ECON 2015-1
Final Exam - A
Carlos Hurtado
Game Theory
1
(MWG 8.D.9) Nash Equilibrium
Consider the following game
1/2
LL
L
M
R
U
100,2
-100,1
0,0
-100,-100
D
-100,-100
100,-49
1,0
100,2
1. [1 pt] If you were player 2 in this game and you were playing it once without the ability to engage in preplay communication
with player 1, what strategy would you choose? Explain your reasons.
2. [1 pt] Find the Pure Strategies Nash Equilibrium (PSNE) of the game.
3. [1 pt] Find the Mix Strategies Nash Equilibrium (MSNE) of the game.
4. [1 pt] Is your strategy choice in 1 a component of any Nash equilibrium? Is it a rationalizable strategy?
5. [1 pt] Suppose now that preplay communication were possible. Would you expect to play something dierent from your choice
in 1? Explain your reasons.
Answer:
1. Playing
L, R
or
LL
is quite risky for player 2 since the positive payo is small compare to the negative payo in each
strategy. Therefore, it seems reasonable to play
2. The two PSNE are
(U, LL)
and
M.
(D, R).
3. To check for MSNE player 1 must mix between
U
probabilities for player 2 mixing between LL and
1
and E2 (LL) = E2 (L) = 26 . For player 1 E1 (U ) =
cases yield to a contradiction.
4. The choice in 1 is not part of any NE. Strategy
M
(with probability p) and D (with probability 1 − p). Let q and 1 − q be the
51
, with E2 (R) < 0, E2 (M ) = 0,
L. Equating E2 (LL) = E2 (L) we have p = 52
E1 (D) which implies q = 21 . Then, p = 51
and
q = 21 is a MSNE. All other
52
p=
is rationalizable: If player 1 plays
1
then
2
M
is the unique best response
of player 2.
5. If preplay communication is possible, the players can agree to play one of the PSNE. Note that this, by denition of NE, is
Pareto dominant for both players.
2
(MWG 9.B.9) Subgame Perfection
[5 pt] Consider the game in which the following simultaneous-move game is played twice:
a1
a2
a3
b1
b2
b3
10,10
2,12
0,13
12,2
5,5
0,0
13,0
0,0
1,1
The players observe the actions chosen in the rst play of the game prior to the second play. What are the Subgame Perfect
Nash Equilibria (SPNE) of this game?
Answer:
The PSNE of the one-shot game are
(a2 , b2 )
and
(a3 , b3 ).
Thus, playing either of these strategies in both periods constitutes a
SPNE. Additionally, the players could use them in any combination in the two periods. This results in the following two classes of
SPNE
1. Player 1 plays
ai
and player 2 plays
ai in the rst period
i, j ∈ {2, 3}and i 6= j .
2. Player 1 plays
period, with
bi
in both periods, with
and
aj
i ∈ {2, 3}.
in the second period. Player 2 plays
bi
in the rst period and
bj in
However, there exist more SPNE in this game. The reason is that player 1 or 2 can punish the other player by playing
the second
a3
or
b3
in
the second period if the other player did not cooperate in the rst period. Then we have two more cases of SPNE
1. Player 1's strategy: Play
Player 2's strategy: Play
2. Player 2's strategy: Play
Player 1's strategy: Play
ai , i ∈ {1, 2, 3}
b1 in period 1.
bi , i ∈ {1, 2, 3}
a1 in period 1.
in period 1. Play
Play
b2
in period 1. Play
Play
a2
a2
b2
a1
bi , i ∈ {1, 2, 3}
in period 2 if Player 1 played
in period 2 if Player 2 played
1
b1
ai , i ∈ {1, 2, 3}
in period 2 if Player 2 played
in period 2 if Player 1 played
in period 1. Play
in period 1. Play
in period 1. Play
in period 1. Play
a3
b3
otherwise.
b3
a3
otherwise.
otherwise.
otherwise.
3
Weak Perfect Bayesian Nash Equilibria -WPBNEE
Consider the following two person game played by an Entrant
and an Incumbent
I
(where the rst of the two payos given
belongs to the Entrant (E ) and the second to the Incumbent (I )).
1. [2 pt] For what values of
p
will the incumbent
I
choose L over R?
2. [1 pt] Derive all of the pure strategy weak perfect Bayesian Nash equilibria. Structure your answer so it is clear you have
considered all possibilities.
3. [1 pt] Solve for a weak perfect Bayesian Nash equilibrium in mixed strategies. Be sure to fully dene the equilibrium strategies.
4. [1 pt] Is your answer to c a sequential equilibrium? Explain.
4
Folk theorem
Consider the following two person game:
1/2
a1
a2
b1
3, 3
4, 0
b2
0, 4
1, 1
1. Suppose rst that the players discount their payos in the innitely repeated game. Player
∞
X
i
therefore calculates his payo as
δit ρit
t=0
t
where δi is player
i's
(a) [1 pt] Find the
discount factor and
min max
ρit
i
is the payo of player
in the
t-th
repetition of the game.
value of each player
(b) [1 pt] Present a subgame perfect Nash equilibrium strategies in the innitely repeated game that result in
(3, 3)
as the
outcome of each stage on the game.
(c) [1 pt] Explain why your equilibrium is in fact subgame perfect.
2. Now suppose that each player evaluates his payo in the innitely repeated game by taking a limiting average value. Player
i's
payo is therefore:
Pn−1
t=0
lim
where
ρit
is the payo of player
i
in the
t-th
ρit
n
n→∞
repetition of the game.
(a) [1 pt] Plot the region of payos that can be written as the average payo in some Nash equilibrium of the supergame.
(b) [1 pt] Present subgame perfect Nash equilibrium strategies in the innitely repeated game that result in
7 7
,
3 3
as the average payo vector for the two players (Hint: there are innitely many answers).
(c) [1 pt] Explain in words why
[0, 4]
cannot be the average payo vector for a Nash equilibrium of the innitely repeated
game.
2
5
Nash, Dominance and Bayesian
Show the following:
1. [1 pt] (MWG Prop. 8.B.1) Player
i's pure strategy si ∈ Si is strictly dominated in the game ΓN = [I, {∆Si } , {ui (·)}],
σ̃i ∈ ∆Si such that ui (σ̃i , s−i ) > ui (si , s−i ) for all s−i ∈ S−i .
if and
only if, there exist another strategy
2. [1 pt] (MWG 8.B.6) Prove that if pure strategy
is any strategy that plays
si
si
is a strictly dominated strategy in game
3. [1 pt] (MWG 8.B.7) Prove that any strictly dominant strategy in game
ΓN = [I, {∆Si } , {ui (·)}]
4. [2 pt] Consider an economy in which there are two consumers, a public good
contribution game in which each player
the public good is produced. Consumer
i
i
contributes an amount
zi
yi = ŷi − zi .
Consumer
ai = 4
consumer believes that
then so
i's
x
must be a pure strategy.
and a private good
y.
Consider the voluntary
of his private good whereupon an amount
has large initial endowment
ui (x, yi ) = ai x −
where
ΓN = [I, {∆Si } , {ui (·)}],
with positive probability.
ŷi
of
y
x = z1 + z2
of
and a utility function
x2
+ yi
2
ai is either 4 or 6, but only consumer i knows which. The other
ai = 6 with probability 1/2. Compute a Bayesian Nash equilibrium
preference parameter
with probability
1/2
and
of this game in which each player uses the same strategy (i.e., a symmetric Bayesian Nash equilibrium).
6
Second-Price (Vickrey) Auction
[5 pt] An object is auctioned o to
N
bidders.
Bidder
i's
valuation of the object (in monetary terms) is
θi .
The auction rules
are that each player submits a bid (a nonnegative number) in a sealed envelope. The envelopes are then opened, and the bidder
who has submitted the highest bid gets the object but pays the auctioneer the amount of the second-highest bid. If more that one
bidder submits the highest bid, each get the object with equal probability. Show that submitting a bid
dominant strategy for bidder
7
i.
Also argue that this is bidder
i's
θi
with certainty is a weakly
unique weakly dominant strategy.
The Market of Lemons
A seller has a car that he may sell to a buyer. The quality of the car is
θ ∈ [0, 1].
The value of the car to the seller is
θ.
Seller's
utility if he sells the car is p, where p is the money he receives for it, and θ if he retains the car. The buyer's utility for the car if
1
he buys it and pays p is 2 + θ − p, and his utility is 0 if he does not buy the car. The seller alone knows the quality of his car. The
buyer believes that θ is uniformly distributed on [0, 1]. Suppose that the buyer can make an oer p ∈ [0, 1] to the seller, and the
seller can then decide whether to accept or reject the buyer's oer.
1. [1 pt] Given a value of
θ,
what would be the optimal strategy of the seller? Explain why.
2. [2 pt] Given the optimal strategy of the seller, what would be the optimal strategy of the buyer? That is, what would be the
price that the buyer should oer to maximize his expected utility.
3. [2 pt] Which is the eect of asymmetric information in this market? Is there a market for any type of car in this example?
3