1. Chapter 11
Dynamic Games and First and Second Movers

2. Introduction In a wide variety of markets firms compete sequentially
one firm makes a move
new product
advertising
second firms sees this move and responds
These are dynamic games
may create a first-mover advantage
or may give a second-mover advantage
may also allow early mover to preempt the market
Can generate very different equilibria from simultaneous move games

3. Stackelberg Interpret in terms of Cournot
Firms choose outputs sequentially
leader sets output first, and visibly
follower then sets output
The firm moving first has a leadership advantage
can anticipate the follower’s actions
can therefore manipulate the follower
For this to work the leader must be able to commit to its choice of output
Strategic commitment has value

4. Stackelberg Equilibrium: an example
Assume that there are two firms with identical products
As in our earlier Cournot example, let demand be:
P = Q = (q1 + q2)
Total cost for for each firm is:
C(q1) = 10q1; C(q2) = 10q2
Firm 1 is the market leader and chooses q1
In doing so it can anticipate firm 2’s actions
So consider firm 2. Demand is:
P = ( q1) - 2q2
Marginal revenue therefore is:
MR2 = ( q1) - 4q2
Both firms have constant
marginal costs of $10, i.e., c = 10 for both firms

5. Stackelberg equilibrium
This is firm 2’s
best response
function
Stackelberg equilibrium
MR2 = ( q1) - 4q2
But firm 1 knows
what q2 is going
to be q2
MR = ( q1) - 4q2 = 10 = c
q*2 = q1/2
Demand for firm 1 is:
P = ( q2) - 2q1
22.5
P = ( q*2) - 2q1
P = (100 - (45 - q1)) - 2q1 S
11.25
 P = 55 - q1 R2
Marginal revenue for firm 1 is: q1
MR1 = q1
22.5 45
55 - 2q1 = 10
 q*1 = 22.5
 q*2 = 11.25
Equate marginal revenue
with marginal cost

6. 1, Firm 1 knows that this is how firm 2 will react to firm 1’s output choice
2, So firm 1 can anticipate firm 2’s reaction
3, From earlier example we know that 22.5 is the monopoly output. This is an important result. The Stackelberg leader chooses the same output as a monopolist would. But firm 2 is not excluded from the market.

7. Stackelberg equilibrium
3, Leadership benefits
the leader firm 1 but
harms the follower
firm 2
Stackelberg equilibrium
Aggregate output is 33.75 q2
So the equilibrium price is $32.50 45
4, Leadership benefits
consumers but
reduces aggregate
profits
Firm 1’s profit is ( )22.5 R1
 p1 = $506.25
Firm 2’s profit is ( )11.25
 p2 = $
22.5
We know (see slide 28) that the Cournot equilibrium is: C 15 S
11.25
qC1 = qC2 = 15 R2 q1
The Cournot price is $40 15 45
22.5
Profit to each firm is $450
1, Compare this with
the Cournot
equilibrium
2, Firm 1’s best response
function is “like”
firm 2’s

8. Stackelberg and Commitment
It is crucial that the leader can commit to its output choice
without such commitment firm 2 should ignore any stated intent by firm 1 to produce 45 units
the only equilibrium would be the Cournot equilibrium
So how to commit?
prior reputation
investment in additional capacity
place the stated output on the market
Finally, the timing of decisions matters

9. Stackelberg equilibrium
Assume that there are two firms with identical products
As in our earlier Cournot example, let demand be:
P = A – B.Q = A – B(q1 + q2)
Marginal cost for for each firm is c
Firm 1 is the market leader and chooses q1
In doing so it can anticipate firm 2’s actions
So consider firm 2. Residual demand for firm 2 is:
P = (A – Bq1) – Bq2
Marginal revenue therefore is:
MR2 = (A - Bq1) – 2Bq2

10. Stackelberg equilibrium
MR2 = (A - Bq1) – 2Bq2 q2
MC = c
 q*2 = (A - c)/2B - q1/2
Demand for firm 1 is:
P = (A - Bq2) – Bq1
(A – c)/2B
P = (A - Bq*2) – Bq1
P = (A - (A-c)/2) – Bq1/2 S
(A – c)/4B
 P = (A + c)/2 – Bq1/2 R2
Marginal revenue for firm 1 is: q1
MR1 = (A + c)/2 - Bq1
(A – c)/B
(A – c)/2
(A + c)/2 – Bq1 = c
 q*1 = (A – c)/2
 q*2 = (A – c)4B

11. Stackelberg equilibrium
Aggregate output is 3(A-c)/4B q2
So the equilibrium price is (A+3c)/4
(A-c)/B
Firm 1’s profit is (A-c)2/8B R1
Firm 2’s profit is (A-c)2/16B
We know that the Cournot equilibrium is:
(A-c)/2B
qC1 = qC2 = (A-c)/3B C
(A-c)/3B S
The Cournot price is (A+c)/3
(A-c)/4B R2
Profit to each firm is (A-c)2/9B q1
(A-c)/3B
(A-c)/ B
(A-c)/2B

12. Stackelberg and commitment
It is crucial that the leader can commit to its output choice
without such commitment firm 2 should ignore any stated intent by firm 1 to produce (A – c)/2B units
the only equilibrium would be the Cournot equilibrium
So how to commit?
prior reputation
investment in additional capacity
place the stated output on the market
Given such a commitment, the timing of decisions matters
But is moving first always better than following?
Consider price competition

13. With price competition matters are different
first-mover does not have an advantage
suppose products are identical
suppose first-mover commits to a price greater than marginal cost
the second-mover will undercut this price and take the market
so the only equilibrium is P = MC
identical to simultaneous game
now suppose that products are differentiated
perhaps as in the spatial model
suppose that there are two firms as in Chapter 10 but now firm 1 can set and commit to its price first
we know the demands to the two firms
and we know the best response function of firm 2

14. Demand to firm 1 is D1(p1, p2) = N(p2 – p1 + t)/2t
Best response function for firm 2 is p*2 = (p1 + c + t)/2
Firm 1 knows this so demand to firm 1 is
D1(p1, p*2) = N(p*2 – p1 + t)/2t = N(c +3t – p1)/4t
Profit to firm 1 is then π1 = N(p1 – c)(c + 3t – p1)/4t
Differentiate with respect to p1:
π1/p1 = N(c + 3t – p1 – p1 + c)/4t
= N(2c + 3t – 2p1)/4t
Solving this gives:
p*1 = c + 3t/2

15. Stackelberg and price competition
p*1 = c + 3t/2
Substitute into the best response function for firm 2
p*2 = (p*1 + c + t)/2 
p*2 = c + 5t/4
Prices are higher than in the simultaneous case: p* = c + t
Firm 1 sets a higher price than firm 2 and so has lower market share:
c + 3t/2 + t.xm = c + 5t/4 + t(1 – xm) 
xm = 3/8
Profit to firm 1 is then π1 = 18Nt/32
Profit to firm 2 is π2 = 25Nt/32
Price competition gives a second mover advantage.

16. Dynamic games and credibility
The dynamic games above require that firms move in sequence
and that they can commit to the moves
reasonable with quantity
less obvious with prices
with no credible commitment solution of a dynamic game becomes very different
Cournot first-mover cannot maintain output
Bertrand firm cannot maintain price
Consider a market entry game
can a market be pre-empted by a first-mover?

17. Credibility and Predation
Take a simple example
two companies Microhard (incumbent) and Newvel (entrant)
Newvel makes its decision first
enter or stay out of Microhard’s market
Microhard then chooses
accommodate or fight
pay-off matrix is as follows:

18. An Example of predation
1, What is the
equilibrium for this
game?
The Pay-Off Matrix
4, There appear to be
two equilibria to
this game
Microhard
5, But is
(Enter, Fight)
credible?
Fight
Accommodate
(0, 0)
Enter
(0, 0)
(2, 2)
Newvel
(1, 5)
Stay Out
(1, 5)
(1, 5)
2, (Enter, Fight)
is not an
equilibrium
3, (Stay Out, Accommodate)
is not an equilibrium

19. Credibility and Predation
Note that options listed are strategies not actions
Thus, Microhard’s option to Fight is not an action of predatory nature but a strategy that says Microhard will fight if Newvel enters but will otherwise remain placid
Similarly, Accommodate defines what actions to take depending, again, on Newvel’s strategic choice
The question is, are the actions called for by a particular strategy credible—In particular, is the promise to Fight if Newvel enters believable—If not, then the associated equilibrium is suspect
To put it differently, the matrix-form ignores timing. We can see this by representing the game in its extensive form to highlight sequence of moves

20. The Example Again 1, What if Newvel decides to Enter?
3, Fight is
eliminated
2, Microhard is
better to
Accommodate
(0,0)
(0,0)
Fight
Fight
(2,2)
Accommodate
Enter
Enter M2
(2,2)
Newvel N1
Stay Out
(1,5)
4, Newvel will choose
to Enter since Microhard
will Accommodate
5, (Enter, Accommodate) is the unique equilibrium for this game

21. The Chain-Store Paradox
What if Microhard competes in more than one market or with more than one rival?
threatening one may affect the others
But: Selten’s Chain-Store Paradox arises
20 markets established sequentially
will Microhard “fight” in the first few as a means to prevent entry in later ones?
No: this is the paradox
Suppose Microhard “fights” in the first 19 markets, will it “fight” in the 20th?
With just one market left, we are in the same situation as before
“Enter, Accommodate” becomes the only equilibrium
Fighting in the 20th market won’t help in subsequent markets . . There are no subsequent markets
So, “fight” strategy will not be adapted in the 20th market

22. The Chain-Store Paradox
Now consider the 19th market
Taken by itself, we know that the equilibrium for this market would be “Enter, Accommodate”
The only reason to adopt “Fight” in the 19th market is to convince a potential entrant in the 20th market that Microhard is a “fighter”
But as we have just seen, Microhard will not “Fight” in the 20th market regardless as to what has happened in earlier markets
“Fighting” in the 19th market will therefore not convince anyone that Microhard will “fight” in the 20th.

23. What about the 18th market?
With the only possible reason to “Fight” in the 19th now removed, “Enter, Accommodate” becomes the unique equilibrium for this market, too
What about the 18th market?
Here again, the only reason to “Fight” is to influence entrants in the 19th and 20th markets
But we have seen that Microhard’s threat to “Fight” in these markets is simply not credible. “Enter, Accommodate” is again the equilibrium
By repetition, we see that Microhard will not “Fight” in any market