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values (i.e.,
T
tax
,
T
maintain
,
T
cele
,and
T
human
). These four default values are
the desired outcomes. Note that a player does not know these four values. We
can compute the achievement level for each outcome for the player as follows:
a
level
=
−
(
d
j
−
o
j
)
,
(5)
where
d
j
is the desired value of the outcome and
o
j
is the actual outcome.
As soon as we get the achievement level, we apply back propagation to evolve
the neural network.
Let
K
(
L
)
be the number of neurons in the
L
-th layer, i.e., the last layer. The
total error
J
is define as:
K
(
L
)
J
=
1
2
e
j
,
(6)
j
=1
where
e
is:
e
j
=
d
j
−
o
j
.
(7)
o
j
is the output of the
j
-th neuron. To update the hidden layer, we have
K
(
r
+1)
∂J
∂o
(
r
)
j
∂J
∂o
(
r
+1)
k
f
(
s
(
r
+1)
k
w
(
r
+1)
kj
=
∗
)
∗
,
(8)
k
=1
⊡
⊤
K
(
r
+1)
∂J
∂o
(
r
+1)
k
ʔw
(
r
)
ji
⊣
f
(
s
(
r
+1)
k
w
(
r
+1)
kj
⊦
∗
f
(
s
(
r
)
j
o
(
r−
1)
i
=
∗
)
∗
)
∗
(9)
k
=1
where
w
(
r
)
ji
is the change of the weight from
i
-th neuron to the
j
-th neuron of
the
r
-th layer.
o
(
r
+1)
k
is the output of the
k
-thneuroninthe(
r
+ 1)-th layer.
For the last layer, i.e.,
L
-th layer, we have
∂J
∂o
(
L
)
k
=
−
(
d
k
−
o
k
)
.
(10)
The weight changes are computed as follows:
∂J
∂o
(
L
)
k
ʔw
(
L
)
kj
f
(
s
(
L
)
k
o
(
L−
1)
j
=
∗
)
∗
.
(11)
Wecanusethefollowingformulatoupdatealltheweights:
w
(
t
+1)=
w
(
t
)
−
ʷ
∗
ʔ
w
,
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