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(
1/x
1
)
v
1
ˇ
1
A
ˇ
2
q
v
1
ˇ
1
A
2
x
ˇ
1
1
1
2k.
v
1
c
1
:
(4.7)
The same arguments, applied to the preimages of a generic point of the x
1
-axis
x
2
D
2.
v
1
c
1
1/x
1
˙
4
v
1
ˇ
1
A.
v
1
c
1
1/
x
0
1
;0
, can be used to prove that points of the invariant x
1
-axis have preimages in
the positive quadrant
2
C
R
only if
v
1
c
1
>1. Such preimages are located on the curves
with equation
(
kx
ˇ
2
1
2
2.
v
2
c
2
1/x
2
)
v
2
ˇ
2
A
1
ˇ
1
q
v
2
ˇ
2
A
2
x
1
D
2.
v
2
c
2
1/x
2
˙
4
v
2
ˇ
2
A.
v
2
c
2
: (4.8)
1/
These results on the preimages of the invariant axes are of crucial importance to
determine the boundaries of the feasible region, as we will now demonstrate. Let us
first look at the
symmetric
case, where all parameters of the competitors are iden-
tical. Recall that the Nash equilibrium E
given in (4.3) is locally asymptotically
stable if 0<
v
c<2. From the arguments above, we know that 0<
v
c<1is suffi-
cient and necessary for any trajectory to be feasible and bounded, hence it is also a
necessary condition for the global stability of E
. Indeed, we numerically see that
whenever 0<
v
c<1, the basin of attraction of E
is given by the whole positive
quadrant
2
C
, so that the Nash equilibrium is globally stable. On the other hand,
the results given above also show that this is no longer true if 1<
v
c<2.Inthis
case the basin B.E
/ is a proper subset of the positive quadrant
R
2
C
, and this subset
is bounded by the preimages of the coordinate axes. Figure 4.1 illustrates the sit-
uation for
v
c
D
1:05>1. The white region represents the basin of attraction of the
Nash equilibrium E
, and the black region indicates the infeasible set of marketing
efforts. As the figure shows, the rank-1 preimages (denoted by .X
1
/
1
and .X
2
/
1
)
of the axes are curves starting at the origin, they are symmetric with respect to the
diagonal, and join at the rank one preimage of the origin O
1
D
.
v
mBˇ
4.
v
c
R
1/
;
v
mBˇ
1/
/:
Thus, for
v
c>1, the length of the segment OO
1
gives a rough idea of the extent of
the feasible region. If
v
c is decreased below 1,thenE
becomes globally stable. For
v
c
D
1 a global bifurcation occurs which causes the feasible set to be bounded. If
v
c
is further increased, with the other parameters held constant, the feasible set shrinks.
If
v
c is increased beyond the value
v
c
D
2, then the Nash equilibrium loses its sta-
bility and becomes repelling, and we numerically see that the generic trajectory then
becomes infeasible.
To conclude this subsection, we now briefly turn to the question of the robustness
of the results derived for the case of identical competitors. That is, we are trying to
see if the qualitative descriptions given above are still valid if we assume that the
parameters which characterize the two competitors and their effort decisions are dif-
ferent. It turns out that the answer is yes. Also in this case, if
v
i
c
i
<1for i
D
1;2,the
feasible region coincides with the whole positive quadrant
4.
v
c
2
C
R
, because no preim-
2
C
ages of the coordinate axes exist inside
. Our numerical simulations show that the
Nash equilibrium in this case is globally asymptotically stable. Every combination
of initial marketing efforts in
R
2
C
R
generates a sequence of efforts which converges to
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