Working Paper Series No.77, Faculty of Economics, Niigata University

Global Stability of Nash Equilibrium in Aggregative Games under Generalized Hahn Conditions*

Koji Okuguchi Professor Emeritus, Department of Economics, Tokyo Metropolitan University

and Takeshi Yamazaki+ Department of Economics, Niigata University

March 28, 2009

Abstract If an aggregative game satisfies the generalized Hahn conditions, then there exists a unique Nash equilibrium, which may not be interior and is globally stable under two alternative continuous adjustment processes with non-negativity constraints.

JEL Classification Numbers: C72, D43, L13. Key Words: 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.

+ Corresponding author: 8050 Ikarashi 2-no-cho, Nishi-ku, Niigata-shi 950-2181, Japan. e-mail: [email protected]

1. Introduction Many economists have analyzed the local or global stability of the equilibrium in Cournot oligopoly with or without product differentiation.1 Hahn (1962) proved that if the so-called Hahn conditions are satisfied in Cournot oligopoly without product differentiation (hereafter, the Cournot model), the Cournot-Nash equilibrium is globally asymptotically stable under Hahn’s best reply dynamics. Okuguchi (1964) extended Hahn’s result using a more general adjustment system. Corchón (2001) proved that Hahn’s stability result holds in a general aggregative game, which contains the Cournot model as its special case, provided that the game satisfies the generalized Hahn conditions. However, Al-Nowaihi and Levine (1985) presented a counter-example to a certain assertion used in the proof of the Hahn-Okuguchi result.

2

According to Al-Nowaihi and Levine (1985), the

Cournot-Nash equilibrium is globally stable only in the Cournot model with 5 or less firms. Gaudet and Salant (1991) proved that there exists a unique Cournot-Nash equilibrium under the Hahn conditions. Folmer and von Mouche (2004, Corollary 1) proved that an aggregative game possesses a unique Nash equilibrium under the generalized Hahn conditions. Their theorems, however, do not ensure that the unique Nash equilibrium is interior. In other words, some players active out of equilibrium may not be so at the unique Nash equilibrium. The previous works on the global stability of the Nash equilibrium or Cournot-Nash equilibrium do not explicitly describe the dynamics at the point where the trajectory moves out of the non-negative domain. Moreover, the adjustment

See e.g. Okuguchi (1976), Okuguchi and Szidarovszky (1990), and Vives (1999) for comprehensive literature. 2 Corchón’s proof is not free from the defect pointed out by Al-Nowaihi and Levine (1985). 1

2

processes used in the previous works do not necessarily ensure the non-negativity of all strategic variables over time independently of the initial condition. Hence, we will formulate the dynamics at the boundary and prove that if an aggregative game satisfies the generalized Hahn conditions, a unique Nash equilibrium, which may not necessarily be interior, is globally stable under two alternative continuous adjustment processes with the non-negativity constraints.

2. Analysis Let u i  x  be player i’s payoff function in a general n-person game, where

x   x1 , x2 , , xn   Rn  x  R n : x1  0, x2  0, , xn  0 and xi is player i’s choice of strategy.

The actions of the game are strategic substitutes if u i  x  satisfies

 2u i x j xk  0 for all j  k .3

The game is aggregative if player i's payoff function

u i  x  can be written as U i  xi , X  , where X   jN x j .4

Define

hi  xi , X  

  U i  xi , X   U i  xi , X  . xi X

Let us assume that the partial derivatives of hi satisfy the following assumptions. Assumption 1: h2i 

3 4

 i h  xi , X   0 for all feasible strategies and for all i. X

See Bulow et al. (1985). See e.g. Okuguchi (1993) and Corchón (1994, 2001) for examples of aggregative games.

3

Assumption 2: h1i 

 i h  xi , X   0 for all feasible strategies and for all i. xi

Since Assumptions 1 and 2 coincide with the so-called Hahn conditions for the Cournot model, call them the generalized Hahn conditions.5 Note that, under the generalized Hahn conditions, the actions of the game are assumed to be strategic substitutes. Folmer and von Mouche (2004, Corollary 1) prove that an aggregative game possesses a unique Nash equilibrium under the generalized Hahn conditions.6 As already mentioned, the unique Nash equilibrium may not be interior under the generalized Hahn conditions. Because of this reason, for X  i   j i x j  0 , construct a new function from hi  xi , X  i  . hi  xi , X  i  h  xi , X    i max h  0, X  i  , 0

for

xi  0

for

xi  0.

Note that, at the unique Nash equilibrium x*   x1* , x2* , , xn*   Rn , all i  N , where X *   jN x*j . 7

(1)

h i  xi* , X *   0 for

Equivalently, at the unique Nash equilibrium x* ,

xi*  Ri  X *i  , where X *i   j i x*j and Ri  X  i  is player i’s best reply function, that is, Ri  X  i  is a unique non-negative solution to the problem of maximization problem of

5

In the Cournot model U i  xi , X   xi P  X   Ci  xi  and hi  xi , X   P  X   xi P  X   Ci  xi  , 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. The so-called Hahn conditions are

h1i  xi , X   P  X   Ci  xi   0

and

h2i  xi , X 

 P  X   xi P  X   0 . Assumption 1 with strict inequality and Assumption 2 are introduced in Corchón

(1994). 6 As for the Cournot model, Gaudet and Salant (1991) prove that there exists a unique Cournot-Nash equilibrium under the Hahn conditions. 7 Gaudet and Salant (1991) describe the Cournot-Nash equilibrium, which may not be interior, in a similar but slightly different way.

4

U i  xi , X  with respect to xi , given X i .8

There can be more than one way to model how xi is adjusted as a continuous function of time t.

The following two dynamics are the most representative in the

literature. Assumption 3: xi as a non-negative function of continuous time t is adjusted according

to d xi   i h i  xi , X  , dt

(2)

where a positive number  i denotes speed of adjustment. Assumption 4: xi as a non-negative function of continuous time t is adjusted according

to d xi  i  Ri  X  i   xi  , dt

(3)

where a positive number  i denotes speed of adjustment. First consider the gradient dynamics in Assumption 3, which is assumed by Dixit (1986), Furth (1986), Dastidar (2000) among others. Define

I  i  N : xi  xi* , J   j  N : x j  x*j  and K  k  N : xk  xk*  .

(4)

For the sake of notational simplicity, define

X A   iA xi and X A*   iA xi* ,

(5)

Assumptions 1 and 2 ensure that Ri  X  i  is a unique positive solution to the first order condition

8

h  xi , xi  X i   0 for any X  i such that hi  0, X i   0 and that i

that hi  0, X i   0 .

5

Ri  X  i   0 for any X  i such

where A is a subset of N.

Since an increase in t may change the index sets I and J, we

occasionally write I  t  and J  t  instead of I and J, respectively. By the same reason, we may write X A as X A  t  . The following theorem gives the complete proof to Corchón (2001, Proposition 1.6.), taking into account the non-negativity of players’ strategic variables during the adjustment periods.9 Theorem 1: Under Assumptions 1, 2 and 3, the unique Nash equilibrium in the aggregative game is globally stable. Proof: For x   x1 , x2 , , xn   Rn , define a Lyapunov function as follows. 1 I I* 2  2  X  X  V  x    1  X J  X J * 2  2

if

X  X *,

if

XX ,

(6) *

It is clear that V  x  in (6) is zero for x  x* and positive for x  x* .10 It is also clear that

X

I

V  x

is continuous in

 X I*     X J  X J*  .

X  X* ,

the fact

continuous in x .

X

I

x

X  X*

for

and

X  X* .

If

Since V  x  is continuous in x for

X  X* ,

X  X * and

 X I *     X J  X J *  at X  X * implies that V  x  is

Since xi is non-negative for all i, x   i 1 xi2 is infinite if and only

if xi is infinite for some i.

n

Hence, x is infinite if and only if

9

X

I

 X I *  and/or

Footnote 2 applies. If x  x* and X  X * , the set I is not empty. If x  x* and X  X * , the sets I and J are not empty. Hence, if x  x* and X  X * , V  x  in (6) is positive. Similarly, if x  x* and X  X * , 10

V  x  in (6) is positive.

6

X

J

 X J *  is infinite.

Therefore, V  x    as x   .

V  x  may not be differentiable with respect to t.11

Following Hale (1969,

p.293), define 

V x

 lim sup t  0

1 V  x  t  t    V  x  t    . t  

If X  t   X * , as we prove in the Appendix, V  x  satisfies V  x   X I t   X I* t  

Since xi  xi* for all i  I ,

  h

iI  t 

 X t   X t   0 . I

i

i

.

(7)

Since xi  xi* for all i  I and

I*

X  X * , Assumptions 1 and 2 ensure h i  xi , X   h i  xi* , X *   0 for all i  I .12

Hence,

if X  t   X * , V  x   X I t   X I* t  

  h

iI  t 

i

i

 0.

(7’)

If X  t   X * , the arguments similar to the ones in the Appendix lead to V  x   X J t   X J* t  

11

  h

jJ  t 

j

j

.

(8)

If X  t   X * and the set K  t  in (4) is empty, V  x  is differentiable with respect to t and its time

derivative can be calculated as V  x    X I  t   X I *  t     h i iI  t  i 

for



X  t   X * and V  x 

  X J  t   X J *  t    jJ  t   j h j for X  t   X * .

12

Since h i  xi* , X *   hi  xi* , X *   0 for xi*  0 , Assumptions 1 and 2 directly imply that h i  xi , X 

 h i  xi* , X *   0 for xi*  0 .

If xi*  0 , h i  xi* , X *   0 and h i  xi* , X *   0 .

imply that hi  xi , X   hi  xi* , X *   0 for xi*  0 .

Assumptions 1 and 2

If xi*  0 , the fact i  I implies xi  0 .

by (1), h i  xi , X   hi  xi , X   0 for xi  xi*  0 .

7

Hence,

Since x j  x*j for all j  J ,

 X t   X t   0 . J

Since x j  x*j for all j  J and

J*

X  X * , Assumptions 1 and 2 ensure h j  x j , X   h j  x*j , X *   0 for all j  J . 13

Hence, if X  t   X * , V  x   X J t   X J* t  

  h j

jJ  t 

j

0.

(8’)

Consider the case of X  t   X * . If there exists   0 such that X  t '   X *

for any t '   t   , t    , V  x   X I t   X I* t  

  h

i

i

iI  t 

 0.

If there is no   0 such that X  t '   X * for any t '   t   , t    , then, without loss of generality, X  t  t   X * and X  t  t   X * for t  0 small enough. 











If we define 





V  x   lim V  x  t  t   and V  x   lim V  x  t  t   , V  x   max V  x  , V  x  . t  0

t  0

By (7’) and (8’), the following two inequalities must hold.14 V  x   lim  X I  t  t   X I *  t  t   

t  0

V  x   lim  X J  t  t   X J *  t  t   

t  0



 i h i ,



 j h j .

iI  t t 

jJ  t t 

(9)

(10)

13

The fact j  J implies that x*j is positive for all j  J , since x*j  x j  0 . Assumptions 1 and 2 imply that h j  x , X   h j  x* , X *   0 . Hence, by (1), h j  x , X   h j  x , X   0 for x  x* , even j

j

j

j

j

j

if x j  0 . (A2) in Appendix is proved for X  t   X * . If X  t   X * , (A2) holds with equality. must hold with equality. Similarly, (10) holds with equality. 14

8

Hence, (9)

Since the sets I and J are not empty for any x  x* such that X  X * , as in (7’), the right hand side of (9) is negative. 









V  x   max V  x  ,V  x 

Similarly, the right hand side of (10) is negative. Hence, is negative for any x  x* such that X  X * .

In any case,



V  x  is negative for any x  x* .  Now consider Hahn’s best reply dynamics in Assumption 4, which is assumed by Hahn (1962), Okuguchi (1964), Seade (1980) among others. The following theorem gives the complete proof to Corchón (2001, Proposition 1.5.).15 Theorem 2: Under Assumptions 1, 2 and 4, the unique Nash equilibrium in the aggregative

game is globally stable. Proof: Consider the Lyapunov function in (6). Since for all i  N , by Assumptions 1 and

2, hi is strictly concave in xi for X  i  t    j i x j  t  , the definition of h i in (1) ensures that R j  X  j  t    x j  t   0 if and only if h j  x j  t  , X  t    0 . 16

Similarly,

Ri  X  i  t    xi  t   0 (  0 ) if and only if h i  xi  t  , X  t    hi  xi  t  , X  t    0 (  0 ,

respectively). Hence, if X  X * , V  x   X I t   X I* t  

   R  X t   x t   0 .

iI  t 

i

i

i

i



Arguments are similar for other two cases. Hence, even under Assumption 4, V  x  is negative for any x  x* .  15 16

Footnote 2 applies. See Footnote 8.

9

3. Conclusion

As Folmer and von Mouche (2004) proved, if an aggregative game satisfies the generalized Hahn conditions, then there exists a unique Nash equilibrium. However, their theorem does not ensure that the unique Nash equilibrium is interior. We have therefore formulated dynamics at the point where the trajectory moves out of the non-negative domain and proved that if an aggregative game satisfies the generalized Hahn conditions, the unique Nash equilibrium is globally stable under two alternative continuous adjustment processes with non-negativity constraints. This implies the global stability result of Hahn (1962), Okuguchi (1964) and Corchón (2001) under the original or generalized Hahn conditions, whose proofs contain a defect as pointed out by Al-Nowaihi and Levine (1985).

10

References

Al-Nowaihi, A., and Levine, P. L. 1985. ‘‘The Stability of the Cournot Oligopoly Model: A Reassessment.’’ Journal of Economic Theory, 35, 307-321. Bulow, J., Geanakoplos, J. and Klemperer, P. 1985. ‘‘Multimarket Oligopoly: Strategic Substitutes and Complements.’’ Journal of Political Economy, 93, 488-511. Corchón, L. 1994. ‘‘Comparative Statics for Aggregative Games. The Strong Concavity Case’’ Mathematical Social Sciences, 28, 151-165. Corchón, L. 2001. Theories of Imperfectly Competitive Markets. (Second Edition) Springer-Verlag: Berlin, Heidelberg and NY. Dastidar, K. G. 2000. ‘‘Is a Unique Cournot Equilibrium Locally Stable?’’ Games and Economic Behavior, 32, 206-218. Dixit, A. 1986. ‘‘Comparative Statics for Oligopoly.’’ International Economic Review, 27, 107-122. Folmer, H., and von Mouche, P. 2004. ‘‘On a Less Known Nash Equilibrium Uniqueness Result.’’ Journal of Mathematical Sociology, 28, 67-80. Furth, D. 1986. “Stability and Instability in Oligopoly,” Journal of Economic Theory, 40, 197-228. Gaudet, G., and Salant, S. W. 1991. ‘‘Uniqueness of Cournot Equilibrium: New Results from Old Methods.’’ Review of Economic Studies, 58, 399-404. Hahn, F. H. 1962. ‘‘The Stability of the Cournot Oligopoly Solution,’’ Review of Economic Studies, 29, 329-333. Hale, J. K. 1969. Ordinary Differential Equations. Wiley-Interscience: New York.

11

Okuguchi, K. 1964. ‘‘The Stability of the Cournot Oligopoly Solution: A Generalization,’’ Review of Economic Studies, 31, 143-146. Okuguchi, K. 1976. Expectations and Stability in Oligopoly Models. Springer-Verlag: Berlin, Heidelberg and NY. Okuguchi, K. 1993. ‘‘Unified Approach to Cournot Models: Oligopoly, Taxation and Aggregate Provision of a Pure Public Good,’’ European Journal of Political Economy, 9, 233-245. Okuguchi, K., and Szidarovszky, F. 1990. The Theory of Oligopoly with Multi-Product Firms. Springer-Verlag: Berlin, Heidelberg and NY. Seade, J. 1980. “The Stability of Cournot Revisited,” Journal of Economic Theory, 23, 15-27. Vives, X. 1999. Oligopoly Pricing, Old Ideas and New Tools. MIT Press: Cambridge, USA and London.

12

Appendix

This appendix proves the inequality in (7), that is, V  x   X I t   X I* t  

  h

iI  t 

i

i

.

Let K   t  and K   t  be subsets of K  t  in (4) such that I  t  t   I  t   K   t  and I  t  t   I  t   K   t  for any t  0 small enough. 1 V  x  t  t    V  x  t    t  

1 1 t  2

 X

I t 

 t  t   X I t *    X K t   t  t   X K t *  



2







2 1 I t  I t * X t   X    2 

where X At   t     iA t  xi  t   and X At *   iAt  xi* . A simple calculation shows 1 V  x  t  t    V  x  t    t  





1  1 I t  I t * X  t  t   X   t  2



1 X K   t   t  t   X K  t * t



2







2 1 I t  I t * X t   X    2 

  X    t  t   X     1t 12  X I t

I t *

K t 

 t  t   X K t *  

2

.

It is clear that



1  1 I t  I t * X  t  t   X   t  0 t  2  lim



2







2 1 I t  I t * X  t   X      X I  t   X I *  t     i h i . 2  iI  t 

By L’Hospital’s rule,



1 1 K t  X  t  t   X K t * t  0 t 2 lim



2

13



 lim X K   t   t  t   X K   t * t  0

k

kK  t



1 X K  t   t  t   X K t * t  0 t lim

    h  x  t  t  , X  t  t   0 , k

k

  X    t  t   X    I t

I t *

  It I t * K t K t *  lim    k h k  X    t  t   X    X     t  t   X     t  0   kK  t  



 





     h  i

 iI t

i



      k h k  xk  t  , X  t    X I t   t   X I  t * .  kK  t     





Hence,

lim

t  0

Since

1 V  x  t  t    V  x  t      X I  t   X I *  t     i h i . t iI  t  K   t 

xk  t   xk*

for

any

k  K   t  and

X t   X * ,

Assumption

(A1)

1

implies

h k  xk  t  , X  t    0 for any k  K   t  so that

 X  t   X  t     h   X  t   X  t     h    I

I*

i

iI t  K  t

I

I*

i

iI t

i

i

.

(A2)

Similarly, we can show 1 V  x  t    V  x  t  t      X I  t   X I *  t     i h i , t  0 t  iI  t  K   t 

(A3)

 X  t   X  t     h   X  t   X  t     h   

(A4)

lim

I

I*

i

iI t  K  t

i

I

I*

iI t

Hence, by (A1)-(A4), the inequality in (7) holds.

14

i

i

.