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ECO Game Theory Problems




Question;QUESTION 1. (Flat Tire and Coordination) Two college students, Al and Bob, veryconfident about their mid-term exam performance, decided to attend a party the weekendbefore the final exam. The party was so good that they overslept the whole Sunday. Insteadof taking the final exam unprepared on Monday, they pleaded to the professor to give them amake-up exam. Their excuse was a flat tire without a spare and any help. The professor,surprisingly, agreed. On Tuesday morning, the professor placed them in separate rooms andhanded them the test. The test had just one question: which tire?.(a) Suppose the final score is entirely based on the make-up test. If Al and Bob give thesame answer on the flat tire question, both pass the exam, yielding a payoff of 10 for each ofthem. If Al and Bob, however, give different answers, both fail the exam, getting payoffzero. Write down the game table of this simultaneous-move game between Al and Bob..(b) Find all pure-strategy Nash equilibria of this game..(c) Find two mixed-strategy Nash equilibria of this game with the following requirements:in one of the mixed Nash equilibrium, each player randomizes using two pure strategies,and in the other mixed Nash equilibrium, each player randomizes using three or four purestrategies.QUESTION 2. (The Employee Monitoring Game) Consider the employee monitoringgame discussed in class (Lecture Notes 3, page 18): Suppose an employee in a company caneither work or shirk. The manager of the company cannot directly observe whether theemployee is working or not. However, the manager can pay a cost of $10 to find out theemployees behavior. If the manager finds out that the employee shirks, the manager canlegally pay nothing to the employee. If, however, the manager has no hard evidence aboutthe employees shirking, the manager has to pay the employee $100. We assume that themanager will get a payoff of $200 if the employee works and $0 if the employee shirks. Inaddition, we assume that it costs the employee $50 if he works.1. Write down the game table of the above simultaneous-move game. What are the(complete) strategy sets of the players?2. Find all (pure-strategy and mixed-strategy) Nash equilibria for the above game.QUESTION 3. (First-Price Sealed-Bid Auction) Alice is selling her 2000 Chevroletcavalier to her friends, Bob and Charles. It is commonly known that Bob attaches a value of$6000 to Alices old car, while Charless value of the car is $6500. Alice designs thefollowing auction to sell her car: First, she asks each of them to write his bid on a piece ofpaper. Then Bob and Charles give their bids to Alice. Notice that when Bob and Charleswrite down their bids, they dont know each others bid (hence the name "sealed bid"). AfterAlice receives the sealed bids, the bids are shown to everyone and the car will be sold to theperson who has a higher bid. When there is a tie (Bob and Charles bid the same amount),then Alice would flip a fair coin to decide who will get the car..(a) Write down the normal form representation of this game..(b) Find weakly dominated strategies for both Bob and Charles. Is bidding 0 a weaklydominated strategy for either player? Explain..(c) Find the set of all pure-strategy Nash equilibria for this game.QUESTION 4. (Cournot Duopoly Revisited) Consider the Cournot duopoly model wherethe (inverse) demand is P (Q) = a - Q. The two firms now have asymmetric marginal costs:c1 for firm 1 and c2 for firm 2.(a) What is the Nash equilibrium if 0 < ci < a/2, i {1,2}for each firm?(b) What is the Nash equilibrium if 0 < c1 < c2 a+c1?QUESTION 5. (Dominance by Mixed Strategies and Mixed Strategy Nash Equilibrium)Consider the following 3 x 3 simultaneous-move game ("3 x 3" means that both playershave three pure strategies).LCRT`1, 10, 02, 0.3M0, 01, 11, 0.3B0.3, a 0.3, b0.5, c(a) Given that all you know about the game is the above game table and that a, b and c isthat they are three arbitrary constants, find all strictly dominated strategies for the players(Hint: One strategic consequence of introducing mixed strategies to a game is that now apure strategy can be strictly dominated by a mixed strategy, which is the case for one of theplayers in the above game.)(b) Find all (pure-strategy and mixed-strategy) Nash equilibria for the above game.


Paper#55921 | Written in 18-Jul-2015

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