bertrand model game theory prisoner’s dilemma dominant strategies repeated games oligopoly and...

• Bertrand Model

• Game Theory• Prisoner’s Dilemma• Dominant Strategies• Repeated Games

Oligopoly and Game Theory

Bertrand Model

• Competition based on setting prices—not quantities (like Cournot)

• Two variants:• Homogeneous goods• Differentiated goods

Bertrand Model with Homogeneous Products

Market demand curve: P = 30 – Q

Q = q1 + q2

MC1 = MC2 = 3

Good is homogeneous Buyers only careabout price

Outcome: Both firms will charge $3Total output will be Q = 27 q1 = q2 = 13.5

π1 = π2 = 0

Bertrand Model with Differentiated Products

Firm 1’s Demand: Q1 = 12 – 2P1 + P2

Firm 2’s Demand: Q2 = 12 – 2P2 + P1

TFC = $20

TVC = 0

π1 = P1Q1 – 20 = 12P1 – 2P12 + P1P2 – 20

Δπ1 / ΔP1 =12 – 4P1 + P2 = 0

Firm 1’s reaction curve: P1 = 3 + ¼P2 Firm 2’s reaction curve: P2 = 3 + ¼P1

Bertrand Model with Differentiated Products

Firm 1’s reaction curve: P1 = 3 + ¼P2 Firm 2’s reaction curve: P2 = 3 + ¼P1

To find the Nash Equilibrium:

P1 = 3 + ¼P2 = 3 + ¼(3 + ¼P1)

3 1

4 16 13 P 15 15

16 41P 1P 4

Nash Equilibrium in a Bertrand Model with Differentiated Products

P1

P2 $4

$4

Firm 2’s reaction curve

Firm 1’s reaction curve

Nash Equilibrium

What if Firm 1 and 2 Could Collude?

π1 = 12P1 – 2P12 + P1P2 – 20

π2 = 12P2 – 2P22 + P1P2 – 20

πT = 24P – 4P2 + 2P2 – 40

= 24P – 2P2 – 40ΔπT / ΔP = 24 – 4P

= 0

P* = 6πT = 24(6) – 2(62) – 40

πT

= 32 π1 = π2 = 16

Components of a Game

• Players

Example: Coke and Pepsi

Components of a Game

• Players

Example: Coke and Pepsi

• Strategies for Each PlayerExample: Spend a little (small) or a

lot

(large) on advertising

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

Small

Large

Components of a Game

• Players

Example: Coke and Pepsi

• Strategies for Each PlayerExample: Spend a little (small) or a

lot

(large) on advertising

• Payoffs

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

Large

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

Large

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

Large

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

Large

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Prisoner’s Dilemma Situation

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Nash Equilibrium

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Equilibrium if Players Could Collude

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Coke’s Dominant Strategy

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Pepsi’s Dominant Strategy

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Outcome of the Game

Pepsi’s SpendingOn Advertising

Small Large

Coke’s Spending on Advertising

SmallπC = +8

πP = +8

πC = −2

πP = +13

LargeπC = +13

πP = −2

πC = +3

πP = +3

Repeated Games● repeated game Game in which actions are taken and payoffs received over and over again.

PRICING PROBLEM

Firm 2

Low price High price

Firm 1Low price 10, 10 100, –50

High price –50, 100 50, 50

Suppose this game is repeated over and over again—for example, you andyour competitor simultaneously announce your prices on the first day of everymonth. Should you then play the game differently? 13.8

TIT-FOR-TAT STRATEGY

In the pricing problem above, the repeated game strategy that works best is the tit-for-tat strategy.

● tit-for-tat strategy Repeated-game strategy in which a player responds in kind to an opponent’s previous play, cooperating with cooperative opponents and retaliating against uncooperative ones.