Game Theory
ECON 3208
Spring 2024
PRACTICE FINAL EXAM
QUESTION 3
Consider the following two-player game.
a. [5 pts] Which player is unaware of the type of game he or she is playing? Please explain.
b. [15 pts] What is the Bayes’ Nash Equilibria of this game? Show all work for full credit.
QUESTION 4
There are two players in the game: a company and a potential worker. The company first offers the worker a contract that pays w. The worker accepts the contract, or rejects it and receives his reservation income of 10. The worker exerts effort e, making his return w − e if accepts the contract. The nature then chooses output q(e) by the probabilities assigned in the following table. The company’s return is q(e) − w.
The company can pay the worker a fixed payment w! and a bonus bp(e), where p(e) is the probability of success.
a. [10 pts] What is the optimal combination of fixed wage and bonus to entice high effort? Carefully show or explain each step.
b. Suppose that the rate of success from low effort increases slightly from 40%.
i. [5 pts] In part (a) above, would the bonus for high effort increase, decrease, or stay the same? [No need to provide calculations but do explain your answer carefully.]
ii. [5 pts] In part (a) above, would the fix wage for high effort increase, decrease or stay the same? [No need to provide calculations but do explain your answer carefully.]
QUESTION 5
A university is selling two types of parking permits — Permit A and Permit B —to its students and workers. The following table lists the cost of maintaining each parking spot, the price of each permit, and the consumers’ willingness to pay for each permit.
Suppose the university cannot identify the type of consumers, and the students comprise b share of the consumers.
a. [8 pts] Suppose the university wants to sell Permit A to workers and Permit B to students. What should x andy be? Show all work for full credit.
b. [4 pts] How much money would the university make from selling Permit A to workers and Permit B to students? Show all work for full credit.
c. [8 pts] The university decided to sell Permit A only instead. Its goal is to have both students and workers buy Permit A at the highest price possible. How much money would the university make from selling the parking permits? Show all work for full credit.
QUESTION 7
a. [12 pts] In a private-value auction with three bidders, David values the object being auctioned at
$15,000, John values the object at $9,000, and Sam values the object at $5,000, though none of the bidders is aware of the other’s values. For each of the following auction formats, list and carefully explain what price you expect in equilibrium:
i. English auction
ii. Dutch auction
iii. First-price sealed-bid auction
iv. Second-price sealed-bid auction
b. [8 pts] How does risk aversion change bidding in a second-price sealed-bid auction? How does risk aversion change bidding in a first-price sealed-bid auction? Explain.