Transcript Dynamic Games and First and Second Movers Introduction • In a wide variety of markets firms compete sequentially – one firm makes a.

Dynamic Games and First and
Second Movers
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
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
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
Stackelberg
equilibrium 2
This isEquate
firm 2’smarginal revenue
with marginal cost
MR2 = (A - Bq1) – 2Bq
best 2response
But firmq21 knows
function
MC = c
what q2 is going
Firm 1 knows that
From
earlier
example
we
know
to be
this is how firm 2
 q*2 = (A - c)/2B - q1/2
that this is the monopoly output. This is an
will reactSotofirm
firm11’s
can
Demand
forresult.
firm 1The
is: Stackelberg leader
important
output
choicefirm 2’s
anticipate
chooses
the
same
output
as
a
monopolist
would.
P = (A - Bq2) – Bq1
(A – c)/2B
reaction
But
firm
2
is
not
excluded
from
the
market
P = (A - Bq*2) – Bq1
Equate marginal revenue
P = (A - (A-c)/2) – Bq1/2
S
with marginal
cost
(A – c)/4B
 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
 q*2 = (A – c)4B
Solve this equation
for output q1
(A – c)/2
R2
(A – c)/B
q1
Stackelberg equilibrium 3
Aggregate output is 3(A-c)/4B
Leadership
benefits
So the equilibrium price is (A+3c)/4 q2
Firm
1’s best
response
(A-c)/B
theLeadership
leader firmbenefits
1 but
Firm 1’s profit is (A-c)2/8B
function
is
“like”
R1 harms the follower
consumers
but
Compare
this with
firm
2’s
2
Firm 2’s profit is (A-c) /16B
firm
2
reduces
aggregate
the
Cournot
profits
We know that the Cournot
equilibrium
(A-c)/2B
equilibrium is:
C
qC1 = qC2 = (A-c)/3B
(A-c)/3B
The Cournot price is (A+c)/3
(A-c)/4B
S
R2
Profit to each firm is (A-c)2/9B
(A-c)/3B (A-c)/2B
(A-c)/ B
q1
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
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 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
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
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.
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?
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:
What is the
An
example
of
predation
But is
equilibrium for this
(Enter, Fight)
The Pay-Off Matrix game?
credible?
There appear to be
Microhard
two equilibria to
(Enter, Fight)
this game
is not an
Fight
Accommodate
equilibrium
Enter
Newvel
Stay Out
(Stay Out,
Accommodate)
(0, 0)
is not an
equilibrium
(2, 2)
(1, 5)
(1, 5)
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
Fight
is
The
example
again
What if Newvel
eliminated
decides to Enter?
(0,0)
(0,0)
Fight
(2,2)
Enter
M2
Microhard is
better to
Accommodate
Accommodate
(2,2)
Newvel
N1
Enter, Accommodate
is the
Stay
Out
unique equilibrium
for(1,5)
this game
Newvel will choose
to Enter since Microhard
will Accommodate
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
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