Midterm with Answers Econ 171, February 24, 2015
Problem 1.
Each of two sellers has one unit of an unusual commodity
available to sell. Seller A posts a price of $40 for its item. Seller B posts
a price of $60 for its item. There are exactly two potential buyers for this
commodity. Each buyer would be willing to pay $100 to get one of these items
and both buyers are aware that there are two buyers and each is willing to pay
$100 for the item. Without knowing what the other buyer has done, the two
buyers simultaneously decide which of the two sellers to approach. If two buyers
approach the same seller, the seller tosses a fair coin to decide which of them
to sell to. The other has to go home without making a purchase. We assume
that a disappointed buyer is not able to approach the other seller. A buyer’s
payoff if he obtains an item for price p is $100-p. Buyers seek to maximize their
expected payoffs.
A) Show the strategic form of the game played between the two buyers.
In thist table, Buyer 1 chooses the row and Buyer 2 chooses the column.
Table 1: Strategic Form
A
B
A
30,30
40,60
B
60,40
2020
B) Find all of the pure and mixed strategy Nash equilibrium profiles for this
game. List the strategy of each player for any equilibrium that you find.
There are two pure strategy Nash equilibria: { Player 1 approaches A, Player
2 approaches B} { Player 2 approaches A, Player 1 approaches B} and one mixed
strategy equilibrium in which each player chooses A with probabiltiy 4/5 and
B with probability 1/5.
C) If the seller’s profit from the sale is the price of the item if at least one
buyer approaches him and zero if no buyers approach him, what is the expected
profit in Nash equilibrium of Seller A? What about Seller B? In the two pure
strategy Nash equilibria, Seller A has a profit of $40 and Seller B has a profit
of $60.
In the mixed strategy Nash equilibrium, Seller A has at least one buyer
with probability 1 − 1/5 × 1/5 = 24/25 and so has expected profits of $40 ×
24/25 = $192/5 = $38.4 and Seller B has has at least one buyer with probabilty
1 − 4/5 × 4/5 = 9/25 and so has expected profits of $60 × 9/25 = 108/5 = $21.6.
D) Suppose that Seller B’s price is $50 and that Seller A’s price is pA . For
what values of pA is there a pure strategy Nash equilibrium for the game between
the two buyers? For what values of pA is the only Nash equilibrium for this
game a mixed strategy equilibrium?
The strategic form for the game is below.
Table 2: strategic form
A
B
A
(100 − pA )/2, (100 − pA )/2
50, (100 − pA )
B
(100 − pA ),50
25,25
There is a Nash equilibrium where both go to B if pA ≥ 75 and there is two
Nash equilibria where one buyer goes to A and one to B if pA ≤ 75.. So there
are no values of pA for which the game has no pure strategy equilibria.
Note: I had intended to ask a slightly easier question here. I meant to ask
For what values of pA is there no symmetric pure strategy equilibrium. The
answer to that is 0 < pA < 75. You should be able to verify this from the
strategic form printed above . Because the question as stated is a little more
difficult than I had planned, I gave full credit for this part to anyone who got
the rest of the question right and made a reasonable effort on this part.
Problem 2. In the game, Rock, Paper, Scissors, two players simultaneously
choose one of the strategies, Rock, Paper, or Scissors. If both choose the same
strategy, both get payoff of zero. If one chooses Rock and the other chooses
Scissors, the Rock player gets a payoff of 1 and the Scissors player gets -1. If
one chooses Scissors and the other chooses Paper, the Scissors player gets a
payoff of 1 and the Paper player gets a payoff of -1. If one chooses Paper and
the other chooses Rock, the Paper player gets a payoff of 1 and the Rock player
gets -1. Consider the following variant of this game. Player 1 is able to play
any of the three strategies, but Player 2 can only play Rock or Paper.
A) Show this game in strategic form. Are there any strictly dominated
strategies?
Table 3: Modified Rock Paper Scissors
Rock
Paper
Scissors
Rock
0,0
1,-1
-1,1
Paper
-1,1
0,0
1,-1
The strategy Rock is strictly dominated for Player 1.
B) How many Nash equilibria does this game have? What is (are) the Nash
equilibrium strategy profile(s)
This game has no pure strategy equilibria. it does have a mixed strategy
equilibrium in which Player 2 chooses rock with probability 1/3 and paper with
probability 2/3 and Player 1 chooses Scissors with probability 1/3 and Paper
with probability 2/3. C) Find the Nash equilibrium payoff for each player.
In Nash equilibrium the expected payoff for Player 1 is 1/3 and for Player 2
is -1/3.
Problem 3. A) Define an independent, private values information setting for
an auction.
B) Define a common values information setting for an auction.
C) In which of these settings, does the “winners curse” occur? Describe the
mistake by bidders that would lead to the so-called winners curse?
See Chapter 10.4 and/or the posted lecture notes for definitions and discussion of these concepts.
Problem 4. Two players are involved in a dispute. Player 1 does not know
whether Player 2 is strong or weak. Player 2 knows the strength of both players.
Each player has two possible strategies, Fight or Yield. A player who chooses
Yield will get a payoff of 0 regardless of the other player’s strength or action.
If a player chooses Fight and the other player chooses Yield, then the payoffs
will be 100 for the player who chose Fight and 0 for the other player. If both
players choose Fight, then if Player 2 is strong, Player 1 gets a payoff of −100
and Player 2 gets a payoff of 100. If both players choose Fight and Player 2
is weak, then Player 1 gets a payoff of 100 and Player 2 gets a payoff of -100.
Suppose that Player 1 believes that with probability p, Player 2 is strong and
with probability 1 − p, Player 2 is weak.
A) Draw an extensive form representation of this game.
B) Find Bayes-Nash equilibrium profiles for both players if p > 1/2. What
is the expected payoff to Player 1 in Bayes-Nash equilibrium?
C) Find Bayes-Nash equilibrium profiles for both players if p < 1/2. What
is the expected payoff to Player 1 in Bayes-Nash equilibrium?
D) Suppose that before play begins, Player 1 has an opportunity to hire a
spy, who will accurately report whether Player 2 is weak or strong. The cost
of hiring the spy is c, so that if Player 1 hires the spy, his payoff, contingent
on choosing either Fight or Yield will be reduced by c. What is the most that
Player 1 would be willing to pay to hire spy if p > 1/2. What is the most that
Player 1 would pay to hire the spy if p < 1/2? (Your answers will be functions
of p.)
See Lecture Notes for Lecture 14 for answers to this question.
Problem 5.
In the game below, the top number is Player 1’s payoff, the
middle number is Player 2’s payoff, and the bottom number is Player 3’s payoff.
A) How many strategies are possible for Player 1? How many strategies are
possible for Player 2? How many strategies are possible for Player 3? There are
3 possible strategies for Player 1, 4 for Player 2, and 8 for Player 3. Give an
example of a possible strategy for Player 2.
B) How many subgames does this game have? How many proper subgames
does this game have? medskip
The game has 4 subgames and 3 proper subgames.
C) For each proper subgame of this game, find all of the subgame perfect
Nash equilibrium profiles.
For the subgame starting at Player 2’s decision node after Player 1 has played
a1 , the subgame perfect Nash equilibrium profile is the strategy a2 for Player 2
and b3 for Player 3.
For the subgame starting at Player 3’s decision node after Player 1 has played
c1 and Player 2 has played a2 , the equilibrium strategy profile has Player 3
playing a3 .
For the subgame starting at Player 3’s decision node after Player 1 has
played c1 and Player 2 has played b2 , the equilibrium strategy profile has Player
3 playing b3 .
D) Find all of the subgame perfect Nash equilibrium strategy profiles for
this game.
The strategy profile in which Player 1 plays b1 , Player 2 plays a2 in both of
her information sets, and Player 3 plays b3 in her leftmost information set, a3 in
her middle information set and b3 in her rightmost information set is a SPNE.
There is one other SPNE. In this one, Player 1 plays a1 , Player 2 plays the
strategy “a2 if 1 plays a1 and b2 if 1 plays b1 or c1 . Player 3 plays the straegy
b3 in her leftmost information set, a3 in her middle information set and b3 in
her rightmost information set.
These are the only two SPNE’s. I gave full credit if you found one but not
the other one.
Problem 6. Consider a first-price sealed bid auction in which there are two
bidders. The bidders’ valuations for the object being sold are independently
drawn by Nature. Each bidder’s valuation must take one of the three values 3,
5, or 8. The probability is .3 that a bidder’s value is 3, the probability is .3 that
the bidder’s value is 5, and the probability is .4 that the bidder’s value is 8.
A) Determine whether there is a symmetric Bayes-Nash equilibrum in which
a bidder bids 2 when her valuation is 3, bids 3 when her valuation is 5 and bids
4 when her valuation is 8. Explain how you reached your answer.
B) Determine whether there is a symmetric Bayes-Nash equilibrum in which
a bidder bids 2 when her valuation is 3, bids 3 when her valuation is 5 and bids
5 when her valuation is 8. Explain how you reached your answer.
See Lecture Notes for Lecture 14 for answers to this question.