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