and their Chapter 5 equilibria

aThe concept of subgames Extensive Form aEquilibrium of a aCredibility problems: threats you have no with incentives to carry out when the time comes Perfect aTwo important examples `Telex vs. IBM Information `Centipede

1 2

Subgame Perfection in Extensive Form (Selten, 1965) aWho plays when? : each player must act aWhat can they do? optimally given the other players' aWhat do they know? strategies, i.e., play a to the others' strategies. aWhat are the payoffs? Problem: Optimality condition at the beginning of the game. Hence, some Nash equilibria of dynamic games involve incredible threats.

3 4

Game in Extensive Form: Game in Normal Form

1 2 L R ll lr rl rr

2 2 L 1, 3 1, 3 2, 0 2, 0 1 l r l r R 4, 2 0, 1 4, 2 0, 1

1: 1 2 4 0 a Three Nash equilibria in pure strategies: {R,ll}, {L,lr}, 2: 3 0 2 1 and {R,rl}. a {L,lr}, and {R,rl} involve incredible threats. Unique equilibrium path 5 6 Subgame Perfection with Definition

Consider a game Γ of perfect information consisting of a tree T linking the A Nash equilibrium of Γ is subgame perfect information sets i ∈ I (each of which if it specifies Nash equilibrium strategies consists of a single node) and payoffs at in every subgame of Γ. In other words,

each terminal node of T. A subtree Ti is the players act optimally at every point the tree beginning at information set i, during the game.

and a subgame Γi is the subtree Ti and the payoffs at each terminal node of Ti.

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Subgame Perfection with Telex vs. IBM, extensive form: Imperfect Information subgame, perfect information

1 Subgame L R 0, 0 2 Smash

l r l r Enter IBM

Telex Accommodate 1: 3 1 2 4 2, 2 2: 1 4 3 2 Stay Out With imperfect information, each information set consisting of a single node determines a subgame. Hence, there are 1, 5 no (proper) subgames in this example. 9 10

Telex vs. IBM, extensive form: Telex vs. IBM, normal form: no subgame The payoff matrix

0, 0 IBM Smash Telex Smash Accommodate

Enter Enter 0, 0 2, 2 Accommodate 2, 2 IBM Telex Stay Out Smash 1, 5 Stay Out 1, 5 1, 5

Accommodate 1, 5 11 12 Telex vs. IBM, normal form: Telex vs. IBM, normal form: for IBM Strategy for Telex

IBM IBM Telex Smash Accommodate Telex Smash Accommodate

Enter 0, 0 2, 2 Enter 0, 0 2, 2

Stay Out 1, 5 1, 5 Stay Out 1, 5 1, 5

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Telex vs. IBM, normal form: Telex vs. IBM, extensive form: Two equilibria noncredible equilibrium

IBM Telex Smash Accommodate 0, 0 Smash

Enter IBM 0, 0 2, 2 Enter

Telex Accommodate 2, 2 Stay Out 1, 5 1, 5 Stay Out 1, 5

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Telex vs. IBM, extensive form: credible equilibrium Centipede, extensive form

Take the money 1, 0 0, 0 Smash

IBM 1 Take the Enter 0, 4 money Wait Telex Accommodate 2, 2 2 Stay Out Split the 1, 5 money 2, 2

17 18 Centipede, normal form: Centipede, extensive form The payoff matrix

Take the Player 2 money 1, 0 Take the Split the Player 1 money money

1 Take the Take the 0, 4 1, 0 1, 0 money money Wait

2 Wait 0, 4 2, 2 Split the money 2, 2

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Centipede, normal form: Centipede, normal form: Strategy for player 1 Strategy for player 2

Player 2 Player 2 Take the Split the Take the Split the Player 1 money money Player 1 money money

Take the Take the money 1, 0 1, 0 money 1, 0 1, 0

Wait 0, 4 2, 2 Wait 0, 4 2, 2

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Centipede, normal form: The equilibrium Look Ahead and Reason Back

Player 2 Take the Split the aThis is also called Backward Induction Player 1 money money aBackward induction in a leads to a subgame perfect equilibrium Take the money 1, 0 1, 0 aIn a subgame perfect equilibrium, best responses are played in every subgames

Wait 0, 4 2, 2

23 24 Credible Threats and Promises Telex vs. Mean IBM aThe variation in credibility when money is all that matters to payoff 0, 4 Smash aTelex vs. Mean IBM aCentipede with a nice opponent Enter IBM aThe potential value of deceiving an opponent about your type Accommodate Telex 2, 2 Stay Out 1, 5

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Centipede with a nice opponent, Centipede with a nice opponent, extensive form normal form: The payoff matrix

Take the Player 2 money 1, 0 Take the Split the Player 1 money money

1 Take the Take the 0, 0 1, 0 1, 0 money money Wait

2 Wait 0, 0 2, 2 Split the money 2, 2

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Centipede with a nice opponent, Centipede with a nice opponent, normal form: Strategy for player 1 normal form: Strategy for player 2

Player 2 Player 2 Take the Split the Take the Split the Player 1 money money Player 1 money money

Take the Take the money 1, 0 1, 0 money 1, 0 1, 0

Wait 0, 0 2, 2 Wait 0, 0 2, 2

29 30 Centipede with a nice opponent, normal form: The equilibrium Mutually Assured Destruction

Player 2 Take the Split the aThe credibility issue surrounding weapons Player 1 money money of mass destruction aA game with two very different subgame Take the perfect equilibria money 1, 0 1, 0 aSubgame perfection and the problem of mistakes

Wait 0, 0 2, 2

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MAD, extensive form: path to final MAD, extensive form: entire game backing down Player 2 Player 1 Doomsday Back down

Doomsday -L, -L -L, -L -0.5, -0.5 Escalate Back down -L, -L -0.5, -0.5 Escalate 2 Escalate Back down 1 Escalate 1, -1 2 Ignore Back down 1 1, -1 0, 0 Ignore 0, 0 33 34

MAD, normal form: b = Back down; e = Escalate; MAD, extensive form: path to D = Doomsday; i = Ignore; = equilibrium; Doomsday = subgame perfect equilibrium

Country 2 Country 1 e, D e, b b, D b, b -L, -L Escalate e, D -L, -L -L, -L 1, -1 1, -1

Escalate 2 e, b -L, -L -0.5, -0.51, -1 1, -1 Back down 1 1, -1 i, D 0, 0 0, 0 Ignore 0, 0 0, 0

0, 0 i, b 0, 0 0, 0 0, 0 0, 0 35 36 Credible Quantity : Cournot-Stackelberg Equilibrium: Cournot-Stackelberg Equilibrium firm 2’s best response aThe first mover advantage in Cournot- X = q(x ) 2 1 60 aOne firm sends its quantity to the market Cournot Point first. The second firm makes its moves 40 subsequently. Stackelberg Point aThe strategy for the firm moving second is 30 a function 20 aIncredible threats and imperfect equilibria Stay out X 04060 120 1 37 38

Cournot-Stackelberg Equilibrium Cournot-Stackelberg Equilibrium for for two firms two firms: Firm 2 maximizes its profits Market Price, P = 130 - Q Firm 2 faces the demand curve,

P = (130 - x1) - x2 Market Quantity, Q = x1 + x2 Firm 2 maximizes its profits, Constant average variable cost, c = $10 max u2(x) = x2(130 - x1 - x2 - 10) Firm 1 ships its quantity, x1, to market first Differentiating u2(x) with respect to x2: Firm 2 sees how much firm 1 has shipped and 0 = ∂u2/∂x2 = 120 - x1 -2x2 then ships its quantity, x2, to the market ⇒ x2 = g(x1) = 60 - x1/2

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Games like : Tic-Tac-Toe Player draft

aEach team takes turn choosing players aIs it best to always chose most preferred player?

41 42 Cournot-Stackelberg Equilibrium: Firm 1 The Cournot-Stackelberg also wants to maximize its profits Equilibrium for two firms Firm 1’s profit function is given by: The Cournot-Stackelberg equilibrium value of

u1(x) = [130 - x1 - g(x1) - 10] x1 firm 1’s shipments, x1* = 60

Substituting g(x1) into that function: Firm 2’s shipments, x2* = 60 - 60/2 = 30 u (x) = (120 - x - 60 + x /2) x 1 1 1 1 Market Quantity, Q = 60 + 30 = 90 ∴ Firm 1’s profits depend only on its Market Price, P = 130 - 90 = $40 shipment This equilibrium is different from Cournot Taking the first order condition for u (x): 1 competition’s equilibrium, where x * = x * = 40, 0 = 60 - x 1 2 1 Q = 80 and P = $50 43 44

Credible Price Competition: Bertrand -Stackelberg Equilibrium Bertrand-Stackelberg Equilibrium for two firms Market Price, P = 130 - Q and aPrice is the strategic behavior in Bertrand- Constant average variable cost, c = $10 Stackelberg competition Firm 1 first announces its price, p1 aThe strategy for the firm moving second is a function Firm 2’s profit maximizing response to p1: aFirm 2 has to beat only firm 1’s price which p2 = $70 if p1 is greater than $70 is already posted p2 = p1 - $0.01 if p1 is between $70 and $10.02 aThe second mover advantage in Bertrand- Stackelberg competition p2 = p1 if p1 = 10.01

p2 = $10 otherwise 45 Get competitive ; no extra profits! 46

Market Games with Differentiated Products Differentiated Products aPrice and quantity competition when All differentiated products products are differentiated have one thing in common: aCournot and Bertrand equilibrium still if the price is slightly above different, but the difference is muted aMonopolistic competition as the limit of the average price in the market game equilibrium market, a firm doesn’t lose all its sales

47 48 Two firms in a Two firms in a Bertrand competition

aThe demand function faced by firm 1: aFirm 1 has profits x (p) = 180 - p -(p - average price) 1 1 1 u1(p1,p2) = (p1 -20) x1

= (p1 - 20) (180 - 2p1 + average price) aThe demand function faced by firm 2: = (p1 - 20) (180 - 1.5p1 + 0.5p2) x (p) = 180 - p -(p - average price) 2 2 2 aFirm 2’s profit function

u2(p1,p2) = (p2 - 20) (180 - 1.5p2 + 0.5p1)

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Bertrand best responses, two firms, Maximizing profits differentiated products Firm 1 maximizes its profits when its p1 = f1(p2) = 70 + p2/6 marginal profit is zero: p2 p2 = f2(p1) = 70 + p1/6 0 = ∂u1/∂p1

= (p1 - 20) (-1.5) + (180 - 1.5p1 + 0.5p2) p* = (84, 84) ⇒ 0 = 210 - 3p1 + 0.5p2 Firm 1’s best response function:

p1 = f1(p2) = 70 + p2/6 Similarly, Firm 2’s best response function: p1 p2 = f2(p1) = 70 + p1/6 51 52

Bertrand Equilibrium Bertrand competition with n firms aThe Bertrand equilibrium of the market game aFirm 1’s market demand is located at (84, 84) x1= 180 - p1 -(n/2) ( p1 - average price) aThe market price is $84, significantly higher aFirm 1’s profit function than the marginal price which is given at $20 u1(p) = (p1 -20) x1 aEach firm sales (180 - 84) units = 96 units aWhen the first order condition is satisfied aEach firms profit = (84 - 20) × 96 0 = ∂u1/∂p1 = (p1 - 20)(-1-n/2+1/2)+ x1

= $6144 ⇒ K (p1 - 20) = 180 - p1 aTherefore, each firm could spend over $6000 where K = (n + 1)/2

in differentiating its products and can still ∴ p1* = 180 /(K+1) + 20K/(K+1) come out ahead 53 54 Bertrand competition with infinite number of firms Differentiated Products

an →∞⇒K →∞ and 1/K → 0 aProduct differentiation mutes both types of aTaking limit of p1* as n goes to infinity mover advantage

lim p1* = lim 20/(1 + 1/K) = 20 aA mover disadvantage can be offset by a aIn this limit, price is equal to marginal large enough cost advantage cost and profits vanish. aThis limit is monopolistic competition

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Two firms in a Bertrand-Stackelberg Two firms in a Bertrand-Stackelberg

competition competition: Determining optimum p2

The demand function faced by firm 1: Firm 2 wants to maximize its profits, given p1: x1(p) = 180 - p1 -(p1 - average p) max (p2 - 20)(180 + 0.5p1 -1.5p2)

⇒ x1 = 180 - 1.5p1 + 0.5p2 Profit maximizes when the first order condition is Similarly, the demand function faced by firm 2: satisfied: 0 = 180 + 0.5p1 -3p2 + 30

x2 = 180 + 0.5p1 -1.5p2 Solving for optimal price p2, we get Constant average variable cost, c = $20 p2* = g(p1) = 70 + p1/6

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Two firms in a Bertrand-Stackelberg Two firms in a Bertrand-Stackelberg competition: Equilibrium prices competition: Profits for the two firms

Knowing that firm 2 will determine p2 by using Firm 1 sells less than firm 2 does: g(p ), firm1 tries to maximize its profit: 1 x1* = 93.34 and x2* = 96.48 max (p1 - 20)[180 - 1.5p1 + 0.5(70 + p1/6)] Firm 1’s profit, u1* = (93.34)(85.88 - 20) Profit maximizes when the first order condition is = $ 6149.24 satisfied: 0 = 215 - (17/12)p1 + (p1 - 20) (-17/12) Firm 2’s profit, u2* = (96.48)(84.31 - 20) ∴ p1* = 2920/34 = $85.88 = $ 6204.63 Firm 2, which moves last, charges slightly lower Firm 2, the second mover, makes more price than p *: 1 money p2* = 70 + p1* /6 = 70 + $14.31 = $84.31 59 60 This Offer is Good for a An example of “This offer is good Limited Time Only for a limited time only” aThe credibility problems behind the aExploding job offers marketing slogan `An early job offer with a very short aThe principle of costly commitment time to decide on whether to take the aIndustries where the slogan is credible job. `Risk-averse people often end up accepting inferior job offers

61 62

Ultimatum Games in the Laboratory aGames with take-it-or-leave-it structure aIn experiments, subjects playing such games rarely play subgame perfect equilibria aThe nice opponent explanation vs. the expected payoff explanation

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