Lying students and games of coordination

It’s sad, but many students lie to teachers and get away with it. That’s why I love to read about stories where teachers catch the students red-handed. It’s even better when the teacher has a little fun with it.

Here’s one story I especially enjoyed:

My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were partying in another state and didn’t get back to Duke until it was over. Their excuse to the professor was that they had a flat tire, and they asked if they could take a make-up test. The professor agreed, wrote out a test and sent the two to separate rooms to take it. The first question (on one side of the paper) was worth 5 points, and they answered it easily. Then they flipped the paper over and found the second question, worth 95 points: ‘Which tire was it?’

Source: excerpted from Marilyn vos Savant, Parade Magazine, 3 March 1996, p 14.

Besides being a nice story, the situation raises some interesting questions in game theory.

Coordination games

The students found themselves in an interesting situation. It mattered less what they answered and more that their answers matched. Such a situation is generally classified as a coordination game.

A coordination game is one where all parties stand to gain and all wish to coordinate. Common examples include the convention of a driving side (left or right), the arrangement of letters on a standard keyboard (qwerty or dvorak).

In such games, the worst choice is for players to miscoordinate and the best is for them to match. Thus, the strategy in coordination games is simply to match what other players might do. If possible, you may signal try to announce your intentions in advance.

Sometimes players are helped because some choices seem natural or attractive. I wrote about topic before in coordinating bike traffic but let me recap the main idea.

Suppose, for example, that I surveyed people today to name a Nobel Peace Prize winner. While I would get a variety of answers, I bet I would find many answers of Barrack Obama since this award is topical and controversial. Obama is a natural answer to the question, and such information is useful when coordinating. More generally, there are choices people consider more prominent for things like naming a person, a number, or a place. This was an observation developed by Harvard economist Thomas Schelling, and the concept is named Schelling points in his honor (or sometimes called focal points).

Now that the stage is set, let us consider how the students might fare in guessing the same tire.

What is the probability they guess correctly?

There are four possible tires to pick: the front-right, front-left, rear-right, and rear-left. The probability they guess correctly is the sum of the probabilities of matching on any of these choices.

To get an estimate, suppose each choice has an equal chance of being picked. Then the probability of guessing correctly will equal to one-forth. That’s because there are four correct matches divided by sixteen pairs of answers. (Alternately, conditional on the first student making a choice, the second student has a one in four chance of making the same selection).

But lucky for the students, the actual chance of matching will likely be even higher. The reason for this is that one of the tires is like a Schelling point and more often to be picked. As discussed in the free textbook Introduction to Probability, an informal classroom survey found that 58 percent chose front-right, 11 percent front-left, 18 percent rear-right, and 13 percent for rear-left.

If we take these probabilities, then the chance of matching is about 40 percent (equal to 0.58 squared plus 0.11 squared plus 0.18 squared plus 0.13 squared). That’s a nice increase from random chance!

Discussion questions

(I am trying something new in this article. I’m adding several discussion questions to help teachers and students. So, here goes–and let me know if you like this section.)

1. The story takes it for granted the students are lying, and the teacher would punish them even if they matched on their alibi. But suppose the students were telling the truth. What steps could they take to signal their honesty?

2. Let’s be honest: a flat tire is not a great excuse. A flat tire is easy to fix and roadside assistance can be relatively quick. What alibis would be more believable? What makes them better?

3. Perhaps the professor was being too nice. The test question actually allowed the students a chance to match by pure chance. What other questions might the professor have asked?

4. Suppose the professor was trained in game theory. Perhaps the test question might have been the following:

“You may pick one of the following options.

• Option 1: Which tire was flat?
• Option 2: Name as many elements from the periodic table as you know.

Now before you answer, let me tell you how I’m grading this question. If you both pick option 1, then I’ll give you full credit only your answer matches what the other student writes. Heck, I’ll even be generous–I’ll give you half credit if your answers don’t match. But there is a catch: if you pick option 1, and the other person does not, then I will give you no credit.

If you pick option 2, I’ll give you one point per element correctly listed. I’ll give you points regardless of what the other person does.

Choose carefully, and good luck!”

What are the Nash equilibria of this game? Hint: how does this game compare to the stag-hunt game?