Student ID: ______________
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Student Name: ___________________
ECON N110: Game Theory in the Social Sciences
Practice Final Exam A
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Question Booklet
Instructor: Zheng Huang
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Summer 2023
Time Allowed: 80 Minutes
Total Number of Questions: 4
Instructions:
Total Points: 80
• Do not turn the page containing the instructions until the proctor instructs you to do so.
• This is a closed book exam. You may not use any notes during the exam.
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• You are allowed to use a non-programmable, non-graphing calculator during the exam.
• Please write your answers in the space provided in the answer booklet.
• To receive full credit, you need to clearly show all essential derivations to demonstrate your
knowledge of the appropriate techniques.
• Submit both the question booklet and the answer booklet. (If your question booklet is not
submitted, your answers will not be graded.)
• You are expected to maintain the highest standard of academic integrity at the University of
California, Berkeley. Please sign the UC Berkeley Honor Code on the next page.
Good luck!
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UC Berkeley Honor Code
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As a member of the UC Berkeley community, I act with honesty, integrity,
and respect for others.
• I alone am taking this exam.
• I will not receive assistance from anyone while taking the exam, nor will I provide
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assistance to anyone while the exam is still in progress.
• Other than with the instructor, I will not have any verbal, written, or electronic
communication with anyone else while I am taking the exam or while others are
taking the exam.
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share with others.
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• By signing below, I agree to abide by the UC Berkeley Honor Code as stated above.
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I also agree to come forward if I suspect or witness violations of the Honor Code.
Signature: ______________________________
Printed Name: ___________________________
Student ID: ______________________________
Date: ____________________________________
Student Name: __________________
ECON N110, Summer 2023
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1. (20 points) Consider the following game.
(a) How many proper subgames does this game have?
(b) Find the Nash equilibrium(a) of this game.
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(c) Find the subgame perfect Nash equilibrium(a) of this game.
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2. (20 points) Consider the following stage game to be played twice. Assume that players do not discount
future payoffs.
Is there a subgame perfect Nash equilibrium in which (A, X) is played in the first period? If so, describe
this strategy. If not, explain why not.
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Student Name: __________________
ECON N110, Summer 2023
3. (20 points) Consider the following static game of incomplete information. Nature selects the type (𝑚) of
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player 1, where 𝑚 = 1 with probability 3 and 𝑚 = −1 with probability 3. Player 1 observes 𝑚 (player
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(a) Draw the normal-form matrix of this game.
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1 knows their own type), but player 2 does not observe 𝑚. Then the players make simultaneous and
independent choices and receive payoffs as described by the following matrix.
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(b) Compute the Bayesian Nash equilibrium.
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4. (20 points) Consider an “all-pay auction” with two players (the bidders). Player 1’s valuation 𝑣1 for the
object being auctioned is uniformly distributed between 0 and 1. That is, for any 𝑥 ∈ [0, 1], player 1’s
valuation is below 𝑥 with probability 𝑥. Player 2’s valuation is also uniformly distributed between 0 and
1, so the game is symmetric. After nature chooses the players’ valuations, each player observes his/her
own valuation but not that of the other player. Simultaneously and independently, the players submit
bids. The player who bids higher wins the object, but both players must pay their bids. That is, if player
𝑖 bids 𝑏𝑖 , then his/her payoff is −𝑏𝑖 , if he/she does not win the auction; his/her payoff is 𝑣𝑖 − 𝑏𝑖 if he/she
wins the auction. Calculate the Bayesian Nash equilibrium strategies (bidding functions). (Hint: The
equilibrium bidding function for player 𝑖 is of the form 𝑏𝑖 (𝑣𝑖 ) = 𝑘𝑣𝑖2 for some number 𝑘.)
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