ECONS 424 – STRATEGY AND GAME THEORY

HOMEWORK #3 – ANSWER KEY

Exercise #1: Harrington Chapter 7 – Exercise 12

Part A

UN

a b = (1-a)

Saddam

x

y

z = (1-x-y)

X&Y Z

X 2, 9 5, 4

Y 2, 9 5, 4

Z 5, 4 2, 9

1

2

Part B

UN

a b c = (1-a-b)

Saddam

x

y

z = (1-x-y)

X&Y X&Z Y&Z

X 2, 9 2, 9 5, 4

Y 2, 9 5, 4 2, 9

Z 5, 4 2, 9 2, 9

3

4

Exercise 5 – Chapter 9 Harrington

A: In this game of “Hunt for Red October” we can only divide the game into two subgames. There is

one subgame encompassing the decisions of both Borodin and Melekhin when Ramius sends the

letter, and one subgame when Ramius does not send the letter. We can then construct the normal

form of these games and determine the best responses, finally analyzing Ramius’ decisions from

there.

Having determined the Nash equilibria of the final subgames, we can now look at the last subgame,

which is the entire game.

5

B:

Or in very simple terms, Ramius must send the letter in order to ensure the defection of his offices

with him.

Exercise 2 – Mixed strategy Nash equilibrium

a) The normal form representation of the game for n=2 players is given below. Player 2

Player 1 X Y

X 3,3 4,3

Y 3,4 2,2

There are three pure strategy Nash equilibria in this game, (X,X), (X,Y) and (Y,X).

b) When introducing n=3 players, the normal form representation of the game is:

• First, if Player 3 chooses X, Player 2

Player 1 X Y

X 0,0,0 3,3,3

Y 3,3,3 2,2,4

• And if Player 3 chooses Y, Player 2

Player 1 X Y

6

X 3,3,3 4,2,2

Y 2,4,2 1,1,1

Hence, the pure strategy Nash equilibria of the game with n=3 players are (X,Y,X), (Y,X,X) and (X,X,Y).

c) If every player is choosing X with probability p and Y with probability 1-p, the expected utility that player 1 obtains by playing X is:

• EU1(X)=p20+p(1-p)3+ (1-p)p3+(1-p)24=p(1-p)6+4(1-p)2 And player 1’s utility from playing Y is:

• EU1(Y)=p23+p(1-p)2+(1-p)p2+(1-p)21=3p2+4(1-p)p+(1-p)2 Player 1 is indifferent between choosing strategy X and Y for values of p such that EU1(X)= EU1(Y). That is,

• p(1-p)6+4(1-p)2=3p2+4(1-p)p+(1-p)2, and simplifying, 2p2+4p-3=0 • Solving for p, we find that either p= -1- <0 (which cannot be a solution to our

problem, since , or p= -1+ =0.58, which is the solution to our problem.

Hence, every player in this game randomizes between X and Y (using mixed strategies) assigning probability p=0.58 to strategy X, and 1-p=0.42 to strategy Y.

Exercise 3 – Cournot competition with N firms

Consider three firms competing a la Cournot, in a market with inverse demand function 𝑃𝑃(𝑄𝑄) =1 − 𝑄𝑄, and production costs normalized to zero.

a. Find the psNE of the game when firms simultaneously and independently choose quantities. Determine the equilibrium profit level for each firm.

b. Consider now that two (out of three) firms merge, and thus choose their output decision in order to maximize their joint profits. Find the psNE in this game for the merged firms and the unmerged firms. Identify the equilibrium profits for each firm, and compare them with your results pre-merger in part (a)

c. Consider now that all three firms merge. Find their profit maximizing output and profits, comparing them with your results in (a) and (b).

d. Repeat parts (a)-(b), but considering that firms compete in a Bertrand model of price competition with differentiated products, with demand function 𝑞𝑞𝑖𝑖 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏𝑖𝑖 + 𝑑𝑑𝑏𝑏𝑗𝑗 + 𝑑𝑑𝑏𝑏𝑘𝑘, for any firm i and its rival k≠i. Show that in this case, the merger between firm 1 and 2 would yield profits above those these firms obtain before the merger.

2.5[0,1]p∈ 2.5

7

Answer:

a. The profits for firm i are

𝜋𝜋𝑖𝑖 = (1 − 𝑄𝑄)𝑞𝑞𝑖𝑖 = �1 − 𝑞𝑞𝑖𝑖 − 𝑞𝑞𝑗𝑗 − 𝑞𝑞𝑘𝑘�𝑞𝑞𝑖𝑖 Taking first order conditions with respect to 𝑞𝑞𝑖𝑖, we obtain:

1 − 2𝑞𝑞𝑖𝑖 − 𝑞𝑞𝑗𝑗 − 𝑞𝑞𝑘𝑘 = 0 ⟹ 𝑞𝑞𝑖𝑖�𝑞𝑞𝑗𝑗 , 𝑞𝑞𝑘𝑘� =1 − 𝑞𝑞𝑗𝑗 − 𝑞𝑞𝑘𝑘

2

And in a symmetric Nash equilibrium in which all firms are producing the same output, i. e., 𝑞𝑞𝑖𝑖 = 𝑞𝑞𝑗𝑗 = 𝑞𝑞𝑘𝑘 = 𝑞𝑞, we find 𝑞𝑞𝑖𝑖 = 1

4 for every firm 𝑖𝑖.

Hence, equilibrium prices are 𝑏𝑏 = 1 − 𝑄𝑄 = 1 − 3 1

4= 1

4

And, therefore, profits for every firm are (recall that there are no production costs)

𝜋𝜋𝑖𝑖 = 𝑏𝑏𝑞𝑞𝑖𝑖 =14∙

14

=1

16

b. There are only two firms in the market now: the merge of firms 1 and 2, and the

(unmerged) firm 3. The profits for either of these two firms are:

𝜋𝜋𝑖𝑖 = (1 − 𝑄𝑄)𝑞𝑞𝑖𝑖 = �1 − 𝑞𝑞𝑖𝑖 − 𝑞𝑞𝑗𝑗�𝑞𝑞𝑖𝑖 And taking FOCs with respect to 𝑞𝑞𝑖𝑖, we obtain:

1 − 2𝑞𝑞𝑖𝑖 − 𝑞𝑞𝑗𝑗 = 0 ⟹ 𝑞𝑞𝑖𝑖�𝑞𝑞𝑗𝑗� =1 − 𝑞𝑞𝑗𝑗

2

and in a symmetric Nash equilibrium in which all firms are producing the same output, i. e., 𝑞𝑞𝑖𝑖 = 𝑞𝑞𝑗𝑗 = 𝑞𝑞, we find 𝑞𝑞𝑖𝑖 = 1

3 for every firm 𝑖𝑖.

Hence, equilibrium prices are 𝑏𝑏 = 1 − 𝑄𝑄 = 1 − 2 1

3= 1

3

And, therefore, profits for every firm are

8

𝜋𝜋𝑖𝑖 = 𝑏𝑏𝑞𝑞𝑖𝑖 =13∙

13

=19

This implies that

Firms 1 and 2 obtain profits of 192

= 118

after the merger, which are lower than the

pre-merger profits of 116

Firm 3 obtains profits of 19, which exceed its pre-merger profits of 1

16

Intuition: the merged firms internalize part of price reduction that an increase in their aggregate production entails, i. e., they consider the profit loss that the increase in production by one of the firm participating in the merger entails on the other firm that joined the merger. As a consequence, the merged firms reduce their individual production relative to pre-merger levels (in the standard Cournot competition analyzed in part a). However, the unmerged Firm 3 does not take into these price effects, and must responds to a lower output level from both of its competitors by increasing its own production. Ultimately, the firms that merged obtain a lower profit than before the merger, while the merged firm earns a larger profit. This result is often referred as the “merger paradox”.

c. If all firms merge, they form a cartel, acting as a monopolist. [Note that this is only true when they all merge, not when only two of them merge, as we examined in the previous section]. When they all merge their joint profits are

𝜋𝜋𝑖𝑖 = (1 − 𝑄𝑄)𝑄𝑄 = 𝑄𝑄 − 𝑄𝑄2 Taking first order conditions with respect to Q, we obtain

1 − 2𝑄𝑄 = 0 ⟹ 𝑄𝑄 =12

Which implies that equilibrium price is

𝑏𝑏 = 1 − 𝑄𝑄 = 1 −12

=12

And equilibrium profits are

𝜋𝜋 = 𝑏𝑏𝑄𝑄 =12∙

12

=14

9

Therefore, the individual profits of every firm participating in the merger are 143

= 112

, which are clearly higher than their profits pre-merger (when all firms compete as Cournot oligopolists) of 1

16.

d. First, note that the inverse demand function is

𝑞𝑞𝑖𝑖 = 𝑎𝑎 − 𝑏𝑏𝑏𝑏𝑖𝑖 + 𝑑𝑑𝑏𝑏𝑗𝑗 + 𝑑𝑑𝑏𝑏𝑘𝑘

No merger (Bertrand competition with product differentiation):

max𝑝𝑝𝑖𝑖

𝑏𝑏𝑖𝑖𝑞𝑞𝑖𝑖 = 𝑏𝑏𝑖𝑖�𝑎𝑎 − 𝑏𝑏𝑏𝑏𝑖𝑖 + 𝑑𝑑𝑏𝑏𝑗𝑗 + 𝑑𝑑𝑏𝑏𝑘𝑘�

Taking FOCs with respect to 𝑏𝑏𝑖𝑖, we obtain

𝑎𝑎 − 2𝑏𝑏𝑏𝑏𝑖𝑖 + 𝑑𝑑𝑏𝑏𝑗𝑗 + 𝑑𝑑𝑏𝑏𝑘𝑘 = 0 And at a symmetric equilibrium where all firms set the same price, 𝑏𝑏𝑖𝑖 = 𝑏𝑏𝑗𝑗 =𝑏𝑏𝑘𝑘 = 𝑏𝑏 we have

𝑎𝑎 − 2𝑏𝑏𝑏𝑏 + 𝑑𝑑𝑏𝑏 + 𝑑𝑑𝑏𝑏 = 0 ⟹ 𝑏𝑏 =𝑎𝑎

2(𝑏𝑏 − 𝑑𝑑)

With associated equilibrium profits of

𝜋𝜋 = 𝑏𝑏𝑞𝑞 =𝑎𝑎

2(𝑏𝑏 − 𝑑𝑑) �𝑎𝑎 − 𝑏𝑏𝑎𝑎

2(𝑏𝑏 − 𝑑𝑑) + 2𝑑𝑑𝑎𝑎

2(𝑏𝑏 − 𝑑𝑑)�

And rearranging

𝜋𝜋 =𝑏𝑏𝑎𝑎2

4(𝑏𝑏 − 𝑑𝑑)2

Merger of two firms: firms 1 and 2 merge, and choose 𝑏𝑏1 and 𝑏𝑏2 to maximize

their joint profits, as follows

max𝑝𝑝1,𝑝𝑝2

𝑏𝑏1(𝑎𝑎 − 𝑏𝑏𝑏𝑏1 + 𝑑𝑑𝑏𝑏2 + 𝑑𝑑𝑏𝑏3) + 𝑏𝑏2(𝑎𝑎 − 𝑏𝑏𝑏𝑏2 + 𝑑𝑑𝑏𝑏1 + 𝑑𝑑𝑏𝑏3)

Taking FOCs with respect to 𝑏𝑏1 and 𝑏𝑏2 respectively, we obtain:

𝑎𝑎 − 2𝑏𝑏𝑏𝑏1 + 2𝑑𝑑𝑏𝑏2 + 𝑑𝑑𝑏𝑏3 = 0 (1)

10

𝑎𝑎 − 2𝑏𝑏𝑏𝑏2 + 2𝑑𝑑𝑏𝑏1 + 𝑑𝑑𝑏𝑏3 = 0 And the unmerged firm 3 chooses 𝑏𝑏3 to maximize its own profits:

max𝑝𝑝3

𝑏𝑏3(𝑎𝑎 − 𝑏𝑏𝑏𝑏3 + 𝑑𝑑𝑏𝑏1 + 𝑑𝑑𝑏𝑏2)

And taking FOCs with respect to 𝑏𝑏3

𝑎𝑎 − 2𝑏𝑏𝑏𝑏3 + 𝑑𝑑𝑏𝑏1 + 𝑑𝑑𝑏𝑏2 = 0 (2) Solving (1) and (2), we obtain equilibrium prices of:

𝑏𝑏1 = 𝑏𝑏2 =𝑎𝑎(2𝑏𝑏2 − 𝑏𝑏𝑑𝑑 − 𝑑𝑑2)

2(𝑏𝑏 − 𝑑𝑑)(2𝑏𝑏2 − 2𝑏𝑏𝑑𝑑 − 𝑑𝑑2)

And for firm 3:

𝑏𝑏3 =𝑎𝑎𝑏𝑏

(2𝑏𝑏2 − 2𝑏𝑏𝑑𝑑 − 𝑑𝑑2)

The joint profits post-merger for firms 1 and 2 are

(2𝑏𝑏2 − 𝑏𝑏𝑑𝑑 − 𝑑𝑑2)𝑎𝑎2

(𝑏𝑏 − 𝑑𝑑)2(2𝑏𝑏2 − 2𝑏𝑏𝑑𝑑 − 𝑑𝑑2)2

While the pre-merger profits for firms 1 and 2 were only

2𝑏𝑏𝑎𝑎2

4(𝑏𝑏 − 𝑑𝑑)2

Hence, the profits post-merger are higher than pre-merger.

Exercise 4 (Bonus Exercise) – Cournot mergers with efficiency gains

a. Each firm i={1,2,3} has a profit of

11

𝜋𝜋𝑖𝑖 = (1 − 𝑄𝑄 − 𝑐𝑐)𝑞𝑞𝑖𝑖.

Hence, since 𝑄𝑄 ≡ 𝑞𝑞1 + 𝑞𝑞2 + 𝑞𝑞3, profits can be rewritten as:

𝜋𝜋𝑖𝑖 = �1 − (𝑞𝑞𝑖𝑖 + 𝑞𝑞𝑗𝑗 + 𝑞𝑞𝑘𝑘) − 𝑐𝑐�𝑞𝑞𝑖𝑖,

The first-order conditions are given by

1 − 2𝑞𝑞𝑖𝑖 − 𝑞𝑞𝑗𝑗 − 𝑞𝑞𝑘𝑘 − 𝑐𝑐 = 0

since firms are symmetric i j kq q q q= = = in equilibrium, that is

1 − 2𝑞𝑞 − 𝑞𝑞 − 𝑞𝑞 − 𝑐𝑐 = 0

or

1 4 0q c− − = .

Solving for q at the symmetric equilibrium yields a Cournot output of,

14c

cq −=

Hence, equilibrium prices are

1 1 1 1 1 31 1 34 4 4 4 4c

c c c c cp − − − − − = − − − = − =

and equilibrium profits are( )211 3 1

4 4 16c

cc ccπ−− − = − =

b.

1. After the merger, two firms are left: firm 1, with cost 𝑒𝑒 ∙ 𝑐𝑐, and firm 3, with cost 𝑐𝑐.

Hence, the two profit functions are now given by:

𝜋𝜋1 = (1 − 𝑄𝑄 − 𝑒𝑒𝑐𝑐)𝑞𝑞1

𝜋𝜋3 = (1 − 𝑄𝑄 − 𝑐𝑐)𝑞𝑞3

Taking first order conditions of 1π with respect to 1q yields

12

1 31 2 0q q ec− − − = and, solving for 1q , we obtain firm 1’s best response function

( )1 3 31 1

2 2ecq q q−

= −

Similarly taking first-order conditions of firm 3’s profits, 3π , with respect to 3q yields

3 11 2 0q q c− − − = which, solving for 3q , provides us with firm 3’s best response function

( )3 1 11 1

2 2cq q q−

= −

Plugging ( )3 1q q into ( )1 3q q , yields

* *1 1

1 1 1 12 2 2 2ec cq q− − = − −

Rearranging and solving for *1q , we obtain firm 1’s equilibrium output

( )*

1

1 2 13

c eq

− −=

Plugging this output level into firm 3’s best response function yields an equilibrium output of

*3

1 (2 )3

c eq − −=

Note that the outsider firm can sell a positive output at equilibrium only if the merger does not give

rise to strong cost savings: that is 𝑞𝑞3 ≥ 0 if 2 1cec−

≥ (if 𝑐𝑐 < 1/2, then the previous payoff becomes

2 1 0cc−

< , implying that 2 1ce

c−

≥ holds for all 0e ≥ , ultimately implying that the outsider firm

will always sell at the equilibrium. We hence concentrate on values of c that satisfy 𝑐𝑐 > 1/2.) Figure

1 illustrates cutoff 2 1ce

c−

> , where 𝑐𝑐 > 1/2, and the region of ( ),e c -combinations above this

cutoff indicate parameters for which the merger it is sufficiently cost saving to induce the outside firm to produce positive output levels.

13

Figure 1. Positive production after the merger.

• The equilibrium price is ( )1 11 (2 1) 1 (2 )1

3 3 3m

c ec e c ep+ +− − − −

= − − = , and equilibrium profits

are given by 2

1(1 (2 1))

9c eπ − −

= and 2

3(1 (2 ))

9c eπ − −

= .

2. Prices decrease after the merger only if there are sufficient efficiency gains: that is, 𝑏𝑏𝑚𝑚 ≤ 𝑏𝑏𝑐𝑐 can

be rewritten as 5 1

4ce

c−

≤ . Note that if 𝑐𝑐 < 1/5, then 5 1 0

4c

c−

< , implying that 5 1

4ce

c−

≤

cannot hold for any 0e ≥ . As a consequence, m cp p> , and prices will never fall no matter how strong efficiency gains, e, are.

3. To see if the merger is profitable, we have to study the inequality 𝜋𝜋1 ≥ 2𝜋𝜋𝑐𝑐 , which after some algebra can be seen to correspond to an inequality of the second order whose relevant solution is

𝑒𝑒 ≤4(1 + 𝑐𝑐) − 3√2(1 − 𝑐𝑐)

8c

In other words, the merger is profitable only if it gives rise to enough cost savings. Figure 2

depicts the cutoff of e where costs are restricted to 1 ,15

c ∈ . Notice that if cost savings are

sufficiently strong, i.e., parameter e is sufficiently small as depicted in the region below the cutoff, the merger is profitable.

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Figure 2. Profitable mergers if e is low enough.

15