Simon Fraser University Spring 2015
Econ 302 D200 Final Exam Solution
Instructor: Songzi Du
Tuesday April 21, 2015, 12 – 3 PM
The brief solutions suggested here may not have the complete explanations necessary for full marks.
NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
Bayesian equilibrium
1.
(10 points) Consider the following simultaneous-move game.
W X Y Z
A 0, 2 4, 1 4, 0 4, 0
B 1, 0 2, 1 2, 0 7, 0
C 0, 1 2, 3 3, 1 3, 2
D 0.35, 1 3, 3 4, 2 3, 1
(i): Find the strategies that survive iterative deletion of strictly dominated strategies
(ISD). For each strategy that you delete, write down the strategy that strictly dominates it.
Solution:
Iteration 1: delete Y (strictly dominated by X) and Z (strictly dominated by X).
Iteration 2: delete C (strictly dominated by 1 / 2 A + 1 / 2 B ) and D (strictly dominated by
0 .
6 A + 0 .
4 B ).
The set of strategies that survive: { A, B } of player 1 and { W, X } of player 2.
(ii): Find all NE (pure and mixed) in this game, and for each NE that you find, calculate the two players’ expected payoffs in the equilibrium.
Solution:
Clearly, there is no pure-strategy NE. For the mixed-strategy NE, suppose player 1 uses pA + (1 − p ) B and player 2 uses qW + (1 − q ) X . We must have both A and B being the
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best responses for player 1: 4(1 − q ) = q + 2(1 − q ), i.e., q = 2 / 3; and both W and X being the best responses for player 2: 2 p = 1, i.e., p = 1 / 2. In this equilibrium, player 1 gets
4(1 − q ) = 4 / 3 and player 2 gets 1.
2.
(15 points) Consider a game in which there is a prize worth $30. There are three players: Alice, Bob, and Clair. Each can buy a ticket worth $15 or $30 or not buy a ticket at all. They make these choices simultaneously and independently. Then, knowing the ticketpurchase decisions, the game organizer awards the prize. If no one has bought a ticket, then prize is not awarded. Otherwise, the prize is awarded to the buyer of the highest-cost ticket if there is only one such player or is split equally between two or three if there are ties among the highest-cost ticket buyers. (For example, if Alice buys $15 ticket, Bob buys $30 ticket and Clair does not buy, then Bob gets the prize of $30; if Alice, Bob and Clair all buy $15 ticket, then each gets a prize of $10.) Suppose the final payoff of a player is the prize that he gets minus the money he pays.
(i) Find all pure-strategy NE of this game (remember, a pure-strategy profile is of the form: Alice chooses X, Bob chooses Y, and Clair chooses Z, not necessarily distinct choices).
Solution:
There are six pure-strategy NE:
1. Alice buys $15, Bob does not buy, Clair does not buy.
2. Alice does not buy, Bob buys $15, Clair does not buy.
3. Alice does not buy, Bob does not buy, Clair buys $15.
4. Alice buys $15, Bob buys $15, Clair does not buy.
5. Alice buys $15, Bob does not buy, Clair buys $15.
6. Alice does not buy, Bob buys $15, Clair buys $15.
(ii) Each player has a weakly dominated strategy. What is it? (A strategy is weakly dominated if there is another strategy that is always weakly better (weakly better = either the same or strictly better).) Consider the remaining two pure strategies. Suppose each player uses the same mixed strategy over these two pure strategies. Calculate this symmetric mixed-strategy NE. What is a player’s expected payoff in this mixed strategy equilibrium?
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Solution:
For each player, buying the $30 ticket is a weakly dominated strategy. For a mixedstrategy NE, suppose each player buys $15 ticket with probability p and does not buy with probability 1 − p . Then we must have that each player is indifferent between buying the $15 ticket and not buying, if others are using the mixed strategy:
0 = p
2 · 10 + 2 p (1 − p ) · 15 + (1 − p )
2 · 30 − 15 = p
2 · ( − 5) + (1 − p )
2 · 15
= − 5 p
2
+ 15 − 30 p + 15 p
2
= 10 p
2
− 30 p + 15 , i.e., (by the quadratic equation): p =
3 −
√
3
≈ 0 .
63 .
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In this equilibrium, each player gets an expected payoff of 0.
(1)
3.
(10 points) Consider the price competition between two firms with differentiated products (for example, iPhone and Samsung Galaxy). Firm 1 and 2 simultaneously set p
1 and p
2
, where p i is q
1
= 22 − 2 p
1
+ p
2 is the price of firm i ’s product. The demand for firm 1’s product
(the demand for firm 1’s product increases with p
2
(the competitor’s price) and decreases with p
1
). The demand for firm 2’s product is similarly q
2
= 22 − 2 p
2
+ p
1
.
The payoff of firm i is p i q i
− 10 q i
, i = 1 , 2 (each firm produces its product at a constant marginal cost of 10).
Calculate the Nash equilibrium ( p
1
, p
2
). What are the quantities ( q
1
, q
2
) at this equilibrium?
Solution:
For the best response of firm 1, we have: max p
1
(22 − 2 p
1
+ p
2
)( p
1
− 10) , i.e., the first order condition:
(22 − 2 p
1
+ p
2
) − 2( p
1
− 10) = 0 .
Since firm 1 and firm 2 are symmetric, we solve for a symmetric NE p
1
= p
2
= p , so the
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first order condition becomes:
22 − p − 2 p + 20 = 0 , i.e., p = 14.
In this equilibrium, q
1
= q
2
= 22 − 2 p + p = 22 − 14 = 8.
4.
(15 points) Player 1 and 2 are bargaining about the division of $100 over two rounds.
In round 1, Player 1 proposes p
1
∈ { 9 , 44 , 94 , 100 } in which he gets p
1 and player 2 gets
100 − p
1
; Player 2 responds yes or no to p
1
: if yes this division is implemented and the game ends. If no, the game proceeds to round 2, in which Player 2 proposes p
2
∈ { 0 , 45 , 90 } , where he gets p
2 and player 1 gets 90 − p
2
(by going to round 2 there is a delay cost of 10); Player
1 responds yes or no to p
2
: if yes this division is implemented, and if no each player gets 0.
The game ends after round 2.
(i) Draw the game tree.
Solution:
(ii) Find and report two distinct pure-strategy SPE.
Solution:
SPE #1: Player 1 says Yes to all p
2 in round 2, and proposes p
1
= 9 in round 1; Player
2 proposes p
2
= 90 in round 2, and says Yes to p
1
= 9 and No to p
1
∈ { 44 , 94 , 100 } in round
1.
SPE #2: Player 1 says Yes to p
2
∈ { 0 , 45 } and No to p
2
= 90 in round 2, and proposes
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p
1
= 44 in round 1; Player 2 proposes p
2
= 45 in round 2, and says Yes to p
1
∈ { 9 , 44 } and
No to p
1
∈ { 94 , 100 } in round 1.
5.
(15 points) Consider the following game between a firm (Player 1) and a worker
(Player 2). Nature first chooses the type of the firm, which is high type with probability 0.5
and is low type with probability 0.5. The firm (of either type) chooses to offer a job to the worker or not offer a job. If no job is offered, the game ends and both players receive 0. If the firm offers a job, the worker either accepts or rejects the offer (without observing the type of the firm). The worker’s acceptance of the offer brings a payoff of 2 to the firm; the worker’s rejection brings − 1 to the firm. Rejecting the offer yields 0 to the worker. Accepting the offer of a high-type firm yields 2 to the worker, and accepting the offer of a low-type firm yields − 1 to the worker.
(i) Draw the game tree.
Solution:
(ii) Find all pure-strategy PBE (if any exists), and explain. Include conditional beliefs
(for example,
P
( H | Offer) ≤ 1 / 2) in your description of PBE.
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Solution:
We first find the worker’s best response as a function of
P
( H | Offer): Accept leads to
2
P
( H | Offer) − 1(1 −
P
( H | Offer)) = 3
P
( H | Offer) − 1; Reject leads to 0. So Accept is a best response for the worker if
P
( H | Offer) ≥ 1 / 3, and Reject is a best response for the worker if
P
( H | Offer) ≤ 1 / 3.
Next, we go through the firm’s strategies.
1. Suppose the firm plays (H-Offer, L-Offer). Then
P
( H | Offer) = 1 / 2 > 1 / 3 so the worker best response is to Accept. Therefore, (H-Offer, L-Offer) are best responses of the firm. So this is a PBE: the firm plays (H-Offer, L-Offer), the worker plays Accepts, the worker believes
P
( H | Offer) = 1 / 2.
2. Suppose the firm plays (H-Offer, L-Not). Then
P
( H | Offer) = 1 > 1 / 3 so the worker best response is to Accept. Therefore, L-Not is not a best response of the low-type firm. So this scenario cannot be an equilibrium.
3. Suppose the firm plays (H-Not, L-Offer). Then
P
( H | Offer) = 0 < 1 / 3 so the worker best response is to Reject. Therefore, L-Offer is not a best response of the low-type firm. So this scenario cannot be an equilibrium.
4. Suppose the firm plays (H-Not, L-Not). Then anything goes for
P
( H | Offer). Suppose
P
( H | Offer) ≤ 1 / 3 so the worker best response is to Reject. Therefore, (H-Not, L-Not) are best responses of the firm. So this is a PBE: the firm plays (H-Not, L-Not), the worker plays Reject, the worker believes
P
( H | Offer) ≤ 1 / 3.
6.
(10 points)
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In the above game, player 1 is either a cooperative ( C ) type or an ordinary ( O ) type.
Player 1 and Player 2 take turns to decide whether to invest on a motorcycle. After both have invested, player 1 decides whether to share the motorcycle with player 2 (self = not share). Find all pure-strategy PBE (if any exists), and explain. Include conditional beliefs
(for example,
P
( C | Invest) ≤ 1 / 2) in your description of PBE.
Solution:
Clearly, in any PBE, player 1 must play C-Share and O-Self (i.e., the C type shares, and the O type does not share).
We next find the best response of player 2 as a function of
P
( C | Invest): invest leads to
2
P
( C | Invest) − 2(1 −
P
( C | Invest)) = 4
P
( C | Invest) − 2; not invest leads to 0. So invest is a best response for player 2 if
P
( C | Invest) ≥ 1 / 2, and not invest is a best response for player 2 if
P
( C | Invest) ≤ 1 / 2.
Finally, we go through player 1’s strategies. Clearly C-Invest is better than C-Not for the C type of player 1.
1. Suppose player 1 plays (C-Invest, O-Invest). Then
P
( C | Invest) = 1 / 4 < 1 / 2 so player
2’s best response is to not invest. Therefore, O-Invest is not a best response for the O type of player 1. So this scenario cannot be an equilibrium.
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2. Suppose player 1 plays (C-Invest, O-Not). Then
P
( C | Invest) = 1 > 1 / 2 so player 2’s best response is to invest. Therefore, O-Not is not a best response for the O type of player 1. So this scenario cannot be an equilibrium.
3. (C-Not, O-Invest). Clearly not an equilibrium, since the C type prefers to invest.
4. (C-Not, O-Not). Clearly not an equilibrium, since the C type prefers to invest.
Therefore, there is no pure-strategy PBE. For get a PBE we have to go to the mixed strategy.
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