Game theory exam 2

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Exam 2

Econ 431

  1. This exam is due by 7:59 AM on 4/30/21. Turn in your exam by submitting it through the “Assignments” tab in d2l.
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Explain your answers and show work whenever possible. There are a total of 105 points possible out of 100 total points. Good luck!

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  1. (20) Consider the following game:

2

2

2,2

-1,0

0,0

0,2

UL

UL

DL

DL

[r] [1-r]

N

3 4

1 4

RG

RB

1

1

LG

LB

2

[q]

2

[1-q]

4,1

2,0

3,1

0,3

UR

UR

DR

DR

G

B

(a) Find all separating perfect Bayesian equilibria. (10)

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(b) Find all pooling perfect Bayesian equilibria. (10)

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  1. (25) Consider the following game:

1,1

1,1

N

p

1-p

HA

HB

1

1

HB

GB

2

[q]

2

[q′]

4,4

1,1

1,1

4,4

C

C

R

R

A

B

(a) Let p= 2 3 . Find all Bayesian Nash Equilibria. (Note: I have included a table below

for you to show your work. You should still write out your answer.) (10)

Player 2

C R

Player 1

HAHB HAGB GAGB GAHB

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(b) Let p= 3 4 . Will this result in different equilibria than you found in (a)? (5)

(c) Let p= 1 3 . Will this result in different equilibria than you found in (a)? (5)

(d) Suppose that player 1 has the option to send a message (A or B) about their type to player 2 after observing their type but before choosing their strategy. Do you think player 1 has an incentive to send a truthful message to player 2? Why or why not? Discuss your logic in a few sentences. (5)

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  1. (20) Two firms compete in a market, simultaneously choosing quantities q1 ≥ 0 and q2 ≥ 0. The market price is given by p = 55 − q1 − 6q2. Firm 1 faces a marginal cost of 3 and firm 2 faces a marginal cost of 2.

(a) Find all pure strategy Nash equilibria. (12)

(b) Calculate each firm’s equilibrium profit. (4)

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(c) Is there a first-mover advantage in this game? Explain. (4)

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  1. (15) Suppose 7 players participate in a second-price sealed-bid auction. The values for each player are listed below, with vi corresponding to player i. Here, consider the equilibrium in which no player used a weakly dominated strategy. Remember to explain your answers.

v1 v2 v3 v4 v5 v6 v7 13 4 4 9 14 3 8

(a) Which of these players wins the auction? (3)

(b) What price does the winner pay for the item? (3)

(c) What bid will the bidder with value v5 submit? (3)

(d) What is the highest bidder’s utility? (3)

(e) What is the second highest bidder’s utility? (3)

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  1. (20) Consider a twist on the ultimatum bargaining game we discussed in class. First, player A chooses a value xA ∈ [0, 1]. Upon seeing xA, player B may then Accept or Reject A’s offer. If B Accepts, player A receives 1 −xA and B receives xA. If player B rejects, both players evenly split the pie and receive a payoff of 0.5.

(a) Find all subgame perfect equilibria of this game. (10)

(b) Suppose player B’s strategy is to Accept when xA ≥ 0.75 and Reject when xA < 0.75. What is the set of all best responses for player A? Can this be supported in a Nash equilibrium? (5)

(c) Consider player B’s strategy in (b). Suppose player B were to reach the subgame in which A has played xA = 0.6. Can B commit to his strategy? Discuss this result in relation to your previous answers. (5)

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  1. (5 points extra credit) On the morning of April 29th, a farm nursery has 3 brown calves (baby cows) and some number of black calves (baby cows). Brown calves and black calves are born with equal probability. That night, a mother cow gives birth to a new calf, and the calf is placed in the nursery.

On April 30th, a statistician conducts a survey and selects a calf at random from the nursery (including the newborn calf and every calf from April 29th). The calf is brown. What is the probability the calf born on April 29th was a brown calf? (Hint: use Bayes’ rule).

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Last Updated on April 29, 2021 by EssayPro