In the first PhD game theory class, the professor presented a puzzle for all of us. The story goes like this:
A professor comes to the class with a couple of white and red hats. He places a hat on each of his students and said: “I’m gonna place the same hat on each of you in every class. You can guess the color of your hat after class but you only have one chance. If you guess it right you will get an A, but if your guess is wrong you will get an F for the course. You are not allowed to communicate with each other on this. There is at least one white hat.” Will any student guess his/her hat right and get an A?
I do not attempt to solve the problem by traditional method of drawing the normal form and solving the matrix. There are a couple of underlying assumptions that are easily ignored in this setting:
1. Students may approach the professor at the end of class to report their hat color and all other students see whether any students approach the professor on a given day.
2. Each player is rational and knows the others are all rational (mutual knowledge of order infinity).
3. Students strictly prefer any other grade to an F.
Here’s my thinking:
The simplest case is when there is only one white hat. The student wearing the white hat, denoted by student A, would go to the professor and guess his hat is white, because there is at least one white hat and he observes all others wearing red hats. Having observed student A’s behavior, the other students know that student A must be the only one with white hat (because otherwise he wouldn’t have guessed) and they will all guess their hats to be red. So everyone in the class will get an A.
When there is more than one white hat, however, things get more complicated. Since a student will get an F if he guesses wrong, he will remain silent until he is perfectly sure about the color of his hat. But can this happen? I don’t think so. Since there are more than one white hat in the class, each student will see at least one other student wearing a white hat. Without any information on the number of white and red hats, students cannot draw any conclusion on the color of their hats. So no one will get an A.
Update: My deduction of the simplest case is correct. The rest is really an extension to multiple white hats.
If only student A and student B wear white hats, i.e. there are only two white hats, student A will observe on the first day that there is only one other white hat (which B is wearing) in the class. She also observes that B does not go to the professor after class. Since there is at least one white hat in the class, B must know at least another student in the class wearing white hat. Now A knows she is wearing white hat. Therefore on the second day of class, both A and B will go to the professor and say they have white hats, and they both get A.
For generaliztion, suppose there are k white hats in the class (and n students), on the kth day everyone wearing a white hat will go to the professor and say they are wearing white hats. So k students will get an A.