2 students need to write a paper for a class that they're taking. The paper is either "good", in which case each student gets a payoff of 20, or "not good", where each student gets 0. Each student has 3 levels of effort for the paper: None, Low, and High effort.

There is a cost associated with each level of effort: No effort costs nothing Low effort costs 1 High effort costs 8

Exerting more effort increases the likelihood that the project will be a success.

If both players exert low effort,the project succeeds 20% of the time.

If both players exert high effort, the project succeeds 100% of the time.

If one exerts high effort while the other exerts low effort, the project succeeds 50% of the time.

If either player exerts no effort, then the project fails (and those that exerted effort still pay their costs).

The success of the project is worth 20 to each player. If the project has a 20% chance of succeeding, then each player values that at 20 x 0.2 = 4. If the project has a 50% chance of success, then they value that at 20 x 0.5 = 10.

(a) Set this up as a game in matrix form. Label the players, strategies, and payoffs.

(b) Does either player have a dominant strategy in this game? That is, does either player have a strategy that is always a best response? Briefly explain.

(c) Find all of the pure strategy Nash equilibria of this game.

(d) Suppose that student 2 randomizes according to 2 = ( 1/2H, 1/2 L, 0N ) so that they choose high effort half of the time and low effort half of the time. What is student 1's expected payoff of choosing high effort? What is their expected payoff of choosing low effort?

(e) Find a mixed strategy Nash equilibrium where both players randomize over high and low effort (so they put probability 0 on "no effort")

(f) Is there a mixed strategy Nash equilibrium where both players randomize over high effort and no effort? Explain.