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ECON 3073 Games and Economic Decisions

Assignment 1

Semester 1, 2022-23

Instruction: Please answer ALL questions and upload your PDF file to ispace dropbox by

1. [20%] Two people enter a bus. Two adjacent seats are free. Each person must decide whether to sit or stand. Sitting alone is more comfortable (payoff 3) than sitting next to the other person (payoff 2), which is more comfortable than standing (payoff 1).

[8%] Suppose that each person cares only about own comfort (selfish). Represent the game in the normal form. Find its Nash equilibrium (equilibria).

[8%] Suppose that each person is altruistic, ranking the outcomes according to the other person’s payoff. However, out of politeness, each person prefers to stand (payoff 1) than to sit (payoff 0), when the other person stands. One also prefers to stand (payoff 3) than to sit (payoff 2) when the other sits. Represent the game in the normal form. Find its Nash equilibrium (equilibria).

[4%] Compare people’s comforts in the Nash equilibria of a) and b). Which one of the two games can be considered as an example of the Prisoners’ Dilemma? Explain.

2. [20%] Consider the following normal form game:

a) [6%] Find all dominant strategy equilibria, if any.

b) [8%] Find all weakly dominated strategy and strictly dominated strategy.

c) [6%] Find equilibria survive the iterated elimination of strictly dominated strategy.

3. [20%] There are two players. Each player has to write down a real number greater than or equal to 1; thus the strategy sets are . Payoffs are as follows (x is the number written by Player 1 and y is the number written by Player 2):

a) [6%] Show that (x, y) = (1, 1) is a Nash equilibrium.

b) [6%] Find the best response function for each player.

c) [8%] Show that there is no other Nash equilibrium besides (1, 1).

4. [20%] There are two companies each with exactly one job opening. Suppose firm 1 offers the pay p₁ and firm 2 offers pay p₂ with p₁ < p₂ < 2p₁. There are two prospective applicants each of whom can apply to only one of the two firms. They make simultaneous and independent decisions. If exactly one applicant applies to a company, that applicant gets the job. If both apply to the same company, the firm flips a fair coin to decide who is hired and the other is unemployed (payoff zero).

a) [6%] Describe the game in normal form.

b) [10%] Find all Nash equilibriums.

c) [4%] Compute the expected payoff at the mixed strategy Nash equilibrium in b).

5. [20%] 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 does not (in which case again nothing is accomplished). Let c be a positive number less than 1 that can be interpreted as the cost of exerting effort.

a) [10%] Derive each player’s best response function.

b) [10%] Find all the mixed strategy Nash equilibriums of this game. How do the equilibriums change as c increases?