Another Application of Game Theory

From Freakonomics blog. Game theory and test taking, from Dixit and Nalebuff's new book: The Art of Strategy.

Consider the following question for the GMAT (the test given to MBA applicants). Unfortunately, issues of copyright clearance have prevented us from reproducing the question, but that shouldn’t stop us.

Which of the following is the correct answer?
a) 4π sq. inches
b) 8π sq. inches
c) 16 sq. inches
d) 16π sq. inches
e) 32π sq. inches

O.K., we recognize that you’re at a bit of a disadvantage not having the question. Still, we think that by putting on your game-theory hat you can still figure it out.

Before reading their analysis, take a shot at trying to reason your way to the correct
answer.

Here’s what they said:
The odd answer in the series is c. Since it is so different from the other answers, it is probably not right. The fact that the units are in square inches suggests an answer that has a perfect square in it, such as 4π or 16π.

This is a fine start and demonstrates good test-taking skills, but we haven’t really started to use game theory. Think of the game being played by the person writing the question. What is that person’s objective?

He or she wants people who understand the problem to get the answer right and those who don’t to get it wrong. Thus wrong answers have to be chosen carefully so as to be appealing to folks who don’t quite know the answer. For example, in response to the question: “How many feet are in a mile?” an answer of “Giraffe,” or even 16π, is unlikely to attract any takers.

Turning this around, imagine that 16 square inches really is the right answer. What kind of question might have 16 square inches as the answer but would lead someone to think 32π is right? Not many. People don’t often go around adding π to answers
for the fun of it. “Did you see my new car — it gets 10π miles to the gallon.” We think not. Hence we can truly rule out 16 as being the correct solution.

Let’s now turn to the two perfect squares, 4π and 16π. Assume for a moment that 16π square inches is the correct solution. The problem might have been: “What is the area of a circle with a radius of 4?” The correct formula for the area of a circle is πr^2. However, the person who didn’t quite remember the formula might have mixed it up with the formula for the circumference of a circle, 2πr. (Yes, we know that the circumference is in inches, not square inches, but the person making this mistake would be unlikely to recognize this issue.)

Note that if r = 4, then 2πr is 8π, and that would lead the person to the wrong answer of b. The person could also mix and match and use the formula 2πr^2, and hence believe that 32π or e was the right answer. The person could leave off the π and come up with 16 or c, or the person could forget to square the radius and simply use πr as the area, leading to 4π or a. In summary, if 16π is the correct answer, then we can tell a plausible story about how each of the other answers might be chosen. They are all good wrong answers for the test maker.

What if 4π is the correct solution (so that r = 2)? Think now about the most common mistake: mixing up circumference with area. If the student used the wrong formula, 2πr, he or she would still get 4π, albeit with incorrect units. There is nothing worse, from a test maker’s perspective, than allowing the person to get the right answer for the wrong reason. Hence 4π would be a terrible right answer, as it would allow too many people who didn’t know what they were doing to get full credit.

At this point, we are done. We are confident that the right answer is 16π. And we are right. By thinking about the objective of the person writing the test, we can suss out the right answer, often without even seeing the question.

Now, we don’t recommend that you go about taking the GMAT and other tests without bothering to even look at the questions. We appreciate that if you are smart enough to go through this logic, you most likely know the formula for the area of a circle. But you never know. There will be cases where you don’t know the meaning of one of the answers or the material for the question wasn’t covered in your course. In those cases, thinking about the testing game may lead you to the right answer.