Working Paper Series No.66, Faculty of Economics, Niigata University
Global Stability of Unique Nash Equilibrium in Cournot Oligopoly and Rent-Seeking Game Koji Okuguchi* Department of Economics and Information Gifu Shotoku Gakuen University 1-38 Nakauzura, Gifu-shi 500-8288, Japan email:
[email protected]
and Takeshi Yamazaki* Department of Economics, Niigata University 8050 Ikarashi 2-no-cho, Niigata-shi 950-2181, Japan email:
[email protected]
February 3, 2007
Abstract A sufficient condition is derived for the global stability of unique Nash equilibrium in aggregative game. The condition is applied to investigate the global stability of Nash equilibrium in rent-seeking game and that of Nash-Cournot equilibrium in Cournot oligopoly without product differentiation. JEL Classification Numbers: C72, D43, L13. Key Words: rent-seeking, Nash equilibrium, global stability, Cournot oligopoly
* We would like to thank Ferenc Szidarovszky for invaluable comments and suggestions on an earlier version of this paper.
All remaining errors are of course ours.
1. Introduction In spite of abundant literatures on the stability of the Cournot oligopoly equilibrium since the seminal work of Theocharis (1960), only restrictive conditions have been found for the global stability.
Hahn (1962) has derived a set of general stability conditions on
the basis of a continuous output adjustment system in which a firm’s rate of change of actual output is proportional to the difference between its profit-maximizing and actual outputs.
Okuguchi (1964) has extended his result using a more general adjustment system.
Al-Nowaihi and Levine (1985) have presented a counter-example to the Hahn-Okuguchi result.
Okuguchi (1976), on the other hand, has given a proof of the global stability which
is exempt from the criticism.
Al-Nowaihi and Levine (1985) have derived a set of
conditions which is sufficient for the global stability of the Nash-Cournot equilibrium. Their conditions imply that all firms’ reaction functions are upward sloping, that is, the game is submodular.1
According to them the equilibrium is globally stable if the number
of firms less than or equal to 5.
Corchón (2001) has proved the global stability of the
equilibrium adopting Hahn’s method of proof as well as taking into account the property of aggregative game for Cournot oligopoly without product differentiation.
His proof,
however, is not free from the same defect as in Hahn (1962) and Okuguchi (1964). It is well known that the Cournot oligopoly without product differentiation (hereafter, Cournot game) is a submodular aggregative game under usual or traditional assumptions. 2
Most works mentioned above have studied the global stability of the
1
In other words, firms’ actions are strategic substitutes in the terminology of Bulow et al. (1983). If the inverse demand function is linear and each firm’s cost function is convex, the Cournot oligopoly without product differentiation is a submodular aggregative game. 2
2
Nash-Cournot equilibrium mainly in the submodular Cournot game. Cournot game can be supermodular.3
However, the
Recently, some researchers including Vives (1990)
and Amir (1996) have analyzed the supermodular Cournot game. Vives (1999, Theorem 2.11) implies that the unique Nash equilibrium in the supermodular Cournot game is globally stable under Hahn’s best reply dynamics.
As for the traditional submodular
Cournot game, as already mentioned, Al-Nowaihi and Levine (1985) have proved that the Nash-Cournot equilibrium is globally stable if the number of firms is less than or equal to 5. However, the Cournot game may not be either supermodular or submodular.
That is, it is
possible that some firms have downward sloping reaction functions and at the same time other firms have upward sloping reaction functions in the Cournot game.
It is also
possible that a Cournot firm has a non-monotone reaction function. This paper examines the global stability of the equilibrium in the general Cournot game which allows such cases.4
To do so, this paper derives a set of conditions which ensures that a unique
(interior) Nash equilibrium in a general aggregative game is globally stable, and then applies this general result to the Cournot game and a rent-seeking game.
The rent-seeking
game is an aggregative game where players’ reaction functions are non-monotone under the standard assumptions in the literature of rent- seeking.
3
In the terminology of Bulow et al. (1983), the game can have strategic complementarities. As for local stability, Dastidar (2000, Proposition 3) derives conditions for the Nash equilibrium to be locally stable in the Cournot game which has firms with downward sloping reaction functions and firms with upward sloping reaction functions. 4
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2. Aggregative Game Consider a general aggregative game with n players.5
Player i's payoff function is
given as U i ( xi , X ) , where a non-negative number xi is player i's choice of strategy and X = ∑ j =1 x j . n
Define hi ( xi , X ) ≡
∂ ∂ U i ( xi , X ) + U i ( xi , X ) . ∂xi ∂X
At an interior Nash equilibrium x* = ( x1* , x2* ,L , xn* ) ,
X * = ∑ j =1 x*j . n
hi ( xi* , X * ) = 0 for all i, where
Assume that xi as a function of continuous time t is adjusted according
to
d xi = ki hi ( xi , X ) , dt where a positive number ki denotes speed of adjustment. hi as h1i ≡
∂ i ∂ i h ( xi , X ) and h2i ≡ h ( xi , X ) . ∂X ∂xi
(1) Define partial derivatives of
We then have the following stability
theorem. Theorem 1: The unique Nash equilibrium in the aggregative game is globally stable if
h1i + h2i < 0 , n
2ki h1i + h2i >
∑
k j h2i + ki h2j
j =1, j ≠ i
and
5
See Corchón (2001) and Okuguchi (1993) for more details on the aggregative game.
4
(2)
(3)
lim hi ( xi , xi + X − i ) = −∞
(4)
xi →∞
are satisfied for all feasible strategies and for all i and j ≠ i , where X − i = ∑ j ≠i x j . Proof: The following proof assumes the uniqueness of the equilibrium and all
mathematical regularities required for applying the Lyapunov method.
Define a
Lyapunov function by V ( x) ≡
1 n i ∑ h ( xi , X 2 i =1
)
2
,
(5)
where x = ( x1 , x2 ,L , xn ) . Since xi is non-negative for all i, x ≡ ∑ i =1 xi2 is infinite if n
and only if xi is infinite for some i.
Hence, condition (4) implies that V ( x ) → ∞ as
x → ∞ . Differentiation of (5) with respect to t yields dV 1 = hJh t = h ( J + J t ) h t dt 2
(
where h ≡ h1 , h 2 ,L , h n
)
(6)
and
k1 ( h11 + h21 ) k2 h21 k1h22 k2 ( h12 + h22 ) J≡ M M n k2 h2n k1h2
L kn h22 . O M n n L kn ( h1 + h2 ) L
kn h21
(7)
If the symmetric matrix J + J t in (6) is negative definite, the unique Nash equilibrium is globally stable.
A sufficient condition for J + J t to be negative definite is conditions (2)
and (3), which ensure that J + J t
6
is negative dominant diagonal.6 g
See McKenzie (1960).
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3. Applications 3.1 Cournot Oligopoly without Product Differentiation In Cournot oligopoly without product differentiation, U i ( xi , X ) = xi P ( X ) − Ci ( xi ) ,
(8)
hi ( xi , X ) = P ( X ) + xi P′ ( X ) − Ci′ ( xi ) ,
(9)
where xi is firm i's output, P ( X ) with P′ ( X ) < 0 is the inverse demand function and Ci is firm i's cost function.
Al-Nowaihi and Levine (1985, Theorem 6) prove that if
h1i ( xi , X ) = P′ ( X ) − Ci′′ ( xi ) 0 and f i " < 0 for all i.
(14)
In addition, assume lim fi ' = 0 for all i.
yi →∞
(15)
If fi ( yi ) = α i yir with α i > 0 and r ∈ ( 0,1) , assumptions (14) and (15) are satisfied. Agent i's expected utility is
ui ( y ) =
fi ( yi )
∑
f ( yj ) j =1 j
n
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− yi ,
(16)
where y = ( y1 , y2 ,L , yn ) .
If all yi = 0 , then ui is defined to be zero.
Agent i is
assumed to maximize his or her expected utility (16) with respect to yi .
Define
xi = fi ( yi ) to transform the expected utility of agent i into a function of xi and X = ∑ j =1 x j : n
U i ( xi , X ) = where gi = fi −1 .
xi − gi ( xi ) , X
(16’)
Agent i's original maximization problem is equivalent to the one of
maximizing the expected utility (16’) with respect to xi . has made the rent-seeking contest an aggregative game.
Note that this transformation
Note also that assumption (14) on
fi implies g i ( 0 ) = 0 , g i ' > 0 and gi " > 0 for all i
(14’)
and assumption (15) implies lim gi ' = ∞ for all i.
xi →∞
From (16’), it is easy to get hi ( xi , X ) =
x ∂ ∂ 1 U i ( xi , X ) + U i ( xi , X ) = − gi′ ( xi ) − i2 , X X ∂xi ∂X
h1i ( xi , X ) =
∂ i 1 h ( xi , X ) = − gi′′ ( xi ) − 2 , X ∂xi
h2i ( xi , X ) =
∂ i 1 2x h ( xi , X ) = − 2 + 3i . ∂X X X
Now we are ready to state
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(15’)
Theorem 3: If ki = k for all i and assumptions (14) and (15) are satisfied, the unique Nash equilibrium of the rent-seeking game is globally stable if − X 3 gi′′ ( xi ) < ( n − 4 )( xi − X )
(17)
is satisfied.
Proof: Assumption (15) implies condition (4) of Theorem 1. −2 X −1 − X 3 g1 " ( x1 ) ( x1 + x2 ) − X −2 X −2 − X 3 g 2 " ( x2 ) ( x2 + x1 ) − X k J + Jt = 3 X M M ( xn + x2 ) − X ( xn + x1 ) − X
Since
( x1 + xn ) − X ( x2 + xn ) − X
L , O M L −2 X − n − X 3 g n " ( xn )
L
condition (2) of Theorem 1 is satisfied under assumption (14). Furthermore, since
( x + x ) − X ≤ 0 , i ≠ j , condition (3) of Theorem 1 is equivalent to condition (17). i
j
g
As already mentioned, all player’s reaction functions are not monotone in the rent seeking game, since h2i ( xi , X ) = ( xi − X − i ) X 3 is positive for xi < X − i and it is negative for
xi > X − i .
Regardless of such a unique feature, Theorem 3 proves that if the number of
rent-seeking agents is less than or equal to 4, the Nash equilibrium is always globally stable.
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4. Conclusion In this paper we have analyzed the global stability of Nash equilibrium and Nash-Cournot equilibrium in a rent-seeking game and Cournot oligopoly without product differentiation, respectively.
Our simple stability analysis has become possible by
characterizing the rent-seeking game and Cournot oligopoly without product differentiation as aggregative games in which each player’s payoff is a function of its choice of a strategy and the sum of choices of strategies of all players.
It would be interesting to apply our
general result in Theorem 1 to other aggregative games.
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