Boğaziçi University
Department of Economics
Spring 2013
EC 206 MICROECONOMICS II
Problem Set 5 - Answer Key
1. Consider a two player game with a payoff matrix
(1)/(2)
L
R
U
(2,1)
(0,0)
D
(1,θ)
(3,-θ)
where θ ∈ {−1, 1} is a parameter known by player 2 only. Player 1 believes that θ = 1 with probability
1/2 and θ = −1 with probability 1/2. Everything above is common knowledge.
(a) Write down the strategy space of each player, S1 and S2 .
ANSWER: S1 = {U, D} and S2 = {LL, LR, RL, RR} where an element ab ∈ S2 represents the
strategy ”a if type θ = 1, and b if type θ = −1”
(b) Find the set of pure strategy Bayesian Nash equilibria.
ANSWER: We have
(1)/(2)
L
R
(1)/(2)
L
R
U
(2,1)
(0,0)
U
(2,1)
(0,0)
D
(1,1)
(3,−1)
D
(1,−1)
(3,1)
θ=1
with
prob. 1/2
θ = −1
with
prob. 1/2
Note that if type of player 2 is θ = 1, then L strictly dominates R. Thus we can eliminate RL
and RR from S2 . Then the relevant expected payoffs are
Eu1 (U, LL) = 2, and Eu2 (U, LL) = 1
Eu1 (U, LR) = 1, and Eu2 (U, LR) = 1/2
Eu1 (D, LL) = 1, and Eu2 (D, LL) = 0
Eu1 (D, LR) = 2, and Eu2 (D, LR) = 1
So we have
(1)/(2)
LL
LR
U
(2,1)
(1,1/2)
D
(1,0)
(2,1)
which yields two pure Nash equilibrium, (U, LL) and (D, LR). Thus, we get
BN Epure = {(U, LL), (D, LR)} = {(U, L for each type), (D, L if θ = 1 and R if θ = −1}.
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2. Consider a two player game with a payoff matrix
(1)/(2)
L
R
U
(2,1)
(0,0)
D
(1,β)
(α,1)
where α ∈ {−2, 2} is a parameter known to player 1 only, and both −2 and 2 are equally likely. And
β ∈ {0, 2, 3} is a parameter known to player 2 only, and all 0, 2, 3 are equally likely. Everything above
is common knowledge.
(a) Write down the strategy set for each player.
ANSWER:
S1 = {U U, U D, DU, DD} (first one for type -1, second one for type 2)
S2 = {LLL, LLR, LRL, RLL, LRR, RLR, RRL, RRR} (first one for type 0, second one for type
2 and third one for type 3)
(b) Find the set of pure strategy Bayesian Nash equilibria.
ANSWER:
For α = −2, player 1 plays U. For β = 2 and β = 3, player 2 plays L. So we can eliminate DU
and DD for player 1, and eliminate LLR,LRL,LRR,RLR,RRL,RRR for player 2. Thus we have a
reduced game where player 1 has only UU and UD, and player 2 has only LLL and RLL left.
Let the probability of player 2 playing L be p. Then, when α = 2, the expected payoffs are
EU1 (U ) = 2p
EU1 (D) = p + α(1 − p) = p + 2 − 2p = 2 − p
So player 1 plays U if 2p > 2 − p, that is, p > 2/3.
p = (1/3)q + 1/3 + 1/3 = (q + 2)/3 > 2/3 where q is the probability of player 2 playing L when
her type is β = 0. Thus, type α = 2 of player 1 plays U. Thus DU is eliminated as well. Then,
player 2 of type β = 0, also plays L.
Thus the Bayesian Nash equilibrium is (UU, LLL).
3. Consider the Cournot duopoly model in which two firms, 1 and 2, simultaneously choose the quantities
they supply, q1 and q2 . The price each will face is determined by the market demand function p(q1 , q2 ) =
a − b(q1 + q2 ). Each firm has a probability µ of having a marginal unit cost of cL and a probability
1 − µ of having a marginal unit cost of cH , where cH > cL . These probabilities are common knowledge,
but the true type is revealed only to each firm individually. Represent this game as a static Bayesian
game and write down the strategy spaces. Solve for the Bayesian Nash equilibrium. (You might first
want to read the Cournot duopoly example in Gibbons, pages 144-146. Note that the incomplete
information in that example is one sided, whereas in this problem it is two sided.)
ANSWER: A1 = A2 = [0, ∞), T1 = T2 = {H, L}, p1 (L) = p2 (L) = µ, p1 (H) = p2 (H) = (1 − µ),
and the payoffs are given by the expected profits expressed below. The strategy spaces are S1 =
2
S2 = [0, ∞) × [0, ∞), this is because each strategy should specify what quantity to pick for each own
type, for instance, a strategy of player 1 can be ”0 if my type is H, 15 if it is L”, that is, (q1H = 0,
q1L = 15) ∈ [0, ∞) × [0, ∞).
A firm of type t ∈ {H, L}, will maximize its expected profit, taken as given that the other firm will
supply q H or q L depending whether this firm is of type H or L. Firm 1 of type t ∈ {H, L} will maximize
max µ[a − b(q1t + q2L ) − ct ]q1t + (1 − µ)[a − b(q1t + q2H ) − ct ]q1t
q1t
where the FOC yields
µ[a − b(2q1t + q2L ) − ct ] + (1 − µ)[a − b(2q1t + q2H ) − ct ] = 0
Symmety implies q1H = q2H = q H and q1L = q2L = q L . Plugging these into the FOC, we get the following
µ[a − b(2q H + q L ) − cH ] + (1 − µ)[a − 3bq H − cH ] = 0
µ[a − 3bq L − cL ] + (1 − µ)[a − b(q H + 2q L ) − cL ] = 0
And we obtain
1
µ
[a − cH − (cH − cL )]
3b
2
1
1−µ
q1L = q2L = q L = [a − cL +
(cH − cL )]
3b
2
q1H = q2H = q H =
which constitute a Bayesian Nash equilibrium.
4. Consider the following strategic situation. Two rival armies plan to seize a disputed territory. Each
army’s general can choose either to “attack” (A) or to “not attack” (N). In addition, each army is
either “strong” (S) or “weak” (W ) with equal probability, and the realizations for each army are
independent. Furthermore, the type of each army is known only to that army’s general. An army can
capture the territory if either (i) it attacks and its rival does not, or (ii) it and its rival attack, but
it is strong and the rival is weak. If both attack and are of equal strength then neither captures the
territory. As for payoffs, the territory is worth M if captured and each army has a cost of fighting
equal to s if it is strong and w if it is weak, where s < w. If an army attacks but its rival does not,
no costs are bared by either side. Represent this game as a static Bayesian game and write down the
strategy spaces. Identify all the pure-strategy Bayesian Nash Equilibria for the following two cases.
(a) M = 8, s = 2, w = 6.
(b) M = 8, s = 5, w = 6.
ANSWER:
The action spaces are A1 = A2 = {A, N }. The type spaces are T1 = T2 = {S, W }. The probabilities are p(S) = p(W ) = 1/2. The strategy sets are S1 = S2 = {AA, AN, N A, N N }, where AA
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stands for ”Attack if either weak or strong type”, AN stands for ”Attack if strong, Not Attack if
weak”, N A stands for ”Not Attack if strong, Attack if weak”, and N N stands for ”Never Attack”.
(a) The payoff matrix, when M = 8, s = 2, w = 6, is
(1)/(2)
AA
AN
NA
NN
AA
(-2,-2)
(2,1)
(3,-3)
(8,0)
AN
(1,2)
(3/2,3/2)
(7/2,1/2)
(4,0)
NA
(-3,4)
(1/2,7/2)
(1/2,1/2)
(4,0)
NN
(0,8)
(0,4)
(0,4)
(0,0)
For instance, to find the payoffs for (AN,AA), the exected payoffs are calculated as follows:
Eu1 (AN, AA) = 1/4{u1 (A, A, S, S)+ u1 (A, A, S, W )+ u1 (N, A, W, S) +u1 (N, A, W, W )} = 1/4{−s+
(M − s) + 0 + 0} = M/4 − s/2 = 2 − 1 = 1. Solving for the Nash equilibria for this game yields
BN E = {(AA, AN ), (AN, AA)}.
(b) The payoff matrix, when M = 8, s = 5, w = 6, is
(1)/(2)
AA
AN
NA
NN
AA
(-7/2,-7/2)
(5/4,-1/2)
(13/4,-3)
(8,0)
AN
(-1/2,5/4)
(3/4,3/4)
(11/4,1/2)
(4,0)
NA
(-3,13/4)
(1/2,11/4)
(1/2,1/2)
(4,0)
NN
(0,8)
(0,4)
(0,4)
(0,0)
Solving for the Nash equilibria for this game yields BN E = {(AA, N N ), (N N, AA)}.
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