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+
+
+
p
p
+
p
+
p
p
p
c
c
c
(5)
=
(
,
)
with
=
+
Δ
c
,
=
+
Δ
.
0
if
if
i
i
i
i
−
−
−
p
p
−
p
−
p
p
c
c
c
(6)
=
(
,
)
with
=
−
Δ
c
,
=
.
0
if
if
i
i
i
Then, replacing (5) and (6) in (1), we can write the explicit equations for
p
if
+
and
p
if
-
:
+
p
(
c
p
p
p
p
p
=
+
Δ
c
)[(
+
Δ
)
+
(
1
−
−
Δ
)]
if
0
i
i
a
1
i
i
(7)
p
p
p
c
+
(
1
−
−
Δ
c
)(
1
−
−
Δ
)
.
0
i
i
a
2
−
p
(
c
p
p
p
=
−
Δ
c
)[
+
(
1
−
)]
(8)
if
0
i
a
1
i
p
p
c
+
(
1
−
+
Δ
c
)(
1
−
)
.
0
i
a
2
Now, if we apply the properties of an ergodic Markov chain (c.f. [20]), we can
compute a long-run probability that a concept will strengthen (
Π
0
):
p
−
Π
if
(9)
=
.
−
+
0
p
p
1
+
−
if
if
Since (9) is written in terms of
p
if
+
and
p
if
-
, which in turn are given by (7) and (8), it
would be rather cumbersome to write an explicit equation for (9) in terms of
c,
Δ
c, p
i
,
Δ
p
i
,
p(a
1
)
and
p(a
2
).
Thus, we prefer to use the following definitions and write a
simpler expression for
Π
0
:
p
p
p
c
c
a
=
(
−
Δ
c
)
e
=
(
+
Δ
c
)(
+
Δ
)
0
0
i
i
i
p
1
−
p
p
c
c
b
=
(
−
Δ
c
)(
1
−
)
f
=
(
+
Δ
c
)(
−
Δ
)
(10)
0
0
i
i
i
p
1
−
p
p
c
c
d
=
(
−
+
Δ
c
)(
1
−
)
g
=
(
−
−
Δ
c
)(
−
Δ
)
0
0
i
i
i
Then, using (10), we can write:
p
p
a
+
b
+
d
Π
=
a
1
a
2
.
(11)
p
p
p
p
0
1
+
a
+
b
+
d
−
e
−
f
−
g
a
1
a
2
a
1
a
2
Expression (11) can be rearranged so that it looks similar to equation (4), i.e.
represents a line that states the relationship that must exist between
p(a1)
and
p(a2)
for a given
Π
0
:
Π
Π
Π
Π
Π
b
+
f
−
b
a
+
−
a
+
e
p
p
=
0
0
+
0
0
0
.
(12)
Π
Π
Π
Π
d
−
g
−
d
d
−
g
−
d
a
2
a
1
0
0
0
0
Note that (12) is an equation of a line with a slope equal to the expression located to
the left of
p(a1)
and an intercept with the y axis (
p(a2)
axis) equal to the far right
hand expression. If we compare (12) with (4), we can see that the slope in (4) does not
change, depending on the values that
p(a1)
,
p(a2)
and
p
if
take; but in (12) the slope
changes (remember that
p
if
in (4) is equivalent to
Π
0
in (12)). Moreover, if we set in
(10),
c
0
= 0.5,
p
i
= 0.1919
,
p
i
=
0.05;
and put the values
a
through
g
,
defined in (10), in (12), we can get a family of lines that represents the condition that
Δ
c =
0.45
and
Δ
