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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

Both firms have constantmarginal costs of $10,

i.e., c = 10 for both firms

• 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

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Stackelberg equilibrium

MR2 = (100 - 2q1) - 4q2

MR = (100 - 2q1) - 4q2 = 10 = c

q*2 = 22.5 - q1/2

q2

q1

R2

22.5

45

This is firm 2’sbest response

function

Demand for firm 1 is:

P = (100 - 2q2) - 2q1

But firm 1 knowswhat q2 is going

to be

P = (100 - 2q*2) - 2q1

P = (100 - (45 - q1)) - 2q1

P = 55 - q1

Marginal revenue for firm 1 is:

MR1 = 55 - 2q1

Equate marginal revenuewith marginal cost

55 - 2q1 = 10 q*1 = 22.522.5

q*2 = 11.25

11.25 S

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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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2, Firm 1’s best responsefunction is “like”

firm 2’s

Stackelberg equilibrium

Aggregate output is 33.75

So the equilibrium price is $32.50 q2

q1

R2

22.5

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1, Compare this withthe Cournotequilibrium

1, Compare this withthe Cournotequilibrium

22.5

11.25

Firm 1’s profit is (32.50 - 10)22.5

1 = $506.25

Firm 2’s profit is (32.50 - 10)11.25

2 = $253.125

45R1

SCWe know (see slide 28) that the

Cournot equilibrium is:

qC1 = qC

2 = 15

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The Cournot price is $40

Profit to each firm is $450

3, Leadership benefitsthe leader firm 1 butharms the follower

firm 2

4, Leadership benefitsconsumers but

reduces aggregateprofits

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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

MC = c

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

q2

q1

R2

(A – c)/2B

(A – c)/B

Demand for firm 1 is:

P = (A - Bq2) – Bq1

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

q*1 = (A – c)/2

(A – c)/2

q*2 = (A – c)4B

(A – c)/4BS

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

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

q1

R2

(A-c)/2B

(A-c)/ B(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

(A-c)/4B

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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– and we know the best response function of firm 2

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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

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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 = 3/8Profit 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 predationThe Pay-Off Matrix

Microhard

Newvel

Fight

Enter

Accommodate

Stay Out

(0, 0) (2, 2)

(1, 5) (1, 5)

1, What is theequilibrium for this

game?

1, What is theequilibrium for this

game?

2, (Enter, Fight)is not an

equilibrium

2, (Enter, Fight)is not an

equilibrium

(0, 0)

3, (Stay Out, Accommodate)is not an equilibrium

3, (Stay Out, Accommodate)is not an equilibrium

(1, 5)

4, There appear to betwo equilibria to

this game

5, But is (Enter, Fight)

credible?

5, But is (Enter, Fight)

credible?

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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

Newvel

N1

Enter

Stay Out (1,5)

M2

Fight(0,0)

Accommodate

(2,2)

1, What if Newveldecides to Enter?

1, What if Newveldecides to Enter?

2, Microhard isbetter to

Accommodate

2, Microhard isbetter to

Accommodate

(0,0)

3, Fight iseliminated

3, Fight iseliminated

Fight

(2,2)

4, Newvel will chooseto Enter since Microhard

will Accommodate

Enter

5, (Enter, Accommodate) is the unique equilibrium for

this game

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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