# Game Theory - Essay - Economics

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Small Numbers and Strategic Behavior • Fun and games with a duopoly example – Simultaneous vs. sequential choice – One-time vs. repeated game
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Microsoft PowerPoint - Lecture 12 - 2004 - Presentation Handout - Game Theory and Competitive Strategy I.ppt

Overview: Game Theory and Competitive Strategy I

Small Numbers and Strategic Behavior • Fun and games with a duopoly example

– Simultaneous vs. sequential choice – One-time vs. repeated game – Quantity vs. price as the choice variable – Homogeneous vs. differentiated good

• Review of the analytics

Key Ideas

• Know strategic situation (What is the game?).

• Your competitor is just as smart as you are! • Think about the response of others • Nash equilibrium: all participants do the

best they can, given the behavior of competitors.

The Game (a)

• Objective: Max. your profit • # of plays: 1 only • Good: Homogeneous • Choice variable: Quantity • Timing of choice: Simultaneous

Game Payoffs

Firm 2 (competitor)

15 20

Firm 1 15

(you) 20

Game Payoffs Firm 2 (competitor)

15 20 22.5 30

15

20

22.5

Firm 1

(you)

30

450, 450 375, 500 338, 506 225, 450

500, 375 400, 400 350, 394 200, 300

506, 338 394, 350 338, 338 125, 150

450, 225 300, 200 150, 125 0, 0

The Game (a*)

• Objective: Max. your profit • # of plays: 2 • Good: Homogeneous • Choice variable: Quantity • Timing of choice: Simultaneous

The Game (a**)

• Objective: Max. your profit • # of plays: 10 • Good: Homogeneous • Choice variable: Quantity • Timing of choice: Simultaneous

Analytics: Simultaneous Cournot • Homogeneous good, simultaneous choice • Choosing quantity, Q • Objective: Max. your profit • Market demand:

P = 60 - Q • Production:

Q = Q1 + Q2 MC1 = MC2 = 0

What Is the Firm’s Reaction Curve? (Firm 1 example)

• To max profit, set MR = MC R1 = PQ1 = (60 - Q)Q1

= 60Q1 - (Q1 + Q2)Q1 = 60Q1 - (Q1)2 - Q2Q1

Q

MR1 = dR1/dQ1 = 60 - 2Q1 - Q2 Set MR1 = MC = 0, which yields

1 = 30 - ½ Q2 (Firm 1 reaction curve)

Cournot Equilibrium

Q Q

• Symmetric reaction curves:

1 = 30 - 1/2 Q2 (Firm 1)

2 = 30 - 1/2 Q1 (Firm 2) • Equilibrium: Q1 = Q2 = 20 • Total quantity: Q = Q1 + Q2 = 40 • Price: P = 60 - Q = 20 • Profits: Π1 = Π2 = 20·20 = 400

Duopoly: Graphical Version

Q2

Firm 2’s Reaction Curve

15 20 30 60 Collusive Outcomes Q1

30

60

15 20

Firm 1’s Reaction Curve

Cournot Equilibrium

Duopoly Analytics -- Collusion Demand: P = 60 – Q

Π = P · Q - Costs = (60 - Q)·Q

dΠ = 60 − 2Q = 0

dQ

⇒ Q = Q1 + Q2 = 30, P = 30

Total joint Π = 30(30) = 900 If split equally, Π1 = Π2 = 450

The Game (b)

• Objective: Max. your profit • # of plays: 1 • Good: Homogeneous • Choice variable: Q • Timing of choice: Someone goes first

Game Payoffs Firm 2 (competitor)

15 20 22.5 30

15

20

22.5

Firm 1

(you)

30

450, 450 375, 500 338, 506 225, 450

500, 375 400, 400 350, 394 200, 300

506, 338 394, 350 338, 338 125, 150

450, 225 300, 200 150, 125 0, 0

Analytics with a First Mover (Decision variable is Q)

• Suppose Firm 1 moves first • In setting output, Firm 1 should consider

how Firm 2 will respond • We know how Firm 2 will respond! It will

follow its Cournot reaction curve: Q2 = 30 - 1/2 Q1

• So Firm 1 will maximize taking this information into account

First Mover: Max Π given the Reaction of the Follower

• Firm 1 revenue: R1 = Q1P = Q1(60 - [Q1 + Q2])

= 60Q1 - (Q1)2 - Q1Q2 = 60Q1 - (Q1) 2 - Q1 1)

Firm 2’s Reaction

(30 - ½ Q 2= 30Q1 - ½ (Q1)

• Firm 1 marginal revenue: MR1 = dR1/dQ1 = 30 - Q1

First Mover - The Result

• Firm 1 marginal revenue: MR1 = 30 - Q1

Q Q

• Set MR1 = MC (= 0), and 1 = 30 2 = 30 - ½ Q1 = 15

• Price: P = 60 - (Q1 + Q2) = 15

Π • Profits: Π1 = 30 ·15 = 450

2 = 15 ·15 = 225

The Game (c)

• Objective: Max. your profit • # of plays: 1 • Good: Homogeneous • Choice variable: Price • Timing of choice: Simultaneous

Strategic Substitutes vs Complements

• Strategic Complement: reactions match – e.g. lower price is reaction to competitor’s lower price

• Strategic Substitute: opposite reactions – e.g. lower quantity is reaction to competitor’s higher quantity

• Competition tends to be more aggressive with strategic complements than with substitutes.

The Game (c*)

• Objective: Max. your profit • # of plays: 1 • Good: Differentiated • Choice variable: Price • Timing of choice: Simultaneous

Take Away Points

• Game theory allows the analysis of situations with interdependence.

• Nash Equilibrium: Each player doing the best he/she can, given what the other is doing.

• Competition in strategic complements (price) tends to be tougher than in substitutes (quantity).

• Commitment is important since you change the rules of the game. It can lead to a first-mover advantage.

• Repetition can lead to cooperation, but only when the end-game is uncertain or far away.

Preparation for Next Time

Regarding “Lesser Antilles Lines” Case:

• Good case for developing game and payoff analysis (assumptions, payoffs, etc.).

• You do NOT need to prepare this for class (part of Problem Set 5).