Credibility and Subgame Perfect Equilibrium

Transcription

1 Chapter 7 Credibility and Subgame Perfect Equilibrium 1 Subgames and their equilibria The concept of subgames Equilibrium of a subgame Credibility problems: threats you have no incentives to carry out when the time comes Two important examples Telex vs. IBM Centipede 2

2 Telex vs. IBM, extensive form: subgame, perfect information Subgame Smash Enter IBM Telex Stay Out Accommodate 3 Telex vs. IBM, extensive form: no subgame Smash Telex Enter Stay Out Accommodate IBM Smash Accommodate 4

3 Telex vs. IBM, normal form: The payoff matrix IBM Telex Smash Accommodate Enter Stay Out 5 Telex vs. IBM, normal form: Strategy for IBM IBM Telex Smash Accommodate Enter Stay Out 6

4 Telex vs. IBM, normal form: Strategy for Telex IBM Telex Smash Accommodate Enter Stay Out 7 Telex vs. IBM, normal form: Two equilibria IBM Telex Smash Accommodate Enter Stay Out 8

5 Telex vs. IBM, extensive form: noncredible equilibrium Smash Enter 1 Stay Out Accommodate 9 Telex vs. IBM, extensive form: credible equilibrium Smash Enter 2 1 Stay Out Accommodate 10

6 Telex vs. IBM, subgame equilibrium Smash 2 Accommodate 11 Centipede, extensive form 1, 0 1 0, 4 2 Split the 12

7 Centipede, extensive form 1, 0 1 0, 4 2 Split the 13 Centipede, normal form: The payoff matrix Player 1 Player 2 Split the 1, 0 1, 0 0, 4 14

8 Centipede, normal form: Strategy for player 1 Player 1 Player 2 Split the 1, 0 1, 0 0, 4 15 Centipede, normal form: Strategy for player 2 Player 1 Player 2 Split the 1, 0 1, 0 0, 4 16

9 Centipede, normal form: The equilibrium Player 1 Player 2 Split the 1, 0 1, 0 0, 4 17 Maintaining Credibility via Subgame Perfection Subgame perfect equilibria: play equilibria on all subgames They only make threats and promises that a player does have an incentive to carry out Subgame perfection as a sufficient condition for solution of games in extensive form 18

10 Look Ahead and Reason Back This is also called Backward Induction Backward induction in a game tree leads to a subgame perfect equilibrium In a a subgame perfect equilibrium, best responses are played in every subgames 19 Credible Threats and Promises The variation in credibility when is all that matters to payoff Telex vs. Mean IBM Centipede with a nice opponent The potential value of deceiving an opponent about your type 20

11 Telex vs. Mean IBM Smash 0, 4 Enter IBM Telex Stay Out Accommodate 21 Centipede with a nice opponent, extensive form 1, Split the 22

12 Centipede with a nice opponent, normal form: The payoff matrix Player 1 Player 2 Split the 1, 0 1, 0 23 Centipede with a nice opponent, normal form: Strategy for player 1 Player 1 Player 2 Split the 1, 0 1, 0 24

13 Centipede with a nice opponent, normal form: Strategy for player 2 Player 1 Player 2 Split the 1, 0 1, 0 25 Centipede with a nice opponent, normal form: The equilibrium Player 1 Player 2 Split the 1, 0 1, 0 26

14 Reluctant Volunteers: Conscription in the American Civil War, Volunteering vs. waiting to be drafted Volunteering even if the expected value is negative The payoff parameters behind an allvolunteer army 27 Conscription, extensive form Volunteer b-c, 0 1 Volunteer 0, b-c 2 1/2 -c, 0 0 1/2 0, -c 28

15 Conscription, b = $300 and c = $400 Volunteer Volunteer 0, , Mutually Assured Destruction The credibility issue surrounding weapons of mass destruction A game with two very different subgame perfect equilibria Subgame perfection and the problem of mistakes 30

16 MAD, extensive form: entire game Player 2 Player 1 Doomsday Back down Doomsday -L, -L -L, -L Escalate Back down -L, -L -0.5, -0.5 Escalate 2 Back down 1 1, -1 Ignore 31 MAD, extensive form: path to final backing down Escalate -0.5, -0.5 Escalate 2 1 Ignore Back down 1, -1 32

17 MAD, extensive form: path to Doomsday Escalate -L, -L Escalate 2 1 Ignore Back down 1, MAD, normal form: b = Back down; e = Escalate; D = Doomsday; i = Ignore; = equilibrium; = subgame perfect equilibrium Country 2 Country 1 e, D e, D -L, -L e, b b, D b, b -L, -L 1, -1 1, -1 e, b -L, -L -0.5, , -1 1, -1 i, D i, b 34

18 Credible Quantity Competition: Cournot-Stackelberg Equilibrium The first mover advantage in Cournot- Stackelberg competition One firm sends its quantity to the market first. The second firm makes its moves subsequently. The strategy for the firm moving second is a function Incredible threats and imperfect equilibria 35 Cournot-Stackelberg Equilibrium: firm 2 s best response X 2 = q(x 1 ) Monopoly Cournot Point Stackelberg Point Stay out X 1 36

19 Cournot-Stackelberg Equilibrium for two firms Market Price, P = Q Market Quantity, Q = x 1 + x 2 Constant average variable cost, c = $10 Firm 1 ships its quantity, x 1, to market first Firm 2 sees how much firm 1 has shipped and then ships its quantity, x 2, to the market 37 Cournot-Stackelberg Equilibrium for two firms: Firm 2 maximizes its profits Firm 2 faces the demand curve, P = (130 - x 1 ) - x 2 Firm 2 maximizes its profits, max u 2 (x) = x 2 (130 - x 1 - x 2-10) Differentiating u 2 (x) with respect to x 2 : 0 = u 2 / x 2 = x 1-2x 2 x 2 = g(x 1 ) = 60 - x 1 /2 38

20 Cournot-Stackelberg Equilibrium: Firm 1 also wants to maximize its profits Firm 1 s profit function is given by: u 1 (x) = [130 - x 1 - g(x 1 ) - 10] x 1 Substituting g(x 1 ) into that function: u 1 (x) = (120 - x x 1 /2) x 1 Firm 1 s profits depend only on its shipment Taking the first order condition for u 1 (x): 0 = 60 - x 1 39 The Cournot-Stackelberg Equilibrium for two firms The Cournot-Stackelberg equilibrium value of firm 1 s shipments, x 1 * = 60 Firm 2 s shipments, x 2 * = 60-60/2 = 30 Market Quantity, Q = = 90 Market Price, P = = $40 This equilibrium is different from Cournot competition s equilibrium, where x 1 * = x 2 * = 40, Q = 80 and P = $50 40

21 Credible Price Competition: Bertrand-Stackelberg Equilibrium Price is the strategic behavior in Bertrand- Stackelberg competition Firms use prices as the strategic behavior The strategy for the firm moving second is a function Firm 2 has to beat only firm 1 s price which is already posted The second mover advantage in Bertrand- Stackelberg competition 41 Bertrand -Stackelberg Equilibrium for two firms Market Price, P = Q and Constant average variable cost, c = $10 Firm 1 first announces its price, p 1 Firm 2 s profit maximizing response to p 1 : p 2 = $70 if p 1 is greater than $70 p 2 = p 1 - $0.01 if p 1 is between $70 and $10.02 p 2 = p 1 if p 1 = p 2 = $10 otherwise 42

22 Differentiated Products Product differentiation mutes both types of mover advantage A mover disadvantage can be offset by a large enough cost advantage 43 Two firms in a Bertrand-Stackelberg competition The demand function faced by firm 1: x 1 (p) = p 1 -(p 1 - average p) x 1 = p p 2 Similarly, the demand function faced by firm 2: x 2 = p 1-1.5p 2 Constant average variable cost, c = $20 44

23 Two firms in a Bertrand-Stackelberg competition: Determining optimum p 2 Firm 2 wants to maximize its profits, given p 1 : max (p 2-20)( p 1-1.5p 2 ) Profit maximizes when the first order condition is satisfied: 0 = p 1-3p Solving for optimal price p 2, we get p 2 * = g(p 1 ) = 70 + p 1 /6 45 Two firms in a Bertrand-Stackelberg competition: Equilibrium prices Knowing that firm 2 will determine p 2 by using g(p 1 ), firm1 tries to maximize its profit: max (p 1-20)[ p (70 + p 1 /6)] Profit maximizes when the first order condition is satisfied: 0 = (17/12)p 1 + (p 1-20) (-17/12) p 1 * = 2920/34 = $85.88 Firm 2, which moves last, charges slightly lower price than p 1 *: p 2 * = 70 + p 1 * /6 = 70 + $14.31 = $

24 Two firms in a Bertrand-Stackelberg competition: Profits for the two firms Firm 1 sells less than firm 2 does: x 1 * = and x 2 * = Firm 1 s profit, u 1 * = (93.34)( ) = $ Firm 2 s profit, u 2 * = (96.48)( ) = $ Firm 2, the second mover, makes more 47 This Offer is Good for a Limited Time Only The credibility problems behind the marketing slogan The principle of costly commitment Industries where the slogan is credible 48

25 An example of This offer is good for a limited time only Exploding job offers An early job offer with a very short time to decide on whether to take the job. Risk-averse people often end up accepting inferior job offers 49 Appendix. Ultimatum Games in the Laboratory Games with take-it-or-leave-it structure In experiments, subjects playing such games rarely play subgame perfect equilibria The nice opponent explanation vs. the expected payoff explanation 50