Game Theory Homework - Due Friday October 14th

Review Questions

1. Suppose General Motors is trying to hire a small firm to manufacture the door handles for Buick sedans.  The task requires an investment in expensive capital equipment that cannot be used for any other purpose.  Why might the president of the small firm refuse to undertake this venture without a long-term contract fixing the price of the door handles? 
2. Describe the commitment problem that narrowly self-interested diners and waiters would confront at restaurants located on interstate highways.  Given that in such restaurants tipping does seem to assure reasonably good service, do you think people are always selfish in the narrowest sense? 

Problems

1. Consider the following ‘dating game,’ which has two players, A and B, and two strategies, to buy a movie ticket or a baseball ticket.  The payoffs, given in points, are as shown in the following matrix.  Note that the highest payoffs occurs when both A and B attend the same event. 

B

A		Buy Movie ticket	Buy baseball ticket
	Buy movie ticket	2 for A 3 for B	0 for A 0 for B
	Buy baseball ticket	1 for A 1 for B	3 for A 2 for B

Assume that players A and B buy their tickets separately and simultaneously.  Each must decide what to do knowing the available choices and payoffs but not what the other has actually chosen. Each player believes the other to be rational and self-interested. 

1. Does either player have a dominant strategy? 
2. How many potential equilibriums are there? (Hint: To see whether a given combination of strategies is an equilibrium, ask whether either player could get a higher payoff by changing his or her strategy.) 
3. Is this game a prisoner’s dilemma?  Explain. 
4. Suppose player A gets to buy his or her ticket first.  Player B does not observe A’s choice but knows that A chose first.  Player A knows that player B knows he or she chose first.  What is the equilibrium outcome? 
5. Suppose the situation is similar to part d, except that player B chooses first.  What is the equilibrium outcome? 

2. Babbitt and Kaplan are rational, self-interested criminals imprisoned on separate cells in dark medieval dungeon.  They face the prisoner’s dilemma displayed in the following matrix. 

Babbitt	Kaplan
		Confess	Deny
	Confess	5 years for each	0 years for Babbitt 20 years for Kaplan
	Deny	0 years for Babbitt 20 years for Kaplan	1 year for each

Assume that Babbitt is willing to pay $1000 for each year by which he can reduce his sentence below 20 years.  A corrupt jailer tells Babbitt that before he decided whether to confess or deny the crime, she can tell him Kaplan’s decision.  How much is this information worth to Babbitt? 

3. The owner of a thriving business wants to open a new office in a distant city.  If he can hire someone who will manage the new office honestly, he can afford to pay that person a weekly salary of $2000 ($1000 more than the manager would be able to earn elsewhere) and still earn an economic profit of $800.  The owner’s concern is that he will not be able to monitor the manager’s behavior and that the manager would therefore be in a position to embezzle money from the business.  The owner knows that if the remote office is managed dishonestly, the manager can earn $3100 while causing the owner an economic loss of $600 per week. 
1. If the owners that all managers are narrowly self-interested income maximizes, will he open the new office? 
2. Suppose the owners knows that a managerial candidate is a devoutly religious person who condemns dishonest behavior and would be willing to pay up to $15,000 to avoid the guilt she would feel if she were dishonest.  Will the owner open the remote office? 

4. Imagine yourself sitting in your car in the St. Bernard parking lot that is currently full, waiting for somebody to pull out so that you can park your car.  Somebody pulls out, but at the same moment Mr. Craig Brown, who has just arrived, overtakes you in an obvious attempt to park in the vacated spot before you can.  Suppose Mr. Brown would be willing to pay up to $10 to park in that spot and up to $30 to avoid getting into an argument with you.  (That is, the benefit of parking is $10, and the cost of an argument is $30.)  At the same time Mr. Brown guesses, accurately, that you too would be willing to pay up to $30 to avoid a confrontation and up to $10 to park in the vacant spot. 
1. Model this situation as a two-stage decision tree in which Mr. Brown’s bid to take the space is the opening move and your strategies are (1) to protest and (2) not to protest.  If you protest (initiate an argument), the rules of the game specify that Mr. Brown has to let you take the space.  Show the payoffs at the end of each branch of the tree. 
2. What is the equilibrium outcome? 
3. What would be the advantage of being able to communicate credibly to Mr. Brown that your failure to protest would be a significant psychological cost to you? 
5. Consider the following game, called matching pennies, which you are playing with a friend.  Each of you has a penny hidden in your hand, facing either heads up or tails up (you know which way the one in your hand is facing).  On the count of ‘three’ you simultaneously show your pennies to each other.  If the ace-up side of your coin matches the face-up side of your friend’s coin, you get to keep the two pennies.  If the faces do not match, your friend gets to keep the pennies. 
1. Who are the players in this game?  What are each player’s strategies?  Construct a payoff matrix for the game. 
2. Is there a dominant strategy?  If so, what? 
3. Is there an equilibrium?  If so, what? 
6. Consider the following game.  Harry has four quarters.  He can offer Sally from one to four of them.  If she accepts his offer, she keeps the quarters Harry offered her and Harry keeps the others.  If Sally declines Harry’s offer, they both get nothing ($0).  They play the game only once, and each cares only about the amount of money he or she ends up with. 
1. Who are the players?  What are each player’s strategies?  Construct a decision tree for this ultimatum bargaining game. 
2. Given their goal, what is the optimal choice for each player? 

7. Jill and Jack both have two pails that can be used to carry water down from a hill.  Each makes only one trip down the hill, and each pail of water can be sold for $5.  Carrying the pails of water down requires considerable effort.  Both Jill and Jack would be willing to pay $2 each to avoid carrying one bucket down the hill and an additional $3 to avoid carrying a second bucket down the hill. 
1. Given market prices, how many pails of water will each child fetch from the top of the hill? 
2. Jill and Jack’s parents are worried that the two children don’t cooperate enough with one another.  Suppose they make Jill and Jack share their revenue from selling the water equally.  Given that both are self-interested, construct the payoff matrix for the decisions Jill and Jack face regarding the number of pails of water each should carry.  What is the equilibrium outcome?