Cournot game example is provided to illustrate the stability analysis results. Finally, the paper is concluded in Section 5. 2. Preliminaries Consider a market consisting of n sellers. The market price function could be expressed as follows:
λ(t ) = λ0 −
n
αi qi (t )
(1)
i =1
where qi is the output quantity (decision) and αi > 0 represents the market power of the player-i. The scalar λ0 is the cap price of the market which is usually set by the market regulator. Eq. (1) implies that increasing the market supply quantities would result in decreasing the market price. The payoff of player-i is calculated as follows:
πi (t ) = λ(t )qi (t ) − Ci (qi (t )) i = 1, . . . , n (2) where Ci (·) is the supply cost function of player-i with the following quadratic form: Ci (qi (t )) = ai q2i (t ) + bi qi (t ) + ci
(3)
where ai , bi , ci > 0.
2 The basin of attraction is a set of points in the space of system variables such that initial conditions chosen in this set dynamically evolve to a particular attractor.
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As it is clear from Eqs. (1)–(2), each player’s payoff is affected by other players’ output quantities through the market price function. The goal of each player is to maximize her payoff function by taking into account the decision of her rivals; therefore, she needs to estimate other players’ decisions. In oligopolistic games, the players may use different adjustment processes (for updating their output quantity) and different expectations/estimations of the rivals’ outputs. The best reply adjustment process could be used together with naïve expectation (Theocharis, 1960) or with adaptive expectation (Bischi & Kopel, 2001). Adaptive adjustment (Huang, 2001) could be used together with naïve expectation (Fisher, 1961) or with adaptive expectation (Okuguchi, 1976). Players may use adjustment processes based on profit gradient that require less information and consequently lower degree of rationality. Bischi and Naimzada (1999) adopted the adjustment process based on profit gradient (myopic mechanism) with naïve expectation. Moreover, the players may use different adjustment processes (heterogeneous expectations) or similar adjustment processes (homogeneous expectations). Consider a Cournot oligopoly market model, where each player is able to predict its rivals’ decisions one step forward. The optimal output quantities of the players could be determined by solving the following set of n equations: qi (t + 1)
= arg max πi (q1i (t + 1), q2i (t + 1), . . . , q(i−1)i (t + 1) qi (t +1)
× qi (t + 1), q(i+1)i (t + 1), . . . , qni (t + 1)) i = 1, . . . , n (4) where qji (j = 1, . . . , n, j ̸= i) is the predicted output of player-j by player-i. If a unique optimal solution for Eq. (4) exists, it could be described as follows: qi (t + 1) = fi (q1i (t + 1), q2i (t + 1), . . . , q(i−1)i (t + 1)
× q(i+1)i (t + 1), . . . , qni (t + 1))
(5)
where, fi (·) is the reaction function of player-i. Cournot assumed that qji (t + 1) = qj (t ), which implies that each player assumes her rivals will repeat their last step decisions in the next time-step. This is called the naive expectation. Therefore, the Cournot reaction function has the following dynamic discrete-time form: qi (t + 1) = fi (q1 (t ), q2 (t ), . . . , qi−1 (t ), qi+1 (t ), . . . , qn (t )).
(6)
Eq. (6) describes the best reply dynamics in an oligopoly game with naive expectations of the players. In a myopic adjustment mechanism based on profit gradient, if the marginal payoff is positive (negative), the player will increase (decrease) its supply quantity for the next time-step. The myopic adjustment equation together with naive expectation (of rivals) and linear speed of adjustment could be written as follows (Bischi & Naimzada, 1999): qi (t + 1) = qi (t ) + γ qi (t )
∂πi (t ) , ∂ qi (t )
(7)
where γ is a positive scalar that shows the relative behavior adjustment speed. The minimum value of qi (t ) should be bounded by zero. Note that Nash equilibria are fixed points of Eq. (7), but not vice versa.3 Another strategic behavior is adaptive adjustment (Huang, 2001). Through this strategy a player calculates its desired supply quantity for the next time-step using the weighted sum of its reaction function and its output quantity in the previous time-step.
3 The boundary fixed points exist with some zero coordinates, which are not located at the intersections of reaction functions, so they are not Nash equilibria.
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The dynamic equation for the adaptive adjustment for player-i is shown as follows: qi (t + 1) = (1 − v)qi (t ) + v fi (q1 (t ), q2 (t ), . . . , qi−1 (t )
× qi+1 (t ), . . . , qn (t )) (8) where v ∈ [0, 1] is the adjustment factor of player-i. Note that for the special case of v = 1, Eq. (8) reduces to Eq. (3), which represents the best reply dynamics with naive expectations. Another method of dynamic decision making in Cournot games is that each player predicts the output quantities of her rivals for the next period, and then makes a profit maximizing decision based on the predicted outputs (Okuguchi, 1976). A general formation of expectations that is usually used in economy is adaptive expectations. The player-i adapts her expectation from player-j’s output as follows: qji (t + 1) = qji (t ) + βi (qj (t ) − qji (t )) 0 ≤ βi ≤ 1.
(9)
It is easy to see that under this formulation, the estimated output is the weighted average of the past actual outputs, where the weights decrease as time proceeds (Okuguchi, 1976). The generalization of the proposed model by Okuguchi (1976) to multi-product firms was considered by Okuguchi and Szidarovszky (1999). Moreover, it was shown by Bischi, Chiarella, Kopel, and Szidarovszky (2010) that the local stability of the best reply dynamics (4) with adaptive expectation (9) is equivalent to adaptive adjustment process with naive expectation in (8). In the following, an n-dimensional quadratic dynamic model is developed. The n-player Cournot game (Eqs. (1)–(3)) together with the above mentioned adjustment mechanisms and expectations which form a quadratic dynamical system are special cases of the proposed model. In addition, the bounded time varying parameters and disturbances are also considered in the proposed model, which had not been studied in the literature of the dynamic Cournot games. The stability analysis of the proposed stochastic dynamic model is presented for different cases in the next section. 3. Problem formulations 3.1. Time invariant case without disturbance Consider an n-player Cournot game with quadratic dynamic output adjustment. The players’ decision model may be represented by the following dynamical system: y(t + 1) = f (y(t )) = C · y(t ) + Y (t ) · B · y(t ) + e
(10)
where, y(t ) = [q1 (t ), q2 (t ), . . . , qn (t )]T , Y = diag(q1 (t ), q2 (t ), . . . , qn (t )), e is a n × 1 vector, C and B are n × n matrices. If y∗ is an equilibrium vector of the system defined by Eq. (10), then by defining: y˜ = y − y∗ , it could be rewritten as follows: y˜ (t + 1) = [Y ∗ · B + C ] · y˜ (t ) + Y˜ (t ) · B · y˜ (t ) + Y˜ · B · y∗
(11)
where, Y ∗ = diag(y∗ ) and Y˜ (t ) = diag(˜y). In addition we have: Y˜ · B · y∗ = diag(B · y∗ ) · y˜ .
(12)
Therefore, Eq. (10) could be represented as following: x(t + 1) = Ax(t ) + X (t ) · B · x(t )
(13)
where: x(t ) = y˜ (t ),
X (t ) = Y˜ (t )
A = Y ∗ · B + C + diag(B · y∗ ).
∥x(t )∥ =
n
(14)
i=1
(16)
Here λmax (·) is the maximal eigenvalue of the corresponding matrix. Remark 1. The proposed matrix norm is a p-norm (p = 2) (Meyer, 2000) and holds the following property:
∥NM ∥ ≤ ∥N ∥ ∥M ∥ where, N and M are n × n matrices. Theorem 1. Consider the dynamical system of Eq. (13), if there exists two positive definite matrices G and H such that AT HA − H = −G, then the origin, equilibrium point of dynamical system (13), is asymptotically stable within the region Λ defined as:
Λ = x ∈ ℜn | xT Hx ≤ r , r ∈ ℜ+
(17)
where Λ is embedded in Γ , defined as:
Γ = x ∈ ℜn | λmin (G) ≥ ∥B∥2 ∥H ∥ |xi |2 + 2 ∥B∥ ∥HA∥ |xi | for i = 1, 2, . . . , n .
(18)
Proof. Consider the following Lyapunov function for the system of Eq. (13): V (x) = xT Hx > 0.
(19)
Therefore, we have:
1V (x(t )) = V (x(t + 1)) − V (x(t )) = (Ax(t ) + X (t )Bx(t ))T H (A · x(t ) + X (t ) · B · x(t )) − x(t )T Hx(t ) = x(t )T (AT HA − H )x(t ) + x(t )T BT X (t )HAx(t ) + x(t )T BT X (t )HX (t )Bx(t ) + x(t )T AT HX (t )Bx(t ) = x(t )T (AT HA − H )x(t ) + 2x(t )T BT X (t )HAx(t ) + x(t )T BT X (t )HX (t )Bx(t ). Now, using AT HA − H = −G we have: V (x(t + 1)) − V (x(t )) = −x(t )T Gx(t ) + 2x(t )T BT X (t )HAx(t )
+ x(t )T BT X (t )HX (t )Bx(t ) = −x(t )T G − 2BT X (t )HA − BT X (t )HX (t )B x(t ) ≤ −x(t )T G − 2 BT ∥X (t )∥ ∥HA∥ − BT ∥X (t )∥ ∥H ∥ ∥X (t )∥ ∥B∥ x(t ). Since: λmin (G)x(t )T x(t ) ≤ x(t )T Gx(t ) V (x(t + 1)) − V (x(t ))
≤ −x(t )T λmin (G) − 2 BT ∥X (t )∥ ∥HA∥ − BT ∥X (t )∥ ∥H ∥ ∥X (t )∥ ∥B∥ x(t ). λmax (X T (t )X (t )) = maxi Furthermore, we have ∥X (t )∥ = |xi (t )| = xmax (t ); therefore: ≤ −x(t )T λmin (G) − 2 BT ∥HA∥ xmax (t ) − ∥B∥2 ∥H ∥ x2 (t ) x(t ). max
1/2 x2 ( t )
1/2 ∥A∥ = λmax AT A .
V (x(t + 1)) − V (x(t ))
Definitions. Consider the following norm definitions for the n × 1 vector x(t ) and n × n matrix A:
and
(15)
Using Eq. (18) we have:
1V (x(t )) = V (x(t + 1)) − V (x(t )) < 0.
(20)
Eqs. (20) guarantee the stability of system (13), if parameter r is selected small enough, such that the domain described by Eq. (17) is embedded within the domain described by Eq. (18).
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
Remark 2. If all eigenvalues of matrix A lay inside the unit circle, for an arbitrary positive-definite symmetric matrix G there exists a positive-definite symmetric matrix H, which is the unique solution of the so-called Lyapunov equation: AT HA−H = −G (Elaydi, 2005). The obtained matrix H could be used to estimate the stability domain. Remark 3. If there exists the positive-definite symmetric matrix G, according to Eq. (18), then 1V (x(t )) will be negative in a hypersphere neighborhood, Uδ = {x ∈ ℜn | ∥x∥ < δ} ⊂ Γ , of the origin if δ is sufficiently small. Since Uδ exists and is not empty therefore, Γ also exists and is not empty. Remark 4. The parameter δ should be determined first in order to estimate Γ (by Uδ ), and then one may set r = δ 2 λmin (H ) to determine Λ. 3.2. Time invariant case with disturbance In this section a bounded disturbance is considered as an arbitrary input to the system of Eq. (13) as follows: x(t + 1) = A · x(t ) + X (t ) · B · x(t ) + d(t )
(21)
where ∥d(t )∥ < d0 (d0 ∈ ℜ+ ). The sufficient condition for a region to belong to the basin of attraction for the system of Eq. (21), is given by the proposed lemmas and theorem in this section.
1185
Then we have,
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) = ∥A · x(k0 ) + X (k0 )B · x(k0 ) + d(k0 )∥ − ∥x(k0 )∥ ≤ (∥A + X (k0 )B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ ≤ (∥A∥ + ∥X (k0 )∥ ∥B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ . Since x(k0 ) ∈ Ω , we have ∥A∥ + ∥X (k0 )∥ ∥B∥ − 1 < −k. Therefore, 1V (x(k0 )) ≤ −k ∥x(k0 )∥ + ∥d(k0 )∥. V (x(k0 + 1))
= ∥x(k0 + 1)∥ < V (x(k0 )) − k ∥x(k0 )∥ + ∥d(k0 )∥ = (1 − k) ∥x(k0 )∥ + ∥d(k0 )∥ . Since ∥x(k0 )∥
kN and consequently, x(k) ∈ Λ − Ω . Else if x(k0 ) ∈ Ω − Γ then using Lemma 2 we have: x(k0 + 1) ∈ Ω − Γ . Therefore, by induction it could be concluded that x(k) ∈ Ω − Γ for all k > kN .
1−k−∥A∥ Lemma 1. Let Γ be x ∈ ℜn | 1k d0 ≤ ∥x∥ < ∥B∥ < r > 0 , if Γ is not empty and x(k0 ) ∈ Γ then ∥x(k0 + 1)∥ < ∥x(k0 )∥ for some 0 < k < 1 − ∥A∥.
Remark 5. According to Γ in Lemma 1, a sufficient condition for existence of a basin of attraction (Ω ) is: ∥A(t )∥ < 1 − k, which is more restrictive than the case of no disturbance.
Proof. Consider the following Lyapunov function for the system of (21):
Remark 6. Theorem 2 shows that the invariant attracting set of system (21) is conditioned to the bound of disturbance d(t ) (the larger bound of disturbance, the wider bound of convergence).
V (x(k0 )) = ∥x(k0 )∥ .
(22)
3.3. Time variant case with disturbance
Then we have:
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) = ∥A · x(k0 ) + X (k0 )B · x(k0 ) + d(k0 )∥ − ∥x(k0 )∥ ≤ (∥A + X (k0 )B∥ − 1) ∥x(k0 )∥ + ∥d(k0 )∥ .
Consider the time varying version of system (10) as follows: x(t + 1) = f (t , x(t ), d(t ))
= A(t ) · x(t ) + X (t ) · B(t ) · x(t ) + e(t ) + d(t )
Since ∥d(k0 )∥ ≤ d0 ≤ k ∥x(k0 )∥, we have:
1V (x(k0 )) = V (x(k0 + 1)) − V (x(k0 )) ≤ (∥A + X (k0 )B∥ − 1 + k) ∥x(k0 )∥ ≤ (∥A∥ + ∥X (k0 )∥ ∥B∥ − 1 + k) ∥x(k0 )∥ = (∥A∥ + xmax (k0 ) ∥B∥ − 1 + k) ∥x(k0 )∥ . Moreover we have: xmax (k0 ) ≤ ∥x(k0 )∥, then:
1V (x(k0 )) ≤ (∥A∥ + ∥x(k0 )∥ ∥B∥ − 1 + k) ∥x(k0 )∥ . Thus, 1V (x(k0 )) < 0 in Γ . Therefore, V (x(k0 + 1)) = ∥x(k0 + 1)∥ < V (x(k0 )) = ∥x(k0 )∥, and the proof is completed. Lemma 2. If x(k0 ) ∈ Ω − Γ then, x(k0 + 1) ∈ Ω − Γ and thus Ω − Γ is an invariant set for the system of Eq. (21); where:
1 − ∥A∥ − k Ω = x ∈ ℜn | ∥x∥ < 0 . ∥B∥
(23)
Proof. Consider the following Lyapunov function for the system of Eq. (21): V (x(k0 )) = ∥x(k0 )∥ .
(24)
(25)
where, A(t ) and B(t ) are the n × n matrices with bounded elements that deviate around their nominal values; e(t ) is a time varying n × 1 vector with ∥e(t )∥ < e0 and d(t ) is a bounded n × 1 disturbance vector with ∥d(t )∥ < d0 . Since the solutions of x(t ) = f (t , x(t ), d(t )) are time dependent in general, the system of Eq. (25) has no equilibrium points. Nevertheless, since the elements of the above mentioned system matrices and vectors are bounded, there might exist a nominal equilibrium point x∗ and a sufficiently large constant l such that each solution of the quadratic equation: x(t ) = f (t , x(t ), d(t )) lays within a compact set like K = {x| ∥x − x∗ ∥ ≤ l}. In this section, the stability of system (25) with respect to K is analyzed. The basic converse Lyapunov theorems for discrete time varying systems with disturbance, was introduced in Jiang and Wang (2002). Definitions. Let K be a nonempty closed subset of Rn . The distance of x with respect to K is defined as follows (Jiang & Wang, 2002):
∥x∥K = d(x, K ) = inf ∥x − η∥ . η∈ K
(26)
4 Invariant attracting set is a set of points in the phase space, invariant under the dynamics, toward which neighboring states in a given basin of attraction asymptotically approach in the course of dynamic evolution.
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Definitions. For the defined vector norm in Eq. (15), consider the following norm definitions for the n × n matrix M (x, t ):
∥M (x, t )∥ = min {b : ∥M (x, t )ς∥ ≤ b ∥ς ∥ for all ς ̸= 0} .
(27)
−ε ∥x(t )∥ + ∥e(t )∥ + ∥d(t )∥ ≤ −ε ∥x(t )∥ + ε
Clearly, ∥M (x, t )x∥ ≤ ∥M (x, t )∥ ∥x∥. Lemma 3. The defined matrix norm in Eq. (27) is a sub-multiplicative norm. Proof. Suppose that c1 , c2 and c are the norms of matrices M , N and MN, respectively; then we have:
∥MN ς ∥ = ∥M (N ς )∥ ≤ c1 ∥N ς ∥ ≤ c1 c2 ∥ς ∥
for all ς ̸= 0.
According to the norm definition for matrix MN:
∥MN ∥ = c = min {b : ∥MN ς ∥ ≤ b ∥ς ∥ for all ς ̸= 0} . Therefore c ≤ c1 c2 and the proof is completed.
Theorem 3. Consider the nonempty set Ω defined as:
Ω =
x ∈ ℜ | ∥x ∥ < n
1 − ∥A(t )∥ − ε
∥B(t )∥
for some 0 < ε < 1 − ∥A(t )∥
(28)
and the set K : (29)
Then, the system of Eq. (25) is uniformly asymptotically stable with respect to K in Ω . Proof. Consider the following function: V (x(t )) = ∥x(t )∥K
(e0 + d0 ) ε (e0 + d0 ) ∥x(t )∥ ≤ . ε ∥x(t )∥ >
1
Therefore:
(e0 + d0 ) 1V (x(t )) < −ε ∥x(t )∥ − ε
If ∥x(t )∥ ≤
(e0 +d0 ) ε
then ∥x(t )∥K = 0 and we have:
1V (x(t )) = V (x(t + 1)) = ∥A(t )x(t ) + X (t )B(t )x(t ) + e(t ) + d(t )∥ (e0 + d0 ) − ε ≤ (∥A(t )∥ + ∥X (t )∥ ∥B(t )∥) ∥x(t )∥ e0 + d0 + ∥e(t ) + d(t )∥ − ε e0 + d0 (e0 + d0 ) ≤ (1 − ε) + ∥e(t ) + d(t )∥ − ε ε ≤ −e0 − d0 + ∥e(t ) + d(t )∥ ≤ −e0 − d0 + ∥e(t )∥ + ∥d(t )∥ ≤ 0 = −ε ∥x∥K . In addition, if ∥x(t + 1)∥ ≥ 0 ε 0 , then V (x(t + 1)) = 0 and 1V (x(t )) = −V (x(t )) ≤ − ∥x∥K . Therefore V (x(t )) is a continuous Lyapunov function with respect to K in Ω for the system of Eq. (25). As a result, according to Lemma 2.7 in Jiang and Wang (2002), there also exists a smooth Lyapunov function with respect to K in Ω . Now using Theorem 1 in Jiang and Wang (2002), it is concluded that the system is uniformly asymptotically stable with respect to K in Ω and the proof is completed.
In the following, a three-player dynamic Cournot game with disturbance (including three myopic, adaptive and naïve players) is considered. The quadratic cost functions of the players are the same as given by Eq. (3) with bi = 0. The market price function is also defined as in Eq. (2).
Thus based on Definition 2.6 in Jiang and Wang (2002), the first necessary condition for Eq. (30) to be a continuous Lyapunov function with respect to K for the system of Eq. (25) is satisfied. ( e +d ) First assume that ∥x(t + 1)∥ < 0 ε 0 .
1V (x(t )) = = ≤ ≤
≤ −ε ∥x(t )∥K .
4. A Cournot game with three heterogeneous players
∥x∥K ≤ V (x) ≤ 2 ∥x∥K .
If ∥x(t )∥ >
(30)
Then we have: 2
(e0 + d0 ) . ε
(e +d )
(e0 + d0 ) . K = x| ∥x∥ ≤ ε
(e0 + d0 ) ∥x(t )∥ − ε = 0
Since ∥x∥ ≥ xmax and ∥A(t )∥ , ∥B(t )∥ are bounded, according to Eq. (28) there exists a constant 0 < γ < 1 such that 1V (x(t )) < −ε ∥x(t )∥ + ∥e(t )∥ + ∥d(t )∥. Moreover, according to Eq. (28) we have:
(e0 +d0 ) : ε
V (x(t + 1)) − V (x(t ))
∥A(t )x(t ) + X (t )B(t )x(t ) + e(t ) + d(t )∥ − ∥x(t )∥ (∥A(t ) + X (t )B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ (∥A(t )∥ + ∥X (t )∥ ∥B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ .
In addition:
∥X ∥ = min {b : ∥X ς ∥ ≤ b ∥ς ∥ for all ς ̸= 0} = min {b : ∥(x1 ς1 , x2 ς2 , . . . , xn ςn )∥ ≤ b ∥ς ∥ for all ς ̸= 0} = max |xi | = xmax .
q1 (t + 1) = q1 (t ) 1 + β
1V (x(t )) ≤ (∥A(t )∥ + xmax ∥B(t )∥ − 1) ∥x(t )∥ + ∥e(t ) + d(t )∥ .
= (1 + βλ0 )q1 (t ) − β(2α + 2a1 )q21 (t ) − βα q2 (t )q1 (t ) − βα q3 (t )q1 (t ) + d1 cos(t )
(31)
and q2 (t + 1) = (1 − v)q2 (t ) + v
λ0 − α · (q1 (t ) + q3 (t )) 2(α + a2 )
+ d2 sin(t )
(32)
λ0 − α.(q1 (t ) + q2 (t )) + d3 cos(t ) 2(α + a3 )
(33)
and q3 (t + 1) =
where, d1 cos(t ), d2 sin(t ) and d3 cos(t ) are the game disturbances. Considering di = 0 i = 1, 2, 3, the game has a Nash–Cournot equilibrium and one boundary equilibrium as follows:
i
Therefore:
∂π1 (t ) ∂ q1 (t )
λ0 (α3 − α) λ0 (α2 − α) , , α2 α3 − α 2 α2 α3 − α 2 ∗ ∗ ∗ = q1 , q2 , q3
E0 = ENash
0,
(34)
H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
1187
Table 1 The Cournot game parameters. Parameter
a1
a2
a3
α
β
λ0
v
d1
d2
d3
δλ
Value
0.1
0.17
0.27
0.05
0.01
117.83
0.8
10
750.5
15
50
where, αk = 2(α + ak ) k = 1, 2, 3 are positive parameters and: q∗1 = q∗3 =
λ0 (α3 − α)(α2 − α) fα
,
q∗2 =
λ0 (α3 − α)(α1 − α) fα
λ0 (α1 − α)(α2 − α)
,
fα
fα = 2α − α (α1 + α2 + α3 ) + α1 α2 α3 . 3
2
It is easy to show that the obtained boundary and Nash equilibrium points are positive. First, let us analyze the stability of the boundary equilibrium point. As was discussed in Section 3.1 the system dynamic equations could be converted to Eq. (14) ∥X (t )·B·x(t )∥ = 0 in (14) (i.e. and it is easy to verify that lim∥x∥→0 ∥x(t )∥ ∥X (t ) · B · x(t )∥ is O(x(t )) as ∥x(t )∥ → 0). Therefore from Elaydi (2005), if the linear part of system dynamic equations is unstable, then the zero solution of system is unstable too. According to Eqs. (14), matrix A for boundary equilibrium point is defined as in Box I. One could find out that an eigenvalue of A is outside the unit circle as follows:
Fig. 1. The phase plane of the players’ output quantities in the time invariant case in presence of disturbance.
λ0 (α2 − α) λ0 (α3 − α) − βα 2 α2 α3 − α α2 α3 − α 2 βλ0 =1+ (α2 − α)(α3 − α) > 1. α2 α3 − α 2
1 + βλ0 − βα
Therefore, the boundary equilibrium point of the system is unstable even in the time invariant case without disturbance. It is obvious that the boundary equilibrium point would also be unstable in the cases of time varying parameters and in presence of disturbance. Now the stability of Nash equilibrium point of the above game is analyzed. Using Eqs. (14) we have matrices A and B as given in Box II. Moreover, we have: ∥d(t )∥ ≤
d21 + d22 + d23 = l. If l
0} is a basin of attraction for the system. For example, for the dynamic game parameters given in Table 1, we have ∥A∥ = 0.2597, ∥B∥ = 0.0031. The boundary and Nash equilibrium points of the game are: (0, 259.08, 164.64) and (333.57, 213.82, 141.34), respectively. A∥ 1−k−∥A∥ If k is set to 0.2, then 20 < k ∥B∥ = 35.05 and 1−∥k−∥ B∥ = 175.28. Therefore, in the basin of attraction we have: ∥(q1 , q2 , q3 ) − EN −C ∥ < 175.28. The phase plane of the dynamical system (dynamic Cournot game) is depicted in Fig. 1. It can be seen that in the basin of attraction, the trajectories of the players’ output quantities converge. As shown in Fig. 2, if d(t ) is set to zero, then the trajectories of the players’ output quantities converge to the exact Nash equilibrium point. Now to examine the stability of the time-varying case study, it ¯ 0 (t )) belongs is assumed that the intercept of the price function (λ to the interval: [λ0 − δλ , λ0 + δλ ]. Moreover, we have: 1−k−∥A∥
1−k−∥A∥
∥e(t ) + d(t )∥ ≤ (λ0 + δλ ) v 2 /α22 + 1/α32 + l = e0 + d0 . If ε is set to 0.05, then K = {x| ∥x∥ ≤ 2010} , ∥A(t )∥ = 0.45, ∥B∥ = 0.003 and 1−∥A∥(Bt∥)∥−ε = 230.95. Therefore, in the region
Fig. 2. The phase plane of the players’ output quantities in the time invariant case without disturbance.
specified by ∥(q1 , q2 , q3 )∥ < 230.95, the time varying system is ¯ 0 (t ) is set to stable with respect to K . The time varying parameter λ λ0 +δλ sin(t ) in the simulation. The phase plane of this time varying dynamical system is depicted in Fig. 3. It can be seen that within the region specified by ∥(q1 , q2 , q3 )∥ < 230.95, the trajectories of the players’ outputs converge. The resulted convergent bound in the time varying case is bigger than that of the time invariant case and has different shape around the Nash equilibrium point, which is the expected result. 5. Conclusions In a dynamic Cournot game there might be multiple equilibrium points based on the players’ expectations and adjustment mechanisms. In this paper, a quadratic dynamical system was proposed
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H. Kebriaei, A. Rahimi-Kian / Automatica 48 (2012) 1182–1189
1 + βλ0 − βα
A=
λ0 (α2 − α) λ0 (α3 − α) − βα α2 α3 − α 2 α2 α3 − α 2 −α v α2 −α α2
0 1−v
−α α2
0
−α v . α2 0
Box I.
1 + βλ0 −
2βα1 λ0 (α3 − α)(α2 − α) + βαλ0 (α3 − α)(α1 − α) + βαλ0 (α2 − α)(α1 − α) 2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
A= B=
−
βαλ0 (α3 − α)(α1 − α) 2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
−α α2 −α α2
1−v
v
−βα1
−βα
0
0
0
0
−βα
−α α2
−
βαλ0 (α2 − α)(α1 − α)
2α 3 − α 2 (α1 + α2 + α3 ) + α1 α2 α3
v
−α α2 0
0 . 0
Box II.
the stochastic dynamic Cournot game, the estimated domain of stability might be restrictive. More research works could be focused on better estimation of the domain of stability of stochastic dynamic game equilibria. References
Fig. 3. The phase plane of the players’ output quantities in the time varying case in presence of disturbance.
to cover the quadratic dynamic output adjustment of the players in the Cournot game with linear price function and quadratic costs. Moreover, time varying parameters and disturbances were also considered in the proposed model. The stability of the quadratic dynamic adjustment was analyzed and the regions that belonged to the basins of attraction were determined using some converse Lyapunov theorems. In the time invariant case without disturbance, the sufficient condition for existence of a basin of attraction was: ‘‘the norm of matrix A (i.e. absolute value of all eigenvalues of matrix A) should be less than unity’’. It was proved that in presence of additive disturbance (as an input to the system), the norm of matrix A should be less than 1-k for an arbitrary positive constant k. In this case the parameter k limited the domain of stability. In addition, the basin of attraction existed only for some limited norm disturbances. The same condition was obtained for the time varying case with disturbance, where the norm of matrix A(t ) should be less than 1-ε for some positive constant ε . It should be noted that, since the proposed lemmas and theorems offered the sufficient conditions for the stability of
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Hamed Kebriaei was born in Tehran, Iran in 1983. He received his B.Sc. degree in Electrical Engineering from the school of ECE, University of Tehran, Iran, in 2005. He also received his M.Sc. degree in EE at Tarbiat Modares University, Tehran, Iran, in 2007 and received his Ph.D degree in electrical engineering, control systems from the School of ECE, University of Tehran, Iran in 2011. His research interests are game theory and applications to energy and economic systems, estimation theory, reinforcement learning, power system optimization and intelligent control.
Ashkan Rahimi-Kian (SM’08) received the B.Sc. degree in Electrical Engineering from the University of Tehran, Tehran, Iran, in 1992 and the M.S. and Ph.D. degrees in Electrical Engineering from Ohio State University, Columbus, in 1998 and 2001, respectively. He was the Vice President of Engineering and Development with Genscape, Inc., Louisville, KY, from September 2001 to October 2002 and a Research Associate with the School of Electrical and Computer Engineering (ECE), Cornell University, Ithaca, NY, from November 2002 to December 2003. He is currently an Associate Professor of Electrical Engineering (Control and Intelligent Processing Center of Excellence) with the School of ECE, College of Engineering, University of Tehran. His research interests include bidding strategies in dynamic energy markets, game theory and learning, intelligent transportation systems, decision making in multiagent stochastic systems, stochastic optimal control, dynamic stock market modeling and decision making using game theory, smart grid design, operation and control, estimation theory and applications in energy and financial systems, risk modeling, and management in energy and financial systems.
