Assignment:
1. Suppose that two players are playing the following game. Player A can either choose Top or Bottom, and Player B can either choose Left or Right. The payoffs are given in the following table:
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Player B
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Player A
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Left
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Right
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Top
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2 5
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1 4
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Bottom
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0 1
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3 8
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where the number on the left is the payoff to Player A, and the number on the right is the payoff to Player B. Assume the players choose their strategies simultaneously.
A) For each of the following strategy combinations, write "TRUE" if the strategy combination is a Nash Equilibrium, and "FALSE" if it is not a Nash Equilibrium:
B) Now suppose this game is played where Player 1 first chooses either Top or Bottom, and then Player 2 chooses either Left or Right. This type of sequential moves game is called a game in extensive form.
To answer this question, it might be helpful to sketch the game tree, but it is not required.
Using the backward induction method discussed in class, which of the following strategy combinations will be the outcome of the game (circle the correct answer):
C) The sequential game in part B) can be represented by a game in Normal Form, where each player chooses its strategy without knowing what the other player has chosen. To do this, Player 2 must declare at the beginning of the game what strategy Player 2 will choose for each possible strategy chosen by Player 1. Therefore, Player 2's strategy set is now {(L,L), (L,R), (R,L), (R.R)}, where the first element of each pair is Player 2's response to Player 1 choosing Top, and the second element of each pair is Player 2's response to Player 1 choosing Bottom.
Write the payoff matrix for the game in Normal Form which is equivalent to the sequential game in part B).
D) Identify all of the Nash Equilibria of the game given in C) by marking a single star in each box that is a Nash Equilibrium.
E) Identify all of the Subgame Perfect Nash Equilibria of the game given in C) by marking a second star in each box that is a Subgame Perfect Nash Equilibrium.
2. Consider the following payoff matrix:
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High Price
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Low Price
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High Price
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40, 40
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10, 160
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Low Price
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160, 10
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20, 20
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Two firms play this pricing game repeatedly an infinite number of times. Suppose each firm adopts a grim trigger strategy that says they will choose a high price in the first period of the game, and they will continue to choose a high price as long as the other player has chosen a high price in the previous period. If one firm chooses a low price in any period, then the other firm will choose a low price in all subsequent periods until the end of time.
Characterize the values of the discount rate, r, which will lead to the High Price/High Price outcome being played in each period as a non-cooperative Nash Equilibrium.
3. Suppose there are two firms in a market that each simultaneously choose a quantity. Firm 1's quantity is q1, and firm 2's quantity is q2. Therefore the market quantity is Q = q1 + q2. The market demand curve is given by P = 160 - 3Q. Also, each firm has constant marginal cost equal to 16. There are no fixed costs.
The marginal revenue of the two firms are given by:
• MR1 = 160 - 6q1 - 3q2
• MR2 = 160 - 3q1 - 6q2.
A) Write the equations of the Best Response Function for each firm.
B) Graph the Best Response Functions of each firm. Put them both on a single graph and identify the Cournot-Nash Equilibrium. Be sure to label your graph carefully and accurately.
C) How much output will each firm produce in the Cournot-Nash equilibrium?
D) What will be the market price of the good?
E) How much profit does each firm make?
F) Now suppose that the two firms form a cartel and decide to maximize joint profits and split the profits evenly. They agree to each produce half of the profit maximizing quantity. How much output will each firm produce? (Hint: If the two firms form a cartel, they will produce together the same amount of output as a monopolist would produce.)
G) Now suppose that Firm 1 decides to cheat on the agreement. Assuming Firm 2 produces the quantity given in F), write the equation for the residual demand that faces Firm 1.
H) If Firm 1 expects Firm 2 to produce the amount of output in F), how much output should Firm 1 produce to maximize their profit?