1. Dynamic Games and First and Second Movers

2. 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

3. 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

4. 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(q 1 + q 2 ) Marginal cost for for each firm is c Firm 1 is the market leader and chooses q 1 In doing so it can anticipate firm 2’s actions So consider firm 2. Residual demand for firm 2 is: –P–P = (A – Bq 1 ) – Bq 2 Marginal revenue therefore is: –M–MR 2 = (A - Bq 1 ) – 2Bq 2

5. Stackelberg equilibrium 2 MR 2 = (A - Bq 1 ) – 2Bq 2 MC = c Equate marginal revenue with marginal cost  q* 2 = (A - c)/2B - q 1 /2 q2q2 q1q1 R2R2 (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 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 So firm 1 can anticipate firm 2’s reaction Demand for firm 1 is: P = (A - Bq 2 ) – Bq 1 But firm 1 knows what q 2 is going to be P = (A - Bq* 2 ) – Bq 1 P = (A - (A-c)/2) – Bq 1 /2  P = (A + c)/2 – Bq 1 /2 Marginal revenue for firm 1 is: MR 1 = (A + c)/2 - Bq 1 (A + c)/2 – Bq 1 = c Solve this equation for output q 1  q* 1 = (A – c)/2 (A – c)/2  q* 2 = (A – c)4B (A – c)/4B S 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

6. 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 q2q2 q1q1 R2R2 (A-c)/2B (A-c)/ B Compare this with the Cournot equilibrium 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 R1R1 S C We know that the Cournot equilibrium is: q C 1 = q C 2 = (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

7. 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

8. 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

9. Stackelberg and price competition 2 Demand to firm 1 is D 1 (p 1, p 2 ) = N(p 2 – p 1 + t)/2t Demand to firm 2 is D 2 (p 1, p 2 ) = N(p 1 – p 2 + t)/2t Best response function for firm 2 is p* 2 = (p 1 + c + t)/2 D 1 (p 1, p* 2 ) = N(p* 2 – p 1 + t)/2t = N(c +3t – p 1 )/4t Firm 1 knows this so demand to firm 1 is Profit to firm 1 is then π 1 = N(p 1 – c)(c + 3t – p 1 )/4t Differentiate with respect to p 1 :  π 1 /  p 1 = N(c + 3t – p 1 – p 1 + c)/4t = N(2c + 3t – 2p 1 )/4t Solving this gives:p* 1 = c + 3t/2

10. 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 + tx m = c + 5t/4 + t(1 – x m )  x m = 3/8 Profit to firm 1 is then π 1 = 18Nt/32 Profit to firm 2 is π 2 = 25Nt/32 Price competition gives a second mover advantage.

11. 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?

12. 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:

13. 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? What is the equilibrium for this game? (Enter, Fight) is not an equilibrium (Enter, Fight) is not an equilibrium (0, 0) (Stay Out, Accommodate) is not an equilibrium (Stay Out, Accommodate) is not an equilibrium (1, 5) There appear to be two equilibria to this game But is (Enter, Fight) credible? But is (Enter, Fight) credible?

14. 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

15. The example again Newvel N1 Enter Stay Out (1,5) M2 Fight (0,0) Accommodate (2,2) What if Newvel decides to Enter? What if Newvel decides to Enter? Microhard is better to Accommodate Microhard is better to Accommodate (0,0) Fight is eliminated 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

16. 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 20 th ? With just one market left, we are in the same situation as before “Enter, Accommodate” becomes the only equilibrium Fighting in the 20 th market won’t help in subsequent markets.. There are no subsequent markets So, “fight” strategy will not be adapted in the 20 th market

17. The chain-store paradox 2 Now consider the 19 th market –Equilibrium for this market would be “Enter, Accommodate” –The only reason to adopt “Fight” in the 19 th market is to convince a potential entrant in the 20 th market that Microhard is a “fighter” –But Microhard will not “Fight” in the 20 th market –So “Enter, Accommodate” becomes the unique equilibrium for this market, too What about the 18 th market? –“Fight” only to influence entrants in the 19 th and 20 th 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