game theory overheads
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Game Theory Overheads. Games. A game is a formal representation of a situation in which a number of decision makers (players) interact in a setting of strategic interdependence. By that, we mean that the welfare of each decision maker depends not only on her own actions, - PowerPoint PPT PresentationTRANSCRIPT
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Game Theory
Overheads
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A game is a formal representation of a situationin which a number of decision makers (players)interact in a setting of strategic interdependence
Games
By that, we mean that the welfare of each decisionmaker depends not only on her own actions,but also on the actions of the other players
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Structure of a Game
Players or decision makers
firms
consumers
poker or chess players
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Rules of the game
Structure of a Game
Who moves when
What do players know and when
What actions are available at various points
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For each possible set of actions by the players,what is the outcome of the game?
Structure of a Game
Outcomes
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Structure of a Game
Payoffs
What are the payoffs with each outcome?
How do the player's rank the outcomes?
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Players or decision makers
Structure of a Game
Rules of the game
Outcomes
Payoffs
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Nash Equilibrium and Game Theory
A set of strategies in a game is called a Nashequilibrium if, holding the strategies of allother firms constant, no firm can obtain ahigher payoff by choosing a different strategy
In a Nash equilibrium, no firm wants tochange its strategy
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The Prisoner’s Dilemma
Two criminals
Al “Scarface” Capone
Jack “Machine Gun” McGurn
Major crime
Murder of 7 people on St. Valentine’s Day
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Minimal evidence connecting them to massacre
Plenty of evidence linking them to bootlegging
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Police interrogate them in separate rooms
Police offer them each a deal
Finger the other guy, and walk if he don’t talk
You both stay mum, we lock the both of youup for a year for bootlegging
You both talk, its 20 years in the slammer
But if he talks and you give us the silent treatment,its 40 years at hard labor for you
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The Game in Matrix Form
Scarface
Stay Mum
Machine Gun
Al gets 20Jack gets 20
Collaborate
Stay Mum
CollaborateAl walksJack gets 40
Al gets 40Jack walks
Al gets 1Jack gets 1
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Rules of the game
Structure of a Game
Who moves when -- at the same time
What do players know and when -- nothing
What actions are available at various points
Collaborate
Stay mum
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Outcomes
Al gets 20Jack gets 20
Scarface
Stay Mum
Machine Gun
Collaborate
Stay Mum
CollaborateAl walksJack gets 40
Al gets 40Jack walks
Al gets 1Jack gets 1
Structure of a Game
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Structure of a Game
Payoffs
What are the payoffs with each outcome?
How do the player's rank the outcomes?
Years in the slammer for the player
Players prefer less years in the pen
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Best response for Al “Scarface” Capone
What if Jack stays mum?
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Stay Mum
Machine Gun
Scarface
Stay Mum
CollaborateAl walksJack gets 40
Al gets 1Jack gets 1
Jack stays mum
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Al gets the least years by collaborating
What if Jack collaborates?
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Scarface
Stay Mum
Collaborate
Jack collaborates
Machine Gun
Collaborate
Al gets 20Jack gets 20
Al gets 40Jack walks
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Al has found a dominant strategy for this game
Al sells out and collaborates
Al gets the least years by collaborating
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Best response for “Machine Gun” McGurn
What if Al stays mum?
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Scarface
Stay Mum
Al stays mum
Al gets 40Jack walks
Al gets 1Jack gets 1
Machine Gun
Stay MumCollaborate
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Jack gets the least years by collaborating
What if Al collaborates?
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Al collaborates
Machine Gun
Stay MumCollaborate
Scarface
CollaborateAl gets 20Jack gets 20
Al walksJack gets 40
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Jack gets the least years by collaborating
Jack has found a dominant strategy for this game
Jack sells out and collaborates
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In fact
Both Al and Jack sell out and get 20 years
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Dominant strategies
A strategy that is best for a player, no matterwhat strategy is chosen by the competing playeris called a dominant strategy
A dominant strategy is a strategy that is best for aplayer regardless of the strategy of the other player
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If each player in a game has a dominant strategy,it is easy to find the equilibrium of the game
If a player has a dominant strategy in a game, wecan assume that the player will play that strategy
It is simply the outcome that occurs when eachplayer plays the dominant strategy
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Equilibrium in the prisoner’s dilemma
Al and Jack get 20 years
If both stayed mum, each would get 1 year
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The Turkey Pricing game
Two stores
Hy-Vee
Fareway
Rules of the game
Game played 7 days before Thanksgiving
Turkeys are the same quality
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Stores place a weekly advertising supplementin the local paper stating a price for turkeys
Two possible actions for each store
Low Price
High Price
The stores do not know the price that will besubmitted by the other store
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Outcome Matrix
Fareway
High Price
Hy-Vee
F 1000H 1000
Low Price
High Price
Low PriceF 1500H 500
F 500H 1500
F 1200H 1200
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Payoffs
Each store gets the number in the table in dollars
Store managers prefer more dollars to less
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Best response for Fareway
What if Hy-Vee chooses a low price?
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Hy-Vee chooses low
Hy-Vee
Low Price
Fareway
High Price
Low PriceF 1000H 1000
F 500H 1500
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Fareway gets the most dollars with a low price
What if Hy-Vee chooses a high price?
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Hy-Vee chooses high
Fareway
High Price
Low Price
Hy-Vee
High Price
F 1500H 500
F 1200H 1200
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Fareway gets the most money with a low price
Fareway has a dominant strategy for this game
Fareway chooses a low price
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Best response for Hy-Vee
What if Fareway chooses a low price?
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Fareway chooses low
Fareway
Low Price
Hy-Vee
Low Price High Price
F 1000H 1000
F 1500H 500
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Hy-Vee gets the most dollars with a low price
What if Fareway chooses a high price?
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Fareway chooses high
Hy-Vee
Low Price High Price
Fareway
High PriceF 500H 1500
F 1200H 1200
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Hy-Vee gets the most money with a low price
Hy-Vee has a dominant strategy for this game
Hy-Vee chooses a low price
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Equilibrium in the turkey pricing game
Hy-Vee and Fareway choose low prices
If both chose high prices, individual andtotal profits would be higher
The outcome (high, high) is Pareto superiorto the outcome (low, low)
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An outcome is said to be Pareto superiorto another outcome if both both players are better off with the Pareto superior outcome
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Other examples of the prisoner’s dilemma
Arms race
Advertising
tobacco
eye-glasses
Common resources
oil field
pasture
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Noncooperative oligopoly as a game
Zucchini pricing game
2 players or firms -- Hank and Ken
Actions: number of zucchini brought to market
Firm doesn’t know other firm’s quantity
Outcome is market price
Payoff to each firm is its profit given price and costs
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Outcome Matrix
Hank
Ken
H $36K $36
4 boxes
3 boxesH $30K $40
H $40K $30
H $32K $32
3 boxes 4 boxes
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Ken chooses 3
Hank
Ken
H $36K $36
4 boxes
3 boxes
H $40K $30
3 boxes
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Ken chooses 4
Hank
Ken
4 boxes
3 boxesH $30K $40
H $32K $32
4 boxes
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Hank should always choose 4 boxes
Now what about Ken?
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Hank chooses 3
Hank
Ken
H $36K $363 boxes
3 boxes
H $30K $40
4 boxes
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Hank chooses 4
Hank
Ken
3 boxes 4 boxes
H $40K $30
4 boxes H $32K $32
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Ken should always choose 4 boxes
Both of them should always choose 4 boxes
The equilibrium is (4,4)
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Cooperative Oligopoly
Explicit collusion
An association of firms that explicitly agreesto coordinate its activities is called a cartel
A cartel that includes all the firms in anindustry is, in effect, a monopoly
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Zucchini Example with Collusion
Explicit collusion will lead to (3,3)
Profits are ($36, $36) as compared to ($32, $32)
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But the cartel will collapse
Why?
If Hank produces 3, Ken will want to produce 4
And vice versa
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Members of cartels always have an incentive to cheat
Why?
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Explicit collusion is illegal in many countries
Firms must therefore consider the benefits ofcollusion versus the costs of getting caught
OPEC was a very successful cartel for many years
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Tacit Collusion
Any time firms cooperate without an explicitagreement, they are engaging in tacit collusion
Typically, players adopt strategies alongthe following lines:
“In general, I will set a high price”
“If my rival sets a low price this time, I willpunish him by setting a low price next time."
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This is called a tit for tat strategy
The idea is that the rival will catch on and bothplayers will set a high price over time
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Another type of tacit collusion occurs when thereis an acknowledged price leader in the industry
This leader firm acts first and the others follow
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When cheating on the cartel is likely
It is difficult to observe other firms’ prices
Market demand is unstable
There are a lot of other firms in the industry
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The End