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which can be obtained by solving the fourth degree algebraic system (1.46) for
x
i
.t/, upon setting x
i
.t
C
1/
D
0,(i
D
1;2). A simple strategy for obtaining the
preimages of O is to start from the dynamics of T
g
restricted to the axes. Since
x
i
.t/
D
0 implies x
i
.t
C
1/
D
0, starting from an initial condition on a coordinate
axis, the dynamics are “trapped” on this axis for all t. In other words, a monopoly
prevails over time and the one-dimensional “monopoly dynamics” is obtained from
(1.46) with x
i
D
0, namely
x
j
.t
C
1/
D
.1
C
a
j
.A
c
j
//x
j
.t/
2.B
C
e
j
/a
j
x
j
.t/:
(1.51)
We also note that this map is conjugate to the standard logistic map x.t
C
1/
D
x.t/.1
x.t// through the linear transformation x
j
1
C
a
j
.A
c
j
/
.
BCe
j
/
x, from which
the relation
D
1
C
a
j
.A
c
j
/ can be obtained. The following results for our
map can be directly derived from the properties of the logistic map, which is well-
studied in the literature; see for example, Devaney (1989). The rank-1 preimages
O
.j /
D
2a
j
1
given in (1.49) can now be easily derived from (1.51). Along the x
j
-axis (j
D
1;2), the one-dimensional restriction (1.51) gives bounded dynamics for a
j
.A
c
j
/
3 provided that the initial conditions are taken inside the segment !
j
D
OO
.j /
1
. Observe that divergent trajectories along the invariant x
j
axis are obtained
if the initial condition is out of the segment !
j
(j
D
1;2). Let us now turn to
the quadrilateral region bounded by the two segments !
1
and !
2
and their rank-
1 preimages, say .!
1
/
1
and .!
2
/
1
respectively (see Fig. 1.14). The preimages
.!
1
/
1
and .!
2
/
1
can be analytically computed as follows. Let X
D
.x;0/ be a
point of !
1
. Its preimages are the real solutions .x
1
;x
2
/ of the algebraic system
8
<
x
1
Œ1
C
a
1
.A
c
1
/
2a
1
.B
C
e
1
/x
1
a
1
Bx
2
D
x;
(1.52)
:
x
2
Œ1
C
a
2
.A
c
2
/
a
2
Bx
1
2a
2
.B
C
e
2
/x
2
D
0:
From the second equation it is easy to see that the preimages of the points of !
1
are either located on the same invariant axis x
2
D
0 or on the line represented by
the equation
a
2
Bx
1
C
2a
2
.B
C
e
2
/x
2
D
1
C
a
2
.A
c
2
/:
(1.53)
Analogously, the preimages of a point of !
2
belong to the same invariant axis
x
1
D
0 or to the curve represented by equation
2a
1
.B
C
e
1
/x
1
C
a
1
Bx
2
D
1
C
a
1
.A
c
1
/:
(1.54)
It is now straightforward to see that the line (1.53) intersects the x
2
axis in the
point O
.2/
1
and the line (1.54) intersects the x
1
axis in the point O
.1/
1
. Moreover,
the two lines intersect at the point O
.3/
1
. A summary of these observations leads to
the following description of the basin of the asymptotically stable Nash equilibrium
E as shown in Fig. 1.14. The rank-1 preimages of the origin are the vertexes of the
quadrilateral OO
.1/
1
O
.3/
1
O
.2/
1
. The sides of this region are given by !
1
, !
2
and their
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