Production game

Strategic Production Game1

Consider two firms, which have to make production decisions without knowing what the other is doing. For simplicity we shall suppose that the product is essentially identical. If outputs levels are qq12and so = + that the total supply is q q1 q2 , the market-clearing price is

=− =− + p abqabqq()12

= Firm i has a production cost of Ccqiii. Then if firm 2 produces q2 , firm 1’s profit is

Π=−=−−−2 112(,q q ) pq 1 c 11 q ( a c 1 bq 21 ) q bq 1

Next differentiating firm 1's profit by q1 ,

∂Π 1 =−−()22[()/2]a c bq − bq = b a −− c bq b − q ∂ 12 1 12 1 q1

∂Π Hence 1 ==−−0ifqacbqb ( )/2 ∂ 112 q1

Best response:

∂Π Choose output such that marginal profit 1 is zero. ∂ q1

B =−− BR1 : qacbqb112()/2

Similarly for player 2:

B =−− BR2 : qacbqb221()/2

1 The earliest discussion of this game goes all the way back to Augustin Cournot (1848).

1 Production game

q2

BR1

− E ()/2ac2 b BR2

q ()/ac− b 1 ()/2ac− b 2 1

Figure E-2: Best responses

For equilibrium,

BR =−− = BR1 : qacbqbq1121()/2

Similarly for player 2:

BR =−− = BR2 : qacbqbq2212()/2

Rearranging,

+=− × +=− 2bq12 bq a c 1 2 42bq12 bq 2() a c 1 and +=− × +=− qbqac122 2 1 qbqac122 2

=−−− Subtracting: 32()()bq112 ac ac

=− − − Arguing symmetrically: 3()2()bqac21 ac 2

+=11 −−− Dividing by 3 and summing: bq()()()12 q33 a c 1 a c 2

=− + =11 + + 1 Then the equilibrium price p abqq()1233 a c 1 3 c 2

2 Production game

Sequential rather than simultaneous play

The oligopoly game is a natural one to use to introduce games in which moves are sequential, rather than simultaneous. (The classic childhood sequential game is 0’s and X’s or tic-tac-toe)

If firms are very similar in size, the simultaneous model is more likely to be appropriate. However, when one firm is larger, or the incumbent, it may be in a position to make the first move, leaving other firms to follow.

What should the first mover do?

RULE : Look ahead and work backwards

Once firm 1 has made its move, firm 2 simply gets to maximize profit conditional upon firm 1’s move. That is, it will choose its best response:

* =−− qacbqb221()/2 (Best Response by firm 2) has the same best response rule as before, firm 1 is not simply responding to firm 2’s choice. Suppose that firm 1 initially chooses the same output as in the simultaneous move game. Then the outcome is the point E in the figure. But firm 1 knows that firm 2 will choose a response along the heavy response curve. Thus firm 1 optimizes by picking the most profitable point on this curve. As depicted it is the point A.

Example:

==== abcc100, 1,12 10.

Simultaneous moves:

* =−− = − BR1 : qacbqb212( )/2 (90 q 2 )/2

* =−− = − BR2 : qacbqb221( )/2 (90 q 1 )/2

== Solving, qq1230

Sequential moves:

* =−− = − BR2 : qacbqb221( )/2 (90 q 1 )/2

3 Production game

=− + = −BR − p abqq()100()12 q 21 q q 1 Π= − = − = − −BR − 111pqcqpcq()(100()) 1 cqq 2111 qq

Use the product rule

ddqΠ BR 12== − −BR − − − (100cq21 ( q ) q 1 ) 1 dq11 dq

Note that the best response is a decreasing function of q1 . Thus the second last term is positive and so the gain to increasing output is larger than in the simultaneous game.

Proposition: Comparison with the simultaneous move game

The first mover thus produces more. The best response by the second mover is to produce less.

Example:

=− + = − +11 − = − p abqq(12 ) 100 ( q 122 (90 q 1 ) 55 q 1

Π= − = − = −1 111pqcqpcq()(45) 12 qq 11.

Differentiating,

∂Π =−45 q . Thus q* = 45 and qq* =−(90 ) / 2 = 22.5 ∂ 1 1 21 q1

4 Production game

Entry Game

Incumbent monopolist and a potential entrant.

Incumbents payoffs indicated first.

Entrant (6,0) Stay out

Come In

Incumbent

Fight Share

(1,-2) (2,2)

Nash equilibrium 1:

Incumbent announces that he will fight if there is entry. Entrant’s BR is to choose OUT. Given that Entrant stays OUT, the strategy FIGHT is a BR for the Incumbent.

Nash equilibrium 2:

Entrant believes Incumbent will share. Then BR is COME IN. Given that Entrant does come in, Incumbent’s BR is SHARE.

Which equilibrium is more plausible?

If you look down the tree and ask what each player would actually do at a decision node, one equilibrium is eliminated

“Perfect Nash Equilibrium” (or Sub-game Perfect Nash Equilibrium)

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