ECON437_Midterm2_SP23-1

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Name:_______________________________________ NetID:____________________ ECON 437: Game Theory Practice Midterm Exam II Instructions This exam contains 4 sections. Each section is worth 25 points. For this exam, you may only use pen, pencil, and eraser. Everything else (including calculators, computers, and cell phones) are not allowed. Answer as much as possible in order and show all your work. Make sure your handwriting is clean and easy to read. Section I: Sequential Games Consider the following game with two players: Stage 1: The game begins with a move from player 1. Player 1 must choose either to play P or NP. If player 1 chooses NP, then the game ends and both players walk out with a pay-off of $0. If player 1 chooses P, then the game moves into stage 2. Stage 2: In this stage, player 2 chooses either “sequential” or “simultaneous”. If player 2 chooses “simultaneous”, then they play the following normal form game (in the next stage): If player 2 chooses “sequential”, then (in the next stage) player 1 must choose either “up” or “down.” If player 1 chooses “up”, both players walk away of the game with $3 each and the game ends. If player 1 chooses “down”, then player 1 walks away with $5, player 2 walks away with $1, and the game ends. 1. Construct the tree that characterizes the extended form representation of this game. (5 points) 2. Construct the matrix that characterizes the normal form representation of this game. (5 points) 3. How many sub-games are there in this environment (without counting the whole game)? (5 points) 4. What is the set of Nash Equilibria in this game? (5 points) 5. What is the set of Subgame Perfect Nash Equilibrium in this game? (5 points) Section II: Stackelberg Consider a Stackelberg environment in which the market demand is ࠵? = 100 − ࠵? . Both firms have the same cost structure: ࠵?࠵? ) = ࠵? ) + . In this environment, firm 1 moves first. 1. What is the set of strategies for firm 1? (5 points) 2. Find the Subgame Perfect Nash Equilibrium. (5 points) 3. Which is the outcome in a Subgame Perfect Nash Equilibrium? (5 points) 4. Firm 2 threatens to supply ࠵? + = 10 , no matter what. Show that this is not a credible threat. (5 points) 5. What is the best response of firm 1 to the strategy ࠵? + = 10 , no matter what? (5 points) Section III: Bank Runs Consider two investors. Each deposits $100 million dollars at the same small bank. The bank uses the deposits to invest in a long-term project. If any of the money is pulled-out from the bank before the investment matures, the bank can only recover a total of $120 millions. If the investment reaches maturity, the project pays out a total of $240 millions. There are two dates at which investors can withdraw their money: date 1 and date 2. Date 1 is before the investment matures, while date 2 is after. If an investor withdraws their money on date 1, that investor will recover their whole money while the other will receive the rest from the money recovered by the bank. If both investors choose to withdraw at date 1, then both receive $60 millions each. If neither investor withdraws, then the game moves into date 2. L R U ( 3 , 3 ) ( 1 , 5 ) D ( 5 , 1 ) ( 4 , 4 )

Name:_______________________________________ NetID:____________________ At date 2, if both investors withdraw, then they both receive $120 millions each. If only one investor withdraws, then that investor walks out with $140 millions while the other with $100 millions. If they both do not withdraw, the bank splits the $240 millions equally among both investors. 1. Construct the tree that characterizes the extended form representation of this game. (5 points) 2. Construct the matrix that characterizes the normal form representation of this game. (5 points) 3. How many sub-games are there in this environment (without counting the whole game)? (5 points) 4. What is the set of Nash Equilibria in this game? (5 points) 5. What is the set of Subgame Perfect Nash Equilibria in this game? (5 points) Section IV: Consider the following game: 1. How many subgames are there? (5 points) 2. Construct the matrix that characterizes the Normal Form Representation of this game. (5 points) 3. Find the set of Nash Equilibria. (5 points) 4. Find the Subgame Perfect Nash Equilibria. (5 points) 5. Explain whether this game characterizes an environment with perfect information or not. (5 points)

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