Transcript Document 7506304

Chapter 11
Dynamic Games and First and Second
Movers
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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
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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
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Stackelberg Equilibrium: an example
• Assume that there are two firms with identical
products
• As in our earlier Cournot example, let demand be:
– P = 100 - 2Q = 100 - 2(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 = (100 - 2q1) - 2q2
• Marginal revenue therefore is:
– MR2 = (100 - 2q1) - 4q2
Both firms have constant
marginal costs of $10,
i.e., c = 10 for both firms
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This is firm 2’s
best response
function
Stackelberg
equilibrium
MR2 = (100 - 2q1) - 4q2
q2
MR = (100 - 2q1) - 4q2 = 10 = c
q*2 = 22.5 - q1/2
Demand for firm 1 is:
P = (100 - 2q2) - 2q1
P = (100 - 2q*2) - 2q1
P = (100 - (45 - q1)) - 2q1
But firm 1 knows
what q2 is going
to be
22.5
11.25
 P = 55 - q1
Marginal revenue for firm 1 is:
MR1 = 55 - 2q1
55 - 2q1 = 10  q*1 = 22.5  q*2 = 11.25
Equate marginal revenue
with marginal cost
S
R2
22.5
45
q1
5
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.
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Stackelberg equilibrium3, Leadership benefits
the leader firm 1 but
harms the follower
firm 2
Aggregate output is 33.75
q2
So the equilibrium price is $32.50
45
Firm 1’s profit is (32.50 - 10)22.5
R1
 p1 = $506.25
Firm 2’s profit is (32.50 - 10)11.25
22.5
 p2 = $253.125
We know (see slide 28) that the
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Cournot equilibrium is:
11.25
C
4, Leadership benefits
consumers but
reduces aggregate
profits
S
qC1 = qC2 = 15
The Cournot price is $40
Profit to each firm is $450
2, Firm 1’s best response
function is “like”
firm 2’s
R2
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22.5
1, Compare this with
the Cournot
equilibrium
45
q1
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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
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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
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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
P = (A - Bq*2) – Bq1
P = (A - (A-c)/2) – Bq1/2
(A – c)/2B
(A – c)/4B
S
 P = (A + c)/2 – Bq1/2
Marginal revenue for firm 1 is:
MR1 = (A + c)/2 - Bq1
(A + c)/2 – Bq1 = c
 q*1 = (A – c)/2
R2
(A – c)/2
(A – c)/B
q1
 q*2 = (A – c)4B
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Stackelberg equilibrium
Aggregate output is 3(A-c)/4B
So the equilibrium price is (A+3c)/4 q2
(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:
qC1 = qC2 = (A-c)/3B
The Cournot price is (A+c)/3
(A-c)/2B
(A-c)/3B
(A-c)/4B
C
S
R2
Profit to each firm is (A-c)2/9B
(A-c)/3B (A-c)/2B
(A-c)/ B
q1
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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
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• 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
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– and we know the best response function of firm 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
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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.
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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?
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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:
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An Example of predation
1, What is the
The Pay-Off Matrix equilibrium for this
game?
4, There appear to be
two equilibria to
this game
Microhard
5, But is
(Enter, Fight)
credible?
Fight
Accommodate
Enter
(0, 0)
(2, 2)
Stay Out
(1, 5)
(1, 5)
Newvel
2, (Enter, Fight)
is not an
equilibrium
3, (Stay Out, Accommodate)
is not an equilibrium
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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
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The Example Again
3, Fight is
eliminated
1, What if Newvel
decides to Enter?
(0,0)
(0,0)
Fight
(2,2)
Enter
2, Microhard is
better to
Accommodate
Accommodate
M2
(2,2)
Newvel
N1
Stay
Out
(1,5)
5, (Enter, Accommodate) is the
unique equilibrium for
this game
4, Newvel will choose
to Enter since Microhard
will Accommodate
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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
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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.
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– 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
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