Chemistry Reference
In-Depth Information
11
O
x
2
−
1
(
X
1
)
−1
(
X
2
)
−
1
E
*
0
11
x
1
Fig. 4.1
The market share attraction game - the symmetric case. The
white region
represents the
basin of attraction of the stable Nash equilibrium E
,the
black region
represents the infeasible
set. This figure is obtained with parameters A D 20; k D 1,
v
1
D
v
2
D 0:35, ˇ
1
D ˇ
2
D 0:3,
c
1
D c
2
D 3
the equilibrium of the market share game. Like in the symmetric case, a wide range
of parameter values exist such that the Nash equilibrium is stable. If one (or both)
of the above inequalities is (are) reversed, then the Nash equilibrium only attracts
points of the feasible set, which no longer covers the whole area
. The bound-
ary of the feasible set can then be again determined using (4.8) and (4.7). This can
be seen as follows. If
v
1
c
1
<1and
v
2
c
2
>1, then the feasible set is an
unbounded
region
(extending for arbitrarily large x
1
) with the upper boundary formed by the
rank-1 preimage of the x
1
-axis, say .X
1
/
1
(see Fig. 4.2a). The equation of this
preimage is given by (4.8) with the “
C
” sign. The curve .X
1
/
1
is tangent to the
x
1
-axis at the origin. Analogously, if
v
1
c
1
>1 and
v
2
c
2
<1, then the feasible set
is an unbounded region (extending for arbitrarily large x
2
) with the right boundary
formed by the rank-1 preimage of the x
2
-axis, say .X
2
/
1
, whose equation is given
by (4.7) with the “
C
” sign.
If both inequalities are reversed, so that
v
1
c
1
>1 and
v
2
c
2
>1, then the feasi-
ble set is a
bounded region
, whose boundary is formed by the curves .X
1
/
1
and
.X
2
/
1
, starting at the origin O tangent to the axes and intersecting at the preim-
age of the origin O
1
(see Fig. 4.1). Hence, the conditions
v
i
c
i
D
1 and
v
j
c
j
>1,
i
¤
j, denote the occurrence of a global bifurcation, at which the feasible region
is changed from unbounded to bounded. It should be noticed that other bifurca-
tions that change the topological structure of the boundaries of the feasible region
R
2
C
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