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this example, it is worth making some remarks about the extent and the shape of the
feasible region, which is represented by the white area in the figures shown above.
If the parameters are inside the learning region, the extent of the feasible region pro-
vides important information about the robustness of the learning process. Detailed
information about the feasible basin of the steady state " provides an answer to the
fundamental question: how far away from the true demand can the guesses of the
players be in order to still guarantee the success of the learning process? First of
all, it can be noticed that the maximum “distance” of a single subjective scale factor
is not important, as the distance of all the scale factors must be considered. Even
if one firm starts with an initial estimate for "
k
very close to the true value B,the
endogenous dynamics of the global learning process may not lead to convergence to
the true demand in the long run due to the influence of its competitors. Although this
remark may sound obvious ex post, we think that it is worth pointing this out. As a
second and final remark we point out that the boundaries of the feasible region may
be quite complicated. This can be clearly seen in Fig. 5.8b. A study of this kind of
complexity requires an analysis of the global dynamic properties of the map (5.106).
In particular, the creation of complicated topological structures may be related to the
fact that the map (5.106) is noninvertible (as explained in Appendix C).
Now we turn our attention to the continuous time system (5.92). Its Jacobian is
the matrix
J
with eigenvalue equation
.N
C
1/B
2
3
a
k
B
k
"
k
a
k
A
.N
N
N
Y
X
a
k
A
4
1
5
D
0;
(5.113)
C
C
1/B
k
D
1
k
D
1
which can be derived similarly to the discrete case. In this case we have the fol-
lowing stability theorem, which can be proved along the lines of the proof of
Theorem 2.2.
Theorem 5.5.
Under assumption (5.97) the steady state of system (5.92) is always
locally asymptotically stable.
If condition (5.97) is violated, that is, when for at least one firm,
X
N
1
N
C
1
D
c
k
>0;
k
c
l
lD1
then Theorems 5.4 and 5.5 no longer hold, and the asymptotic behavior of the
dynamic systems becomes much more complicated.
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