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In-Depth Information
ð
X
s
0
6¼s
a
q
π
s
ðÞ¼p
ss
a
r
ss
a
þ cs
;
;
p
ss
0
r
ss
0
:
ð
5
:
6
Þ
Expressed in words, for the product
s
, the action value of the recommendation
a
is equal to the probability of the acceptance of this recommendation multiplied by
the corresponding reward plus the weighted sum of the unconditional transition
probabilities multiplied by their rewards. For simplicity's sake, we will occasion-
ally refer to the first term
p
ss
a
r
ss
a
as the
conditional
and the second term (without the
scaling factor)
X
s
0
6¼s
a
p
ss
0
r
ss
0
as the
unconditional action value
.
Despite their apparent simplicity, equations (
5.5
) and (
5.6
) are extremely inter-
esting. As with many fundamental relationship equations, they deserve detailed
study. Consider, for instance, Maxwell's equations. Through their fundamental
interpretation, James Clerk Maxwell was able to predict the existence of electro-
magnetic waves, which were later confirmed experimentally by Heinrich Hertz. By
further studying them, Henri Poincare was able to derive the invariability of the
speed of light, from which Albert Einstein finally developed the special relativity
theory. Of course, (
5.5
) and (
5.6
) are not Maxwell's equations, and the authors are
no Poincare or Einstein; nevertheless, it is rewarding to spend some more time
on them.
5.2.1 Discussion
If we consider (
5.6
) more closely, we see that the conditional action value is the
direct value of the recommendation, whereas the unconditional action value repre-
sents the value of the other product transitions. The scaling factor
c
controls the
weighting between the two. While, therefore, the conditional action value reflects
the success of the recommendation, the unconditional action value gives the
“recommendation-free” potential (i.e., the product transitions that occur even
without a recommendation), whose influence is reduced by the success of the
recommendation.
If the transition probabilities
p
ss
a
and
p
ss
a
are both small,
c
(
s
,
a
) can be regarded
as approximately 1, and both action values have the same weighting. As soon,
however, as
p
ss
a
is significantly greater than
p
ss
a
0
i.e., the recommendation
a
is
strongly accepted,
c
(
s
,
a
)
1 is true, and the influence of the unconditional
action value is reduced as compared to that of the conditional one. If the improve-
ment in the conditional action value overtakes the reduction in the unconditional,
the recommendation is successful. On the other hand, the recommendation may
suffer from down-selling: the popular recommendation captures more reward
overall from the previous product transitions than it generates as new reward.
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