Chemistry Reference
In-Depth Information
L
2
L
2
C
2
C
2
R
2
E
2
E
2
x
2
x
2
R
1
E
E
R
1
R
2
C
2
C
2
x
1
x
1
0
E
1
L
1
0
E
1
L
1
(a)
(b)
L
2
L
2
E
2
E
2
C
x
2
x
2
E
E
C
x
1
x
1
0
E
1
L
1
0
E
1
L
1
(c)
(d)
Fig. 1.11
Basins of attraction for the Cournot duopoly when firms use partial adjustment towards
the best response with linear demand/quadratic cost.
Light grey
basin of E
1
;
dark grey
basin of
E
2
;
white
basin of the 2-cycle C
2
.(
a
) Full adjustment, a
1
D a
2
D 1. The basins are rectangular.
(
b
) Partial adjustment, a
1
D 0:97, a
2
D 0:98. The basins lose their rectangular shape. (
c
) Partial
adjustment, a
1
D 0:93, a
2
D 0:95. The basin of C
2
shrinks. (
d
) Partial adjustment, a
1
D 0:9,
a
2
D
0:92. The 2-cycle C
2
has become unstable, and its basin has disappeared
As a final remark we note that although the cyclic outcome C
2
is an attrac-
tor from a mathematical point of view, it has several shortcomings as a potential
description of real-world economic behavior. First, whereas convergence to a steady
state implies that the players' naive expectations are fulfilled at least in the long
run, a sustained low-periodic oscillation implies that the players' expectations are
permanently wrong. It seems plausible that in such a situation the players would
learn how to improve their forecasts. Second, although profits are always positive
in all Nash equilibria, this is not necessarily true in general for the cycle C
2
.Asan
example consider again the best reply dynamics, where C
2
D
˚
.0;0/
I
x
1
;x
2
.
The corresponding profits along the 2-cycle are '
k
.0;0/
D
0 for firm k, with
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