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The Value of Information and the Internet of Things 157

Fig. 9.4 Surface plot of b o in the nontrivial region for L 2 [0, 2], μ 2 [0, 2].

Of course, all of the above is based upon the fact that we know
L + μ
2
the optimal b o ¼ , which we learned from our above assumptions and
calculations (Fig. 9.4).

9.4.1 Generalization
Let us summarize the above as a generality.
1. We are given distributions on L and C.
2.
B C, if B < L
V¼
0, if B > L
3. L and C are independent of the company’s bid B.
4. The company’s cost C is independent of the lowest bid L.
Given this summary:

EðVjbÞ¼ b EðCފ PðL > bÞ and now; in general
½
b
dhVie ≜max b EðVjbÞ
d d f ðlÞ¼ 0 (which is not a far stretch from the sta-
db db
Assuming that EðCÞ ¼
tistical independence we have assumed for the underlying random variables),