Question

Consider the following strategic situations. The Professor assigns a short assignment at the end of every lecture. On any given lecture day, the Professor has to decide whether or not to collect the homework assignment due that day. Similarly, each student must decide whether or not to do the homework assignment due that day. In particular, consider the problem for a representative student, Brian.

Consider the same basic setup, but with a slightly different, but equally believable, payoff structure. The payoffs are now:

		Brian
		Do Assignment	Don’t
Prof.	Collect	5,10	8,0
	Don’t	12,5	4,7

Find all Nash Equilibria to this game. In equilibrium, with what probability will the Professor collect each assignment?

Suppose that the Professor moves first by announcing whether or not he will collect the assignment. Draw the associated game tree and calculate all the Subgame Perfect Equilibria of that game.

Answer 1

	Do assignment	Don't
Collect	5,10*	8*,0
Don't	12*,5	4,7*

In the above game, there is no pure Nash equilibrium. However, for Professor to collect each assignment (Mixed strategy Nash equilibrium), the probability will be calculated as follows:

Suppose the Brain does the assignment with probability p and not done by 1-p. The professor would collect the assignment if 5p+8(1-p)>12p+4(1-p)

8-3p>8p+4

The prof would collect the assignment has the probability of at least 4/11.

Subgame in I, Brian would choose to do his assignment as 10>0.

Subgame in II. Brian would prefer to let go of assignment as this would yield him 7 which is greater than 5.

Subgame in collect, not collect, the professor would prefer to collect the assignment as it would yield him the payoff of 5 over 4 in case he does not collect.