game theory economics

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Midterm 1: Game Theory and Applications

Prof. Boleslavsky

October 15-31, 2017

Rules. This is a take-home exam. You must submit it to me by the beginning of class on October 31. Late exams will not be accepted.

The contents of this exam, both the questions and your answers, must be kept confiden- tial until the exam is over. Therefore, you may not discuss the contents of this exam with any other individual, including other ECO444 students (present or past), graduate students, private tutors, internet chat boards, family members, or any other individuals. Consistent with business school policies, if the professor uncovers evidence that this rule has been bro- ken, he is required to begin an honor council inquiry.

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Prof. Boleslavsky Eco444, Midterm 1 page 2

1. Three “green energy” startups are independently and simultaneously developing new products to bring to market. The product can be based on a proven technology (P), (for example, wind or solar) or a speculative technology (S) (for example, biofuels). Both the speculative and proven technologies, if developed, will be valued by the market. However, in order to make profit, a firm must be the only one who develops a particular technology: if another firm develops the same technology, then they compete away all profit, which yields payoff normalized to 0. If a firm is the only developer of the proven technology, then its payoff is 1. If a firm is the only developer of the speculative technology, then its payoff is v > 1.

Other Two

Firm (P, P) (P, S) (S, P) (S, S)

P 0 0 0 1 S v 0 0 0

(a) Is it a Nash equilibrium for all firms to develop P? S? Explain in a sentence or two.

(b) Is it a Nash equilibrium for two firms to develop P and the other S? Explain in a sentence or two.

Prof. Boleslavsky Eco444, Midterm 1 page 3

Now consider a mixed strategy equilibrium in which each firm develops the proven product with probability p.

Other Two

Firm (P, P) (P, S) (S, P) (S, S)

P 0 0 0 1 S v 0 0 0

(d) If each other firm develops the proven product with probability p, what is the expected payoff of developing the proven product?

(e) If each other firm develops the proven product with probability p, what is the expected payoff of developing the speculative product?

(f) What is the mixed strategy Nash equilibrium value of p?

(g) (5 Bonus). Suppose that there are N firms in the industry. What is the mixed strategy Nash equilibrium value of p? What happens to p as N grows?

Prof. Boleslavsky Eco444, Midterm 1 page 4

2. King Lear is deciding how to divide his kingdom, worth 1, among his daughters, Regan and Goneril (his youngest daughter, Cordelia, has been disinherited). Each daughter is expected to make a speech in front of King Lear, where she describes her love for him, and describes how she will rule if she becomes queen. Each daughter can either be honest (H) when she makes her speech, or she can lie (L), exaggerating her love for the king and her benevolence toward her future subjects. After listening to both speeches, King Lear chooses one of the daughters to be queen. If both daughters are honest, or both lie, then Lear is equally likely to choose each one (prob. 1/2 for each one). If one daughter is honest and the other lies, then Lear is more likely to choose the daughter who lied, prob 7

10 for the liar and 3

10 for the honest daughter. However, by

lying, a daughter sets expectations for her future subjects which impose a cost of 1/3 if she becomes queen. The simultaneous game between the daughters can be represented in the following table.

Goneril

Regan H L

H 1 2 , 1 2

3 10 , 7 15

L 7 15 , 3 10

1 3 , 1 3

(a) Underline the best responses in the box.

(b) Is it an equilibrium for both sisters to be honest, (H, H)? Explain in a sentence or two.

Prof. Boleslavsky Eco444, Midterm 1 page 5

Goneril

Regan H L

H 1 2 , 1 2

3 10 , 7 15

L 7 15 , 3 10

1 3 , 1 3

Consider a mixed strategy Nash equilibrium in which each sister is honest with prob- ability h ∈ (0, 1).

(d) If Regan expects Goneril to be honest with probability h, what is Regan’s expected payoff if she is honest?

(e) If Regan expects Goneril to be honest with probability h, what is Regan’s expected payoff if she lies?

(f) In a mixed strategy Nash equilibrium, what must be the value of h? Call this value h∗. What is each sister’s payoff?

Prof. Boleslavsky Eco444, Midterm 1 page 6

Suppose that in addition to being honest and lying, each sister can also accuse her sister of lying (A). If a sister accuses the other one of lying and the accuser is wrong, then the kingdom goes to the honest sister. If a sister accuses the other one of lying and the accuser is right, then the kingdom goes to the accuser with probability a ∈ [0, 1]. If both accuse, then neither sister gets the kingdom (King Lear reconciles with the youngest daughter).

Goneril

Regan

H L A H 1

2 , 1 2

3 10 , 7 15

1, 0

L 7 15 , 3 10

1 3 , 1 3

1−a, a

A 0, 1 a, 1−a 0, 0

(g) Consider the mixed strategy Nash equilibrium you found in part (f), i.e. play H with probability h∗, play L with probability 1−h∗, play A with probability 0. For which values of a is it also a mixed strategy Nash equilibrium of the game with accusations (summarized in the table above)?

Prof. Boleslavsky Eco444, Midterm 1 page 7

3. At the University of Umami, Jojo makes money by taking take-home tests for other students. This is, of course, completely against Umami’s code of ethics, but it is worth the risk for the right price. Two students would like to hire Jojo to solve their take- home test in Video Game Theory. The test is scored on a continuous scale from 0 to 1, and it is known that without Jojo’s assistance, each student’s score will be 0. Each student simultaneously makes Jojo a price offer, pi ≥ 1/2, where pi represents the price per point that student i receives on the test. That is, if student i’s score is qi and he offered price pi, then he must pay Jojo piqi. An offer pi < 1/2 isn’t worth the risk for Jojo and would be immediately rejected. Each student seeks to maximize his score on the test, net of Jojo’s fee. Thus, given a final score qi and a price offer pi, each student’s payoff is

ui(qi, pi) = qi − qipi = qi(1−pi).

Each student’s exam score depends on the effort that Jojo puts into the test. In particular, suppose that when Jojo gets price offers (p1, p2), Jojo generates test scores

q1(p1, p2) = p1 − 1

2 p2 q2(p1, p2) = p2 −

2 p1.

Note that offering a price in excess of 1 is strictly dominated for each student. Note also that the smallest price that Jojo would accept is 1/2, and therefore q1 and q2 are always between zero and 1. Thus, by paying more, a student increases his own score on the test, and reduces the other student’s score. It follows that given price offers (p1, p2), the student’s payoffs are

u1(p1, p2) = (p1 − 1

2 p2)(1−p1) = (1 +

2 p2)p1 −p21 −

2 p2

u2(p1, p2) = (p2 − 1

2 p1)(1−p2) = (1 +

2 p1)p2 −p22 −

2 p1

(a) Suppose that it is known that only one student is considering making Jojo an offer. In this case, the student’s payoff is

u1(p1, 0) = p1 −p21.

Find student 1’s optimal offer.

Prof. Boleslavsky Eco444, Midterm 1 page 8

Now suppose that there are two students who make simultaneous offers.

u1(p1, p2) = (1 + 1

2 p2)p1 −p21 −

2 p2

u2(p1, p2) = (1 + 1

2 p1)p2 −p22 −

2 p1

(b) Suppose that student 2 expects student 1 to offer p1 ∈ [1/2, 1]. What is student 2’s best response?

Prof. Boleslavsky Eco444, Midterm 1 page 9

Now suppose that student 1 moves first, and student 2 sees p1 before choosing p2. Consider a Subgame Perfect Nash Equilibrium.

u1(p1, p2) = (1 + 1

2 p2)p1 −p21 −

2 p2

u2(p1, p2) = (1 + 1

2 p1)p2 −p22 −

2 p1

(d) Suppose that student 2 sees that student 1 offered p1 ∈ [1/2, 1]. What is student 2’s best response? (Hint: you found this already).

(e) Explain why student 1’s payoff function can be written

u1(p1) = 9

8 p1 −

8 p21 −

4 .

(e) Find the equilibrium values of p1 and p2. Are they higher or lower than when the students made offers simultaneously? Are they higher or lower than when a single student made the offer?

Prof. Boleslavsky Eco444, Midterm 1 page 10

(f) (Bonus): In the dynamic game, which player or players have a higher payoff than in the simultaneous game? Explain your answer in a sentence or two.

Prof. Boleslavsky Eco444, Midterm 1 page 11

4. Consider the following dynamic game between Mom and Teen. Teen moves first, making a choice between working hard in school (W), or goofing off (G). If Teen works hard, then Teen gets a payoff of 1/9, and Mom gets payoff 2. If Teen goofs off, then Mom must choose whether to punish Teen (P), or not punish (NP). Mom is worried that if she doesn’t punish Teen, then Teen’s behavior will get worse in the future. Thus, if she doesn’t punish, Mom’s payoff is 0. Meanwhile, if Teen goofs off and goes unpunished, Teen’s payoff is 1. If Teen is punished, then Teen’s future behavior may be better, which gives Mom a (gross) payoff of 1. However, punishing Teen is not easy, and imposes a cost of 1/2 on Mom. Thus, if Mom chooses to punish Teen, then Mom’s payoff is 1/2 and Teen’s payoff is 0.

(a) Find the subgame perfect Nash equilibrium of the game by drawing arrows in the game tree.

Prof. Boleslavsky Eco444, Midterm 1 page 12

Now suppose that if Teen goofs off, then Teen can be punished by either Mom or Dad. In particular, Mom and Dad simultaneously between punishing (P) and not punishing (NP). As above, if a parent chooses to punish, then he or she gets payoff 1/2 and Teen gets payoff 0. As above, if both parents choose not to punish (NP), then both parents gets payoff 0 and Teen gets payoff 1. If one parent chooses to punish and the other doesn’t, then the parent who chose to punish gets payoff 1/2 and the one who chose not to punish gets payoff 1. In summary, if Teen goofs off, then the parents play the following simultaneous move game:

Dad

Mom P NP

P 1 2 , 1 2

1 2 , 1

NP 1, 1 2

0, 0

Teen’s payoff is 1 if the outcome is (NP, NP) and 0 otherwise. Consider a mixed strategy Nash equilibrium in which each parent chooses NP with probability n.

(b) If Dad chooses NP with probability n, what is Mom’s expected payoff of P?

(d) What is the mixed strategy Nash equilibrium value of n?

(e) What are Mom and Dad’s payoffs, uM and uD?

Prof. Boleslavsky Eco444, Midterm 1 page 13

(f) What is Teen’s expected payoff if Mom and Dad play this game? (Hint: Teen gets 1 whenever both parents select NP and 0 otherwise. Teen’s expected payoff is just n2).

(g) If Teen anticipates that this mixed strategy Nash equilibrium will be played if Teen goofs off, then Teen’s initial choice is represented in the following tree.

What will Teen choose to do?

(h) (Bonus) What would Teen do if Teen anticipated pure strategy equilibrium (P, NP) would be played? Explain the difference in a sentence or two.