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EconS 424 β Strategy and Game Theory
Homework #5 β Answer Key
Exercise #1 β Collusion among N doctors
Consider an infinitely repeated game, in which there are ππ β₯ 3 doctors, who have created a
partnership. In each period, each doctor decides how hard to work. Let πππππ‘π‘ be the effort chosen by
doctor i in period t, and πππππ‘π‘ = 1, 2, β¦ , 10. Doctor iβs discount factor is πΏπΏππ .
Total profit for the partnership: 2(ππ1π‘π‘ + ππ2π‘π‘ + ππ3π‘π‘ + β―+ πππππ‘π‘ )
A doctor iβs payoff: 1ππ
Γ 2 Γ (ππ1π‘π‘ + ππ2π‘π‘ + ππ3π‘π‘ + β―+ πππππ‘π‘ )β πππππ‘π‘
a. Assume that the history of the game is common knowledge. Derive a subgame perfect NE in
which each player chooses effort ππβ > 1.
To begin, note that doctorβs payoff can be rearranged to:
οΏ½2πποΏ½ (ππ1 + ππ2 + β―+ ππππβ1 + ππππ+1 + β―+ ππππ) β οΏ½
ππ β 2ππ
οΏ½ ππππ
Since a doctorβs payoff is strictly decreasing in her own effort, she wants to minimize it. ππππ = 1 is
then a strictly dominant strategy for doctor i and therefore there is a unique stage game Nash
equilibrium in which each doctor chooses the minimal effort level of 1.
β’ Next, note that each doctorβs payoff from choosing a common effort level of e is:
οΏ½1πποΏ½Γ 2 Γ (ππ + ππ + β―+ ππ) β ππ = οΏ½
1πποΏ½ Γ 2 Γ ππππ β ππ = ππ
β’ To determine a doctorβs best deviation, we must take a partial derivative with respect to ππππ of their payoff function when all other (n-1) players select e, yielding
πππ’π’ππππππππ
= βοΏ½ππ β 2ππ
οΏ½
This suggests a corner solution where doctor i wants to minimize effort by playing the
lowest e possible, i.e., ei=1.
β’ We can now describe a grim-trigger strategy. When conditions are met and the strategy is played symmetrically, that will guarantee cooperation at an effort level e>1.
1
Consider the symmetric grim-trigger strategy:
o In period 1: choose ππππ1 = ππβ o In period t β₯ 2: choose πππππ‘π‘ = ππβ when ππππππ = ππβ for all j, for all ππ β€ π‘π‘ β 1; and
choose 1 otherwise.
This is a subgame perfect Nash equilibrium if and only if
ππβ
1 β πΏπΏππβ₯ οΏ½οΏ½
ππ β 1ππ
οΏ½ 2ππβ β οΏ½ππ β 2ππ
οΏ½ β 1οΏ½ +πΏπΏππ
1 β πΏπΏππ ππππππ ππππππ ππ.
That is to say, the equilibrium will only hold so long as the payoff from remaining in the equilibrium
is greater than or equal to the one period payoff from deviating plus the payoff from the
βpunishmentβ equilibrium played every period thereafter. Solving for πΏπΏππ yields:
πΏπΏππ β₯ππ β 2
2(ππ β 1)
The following figure depicts this cutoff of πΏπΏππ , shading the region of discount factors above πΏπΏππ which
would support collusion.
It is now possible to see how the equilibrium responds to changes in n. Differentiating the about
cutoff of πΏπΏππ with respect to n, we obtain
2
πππΏπΏππππππ
=1
2(ππ β 1)2
This partial is positive, indicating that as the group size n increases, πΏπΏππ has to increase to maintain
the cooperative equilibrium. So it is more difficult to support cooperation as the group size
increases.
b. Assume that the history of the game is not common knowledge, i.e., in each period, only the
total effort is observed. Find a subgame perfect NE in which each player chooses effort ππβ > 1.
Consider the strategy profile in part (a), except that it now conditions on total effort. Let πππ‘π‘denote
total effort for period t.
β’ In period 1: choose ππππ1 = ππβ β’ In period t β₯ 2: choose πππππ‘π‘ = ππβ when ππππ = ππππβ for all j, for all ππ β€ π‘π‘ β 1; and
choose 1 otherwise.
This is a subgame perfect Nash equilibrium under the exact same conditions as in part (a).
Exercise #2 - Collusion when firms compete in quantities
Consider two firms competing as Cournot oligopolists in a market with demand:
ππ(ππ1, ππ2) = ππ β ππππ1 β ππππ2
Both firms have total costs, ππππ(ππππ) = ππππππ where c > 0 is the marginal cost of production.
a. Considering that firms only interact once (playing an unrepeated Cournot game), find the equilibrium output for every firm, the market price, and the equilibrium profits for every firm. Firm 1 chooses q1 to solve
maxq1 ππππ = ππ(ππ β ππππ1 β ππππ2) β ππππ1
Taking first order conditions with respect to q1, we find
ππππ1ππππ1
= ππ β ππππ1 β ππππ2 β ππ = 0
and solving for q1 we obtain firm 1βs best response function
3
ππ1(ππ2) =ππ β ππ
2ππβ
12ππ2
and similarly for firm 2, since firms are symmetric,
ππ2(ππ1) =ππ β ππ
2ππβ
12ππ1
Plugging ππ2(ππ1) into ππ1(ππ2), we find firm 1βs equilibrium output:
ππ1 =ππ β ππ οΏ½ππ β ππππ1 β ππ
2ππ οΏ½ β ππ2ππ
=ππ
2ππβ οΏ½
ππ β ππππ1 β ππ4ππ
οΏ½ βππ
2ππ=
2ππ β 2ππ β ππ + ππ3ππ
=ππ β ππ
3ππ
Similarly, firm 2βs equilibrium output is
ππ2 =ππ β ππ οΏ½ππ β ππ
3ππ οΏ½ β ππ2ππ
=ππ
2ππβ οΏ½
ππ β ππ6ππ
οΏ½ βππ
2ππ=
3ππ β ππ + ππ β 3ππ6ππ
=ππ β ππ
3ππ
Therefore, the market price is
ππ = ππ β ππ οΏ½ππ β ππ
3πποΏ½ β ππ οΏ½
ππ β ππ3ππ
οΏ½
=3ππ β ππ + ππ β ππ + ππ
3=ππ + 2ππ
3
and the equilibrium profits of each firm are
ππ1πππππππππππππ‘π‘ = ππ2πππππππππππππ‘π‘ = οΏ½ππ + 2ππ
3οΏ½ οΏ½ππ β ππ
3πποΏ½ β ππ οΏ½
ππ β ππ3ππ
οΏ½ =(ππ β ππ)2
9ππ
b. Now assume that they could form a cartel. Which is the output that every firm should
produce in order to maximize the profits of the cartel? Find the market price, and profits of
every firm. Are their profits higher when they form a cartel than when they compete as
Cournot oligopolists?
Since the cartel seeks to maximize their joint profits, they choose output levels q1 and q2 that solve
max ππ1 + ππ2 = (ππ β ππππ1 β ππππ2)ππ1 β ππππ1 + (ππ β ππππ1 β ππππ2)ππ2 β ππππ2
which simplifies to
max (ππ β ππππ1 β ππππ2)(ππ1 + ππ2) β ππ(ππ1 + ππ2)
4
Notice that this maximization problem can be further reduced to the choice of aggregate output
Q=ππ1 + ππ2 that solves
max (ππ β ππππ)ππ β ππππ
Interestingly, this maximization problem coincides with that of a regular monopolist. In other
words, the overall production of the cartel of two symmetric firms coincides with that of a standard
monopoly. Indeed, taking first order conditions with respect to Q, we obtain
ππ β 2ππππ β ππ β€ 0
And solving for Q, we find ππ = ππβππ2ππ
, which is an interior solution given that a>c by definition.
Therefore, ach firmβs output level in the cartel is
ππ1 = ππ2 =ππ2
=ππ β ππ
2ππ2
=ππ β ππ
4ππ
And the market price is
ππ = ππ β ππππ = ππ β ππππ β ππ
2ππ=ππ + ππ
2
Therefore, each firmβs profits in the cartel are
ππ1 = ππππ1 β ππππ(ππ1) = οΏ½ππ + ππ
2οΏ½ οΏ½ππ β ππ
4πποΏ½ β ππ οΏ½
ππ β ππ4ππ
οΏ½ =(ππ β ππ)2
8ππ
ππ1πππππππ‘π‘ππππ = ππ2πππππππ‘π‘ππππ =(ππ β ππ)2
8ππ
Comparing the profits that every firm makes in the cartel, (ππβππ)2
8ππ, against those under Cournot
competition, (ππβππ)2
9ππ, we can conclude that
ππ1πππππππ‘π‘ππππ = ππ2πππππππ‘π‘ππππ > ππ1πππππππππππππ‘π‘ = ππ2πππππππππππππ‘π‘
c. Can the cartel agreement be supported as the (cooperative) outcome of the infinitely
repeated game?
1. First, we find the discounted sum of the infinite stream of profits when firms cooperate in the
cartel agreement (they do not deviate).
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β’ Payoff of cartel when they cooperate is (ππβππ)2
8ππ
β’ As a consequence, the discounted sum of the infinite stream of profits from cooperating in
the cartel is
(ππ β ππ)2
8ππ+ πΏπΏ
(ππ β ππ)2
8ππ+ β― =
11 β πΏπΏ
(ππ β ππ)2
8ππ
2. Second, we need to find the optimal deviation that, conditional on firm 2 choosing the cartel
output, maximizes firm 1βs profits. That is, which is the output that maximizes firm 1βs profits, and
which are its corresponding profits from deviating?
Since firm 2 sticks to cooperation (i.e., produces the cartel output ππ2 = ππβππ4ππ
), if firm 1 seeks to
maximize its current profits (optimal deviation), we only need to plug ππ2 = ππβππ4ππ
into firm 1βs best
response function, as follows
ππ1ππππππ β‘ ππ1 οΏ½ππ β ππ
4πποΏ½ =
ππ β ππ2ππ
β12ππ β ππ
4ππ=
3(ππ β ππ)8ππ
which provides us with firm 1βs optimal deviation, given that firm 2 is still respecting the cartel
agreement. In this context, firm 1βs profit is
ππ1 = οΏ½ππ β ππ οΏ½3(ππ β ππ)
8πποΏ½ β ππ οΏ½
ππ β ππ4ππ
οΏ½ β πποΏ½ οΏ½3(ππ β ππ)
8πποΏ½
= 3 οΏ½ππ β ππ
8οΏ½ οΏ½
3(ππ β ππ)8ππ
οΏ½ =9(ππ β ππ)2
64ππ
while that of firm 2 is
ππ2 = οΏ½ππ β ππ οΏ½3(ππ β ππ)
8πποΏ½ β ππ οΏ½
ππ β ππ4ππ
οΏ½ β πποΏ½ οΏ½ππ β ππ
4πποΏ½
= 3 οΏ½ππ β ππ
8οΏ½ οΏ½ππ β ππ
4πποΏ½ =
3(ππ β ππ)2
32ππ
Hence, firm 1 has incentives to unilaterally deviate since its current profits are larger by deviating
(while firm 2 respects the cartel agreement) than by cooperating. That is,
ππ1πππππππππππ‘π‘ππ =9(ππ β ππ)2
64ππ> ππ1πππππππ‘π‘ππππ =
(ππ β ππ)2
8ππ
3. Finally, we can now compare the profits that firms obtain from cooperating in the cartel
agreement (part i) with respect to the profits they obtain from choosing an optimal deviation (part
6
ii) plus the profits they would obtain from being punished thereafter (discounted profits in the
Cournot oligopoly). In particular, for cooperation to be sustained we need
Firm 1:
11 β πΏπΏ
(ππ β ππ)2
8ππ>
9(ππ β ππ)2
64ππ+
πΏπΏ1 β πΏπΏ
(ππ β ππ)2
9ππ
Solving for discount factor Ξ΄, we obtain
18(1βπΏπΏ)
> 964
+ πΏπΏ9(1βπΏπΏ)
, which implies πΏπΏ > 917
Hence, firms need to assign a sufficiently high value to future payoffs, πΏπΏ β οΏ½ 917
, 1οΏ½, for the cartel
agreement to be sustained.
Finally, note that firm 2 has incentives to carry out the punishment. Indeed, if it does not revers to
the NE of the stage game (producing the Cournot equilibrium output), firm 2 obtains profits of 3(ππβππ)2
32ππ, since firm 1 keeps producing its optimal deviation of ππ1ππππππ = 3(ππβππ)
8ππ while firm 2 produces
the cartel output ππ2πππππππ‘π‘ππππ = ππβππ4ππ
. If, instead, firm 2 practices the punishment, producing the Cournot
output ππβππ3ππ
, its profits are (ππβππ)2
9ππ, which exceed 3(ππβππ)2
32ππ for all parameter values. Hence, upon
observing that firm 1 deviates, firm 2 prefers to revert to the production of its Cournot output level
than being the only firm that produces the cartel output.
Exercise #3 β Collusion among N firms
Consider n firms producing homogenous goods and choosing quantities in each period for an
infinite number of periods. Demand in the industry is given by , Q being the sum of
individual outputs. All firms in the industry are identical: they have the same constant marginal
costs , and the same discount factor . Consider the following trigger strategy:
β’ Each firm sets the output that maximizes joint profits at the beginning of the game, and continues to do so unless one or more firms deviate.
β’ After a deviation, each firm sets the quantity , which is the Nash equilibrium of the one-shot Cournot game.
7
(a) Find the condition on the discount factor that allows for collusion to be sustained in this industry. First find the quantities that maximize joint profits (1 )Q Q cQΟ = β β . It is easily
checked that this output level is1
2cQ β
= , yielding profits of
ππ = οΏ½1 β1 β ππ
2οΏ½
1 β ππ2
β ππ1 β ππ
2=
(1 β ππ)2
4
for the cartel.
Therefore, at the symmetric equilibrium individual quantities are 1 1
2m cq
nβ
= and
individual profits under the collusive strategy are 21 (1 )
4m c
nΟ β
= .
As for the deviation profits, the optimal deviation by a firm is given by ( ) argmax 1 ( 1)d m m
qq q n q q q cq = β β β β .
where note that all other n-1 firms are still producing their cartel output 1 1
2m cq
nβ
= .
It can be checked that the value of q that maximizes the above expression is (1 )( ) ( 1)
4d m cq q n
nβ
= + , and that the profits that a firm obtains by deviating from the
collusive output are, hence,
1 1 1 1 11 ( 1) ( 1) ( 1) ( 1)2 4 4 4
d c c c cn n n c nn n n n
Ο β β β β = β β β + + β + ,
which simplifies to 2 2
2
(1 ) ( 1)16
d c nn
Ο β +=
Therefore, collusion can be sustained in equilibrium if 1
1 1m d cndΟ Ο Ο
d dβ₯ +
β β,
which after solving for the discount factor, Ξ΄, yields2
2
(1 )1 6
nn n
d +β₯
+ +. For compactness, we
denote this ratio as 2
2
(1 )1 6
cnnn n
d+β‘
+ +.
Hence, under punishment strategies that involve a reversion to Cournot equilibrium forever after a deviation takes place, tacit collusion arises if and only if firms are sufficiently patient. The following figure depicts cutoff cnd , as a function of the number of firms, n, shading the region of Ξ΄ that exceeds such a cutoff.
8
(b) Indicate how the number of firms in the industry affects the possibility of reaching the tacit collusive outcome. By carrying out a simple exercise of comparative statics using the critical threshold for the discount factor, one concludes that
.
(This could be anticipated from our previous figure, where the critical discount factor increases in n.) Intuition: Other things being equal, as the number of firms in the agreement increases, the more difficult it is to reach and sustain tacit collusion. Since firms are assumed to be symmetric, an increase in the number of firms is equivalent to a lower degree of market concentration. Therefore, lower levels of market concentration are associated β ceteris paribus β with less likely collusion.
9
WATSON CHAPTER 24 βEXERCISE 1
Ex. 1 β Chapter 24 Watson
Player 1Player 1Bet
Bet
Fold
Fold
Ace
King
Prob=.5
Prob=.5
Nature
Player 1
BF
bf
2, -2
1, -1
1, -1
-1,1
-1,1
-2,2
Player 2
Player 2 has only two available strategies ππ2 = {π΅π΅πππ‘π‘,πΉπΉπππππΉπΉ}
But player 1 has four available strategies ππ1 = {π΅π΅ππ,π΅π΅ππ,πΉπΉππ,πΉπΉππ}
This implies that the Bayesian normal form representation of the game is
10
Player 2
Player 1
In order to find the expected payoffs for strategy profile (Bb, Bet) top left-hand side cell of the matrix, we proceed as follows:
πΈπΈππ1 =12β 2 +
12β (β2) = 0
πΈπΈππ2 =12β (β2) +
12β (2) = 0
β (0,0)
Similarly for strategy profile (Bf, Bet),
πΈπΈππ1 =12β (β1) +
12β (β2) = β
32
πΈπΈππ2 =12β 1 +
12β 2 =
32
For strategy profile (Ff, Bet),
πΈπΈππ1 =12
(β1) +12
(β1) = β1
πΈπΈππ2 =12β 1 +
12β 1 = 1
For strategy profile (Bb, Fold),
Bet Fold
Bb
Bf
Fb
Ff
11
πΈπΈππ1 =12β 1 +
12β 1 = 1
πΈπΈππ2 =12
(β1) +12
(β1) = β1
For strategy profile (Bf, Fold),
πΈπΈππ1 =12β 1 +
12
(β1) = 0
πΈπΈππ2 =12
(β1) +12β 1 = 0
Similarly for (Fb, Fold),
πΈπΈππ1 =12β (β1) +
12β 1 = 0
πΈπΈππ2 =12β 1 +
12
(β1) = 0
Finally, for (Ff, Fold),
πΈπΈππ1 =12β (β1) +
12
(β1) = β1
πΈπΈππ2 =12β 1 +
12β 1 = 1
We can now insert these expected payoffs into the Bayesian normal form,
Player 2
Player 1
Bet Fold
Bb 0, 0 1, -1
Bf 12
,β12
0, 0
Fb β32
,32
0, 0
Ff -1, 1 -1, 1
12
WATSON CHAPTER 26 βEXERCISE 6
Following the methods used above to convert this game into normal form, we see that players 1&2 have the following strategies spaces:
Player 2 has only two available strategies ππ2 = {ππ,π·π·}
But player 1 has four available strategies ππ1 = {πΏπΏπΏπΏβ², πΏπΏπ π β²,π π πΏπΏβ²,π π π π β²}
The Bayesian normal form game would be constructed as such:
Player 2
Player 1
Making similar calculations as before, we may find the expected values of the different strategy profiles for each person and fill in the normal form table as such:
Player 2
Player 1
U D
LLβ
LRβ
RLβ
RRβ
U D
LLβ 2, 0 2, 0
LRβ 1, 0 3, 1
RLβ 1, 2 3, 0
RRβ 0, 2 4, 1
13
It is evident by finding each playerβs Best Responses that the BNE is at {LLβ, U}
WATSON CHAPTER 26 βEXERCISE 7
Watson Chapter 26 β Ex. 7
Player 1
c=2
Prob=2/3 Prob=1/3
Nature
B
0, 1
Player 2
c=0
A
0, 1
1, 0
1, 0
Player 1
B
0, 1
Player 2
A
2, 1
1, 0
1, 0
A) Player 2 (uninformed) for only two strategies ππ2 = {ππ,ππ} Player 1 (informed) has four strategies, depending on his type,
ππ1 = {π΄π΄π΄π΄β²,π΄π΄π΅π΅β²,π΅π΅π΄π΄β²,π΅π΅π΅π΅β²} Where the first component of every strategy pair denotes what player 1 does when his type is c=2 while the second component reflects his choice when his type is c=0.
14
The Bayesian normal form representation is:
Player 2
Player 1
B) Underlining playersβ best responses as usual, we find a unique BNE: {BAβ, Y}
X Y
AAβ 0, 1 1, 0
ABβ 1/3, 2/3 2/3, 1/3
BAβ 2/3, 1/3 5/3, 2/3
BBβ 1, 0 4/3, 1
15