General Oligopolies - Nonlinear Oligopolies: Stability and Bifurcations - page 127

In the case of >1there is always an interior Nash equilibrium E S which belongs
to the diagonal D f .x;x/;x 2 Rg . Its coordinates are given by

1 ;

1
;1

1

E S D

and it is characterized by identical production quantities of the two firms. At >3
two further Nash equilibria exist. They are given by

C 1 C p . C 1/. 3/
2

! ;

; C 1 p . C 1/. 3/
2

E 1 D

(3.21)

C 1 p . C 1/. 3/
2

! ;

; C 1 C p . C 1/. 3/
2

E 2 D

and they are located in symmetric positions with respect to the diagonal . Notice
that for D 3, E 1 ;E 2 and E S coincide. These Nash equilibria are characterized by
different production quantities of the two players. It is easy to see that the market
shareoffirm1(firm2)islargerinE 1 (E 2 ). Obviously, in a situation where multi-
ple Nash equilibria coexist, a coordination problem for the two firms arises. It is not
clear which of the Nash equilibria the firms can agree upon as an outcome of the
game. One possibility to discriminate among the equilibria is to assume that players
start with quantity pairs out of equilibrium and adjust their production decision to
evolving changes in their environment, for example, using their best replies or esti-
mates of the gradient of the profit functions. Then we can use local stability, global
dynamics, or for example, the extent of the basins of attraction in the case of mul-
tiple locally stable equilibria to obtain insights into the question about which of the
equilibria is more likely to be a long run outcome of the game (see Kopel (2009)
and Cox and Walker (1998)).
We will assume that in order to update their production decisions, the duopolists
use partial adjustment towards the best response with naive expectations. Recall,
however, that in Chap. 1 we have shown that in the duopoly case the best reply
dynamics with adaptive expectations is identical to the dynamical system obtained
by partial adjustment towards the best response with naive expectations (see (1.20)
and (1.21)). Consequently, for our duopoly model with symmetric cost externalities,
in either case the dynamical systems which generates the sequences of (expected)
production quantities is given by

x 1 .t C 1/ D .1 a 1 /x 1 .t/ C a 1 R 1 .x 2 .t// D .1 a 1 /x 1 .t/ C a 1 x 2 .t/.1 x 2 .t//;

x 2 .t C 1/ D .1 a 2 /x 2 .t/ C a 2 R 2 .x 1 .t// D .1 a 2 /x 2 .t/ C a 2 x 1 .t/.1 x 1 .t//:
(3.22)