40.316-Homework 1 Solved

Question 1: Each of two players has two possible actions, Quiet and Fink; each action pair results in the players’ receiving amounts of money equal to the numbers corresponding to that action pair in the following game matrix:

                                                                                            Quiet            Fink

2,2
0,3
3,0
1,1
Quiet Fink

For example, if player 1 chooses Quiet and player 2 chooses Fink, then player 1 receives nothing, whereas player 2 receives $3.

The players are not “selfish”; rather the preferences of each player i are represented by the payoff function mi(a) + αmj(a), where mi(a) is the amount of money received by player i when the action profile is a, j is the other player, and α is a given non-negative number. Player 1’s payoff to the action pair (Quiet,Quiet), for example, is 2 + 2α.

(a)    Formulate a strategic game that models this situation in the case α = 1. Is this game the Prisoner’s Dilemma?

(b)   Find the range of values of α for which the resulting game is the Prisoner’s Dilemma.

Question 2: Two players (player 1 and player 2) are bargaining over how to split one dollar. Both of them simultaneously name shares they would like to have, s1 and s2, where 0 ≤ s1,s2 ≤ 1. If s1 + s2 ≤ 1, then the players receive the shares they named; if s1 + s2 1, then both players receive zero. What are the pure strategy Nash equilibria of this game?

Question 3: Two candidates are competing in a political race. Each candidate i can spend si ≥ 0 on ads which affect the probability of winning the election. Given a strategy profile

(s1,s2), candidate 1 wins the election with probability

 

and candidate 2 wins with probability

 .

If both candidates spend zero, then each wins with probability one half. The payoff to candidate i is the probability of winning minus the amount spent on ads.

(a)    Write an expression for the payoff function of candidate 1.

(b)    Solve for the best response function of each candidate.

(c)    Find the pure strategy Nash equilibrium.

Question 4: An employer hires an employee and promises him wage w. The employee can work (W) or shirk (S). Working is associated with cost of effort, e, where w e 0. If the employee works, the employer obtains revenue r and otherwise  .

Employer cannot tell whether the employee is working or shirking unless she chooses to run inspection (I) at a cost of c 0. Inspection reveals whether the employee is working or not and in the latter case the wage is withheld. The decisions of the two players (W/S and I/NI) are taken simultaneously. The payoff matrix is as follows:

                                                                                        I                               NI

w-e, r-c-w
w-e,r-w
0,r/2-c
w,r/2-w
W

S

Employee is choosing a row (row player) and the employer is choosing a column (column player).

(a)    Under what condition(s) will this game have a pure strategy Nash equilibrium?

(b)   Assume the aforementioned condition(s) is (are) not satisfied. Derive the formulas forprobability of working, p and probability of inspection, q in the mixed strategy Nash equilibrium.

Question 5: Consider the Cournot’s game with n number of firms. Assume that the inverse demand function is given by:

P(Q) = max{α − Q,0},

where α 0. The cost function of each firm i is C(qi) = cqi for all qi, with 0 ≤ c ≤ α. Assuming that all firms produce the same output in Nash equilibrium, solve for the equilibrium output for each firm. Find the market price at which output is sold. Does the equilibrium price increase or decrease as n increases?

Question 6: Two people can perform a task if, and only if, they both exert effort. They are both better off if they both exert effort and perform the task than if neither exerts effort (and nothing is accomplished); the worst outcome for each person is that she exerts effort and the other person does not (in which case again nothing is accomplished). Specifically, the players’ preferences are represented by the expected value of the payoff functions in the following figure (where c is a positive number less than 1 that can be interpreted as the cost of exerting effort):

                                                                                   No effort                Effort

0,0
0,−c
−c,0
1 − c,1 − c
No effort

Effort

Find all the mixed strategy Nash equilibria of the game. How do the equilibria change as c increases?