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 2MR2 = (A - Bq1) – 2Bq2

MC = c

Equate marginal revenuewith marginal cost

q*2 = (A - c)/2B - q1/2

q2

q1

R2

(A – c)/2B

(A – c)/B

This is firm 2’sbest response

function Firm 1 knows that

this is how firm 2will react to firm 1’s

output choice

Firm 1 knows thatthis is how firm 2

will react to firm 1’soutput choice So firm 1 can

anticipate firm 2’sreaction

So firm 1 can anticipate firm 2’s

reaction

Demand for firm 1 is:

P = (A - Bq2) – Bq1

But firm 1 knowswhat q2 is going

to be

P = (A - Bq*2) – Bq1

P = (A - (A-c)/2) – Bq1/2

P = (A + c)/2 – Bq1/2Marginal revenue for firm 1 is:

MR1 = (A + c)/2 - Bq1

(A + c)/2 – Bq1 = c

Solve this equationfor output q1

q*1 = (A – c)/2

(A – c)/2

q*2 = (A – c)4B

(A – c)/4BS

Equate marginal revenuewith 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

Firm 1’s best responsefunction is “like”

firm 2’s

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

So the equilibrium price is (A+3c)/4 q2

q1

R2

(A-c)/2B

(A-c)/ B

Compare this withthe Cournotequilibrium

Compare this withthe Cournotequilibrium

(A-c)/2B

Firm 1’s profit is (A-c)2/8B

Firm 2’s profit is (A-c)2/16B

(A-c)/BR1

SC

We know that the Cournot equilibrium is:

qC1 = qC

2 = (A-c)/3B

(A-c)/3B

(A-c)/3B

The Cournot price is (A+c)/3

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

Leadership benefitsthe leader firm 1 butharms the follower

firm 2

Leadership benefitsconsumers but

reduces aggregateprofits

(A-c)/4B

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

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

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/32Profit 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:

An example of predationThe Pay-Off Matrix

Microhard

Newvel

Fight

Enter

Accommodate

Stay Out

(0, 0) (2, 2)

(1, 5) (1, 5)

What is theequilibrium for this

game?

What is theequilibrium for this

game?

(Enter, Fight)is not an

equilibrium

(Enter, Fight)is not an

equilibrium

(0, 0)

(Stay Out, Accommodate)

is not anequilibrium

(Stay Out, Accommodate)

is not anequilibrium

(1, 5)

There appear to betwo equilibria to

this game

But is (Enter, Fight)

credible?

But is (Enter, Fight)

credible?

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

The example again

Newvel

N1

Enter

Stay Out (1,5)

M2

Fight(0,0)

Accommodate

(2,2)

What if Newveldecides to Enter?What if Newvel

decides to Enter?Microhard is

better toAccommodate

Microhard isbetter to

Accommodate

(0,0)

Fight iseliminated

Fight iseliminated

Fight

(2,2)

Newvel will chooseto Enter since Microhard

will Accommodate

Enter

Enter, Accommodate is the unique equilibrium for

this game

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