Page 176 - Artificial Intelligence for the Internet of Everything

162 Artificial Intelligence for the Internet of Everything

Unlikewith the clairvoyant knowledge of C, with knowledge of L, EðVjbÞ
may be negative, but the profit may be much larger. Knowledge of C gives the
company nonnegative profit, whereas knowledge of L gives it a larger poten-
tial profit. This result is in-line with the results Howard obtained.

9.7 CLAIRVOYANCE ABOUT C AND L
After we combine both pieces of information, the bid b will never be
less than c, and it will always match l, meaning that we must modify
Eq. (9.1) again, now different from Eq. (9.25)because L is more restricted,
resulting in:

(
l c, if c l, and l 2 support of L
(9.31)
EðVjb,c,lÞ¼
0, otherwise
So we have an assumption of independence between C and L:
Z
EðVjc,lÞ f ðcÞdc (9.32)
EðVjlÞ¼

Z l
ðl cÞ f ðcÞdc (9.33)
¼
∞
Z l
c f ðcÞdc (9.34)
¼ l PðC < lÞ
∞
when l 2 support of L.
Example 9.4 [L¼ U½0,2Š and C¼ U½0,1Š]
8
Z l Z l
cdc if 0 l < 1
>
>
> l dc
>
0 0
<
Z 1 Z 1
EðVjbÞ¼ (9.35)
cdc, if 1 l 2,
>
l dc
>
>
0 0
>
:
0, otherwise
The results:
8 2
l
, if 0 l < 1
>
>
>
2
<
(9.36)
l :5, if 1 l 2,
EðVjbÞ¼
>
>
>
0, otherwise
: