While I was doing research for this assignment, I stumbled upon a wonderful blog titled “Mind Your Decisions”, which is essentially a collection of neat articles related to game theory. It features a variety of puzzles, jokes, paradoxes, as well as real-life applications, with topics ranging from rigging Super Bowl bets to negotiating a pay raise. There is one article on the blog that I found particularly relevant to the CS 2850 course material, so I’ll discuss it here.

The story goes as follows:

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.

It turns out we can model this story as a coordination game played by two students. Each student has four different answers to choose from as to which tire was flat: front-right (FR), front-left (FL), rear-right (RR), or rear-left (RL). Presumably, if their answers match, they’ll both receive a score of 100 on the exam. Otherwise, they’ll both receive a score of 0 for the second question and end up with a score of 5 on the exam. The following table illustrates the outcomes:

The above game has four Nash equilibriums, (FR, FR), (FL, FL), (RR, RR), and (RL, RL), since they are mutual best responses to each other, but the problem is that neither student knows what the other will pick. If both students choose arbitrarily from the four answers, then the students have a 1/4 probability of matching (four pairs match out of sixteen pairs of answers).

However, as the article points out, students have biased preferences as to which wheel to pick. In fact, a survey showed that 58% would choose front-right, 11% front-left, 18% rear-right, and 13% rear-left. This indicates that front-right is a “focal point” in this game, so the students will more likely settle into the (FR, FR) Nash equilibrium.

Now here is where it gets interesting. The author of the article posed the following question to the reader:

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.”

What are the Nash equilibriums of this game?

It turns out we can model this scenario as a stag-hunt game by adding a new option to the original game. Each student can now choose to name elements (N) and receive say 80 points regardless of what the other student does. The following table illustrates the new outcomes:

The game now has five Nash equilibriums, (FR, FR), (FL, FL), (RR, RR), (RL, RL), and (N, N). If the students coordinate on picking a wheel, they are able to receive 100 points each (the stag). But due to the risk involved, there is a good chance that they will both settle for naming the elements and receive 80 points each (the rabbit).

It is interesting to note that even if the students are still hung-over from the party and are only able to receive a score of 40 points when attempting to name the elements, picking a wheel is still not a dominant strategy, since if the other student decides to name the elements, the best response will be to do the same.

Source: http://mindyourdecisions.com/blog/2009/10/13/lying-students-and-games-of-coordination/