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
Game Theory
Overheads
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
Structure of a Game
Players or decision makers
firms
consumers
poker or chess players
Rules of the game
Structure of a Game
Who moves when
What do players know and when
What actions are available at various points
For each possible set of actions by the players,what is the outcome of the game?
Structure of a Game
Outcomes
Structure of a Game
Payoffs
What are the payoffs with each outcome?
How do the player's rank the outcomes?
Players or decision makers
Structure of a Game
Rules of the game
Outcomes
Payoffs
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
The Prisoner’s Dilemma
Two criminals
Al “Scarface” Capone
Jack “Machine Gun” McGurn
Major crime
Murder of 7 people on St. Valentine’s Day
Minimal evidence connecting them to massacre
Plenty of evidence linking them to bootlegging
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
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
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
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
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
Best response for Al “Scarface” Capone
What if Jack stays mum?
Stay Mum
Machine Gun
Scarface
Stay Mum
CollaborateAl walksJack gets 40
Al gets 1Jack gets 1
Jack stays mum
Al gets the least years by collaborating
What if Jack collaborates?
Scarface
Stay Mum
Collaborate
Jack collaborates
Machine Gun
Collaborate
Al gets 20Jack gets 20
Al gets 40Jack walks
Al has found a dominant strategy for this game
Al sells out and collaborates
Al gets the least years by collaborating
Best response for “Machine Gun” McGurn
What if Al stays mum?
Scarface
Stay Mum
Al stays mum
Al gets 40Jack walks
Al gets 1Jack gets 1
Machine Gun
Stay MumCollaborate
Jack gets the least years by collaborating
What if Al collaborates?
Al collaborates
Machine Gun
Stay MumCollaborate
Scarface
CollaborateAl gets 20Jack gets 20
Al walksJack gets 40
Jack gets the least years by collaborating
Jack has found a dominant strategy for this game
Jack sells out and collaborates
In fact
Both Al and Jack sell out and get 20 years
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
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
Equilibrium in the prisoner’s dilemma
Al and Jack get 20 years
If both stayed mum, each would get 1 year
The Turkey Pricing game
Two stores
Hy-Vee
Fareway
Rules of the game
Game played 7 days before Thanksgiving
Turkeys are the same quality
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
Outcome Matrix
Fareway
High Price
Hy-Vee
F 1000H 1000
Low Price
High Price
Low PriceF 1500H 500
F 500H 1500
F 1200H 1200
Payoffs
Each store gets the number in the table in dollars
Store managers prefer more dollars to less
Best response for Fareway
What if Hy-Vee chooses a low price?
Hy-Vee chooses low
Hy-Vee
Low Price
Fareway
High Price
Low PriceF 1000H 1000
F 500H 1500
Fareway gets the most dollars with a low price
What if Hy-Vee chooses a high price?
Hy-Vee chooses high
Fareway
High Price
Low Price
Hy-Vee
High Price
F 1500H 500
F 1200H 1200
Fareway gets the most money with a low price
Fareway has a dominant strategy for this game
Fareway chooses a low price
Best response for Hy-Vee
What if Fareway chooses a low price?
Fareway chooses low
Fareway
Low Price
Hy-Vee
Low Price High Price
F 1000H 1000
F 1500H 500
Hy-Vee gets the most dollars with a low price
What if Fareway chooses a high price?
Fareway chooses high
Hy-Vee
Low Price High Price
Fareway
High PriceF 500H 1500
F 1200H 1200
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
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)
An outcome is said to be Pareto superiorto another outcome if both both players are better off with the Pareto superior outcome
Other examples of the prisoner’s dilemma
Arms race
Advertising
tobacco
eye-glasses
Common resources
oil field
pasture
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
Outcome Matrix
Hank
Ken
H $36K $36
4 boxes
3 boxesH $30K $40
H $40K $30
H $32K $32
3 boxes 4 boxes
Ken chooses 3
Hank
Ken
H $36K $36
4 boxes
3 boxes
H $40K $30
3 boxes
Ken chooses 4
Hank
Ken
4 boxes
3 boxesH $30K $40
H $32K $32
4 boxes
Hank should always choose 4 boxes
Now what about Ken?
Hank chooses 3
Hank
Ken
H $36K $363 boxes
3 boxes
H $30K $40
4 boxes
Hank chooses 4
Hank
Ken
3 boxes 4 boxes
H $40K $30
4 boxes H $32K $32
Ken should always choose 4 boxes
Both of them should always choose 4 boxes
The equilibrium is (4,4)
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
Zucchini Example with Collusion
Explicit collusion will lead to (3,3)
Profits are ($36, $36) as compared to ($32, $32)
But the cartel will collapse
Why?
If Hank produces 3, Ken will want to produce 4
And vice versa
Members of cartels always have an incentive to cheat
Why?
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
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."
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
Another type of tacit collusion occurs when thereis an acknowledged price leader in the industry
This leader firm acts first and the others follow
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
The End