1. Two students sign up to prepare an honours thesis with a professor. Each can invest time in his own project: either no time, one week, or, two weeks (these are the only three options). The cost of time is 0 for no time, and each week costs 1 unit of payoff. The more time a student puts in the better his work will be, so that if one student puts in more time than the other there will be a clear "leader." If they put in the same amount of time then their thesis projects will have the same quality. The professor, however, will give out only one grade of A. If there is a clear leader then he will get the A, while if they are equally good then the professor will toss a fair coin to decide who gets the A. The other student will get a B. Since both wish to continue on to graduate school, a grade of A is worth 3 while a grade of B is worth 0.

(a) Write down the normal form representation of this game as well as the matrix form.

(b) Find the unique mixed-strategy Nash equilibrium

7. Two students sign up to prepare an honours thesis with a professor. Each can invest time in his own project: either no time, one week, or, two weeks (these are the only three options). The cost of time is 0 for no time, and each week costs 1 unit of payoff. The more time a student puts in the better his work will be, so that if one student puts in more time than the other there will be a clear "leader." If they put in the same amount of time then their thesis projects will have the same quality. The professor, however, will give out only one grade of A. If there is a clear leader then he will get the A, while if they are equally good then the professor will toss a fair coin to decide who gets the A. The other student will get a B. Since both wish to continue on to graduate school, a grade of A is worth 3 while a grade of B is worth 0. (a) Write down the normal form representation of this game as well as the matrix form. (b) Find the unique mixed-strategy Nash equilibrium. [5 + 5 = 10]