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x
1
D
R
1
.x
2
/
D
1
x
2
.1
x
2
/; x
2
D
R
2
.x
1
/
D
2
x
1
.1
x
1
/:
(5.31)
As before, the quantities .x
1
;x
2
/ are selected in the strategy space Œ0;1
2
and
k
2
.1;4. For simplicity we restrict our analysis to the best reply dynamics, that is we
set a
1
D
a
2
D
1, and obtain from (5.30) the dynamical system
8
<
x
1
.t
C
1/
D
1
Œ"
1
x
2
.t/.1
x
2
.t//
C
."
1
1/x
1
.t/
1
x
1
.t/
2x
2
.t/
i
;
"
1
1
"
1
T
W
(5.32)
:
x
2
.t
C
1/
D
2
Œ"
2
x
1
.t/.1
x
1
.t//
C
."
2
1/x
2
.t/
1
x
2
.t/
2x
1
.t/
i
;
"
2
1
"
2
where the map T defined above generates the dynamics of the game. Observe that if
both firms know the true demand function so that, "
1
D
"
2
D
1, then (5.32) reduces
to the Cournot best reply dynamics given by
<
x
1
.t
C
1/
D
1
x
2
.t/.1
x
2
.t//;
T
W
(5.33)
:
x
2
.t
C
1/
D
2
x
1
.t/.1
x
1
.t//;
which is a special case of the adjustment dynamics already investigated in Sect. 3.2
(see also Kopel (1996) and Bischi et al. (2000a)). In a similar fashion to the analysis
presented in Sect. 3.2, for (5.32) a complete description of the stability regions of
the emerging equilibria can be obtained for the case of homogenous firms
1
D
2
and "
1
D
"
2
. In fact, analytic expressions of the curves that constitute the bound-
aries of such regions can be obtained from a standard analysis of the eigenvalues
of the Jacobian matrix (for details, see Bischi et al. (2004b)). Here we will instead
focus on the model with heterogeneous firms where, although a rigorous analytical
characterization cannot be given, the global dynamical properties of (5.32) can still
be studied by a mixture of analytical and numerical methods. Figure 5.1a depicts the
full information reaction function in a situation (for a choice of
1
¤
2
)wherea
unique Nash equilibrium exists. The question is, what happens if players misspecify
the demand? Will the adjustment dynamics still converge to a steady state close to
the Nash equilibrium of the true game? As will become clear, this depends on the
global dynamics of the system. First observe that the steady states of the dynamical
system with misspecified beliefs (5.32) are the real solutions of the algebraic system
T.x
1
;x
2
/
D
.x
1
;x
2
/, which yields the equations
1
."
1
1/
2
x
1
C
2
1
"
1
."
1
1/x
1
x
2
C
1
"
1
x
2
C
"
1
.1
1
."
1
1//x
1
1
"
1
x
2
D
0;
2
."
2
1/
2
x
2
C
2
2
"
2
."
2
1/x
1
x
2
C
2
"
2
x
1
C
"
2
.1
2
."
2
1//x
2
2
"
2
x
1
D
0:
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