Suppose we must accept or reject several projects AND choose an appropriate scale/level for each AND we have an overall resource constraint. This is S & Z's "case 4."
In this scenario the '''fundamental rule''' translates to this : '''the last dollar you spend on each accepted project should produce the same amount of benefit.'''
Explanation: if the last dollar spent on a project could produce more net benefit on another project then it should be applied there.
Example. A student has four exams and 12 hours of study time. She has determined that if she does not study at all she can attain a C on all of her exams. In addition, she has calculated how much studying it will take to get various grades on the various exams.
How to proceed? Suppose she has a good idea of what grade she'll get in each course if she does not study at all. Suppose these are
English C Sociology B Chemistry F Arabic D
Suppose further that she has discovered how much she can raise her grade by in each class for each hour she studies. To raise her grade by one letter grade above the no-study level it takes
English 2 hours
Sociology 1 hour
Chemistry 4 hours
Arabic 3 hours
STOP AND THINK: If the goal is to get the best GPA, what should she spend the first hour studying?
The marginal net benefit to an hour of studying is 1 letter grade per hour for sociology, much higher than 0.5, 0.33, and 0.25 letter grades per hour for the other subjects.
Now, let's assume we have further information on what it would take to move her grades up in the different subjects.
What does she study for the next bit of time? Two hours on English. And then? Three hours on Arabic. Total so far? 6. What should she spend the next six hours on?
She's looking at three places where she can raise her grade by a quarter letter grade per hour. If we assume the biggest benefit comes at the beginning of a study session, she should divide her time among the three — two hours each.
What principle is she following? Something like "always spend the next hour of studying where it will do you the most good."
Another way to put it would be : Make sure the last hour you spend on each subject is getting you the same amount of increased grade.