III. HOME WORK PROBLEMS
2. A LIFE-OR DEATH DECISION PROBLEM
Suppose you have a crucial decision problem, say a serious medical
life-or-death decision involving cancer treatment, or a lifetime marriage
partner choice, between courses of action A and B. By spending one million
dollars under A you will find the best decision for sure vs. a cost
$1Million/x dollars under B to get a decisioin which is guaranteed to
be within the top-5% of all decision choices with probability equal to 0.99.
At what value of x will you be indifferent between the two courses of
action.
SOLUTION:
Suppose we agree that a probability of 0.99999 is equivalent to being certain
and a probability of 0.99 being acceptable in the sense all reasonable persons
are willing to take a chance on an event with 0.99 probability of occurring.
Also we are satisfied by being in the top 5% of all performances vs. being
in the top 0.001% which we equate to being the best. We know that the
probability of getting at least one decision within the top-n% of total
N samples is given by 1 - (1 - n%)N where (1 -
n%) is the probability that the picked sample does not belong to the
top-n%). Assume N1 is the total number of samples
needed to get the best for certain and N2 is the total
number of samples needed to get a top 5% decision with probability 0.99. Then
So the total amount of savings from sampling is greater than
1.2x104. Note that the efficiency gains from softening the
certainty and the true optimum are multiplicative rather than additive. Thus,
trading off the best for sure to good enough with high probability is
extremely favorable.