Simon Fraser University
Spring 2016
Econ 302 D200 Final Exam Solution
Instructor: Songzi Du
Thursday April 14, 2016, 8:30 – 11:30 AM
NE = Nash equilibrium, SPE = subgame perfect equilibrium, PBE = perfect
Bayesian equilibrium
1.
(10 points) Two candidates, D and R, are running for mayor in a town with n
residents. A total of d residents support candidate D (0 < d < n), while the remainder,
r = n−d, support candidate R. The value for each resident for having his candidate win is 4,
for having him tie is 2, and for having him lose is 0. Going to vote costs each resident 1; not
voting costs 0. The payoff of each resident is his value minus his cost. The (simultaneous)
actions of each resident are: vote for the his preferred candidate, or do not vote.
(i) Let n = 2 and d = r = 1. Write down this game as a matrix and find all pure-strategy
NE.
(ii) Let n = 10 and d = r = 5. Find all pure-strategy NE. Hint: for a NE you just need
to specify how many of the d voters vote for D, how many of them do not vote, and
likewise for the r voters.
(iii) What are the NE in (i) and (ii) when the cost of voting is equal to 3?
Solution:
Part i:
Vote
Not
Vote
(1,1)
(0, 3)
Not
(3, 0)
(2, 2)
The only pure-strategy NE is (Vote, Vote).
Part ii: Suppose x of the d = 5 voters votes for D, 5 − x of them do not vote; y of the
r = 5 voters votes for R, 5 − y of them do not vote. The only NE is x = y = 5.
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Part iii: When r = d = 1, the only NE is x = y = 0. When r = d = 5, the only NE is
still x = y = 0.
2. (10 points) It is Halloween evening and Nick and Rachel have just returned home
after trick-or-treating. They need to divide four candy bars: Snickers, Milky Way, Kit Kat,
and Baby Ruth. The allocation procedure is that Nick gets to choose one of the four candy
bars first, then Rachel chooses one of the three remaining candy bars, then Nick chooses
one of the two remaining candy bar, and finally Rachel gets the last remaining candy bar.
Nick’s preferences are that he most likes Snickers (assigning it a payoff of 5), then Milky
Way (payoff of 4), then Kit Kat (payoff of 3), and lastly Baby Ruth (payoff of 1). Rachel
most likes Milky Way (payoff of 6), then Kit Kat (payoff of 5), then Baby Ruth (payoff of
4), and finally Snickers (payoff of 2).
(i) Draw the game tree.
(ii) Find and describe a SPE. How are the candy bars allocated in the SPE?
Solution:
Part i: omitted, since it’s straightforward.
Part ii: here is a SPE derived from backward induction:
• Round 3: Nick chooses Snickers (if available), then Milky Way (if available), then Kit
Kat (if available), and lastly Baby Ruth.
• Round 2: If Nick chose Snickers in round 1, then Rachel chooses Milky Way. If Nick
did not choose Snickers in round 1, then Rachel chooses Milky Way if available, and
Kit Kat otherwise.
• Round 1: Nick chooses Milky Way.
In this SPE Nick gets Milky Way and Snickers, and Rachel gets Kit Kat and Baby Ruth.
3. (10 points) Two players, 1 and 2 each own a house. Each player i values his own
house at vi : v1 is either 10 or 100, with probability 1/2 each; v2 is also either 10 or 100, with
probability 1/2 each and independent of v1 . Player 1 values player 2’s house at 3v2 /2, and
likewise player 2 values player 1’s house at 3v1 /2. Each player i knows his own vi but does not
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know vj , j 6= i. Suppose players announce simultaneously whether they want to exchange
their houses (after observing their respective vi ). If both players want the exchange, the
exchange takes place (player 1 gets 3v2 /2, and player 2 gets 3v1 /2). If one or both of them
do not want the exchange, no exchange takes place (player 1 gets v1 and player 2 gets v2 ).
(i) Draw the game tree. (Think carefully about the information sets.)
(ii) Find two different PBE in pure strategy.
Solution:
Part i:
Part ii:
• Suppose both players adopt the strategy of (announcing) no exchange if vi = 10 and
no exchange if vi = 100. Consider any player with any vi , his expected payoff from
announcing no exchange is vi , and his expected payoff from announcing exchange is
also vi (since no exchange will take place). Thus, this is an PBE.
• Suppose both players adopt the strategy of (announcing) exchange if vi = 10 and no
exchange if vi = 100. For a player with vi = 10, his expected payoff from announcing
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exchange is 15/2 + 10/2 = 12.5, and his expected payoff from announcing no exchange
is 10.
For a player with vi = 100, his expected payoff from announcing exchange is 15/2 +
100/2 = 57.5, and his expected payoff from announcing no exchange is 100.
Thus, this is also an PBE.
There are two other conjectures about players’ strategy, and they do not form a PBE.
You do not have to go through them if you have already found the above two equilibria.
4. (10 points) Once there was an evil lord (player 2) that visited a poor serf (player
1) each fall and demanded half of his harvest. Both the lord and the serf knew that a good
harvest produced 10 bushels of potatoes and that a bad harvest produced only 6. In addition,
it was common knowledge that the probability of a good harvest was 0.6. The serf knew
which type of harvest had occurred. The lord was told the type of harvest by the serf, and
had to decide whether or not to believe him. If he believed the serf he took the potatoes he
was offered (5 in a “good harvest” and 3 in a “bad harvest”) and left. If he didn’t believe
the serf, he took what the serf offered and punished the serf for lying by burning down his
house.
The serf kept potatoes in the cellar of his house. Hence, the truth would be revealed
once the house had burnt down and his potato holdings were exposed. Punishing a dishonest
serf gave the lord an added boost of pleasure equal to c > 0. On the other hand, if the fire
revealed that the serf had been telling the truth, the lord’s happiness was diminished by c.
While the lord felt bad when he was wrong, he would not compensate the serf by giving the
serf some of his crop back. Also, neither the serf nor the lord liked baked potatoes, so any
potatoes that were hidden in the cellar were worthless if the house was burned. The payoffs
of this signaling game are shown below (with c = 3).
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(Here I1 is the information of serf claiming a good harvest, and I2 is the information of
serf claiming a bad harvest.)
Find all pure-strategy PBE (if any exists). Include the conditional beliefs P(GoodHarvest |
I1) and P(GoodHarvest | I2) in the descriptions of PBE.
Solution:
At information set I1, the lord’s best response is to believe the serf (I1-Believe) whatever
P(GoodHarvest | I1) is. At information set I2, the lord’s best response is to believe the serf
(I2-Believe) if P(GoodHarvest | I2) ≤ 1/2, and to burn down the serf’s house (I2-Burn) if
P(GoodHarvest | I2) ≥ 1/2.
1. Suppose the serf use the strategy (GoodHarvest-Truth, BadHarvest-Truth). Then
P(GoodHarvest | I2) = 0, so the lord will believe the serf at I2 (when told that the
harvest is bad). So the lord’s best response is (I1-Believe, I2-Believe). But given the
strategy (I1-Believe, I2-Believe) by the lord, GoodHarvest-Truth is not a best response
when the serf has a good harvest. So this conjecture cannot be an equilibrium.
2. Suppose the serf use the strategy (GoodHarvest-Truth, BadHarvest-Lie). Then P(GoodHarvest |
I2) can be anything. The previous reasoning shows that P(GoodHarvest | I2) < 1/2
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cannot work in an equilibrium, since it will leads the lord to believe the serf when told
that the harvest is bad, which takes away the serf’s incentive to tell the truth when
the harvest is good.
Thus, suppose P(GoodHarvest | I2) ≥ 1/2 and the lord burns down the serf’s house
when told that the harvest is bad (I2-Burn). Given the strategy (I1-Believe, I2-Burn)
by the lord, (GoodHarvest-Truth, BadHarvest-Lie) is the best response of the serf.
Thus we have an equilibrium.
3. Suppose the serf use the strategy (GoodHarvest-Lie, BadHarvest-Truth). Then P(GoodHarvest |
I2) = 0.6, so the lord will burn down the serf’s house at I2 (when told that the harvest
is bad). So the lord’s best response is (I1-Believe, I2-Burn). But given the strategy
(I1-Believe, I2-Burn) by the lord, GoodHarvest-Lie is not a best response when the
serf has a good harvest. So this conjecture cannot be an equilibrium.
4. Suppose the serf use the strategy (GoodHarvest-Lie, BadHarvest-Lie). Then P(GoodHarvest |
I2) = 1, so the lord will burn down the serf’s house at I2 (when told that the harvest
is bad). So the lord’s best response is (I1-Believe, I2-Burn). But given the strategy
(I1-Believe, I2-Burn) by the lord, GoodHarvest-Lie is not a best response when the
serf has a good harvest. So this conjecture cannot be an equilibrium.
Thus, the only pure-strategy PBE is: the serf uses (GoodHarvest-Truth, BadHarvest-Lie),
the lord uses (I1-Believe, I2-Burn), and the lord has the conditional beliefs of P(GoodHarvest |
I1) = 0.6 and P(GoodHarvest | I2) ≥ 1/2.
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