The Classical Cournot Model - Nonlinear Oligopolies: Stability and Bifurcations - page 44

respective rank-1 preimages .! 1 / 1 and .! 2 / 1 respectively. All points inside this
quadrilateral region lead to convergence, all points outside cannot generate feasi-
ble trajectories. Points located to the right of .! 2 / 1 are mapped into points with
negative value of x 1 after one iteration, as can be easily deduced from the first com-
ponent of (1.46). Points located above .! 1 / 1 are mapped into points with negative
value of x 2 after one iteration, as can be deduced from the second component of
(1.46). The expressions in (1.53) and (1.54) can be used to determine the impact
of parameter changes on the basin. Finally, observe that for these values of the
parameters the basin of the unique interior Nash equilibrium is a rather simple and
connected set.

1.3.4 Simple Basins and Critical Curves

In this subsection we introduce the concept of critical curves (see also Appendix C).
This subsection uses many concepts about dynamical systems that may not be famil-
iar to some readers (such as noninvertible maps, critical sets, preimages of various
ranks and so on). These concepts are reviewed in Appendix C, which the reader may
need to study before working through this subsection.
Recall that in the previous subsection we have demonstrated how to obtain the
boundaries of the feasible region by taking the preimages .! i / 1 .i D 1;2/ of the
coordinate axes. Since the map T g in (1.46) is a noninvertible map, as can be readily
deduced from the fact that the origin has four preimages, there might be further
preimages of .! i / 1 .i D 1;2/, which have to be also considered in order to obtain
the whole boundary of the feasible region. In order to determine if .! i / 1 .i D 1;2/
have further preimages, we can use the critical curves of the map which can be used
to identify regions in the feasible set (or strategy space) with a different number of
preimages.

To begin with, let us consider a given point x 0 1 ;x 0 2 in the strategy space. Then
its preimages can be calculated by setting x 1 .t C 1/ D x 0 1 ;x 2 .t C 1/ D x 0 2 in (1.46)
and solving with respect to x 1 and x 2 the fourth degree algebraic system,
8
<

x 1 Œ1 C a 1 .A c 1 2.B C e 1 /x 1 Bx 2 / D x 0 1 ;

(1.55)

:

x 2 Œ1 C a 2 .A c 2 2.B C e 2 /x 2 Bx 1 / D x 0 2 :

Clearly, this algebraic system may have up to four real solutions, which are the

rank-1 preimages of x 0 1 ;x 0 2 . We can now use this information to subdivide the
strategy space into regions characterized by a different number of preimages. This
is shown in Fig. 1.15a, which is obtained with the same parameters as Fig. 1.14. The
regions Z k denote the sets of points which have k real and distinct rank-1 preim-
ages. For example, as shown above, the origin O D .0;0/ 2 Z 4 , because it has four