Professor Nash announces that he will auction off a $20 bill in a competition between Jack and Jill, two students chosen randomly at the beginning of class. Each student is to privately submit a bid on a piece of paper; whoever places the highest bid wins the $20 bill. (In the event of a tie, each student gets $10.) The catch, however, is that each student must pay whatever he or she bid, regardless of who wins the auction. Suppose that each student has only two $1 bills in his or her wallet that day, so the available strategies to each student are to bid $0, $1, or $2.
a) Write down a 3 x 3 payoff matrix describing this game.
b) Does either student have any dominated strategies?
c) What is the Nash equilibrium in this game?
d) Suppose that Jack and Jill each could borrow money from the other students in the class, so that each of them had a total of $11 to bid. Would ($11, $11) be a Nash equilibrium?