Page 166 - Artificial Intelligence for the Internet of Everything

152 Artificial Intelligence for the Internet of Everything

Eq. (9.2) only makes sense when 0 b. The company never places a neg-
ative amount bid, so any event involving b < 0 has zero probability. Thus the
conditional probability is not defined in that range.
We are interested in the expected value of profit conditioned on the bid.
That is, we wish to determine EðVjbÞ:
Z ∞
v f ðvjbÞ dv
EðVjbÞ¼
∞
ZZZ
v f ðvjb,c,lÞ f ðc,ljbÞ dc dl dv
¼
 3
∞ (9.3)
ZZ Z
v f ðvjb,c,lÞ dv dc dl
¼ f ðc,ljbÞ
 2 ∞
ZZ
EðVjb,c,lÞ f ðc,ljbÞdc dl
¼
 2
Now, Howard makes two assumptions (Eqs. 9.6, 9.7) Howard (1966) to
simplify the problem.
Assumption 9.1 The joint distribution of cost and lowest bid C, L is inde-
pendent of our company’s bid B. That is:

f ðc,ljbÞ¼ f ðc,lÞ
Assumption 9.2 The company’s cost C is independent of the lowest
bid L. That is:

f ðc,lÞ¼ f ðcÞf ðlÞ
We realize that one could certainly argue the reality of these assumptions
in all cases. Using Assumptions 9.1 and 9.2 we now have that:
ZZ
EðVjb,c,lÞ f ðcÞf ðlÞdc dl (9.4)
EðVjbÞ¼
 2
From Eq. (9.1) we see that once we set the values of B, C, L at b, c, l,
respectively, the density function of V becomes deterministic. That is:
Theorem 9.1
δðv ðb cÞÞ, if b l
f ðvjb,c,lÞ¼
δðvÞ, if b > l
and therefore:

b c, if b l

EðVjb,c,lÞ¼
0, if b > l