Introduction
• In a wide variety of markets firms compete sequentially
– one firm makes a move
• new product
• advertising
Dynamic Games and First and
Second Movers
– 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 11: Dynamic Games
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Chapter 11: Dynamic Games
Stackelberg
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Stackelberg equilibrium
• Assume that there are two firms with identical
products
• As in our earlier Cournot example, let demand be:
• Interpret first in terms of Cournot
• Firms choose outputs sequentially
– leader sets output first, and visibly
– follower then sets output
– P = A – B.Q = A – B(q1 + q2)
• 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
•
•
•
•
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:
• Strategic commitment has value
– MR2 = (A - Bq1) – 2Bq2
Chapter 11: Dynamic Games
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Stackelberg equilibrium 2
MR2 = (A - Bq1) – 2Bq2
Stackelberg equilibrium 3
So the equilibrium price is (A+3c)/4 q2
(A-c)/B
Firm 1’s profit is (A-c)2/8B
R1
 q*2 = (A - c)/2B - q1/2
Firm 2’s profit is (A-c)2/16B
Demand for firm 1 is:
P = (A - Bq2) – Bq1
P = (A - Bq*2) – Bq1
We know that the Cournot
equilibrium is:
qC1 = qC2 = (A-c)/3B
(A – c)/2B
P = (A - (A-c)/2) – Bq1/2
(A – c)/4B
S
 P = (A + c)/2 – Bq1/2
Marginal revenue for firm 1 is:
MR1 = (A + c)/2 - Bq1
The Cournot price is (A+c)/3
R2
(A – c)/2
(A – c)/B
(A-c)/2B
C
(A-c)/3B
S
(A-c)/4B
R2
Profit to each firm is (A-c)2/9B
q1
(A-c)/3B (A-c)/2B
(A + c)/2 – Bq1 = c
 q*1 = (A – c)/2
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Aggregate output is 3(A-c)/4B
q2
MC = c
Chapter 11: Dynamic Games
(A-c)/ B
q1
 q*2 = (A – c)4B
Chapter 11: Dynamic Games
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Chapter 11: Dynamic Games
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1
Stackelberg and commitment
Stackelberg and price competition
• With price competition matters are different
• It is crucial that the leader can commit to its output
choice
– first-mover does not have an advantage
– suppose products are identical
– 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
– now suppose that products are differentiated
• Given such a commitment, the timing of decisions
matters
• But is moving first always better than following?
• Consider price competition
Chapter 11: Dynamic Games
• 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
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Stackelberg and price competition 2
Chapter 11: Dynamic Games
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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
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
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
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
Price competition gives a second mover advantage.
Solving this gives: p*1 = c + 3t/2
Chapter 11: Dynamic Games
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
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Dynamic games and credibility
Chapter 11: Dynamic Games
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Credibility and predation
• The dynamic games above require that firms move in
sequence
• Take a simple example
– two companies Microhard (incumbent) and Newvel (entrant)
– Newvel makes its decision first
– and that they can commit to the moves
• reasonable with quantity
• less obvious with prices
• enter or stay out of Microhard’s market
– Microhard then chooses
– with no credible commitment solution of a dynamic game
becomes very different
• accommodate or fight
– pay-off matrix is as follows:
• 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 11: Dynamic Games
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Chapter 11: Dynamic Games
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Credibility and predation 2
An example of predation
• Options listed are strategies not actions
• Microhard’s option to Fight is not an action
• It is a strategy
The Pay-Off Matrix
Microhard
Fight
– Microhard will fight if Newvel enters but otherwise remains
placid
• Similarly, Accommodate is a strategy
Accommodate
Enter
(0, 0)
(2, 2)
Stay Out
(1, 5)
(1, 5)
– 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
Newvel
• The matrix-form ignores timing.
Chapter 11: Dynamic Games
– represent the game in its extensive form to highlight sequence of
moves
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The example again
Chapter 11: Dynamic Games
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The chain-store paradox
• What if Microhard competes in more than one market?
Fight
– threatening in one market one may affect the others
(0,0)
(0,0)
• But: Selten’s Chain-Store Paradox arises
(2,2)
Enter
– 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
Accommodate
M2
(2,2)
Newvel
N1
Stay
Out
• 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
(1,5)
Chapter 11: Dynamic Games
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Chapter 11: Dynamic Games
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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 11: Dynamic Games
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