Simon Fraser University Fall 2014
Econ 302 D100 Midterm Exam Solution
Instructor: Songzi Du
Tuesday October 28, 2014, 8:30 – 10:20 AM
The brief solutions suggested here may not have the complete explanations necessary for full marks. They may also contain comments that are not asked by the questions and not required for full marks.
1.
(5 points) Alice and Bob are participants in a televised game show, seated in separate booths with no possibility of communicating with each other. Each one of them is asked to submit, in a sealed envelope, one of the following two requests for the show (requests are guaranteed to be honored):
1. Give me $1000.
2. Give the other participant $4000.
Write the payoff matrix for this game. Find the Nash equilibria. Is there any dominated strategy, and why? Why is the Nash equilibrium outcome(s) unfortunate in this game?
Solution:
The payoff matrix is: give me $1000 give other $4000 give me $1000 give other $4000
1000, 1000 5000, 0
0, 5000 4000, 4000
The request of “give other $4000” is clearly strictly dominated by the request of “give me $1000,” because the former is strictly better than the latter given whatever request of the other player. Therefore, the only Nash equilibrium is both players requesting “give me
$1000,” which is unfortunate because if both players would request “give other $4000,” both would be better off than the Nash equilibrium outcome. This game is a prisoner’s dilemma.
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2.
(10 points) Find and describe the pure-strategy subgame perfect equilibria (if any) in the following game. Explain the steps that you use to find the SPE.
The game has random moves by Nature at the beginning of the tree. You can think of the two nodes of Player 1 following the moves of Nature as two distinct players, like the strong and weak types in the Beer-Quiche game.
Solution:
In any SPE, player 1 has to play d (given the right node of player 1), because whatever player 2 does, d is better than c for player 1 given his right node.
Suppose player 2 plays A , then the best response of player 1 given the left node is b . And if player 1 plays ( b, d ), the best response of player 2 is A , because A gives 2 / 5 × 20 + 3 / 5 × 60 which is more than the 40 given by B . Thus, one SPE is player 1 taking ( b, d ) and player 2 taking A .
Suppose player 2 plays B , then the best response of player 1 given the left node is a . And if player 1 plays ( a, d ), the best response of player 2 is A , because A gives 2 / 5 × 30 + 3 / 5 × 60 which is more than 2 / 5 × 30 + 3 / 5 × 40 given by B . Thus, there is no SPE in which player
2 plays B .
In summary, the only pure-strategy SPE is player 1 taking ( b, d ) and player 2 taking A .
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3.
(10 points) The market (inverse) demand function for a homogeneous good is
P ( Q ) = 14 − 2 Q , where Q is the total quantity of the good on the market. There are two firms: firm 1 has a constant marginal cost of 4 for producing each unit of the good, and firm
2 has a constant marginal cost of 1. Firm 1 is the industry leader, so it sets its quantity of production first. Firm 2 sets its quantity of production after observing firm 1’s quantity.
Find (i) the subgame perfect equilibrium of this game. And calculate (ii) the market price and (iii) the firms’ profits in the equilibrium.
Solution:
In the subgame in which firm 1 sets q
1
, the best response of firm 2 can be found by solving: max (14 − 2( q q
2
1
+ q
2
)) q
2
− q
2
.
The first order condition is:
14 − 2 q
1
− 4 q
2
− 1 = 0 ⇐⇒ q
2
=
13 − 2 q
1
.
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Thus in the SPE, firm 2’s strategy is q
2
( q
1
) =
13 − 2 q
1
4
. Given this strategy, firm 1’s best response in the beginning of the game is: max q
1
14 − 2 q
1
+
13 − 2 q
1
4 q
1
− 4 q
1
.
The first order condition is:
14 − 4 q
1
− 13 / 2 + 2 q
1
− 4 = 0 ⇐⇒ q
1
= 7 / 4 .
Thus, the SPE is firm 1 producing 7 / 4, and firm 2 producing
13 − 2 q
1
4 if firm 1 produces q
1
In this equilibrium, the price is 14 − 2 × (7 / 4 + 19 / 8) = 23 / 4, firm 1 gets (23 / 4 − 4) × 7 / 4 ≈
.
3 .
06, and firm 2 gets (23 / 4 − 1) × 19 / 8 ≈ 11 .
28.
4.
(15 points) Elizabeth and Mary appear before King Solomon at his palace, along with an infant. Each woman claims that the infant is her child. The child is “worth” 100 dinars to his true mother, but he is only “worth” 50 dinars to the woman who is not his mother (the imposter). The king knows that one of the two women is the true mother of the child, and he knows the “values” that the true mother and the imposter ascribe to the
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child, but he does not know which woman is the true mother, and which the imposter.
To determine which of the two women is the true mother, the king explains to Elizabeth and Mary that he will implement the following steps:
1. He will ask Elizabeth whether the child is hers. If she answers negatively, the child will be given to Mary. If she answers affirmatively, the king will continue to the next step.
2. He will ask Mary if the child is hers. If she answers negatively, the child will be given to
Elizabeth. If she answers affirmatively, Mary will pay the king 75 dinars, and receive the child, and Elizabeth will pay the king 10 dinars.
(i) Draw the game tree when (a) Elizabeth is the true mother and when (b) Mary is the true mother. (Assume that each woman has a payoff of 0 dinar if she does not receive the child.) (ii) Find and report the subgame perfect equilibrium under (a) and (b), and describe what happens (who gets the child and how much is paid by each woman) in each equilibrium.
Solution:
Let us suppose that Elizabeth is player 1 and Mary is player 2.
(a) When Elizabeth is the true mother:
Clearly, the SPE is Elizabeth saying Y and Mary saying N. The outcome is that Elizabeth gets the child, and each woman pays nothing.
(b) When Mary is the true mother:
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Clearly, the SPE is Elizabeth saying N and Mary saying Y. The outcome is that Mary gets the child, and each woman pays nothing.
In summary, King Solomon has devised a (very clever) mechanism that always assigns the child to his true mother, even though the king does not know who is the true mother.
5.
Part (i) [5 points]: find the Nash equilibria (pure and mixed) of the following simultaneous game, and report the players’ payoffs in each equilibrium:
C D
A 10, 10 1, 10
B 12, 2 0, 0
Part (ii) [10 points]: now suppose the above game is played twice. In each stage the two players move simultaneously, and in the second stage the players observe the actions taken in the previous stage. Assume that the final payoff of a player is the sum of his payoffs from the two stages.
Find and describe a subgame perfect equilibrium (SPE) in which (A, C) is played in the first stage, and report the players’ payoffs in the equilibrium. Explain why it is a SPE.
Solution:
Part (i): the pure-strategy NE are clearly (B, C) and (A, D), with (12, 2) and (1, 10) as payoffs, respectively. Note that D is a weakly dominated strategy of player 2.
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One mixed strategy NE found by solving the indifference conditions is ( A, 1 / 3 C +2 / 3 D ).
(Add calculations/explanations that show this).
In fact, any ( A, p C + (1 − p ) D ) is a NE if 0 ≤ p ≤ 1 / 3, with payoff 9 p + 1 for player 1 and 10 for player 2.
(2 point for this)
Part (ii):
One SPE is:
Player 1: plays A in stage 1; plays B in stage 2 if (A, C) or (A, D) was played previously, and plays A in stage 2 if (B, C) or (B, D) was played previously.
Player 2: plays C in stage 1; plays C in stage 2 if (A, C) or (A, D) was played previously, and plays D in stage 2 if (B, C) or (B, D) was played previously.
(Add some explanations!)
In this SPE, player 1 gets 22 and player 2 gets 12.
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