Game Theory Problem Sets

 

BF – Bierman & Fernandez’s Game Theory with Economic Applications

Q – Additional Questions found on this website

 

Due on:

Problems  (Click on link for non-BF problems)

September 13

Chapter 1 BF 1.4 – 1.6, 10.4, Q1

September 20

Chapter 2 BF 1.11, 2.1, 2.2, 2.3 (part a only), Q1, Q2

Chapter 3 Q1, then BF 3.1, then Q2

September 27

Chapter 4 BF 4.1 – 4.3

Chapter 5 BF 5.1, Q1, Q2

October 4

Chapter 6 BF 6.1, 6.2, 6.4 (parts a & d only), 6.7, Q1

Chapter 7 BF 7.4, Q1, Q2

October 11

Chapter 8 BF 8.1 – 8.4

October 18

Chapter 9 Q1, Q2, Q3

October 25

Nothing due, Midterm Exam in class

November 1

Chapter 10 BF 10.3, 10.5 – 10.7 Q1, Q2

November 8

Chapter 11 BF 11.1 (a only), 11.4 (note error in the text), 11.5 (Hint:  there are two policies)

November 15

Chapter 12 BF 12.1, 12.2 (see additional directions below)

November 22

No Class – Thanksgiving Holiday

November 29

Chapter 13 BF 13.6, Q1, Q2, Q3

December 6

Chapter 15 BF 15.1 – 15.5 (see additional directions below)

December 13

Final Exam

 

 

Chapter 1: 

Q1.  Find all the pure strategy and mixed strategy NE of the following games.  (Hint:  eliminate any strictly dominated strategies on the 3x3 games first.)

 

A

Left

Center

Right

Top

2, 0

1, 1

4, 2

Middle

3, 4

1, 2

2, 3

Bottom

1, 3

0, 2

3, 0

 

B

Left

Right

Top

2, 1

0, 2

Bottom

1, 2

3, 0

 

 

 

 

 

 

 

C

Left

Center

Right

Top

2, 1

0, 0

0, 0

Middle

0, 0

2, 1

0, 3

Bottom

0, 0

0, 3

2, 1

 

 

 

 

 

 

 

 

 

Chapters 2: 

 

Q1.  Consider the basic Bertrand model discussed in class.  If the firms had asymmetric marginal costs:  c1 for firm 1 and c2 for firm 2, what would the likely market outcome be?  (Which firm(s) sells the good?  What is the price that prevails in the market?)

 

Q2. Firm 1 and Firm 2 are competing for a cable television franchise. The present value of the net revenues generated by the franchise is equal to R. Each firm’s probability of winning the franchise is given by its proportion of the total spent by the two firms on lobbying the local government. That is, if  L1 and  L2 represent the lobbying expenditures of firms 1 and 2, respectively, then Firm 1’s probability of winning is given by L1/(L1+ L2), and Firm 2’s probability of winning is  L2/( L1+L2).

A.  Define a profit function for each firm.  Derive their best response functions and find the Nash equilibrium level of spending for each firm (L1*,L2*).

B. The head of Firm 1 meets the head of Firm 2 at the golf club and says, “It’s silly for us to spend (L1*,L2*) on lobbying when in the end, we each only have a 50 percent chance of getting the franchise. Why don’t we each spend zero and the outcome will (on average) be the same but we’ll both be better off by L.” The head of Firm 1 says, “You know, that makes a lot of sense.” How much does each spend on lobbying in the next period, and why? (Hint:  think about possible deviations.)

 

Chapter 3:

Q1.  Derive the equilibrium price and quantity for the US and Britain under autarky, using the profit functions 3.2a and 3.2b.  Calculate equilibrium profits.

Ex 3.1 (Your equilibrium answers should include price, quantity and profit made by each firm.)

Q2.  Compare profits for each firm under autarky (Q1) and trade (3.1)

 

Chapter 4:

Q1. The countries of North and South are in a dispute over fishing rights in George’s Bank, a fertile fishing ground off the coast of New England. The source of disagreement is that each boat sent into George’s Bank reduces the yield of all of the other boats. And while each country, North and South, has control over the number of boats it sends into George’s Bank, it has no control over the number of boats the other country sends. Let B N and B S equal the number of boats sent by North and South respectively, and let Y(BN,BS) = 360 - (BN + BS) equal the annual fishing yield (in tons) per boat from George’s Bank.

 

A. What is the welfare maximizing number of boats (total) that should be sent to George’s Bank each year? What is the total fishing yield in this case? (Note: assume that we are interested in maximizing the fish yield and that we are not taking the welfare of the fish into account.)

 

B. Because there is no fishing treaty, North and South make their fishing decisions noncooperatively by maximizing their national fishing yields. How many boats will each country send to George’s bank (in other words, what is the Nash Equilibrium outcome of this dispute)? Call this quantity of boats B NC , where NC stands for non-cooperative. What is the yield for each country in this case? (Hint: note that the game is symmetric because each country faces exactly the same payoffs.)

 

C. After several years of peaceful cooperation under the treaty, South announces that it has developed a new fishing boat called the “Super Trawler.” The Super Trawler is the equivalent of 150 (!) fishing boats but all in one boat. You could even say that the Super Trawler is exactly like 150 conventional fishing boats all welded together. Moreover, South explains that all of its conventional fishing boats were destroyed by a mysterious fire during the winter. Hence, South can either use the Super Trawler or it can not fish at all. Assuming that South used the Super Trawler, what would be North’s best response in terms of fishing boats deployed? Observe further that if South does not fish at all, North’s best response is to send the social optimum  number of boats that you solved for in A (note: this is because North would have the whole fishing ground to itself). With these facts in mind, what is the equilibrium outcome of this game? How does the number of boats sent  by each country compare to the previous non-cooperative outcome in Part B? Explain the intuition for your result in terms of the game theory tools used in class.

 

Chapter 5:  

Q1. Assume that voters care only about the budget deficit.  Likely voters were polled as to the preferred size of the deficit, and the data is depicted below.  What is the Nash Equilibrium? (ie.  What would the Bush and Kerry camps likely choose?)  Determine a payoff matrix, remove any dominated strategies, and solve for the optimal strategy of each player.  Who is likely to win and with what percentage of the vote?

 

 

Q2.  (Network Externalities)  Consider the following model of demand for a software product such as an operating system that we’ll call OS.  Consumers have heterogeneous preferences for OS.  We’ll index these preferences by x, which is distributed uniformly from 0 to 1, or x ~ U[0, 1].  People with a low x have strong positive preference for OS, whereas people with a high x have much less desire to use OS.  Denote by n, 0 ≤ n ≤ 1 the total proportion of consumers using OS, and let p equal the purchase price.  Finally, define the utility of a consumer as :

Thus, the utility of each purchaser exhibits “network externalities” since it increases with the number of other purchasers. 

 

a.  Write down the equation that shows the utility of a consumer, , who is just indifferent between purchasing and not purchasing OS at a given price p. 

 

b.  Notice that since  is indifferent between buying or not buying OS, this implies that everyone with x <  is strictly happier buying OS at price p since utility is increasing in (1 – x), and conversely everyone with x >  is strictly happier not buying OS.  Moreover, since x is uniformly distributed between 0 and 1, this implies that n in the equation from part a is equal to .  Hence, rewrite your equation from part a solely as a function of p and  and rearrange so that you have p as a function of .  Draw this equation, with є[0, 1] on the horizontal axis.  What is unusual about this demand curve?

 

c.  Now consider the production of OS.  Since it’s a software product, assume for simplicity that the marginal cost of producing the next copy is zero (for software, this is a reasonable approximation).  What would be the price of OS in a competitive industry and what fraction of all potential users, n, would purchase it at that price?

 

d.  Now imagine that OS is owned by a company called MacroSoft that has the sole rights to produce the software.  Using your equation from part b, write down a new equation that represents MacroSoft’s profits as a function of , the number of users of OS.  Draw a figure showing profits as a function of .

 

e.  Solve MacroSoft’s profit maximization problem.  What is *, what are profits, and how does the number of users compare to your answer from part d?

 

Chapter 6: 

Q1.  Consider the following change in the Stackelberg game from class.  Price is P = a – Q, where total quantity, Q = q1 + q2 + q3.  Marginal cost is constant and the same across all firms at c.  In the first stage, Firm 1 chooses q1.  Then, in the second stage, Firms 2 and 3 simultaneously choose q2 and q3 given Firm 1’s choice of q1.  How do the outcomes of the first mover compare across games?  Does total output change in this game, compared to the basic Stackelberg?

 

Chapter 7: 

Q1.  Change the bargaining game in section 7.2 (no impatience) such that Sally moves first and there are three rounds of bargaining.  Draw the game tree, include payoffs, and solve for the SPE.

 

Q2.  We solved a bargaining model in class (with symmetric impatience), where the buyer moved first and there were three periods.  Repeat this game with the seller making an offer first.  Who has the first mover advantage in this bargaining game?

 

Chapter 9:

Q1.  This is based on Q1 from Chapter 4.

a.  North and South sign a 3 year treaty that allows each country to send exactly ½ the socially optimal number of boats that you solved for in Ch4, Q1, part a.  Call this quantity B*.  The treaty stipulates that if either country violates the agreement, the two countries will return the situation that you solved for in Ch4, Q1, part b.  How many boats does each country send in each year of the 3 year treaty?  Explain your answer.

b.  North and South have negotiated a new fishing treaty identical to the previous treaty except that the new treaty lasts forever unless either country violates it.  Assume that each country has an identical discount rate of d, where d = 0 implies that the country discounts the future completely, and d = 1 implies that the country does not discount the future at all.  Under what values of d will the treaty hold?  (Hint, before you solve for d, you must solve for the optimal ‘defection’ in terms of boats sent that a country would make were it to violate the treaty.)

 

Q2

Confess

Remain Silent

Confess

2, 2

0, 3

Remain Silent

3, 0

1, 1

Q2.  Consider the infinitely repeated prisoners’ dilemma game depicted in the table.  For what values of d does the following strategy constitute a subgame perfect equilibria?

 

Strategy (same for both players):  Choose RS in period 1. After the first round, choose RS if the outcome in the preceding period is either (RS, RS) or (C, C). Choose C if the outcome is preceding period is (RS, C) or (C, RS).

 

Q3.  Draw the Folk Theorem region for the Cournot Duopoly solved in class.

 

Chapter 10:

Q1.  Suppose that demand fluctuates randomly in the infinitely repeated Bertrand game:  in each period, the demand intercept is aH with probability p and aL (< aH) with probability 1 – p; demands in different periods are independent.  Suppose that each period the level of demand is revealed to both firms before they choose their prices for that period.  What are the monopoly price levels (pH and p L) for the two levels of demand?  Solve for d*, the lowest value of d such that the firms can use trigger strategies to sustain these monopoly price levels in a subgame perfect Nash equilibrium. (extension: For each value of d between ½ and d*, find the highest price p(d) such that the firms can use trigger strategies to sustain the price p(d) when demand is high and the price p L when demand is low in a subgame perfect Nash equilibrium.)

 

Q2.  This question is about a milkman and a customer.  At any day, with the given order,

  • Milkman puts m ε [0, 1] liter of milk and (1 – m) liter of water in a container and closers the container, incurring cost cm for some c > 0.
  • Customer, without knowing m, decides on whether or not to buy the liquid at some price p.  If she buys, her payoff is (vm – p) and the milkman’s payoff is (p – cm).  If she does not buy, she gets 0, and the milkman gets 

cm .  If she buys, only then does she learn m.

 

a.  Assume that this is repeated for 100 days, and each player tries to maximize the sum of his or her stage payoffs.  Find all the Sub-game Perfect equilibria of this game.

 

b.  Now assume that this is repeated infinitely many times and each placer tries to maximize the discounted sum of his or her stage payoffs, where the discount rate is d ε (0, 1).  What is the range of price p for which there exists a Sub-game Perfect equilibrium such that on the equilibrium path, every day, the milkman chooses m = 1, and the customer buys?

 

Chapter 12:  Ex 12.1 (for part b, derive the terms under which Natalie will accept the firm’s offer), 12.2 (stated another way, using the new profit function for the employer provided in the question, solve for new equilibrium values for w*, L*, E*(w) and U*(E, w).  Differentiate these to determine how each changes when reservation utility increases.)

 

Chapter 13: 

Q1.  Consider the following scenario.  The Row player knows if she is playing Game 1 or Game 2.  The Column player does not know which game they are playing, but does know that there is a 50-50 chance that it will be Game 1.  Row and Column players know each others’ payoffs for both games and move simultaneously.  Find the Bayesian Nash Equilibrium.           

               

Game 1

Left

Right

Top

1, 1

0, 0

Bottom

0, 0

0, 0

 

 

 

 

 

 

 

                   

Game 2

Left

Right

Top

0, 0

0, 0

Bottom

0, 0

2, 2

 

 

 

 

 

 

 

 

Q2.  Consider a Cournot duopoly operating in a market with inverse demand P(Q) = a – Q, where Q = q1 + q2 is the aggregate quantity on the market.  Both firms have total costs ci(qi) = cqi, but demand is uncertain:  it is high (a = aH) with probability q and low (a = aL) with probability 1 – q.  Furthermore, information is asymmetric:  firm 1 knows whether demand is high or low, but firm 2 does not.  All of this is common knowledge.  The two firms simultaneously choose quantities.  What are the strategy spaces for the two firms?  Make assumptions concerning aH, aL, q, and c such that all equilibrium quantities are positive.  What is the Bayesian Nash equilibrium of this game?

 

Q3.  Two individuals are involved in a synergistic relationship. If both individuals devote more effort to the relationship, they are both better off. Specifically, an effort level is a nonnegative number, and player 1’s payoff function is e1 (1 + e2 – e1) ; where e1 is player 1’s effort level and e2 is player 2’s effort level. For player 2 the cost of effort is either the same as that of player 1, and hence her payoff function is given by e2 (1 + e1 – e2) ; or effort is very costly for her in which case her payoff function is given by e2 (1 + e1 – 2e2).  Player 2 knows player 1’s payoff function and if the cost of effort is high for herself or not. Player 1, however, is uncertain about player 2’s cost of effort. He believes that the cost of effort is low with probability p, and high with probability 1 – p, where 0 < p < 1. Find the Bayesian equilibrium of this game as a function of p.

 

Chapter 15

 

15.1 & 15.3.  Skip part a of these questions.  When they mention the ‘Harsanyi transformation’ in part b, this just means the game tree in which Nature moves first. 

 

15.5.  Part a asks for q(p), which they aren’t really clear about.  They are looking for the probability: P(Car is a Lemon|Price = 5000).  Unlike some of the other problems that we did in class, this isn’t just a number.  It’ll be a function that depends on price.