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In-Depth Information
1
1
2
2
y
y
y
2
1
Level 1
1
2
y
y
y
1
2
3
1
x
1
1
2
2
3
3
x
x
x
2
1
x
3
2
x
Level 0
x
x
x
1
2
2
3
x
x
3
1
2
1
3
x
Fig. 6.6 Interpolation operator for state-action aggregations
Example 6.4
We consider the state space of Example 6.1 with the corresponding
iterator
I
1
and restrictor
I
0
. This gives the following preconditioner for the state-
value function:
2
4
3
5
1
2
1
2
1
þ
0
2
4
3
5
¼
2
4
3
5
:
e
x
1
e
x
1
x
2
x
3
C
1
x ¼
1
2
1
2
x
2
e
1
þ
0
x
3
001
þ
1
The example of the state aggregations should now be extended to include the
associated actions, where initially all states are permissible as actions in all states.
The result is shown in Fig.
6.6
. For ease of reading, the actions are shown as
x
at
level 0 and
y
at level 1, where the lower index represents the start nodes and the
upper index the target nodes. For instance,
x
1
is the recommendation of product
2 for product 1 at level 0.
Note that on the finest grid - that is, the product level - the reflexive relation
x
i
is
practically meaningless, since a product cannot recommend itself. At levels
0
these actions are meaningful, however, since they are a measure of the strength of
product recommendations within the same group relative to one another.
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