Value of Information (for Porject IV QI.3)

[Subbarao Kambhampati <rao@asu.edu>]

Some people are having trouble figuring out whether or not Burns
should pay to get the plutonium/heavy water tested. Here is some
discussion on the value of information:

The definition is: In general, the value of a given piece of
 information is defined to be the difference in expected value between
 the best actions before and after the information is obtained.

Here is a Simple Example that explains the concept:
 
Suppose an oil company is hoping to buy one of n indistinguishable
blocks of ocean drilling rights. Let us assume further that exactly
oen of the blocks contains oil worth C dollars, and that the price of
each block is C/n dollars. 

Now suppose that a seismologist offers the company the results of a
survey of block number 3, which indicates definitively wheter the
block contains oil. How much should the company be willing to pay for
the information? 

The way to answer this question is to examine what the company would
do if it had the information:

* with probability 1/n, the survey will show that block 3 contains the
  oil. In this case the company will buy block 3 for C/n dollars, and
  make a profit of  C - C/n = (n-1)C/n dollars

* with probability (n-1)/n, the survey will show that the block
  contains no oil, in which case the company will buy a different
  block. Now the probability of finding oil in one of the other blocks
  changes from 1/n to 1/(n-1), so the company makes an expected profit
  of C/(n-1) - C/n = C/n(n-1) dollars

No we can calculate the expected profit given the survey information:

[1/n* (n-1) C/n  ] + [ n-1/n * C/n(n-1) ]  = C/n

Therefore, the company should be willing to pay the seismologist upto
C/n dollars for the information: the information is worth as much as
the block itself!

The value of information derives from the fact that *with* the
information, one's course of action may change to become more
appropriate to the actual situation. One can discriminate according to
the situation, whereas without the information, one has to do what's
best on average over the possible situations. In general, the value of
a given piece of information is defined to be the difference in
expected value between the best actions before and after the
information is obtained. 

Rao
[Dec 01, 2001]