1

ECONS 424 – STRATEGY AND GAME THEORY

HOMEWORK #3 – ANSWER KEY

Exercises from Harrington: see last pages of this answer key.

Exercise 2 – Mixed strategy Nash equilibrium with N players

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

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:

2

• 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 3 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).

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

2.5[0,1]p∈ 2.5

3

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 𝑝𝑝 = 1 − 𝑄𝑄 = 1 − 3 1

4= 1

4

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

𝜋𝜋𝑖𝑖 = 𝑝𝑝𝑞𝑞𝑖𝑖 =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 𝑝𝑝 = 1 − 𝑄𝑄 = 1 − 2 1

3= 1

3

And, therefore, profits for every firm are

𝜋𝜋𝑖𝑖 = 𝑝𝑝𝑞𝑞𝑖𝑖 =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

4

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

𝑝𝑝 = 1 − 𝑄𝑄 = 1 −12

=12

And equilibrium profits are

𝜋𝜋 = 𝑝𝑝𝑄𝑄 =12∙

12

=14

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.

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

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

𝜋𝜋𝑖𝑖 = (1 − 𝑄𝑄 − 𝑐𝑐)𝑞𝑞𝑖𝑖.

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

5

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

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

6

( )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 1cec−

≥ 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 1cec−

> , 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.

7

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, 𝑝𝑝𝑚𝑚 ≤ 𝑝𝑝𝑐𝑐 can

be rewritten as 5 14ce

c−

≤ . Note that if 𝑐𝑐 < 1/5, then 5 1 04c

c−

< , implying that 5 14ce

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.

8

Figure 2. Profitable mergers if e is low enough.