1. Dynamic Games and First and Second Movers
Chapter 8: Dynamic Games

2. Chapter 8: Dynamic Games
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
Chapter 8: Dynamic Games

3. Chapter 8: Dynamic Games
Stackelberg
Interpret first 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
Chapter 8: Dynamic Games

4. Stackelberg equilibrium
Assume that there are two firms with identical products
Marginal cost for each firm is c
Firm 1 is the market leader and chooses q1
It knows how firm 2 will react and maximizes
P [q1 + R2 (q1)] q1 – cq1
Which gives the condition
P + q1 [1 +
If demand is linear (P = A – B.Q = A – B(q1 + q2), the Residual demand for firm 2 is:
P = (A – Bq1) – Bq2
Chapter 8: Dynamic Games

5. Stackelberg equilibrium 2
Equate marginal revenue
with marginal cost
This is firm 2’s
best response
function
MR2 = (A - Bq1) – 2Bq2
But firm 1 knows
what q2 is going
to be q2
Firm 1 knows that
this is how firm 2
will react to firm 1’s
output choice
MC = c
From earlier example we know that this 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
 q*2 = (A - c)/2B - q1/2
So firm 1 can
anticipate firm 2’s
reaction
Demand for firm 1 is:
P = (A - Bq2) – Bq1
(A – c)/2B
P = (A - Bq*2) – Bq1
Equate marginal revenue
with marginal cost
P = (A - (A-c)/2) – Bq1/2 S
(A – c)/4B
 P = (A + c)/2 – Bq1/2
Solve this equation
for output q1 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
Chapter 8: Dynamic Games

6. Stackelberg equilibrium 3
Leadership benefits
the leader firm 1 but
harms the follower
firm 2
Aggregate output is 3(A-c)/4B
Leadership benefits
consumers but
reduces aggregate
profits q2
So the equilibrium price is (A+3c)/4
Firm 1’s best response
function is “like”
firm 2’s
(A-c)/B
Firm 1’s profit is (A-c)2/8B R1
Compare this with
the Cournot
equilibrium
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
Chapter 8: Dynamic Games

7. 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
Chapter 8: Dynamic Games

8. Stackelberg and price competition
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 7 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
Chapter 8: Dynamic Games

9. Stackelberg and price competition 2
Demand to firm 1 is D1(p1, p2) = N(p2 – p1 + t)/2t
Demand to firm 2 is D2(p1, p2) = N(p1 – p2 + 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
Chapter 8: Dynamic Games

10. Stackelberg and price competition 3
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 + txm = 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.
Chapter 8: Dynamic Games

11. 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?
Chapter 8: Dynamic Games

12. 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:
Chapter 8: Dynamic Games

13. An example of predation
What is the
equilibrium for this
game?
An example of predation
But is
(Stay Out, Fight)
credible?
The Pay-Off Matrix
There appear to be
two equilibria to
this game
Microhard
(Enter, Fight)
is not an
equilibrium
Fight
Accommodate
(Stay Out,
Accommodate)
is not an
equilibrium
(0, 0)
Enter
(0, 0)
(2, 2)
Newvel
(1, 5)
Stay Out
(1, 5)
(1, 5)
Chapter 8: Dynamic Games

14. Credibility and predation 2
Options listed are strategies not actions
Microhard’s option to Fight is not an action
It is a strategy
Microhard will fight if Newvel enters but otherwise remains placid
Similarly, Accommodate is a strategy
defines actions to take depending on Newvel’s strategic choice
Are the actions called for by a particular strategy credible
Is the promise to Fight if Newvel enters believable
If not, then the associated equilibrium is suspect
The matrix-form ignores timing.
represent the game in its extensive form to highlight sequence of moves
Chapter 8: Dynamic Games

15. The example again What if Newvel decides to Enter? Fight is eliminated
Microhard is
better to
Accommodate
(0,0)
(0,0)
Fight
Fight
(2,2)
Accommodate
Enter
Enter M2
Enter, Accommodate is the unique equilibrium for this game
(2,2)
Newvel
Newvel will choose
to Enter since Microhard
will Accommodate N1
Stay Out
(1,5)
Chapter 8: Dynamic Games

16. The chain-store paradox
What if Microhard competes in more than one market?
threatening in one market 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
Chapter 8: Dynamic Games

17. The chain-store paradox 2
Now consider the 19th market
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 Microhard will not “Fight” in the 20th market
So “Enter, Accommodate” becomes the unique equilibrium for this market, too
What about the 18th market?
“Fight” only to influence entrants in the 19th and 20th markets
But the threat to “Fight” in these markets is not credible.
“Enter, Accommodate” is again the equilibrium
By repetition, we see that Microhard will not “Fight” in any market
Chapter 8: Dynamic Games