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Chapter 11: Dynamic Games

Dynamic Games and First and Second Movers

Chapter 11: 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 11: 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 11: Dynamic Games

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

Chapter 11: Dynamic Games

Stackelberg equilibrium 2

MR2 = (A - Bq1) – 2Bq2

MC = c

Equate marginal revenue

with marginal cost

 q*2 = (A - c)/2B - q1/2

(A – c)/2B

(A – c)/B

This is firm 2’s

best response

function

Firm 1 knows that

this is how firm 2

will react to firm 1’s

output choice

So firm 1 can

anticipate firm 2’s

reaction

Demand for firm 1 is:

P = (A - Bq2) – Bq1

But firm 1 knows

what q2 is going

to be

P = (A - Bq*2) – Bq1

P = (A - (A-c)/2) – Bq1/2

 P = (A + c)/2 – Bq1/2

Marginal revenue for firm 1 is:

MR1 = (A + c)/2 - Bq1

(A + c)/2 – Bq1 = c

Solve this equation

for output q1

 q*1 = (A – c)/2

(A – c)/2

 q*2 = (A – c)4B

(A – c)/4B

Equate marginal revenue

with marginal cost

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

Chapter 11: Dynamic Games

Firm 1’s best response

function is “like”

firm 2’s

Stackelberg equilibrium 3

Aggregate output is 3(A-c)/4B

So the equilibrium price is (A+3c)/4

(A-c)/2B

(A-c)/ B

Compare this with

the Cournot

equilibrium

(A-c)/2B

Firm 1’s profit is (A-c)2/8B

Firm 2’s profit is (A-c)2/16B

(A-c)/B

We know that the Cournot equilibrium is:

qC1 = qC2 = (A-c)/3B

(A-c)/3B

The Cournot price is (A+c)/3

Profit to each firm is (A-c)2/9B

Leadership benefits

the leader firm 1 but

harms the follower

firm 2

Leadership benefits

consumers but

reduces aggregate

profits

(A-c)/4B

Chapter 11: Dynamic Games

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 11: Dynamic Games

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

Chapter 11: Dynamic Games

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

D1(p1, p*2) = N(p*2 – p1 + t)/2t = N(c +3t – p1)/4t

Firm 1 knows this so demand to firm 1 is

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 11: Dynamic Games

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 11: Dynamic Games

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 11: Dynamic Games

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 11: Dynamic Games

An example of predation

The Pay-Off Matrix

Microhard

Newvel

Fight

Enter

Accommodate

Stay Out

(0, 0)

(2, 2)

(1, 5)

What is the

equilibrium for this

game?

(Enter, Fight)

is not an

equilibrium

(0, 0)

(Stay Out,

Accommodate)

is not an

equilibrium

(1, 5)

There appear to be

two equilibria to

this game

But is

(Enter, Fight)

credible?

Chapter 11: Dynamic Games

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 11: Dynamic Games

The example again

Newvel

Enter

Stay Out

(1,5)

Fight

(0,0)

Accommodate

(2,2)

What if Newvel

decides to Enter?

Microhard is

better to

Accommodate

(0,0)

Fight is

eliminated

Fight

(2,2)

Newvel will choose

to Enter since Microhard

will Accommodate

Enter

Enter, Accommodate is the
unique equilibrium for
this game

Chapter 11: Dynamic Games

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 11: Dynamic Games

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

Directed reading in Industrial Organization

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