Game%20Theory

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Game Theory Overheads
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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, 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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Structure of a Game
Rules of the game
Who moves when
What do players know and when
What actions are available at various points
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Structure of a Game
Outcomes
For each possible set of actions by the
players, what is the outcome of the 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?
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Structure of a Game
Players or decision makers
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
Nash equilibrium if, holding the strategies of
all other firms constant, no firm can obtain
a higher payoff by choosing a different strategy
In a Nash equilibrium, no firm wants to change
its strategy
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The Prisoners Dilemma
Two criminals
Al Scarface Capone
Jack Machine Gun McGurn
Major crime
Murder of 7 people on St. Valentines 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 dont talk
You both stay mum, we lock the both of you up 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
Machine Gun
Stay Mum
Collaborate
Al gets 20 Jack gets 20
Al walks Jack gets 40
Collaborate
Scarface
Al gets 40 Jack walks
Al gets 1 Jack gets 1
Stay Mum
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Structure of a Game
Rules of the 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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Structure of a Game
Outcomes
Al gets 20 Jack gets 20
Al walks Jack gets 40
Al gets 40 Jack walks
Al gets 1 Jack gets 1
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Structure of a Game
Payoffs
What are the payoffs with each outcome?
Years in the slammer for the player
How do the player's rank the outcomes?
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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Jack stays mum
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Al gets the least years by collaborating
What if Jack collaborates?
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Jack collaborates
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Al gets the least years by collaborating
Al has found a dominant strategy for this game
Al sells out and collaborates
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Best response for Machine Gun McGurn
What if Al stays mum?
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Al stays mum
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Jack gets the least years by collaborating
What if Al collaborates?
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Al collaborates
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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
matter what strategy is chosen by the competing
player is called a dominant strategy
A dominant strategy is a strategy that is best
for a player regardless of the strategy of the
other player
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If a player has a dominant strategy in a game,
we can assume that the player will play that
strategy
If each player in a game has a dominant
strategy, it is easy to find the equilibrium of
the game
It is simply the outcome that occurs when
each player plays the dominant strategy
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Equilibrium in the prisoners 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 supplement in
the local paper stating a price for turkeys
The stores do not know the price that will
be submitted by the other store
Two possible actions for each store
Low Price
High Price
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Outcome Matrix
Hy-Vee
High Price
Low Price
F ? 1000 H ? 1000
F ? 1500 H ? 500
Low Price
Fareway
F ? 500 H ? 1500
F ? 1200 H ? 1200
High Price
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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
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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
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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
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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
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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 and total
profits would be higher
The outcome (high, high) is Pareto superior to
the outcome (low, low)
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An outcome is said to be Pareto superior to
another outcome if both both players are better
off with the Pareto superior outcome
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Other examples of the prisoners 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 doesnt know other firms quantity
Outcome is market price
Payoff to each firm is its profit given price and
costs
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Outcome Matrix
Ken
3 boxes
4 boxes
H ? 36 K ? 36
H ? 30 K ? 40
3 boxes
Hank
H ? 40 K ? 30
H ? 32 K ? 32
4 boxes
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Ken chooses 3
Ken
3 boxes
H ? 36 K ? 36
3 boxes
Hank
H ? 40 K ? 30
4 boxes
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Ken chooses 4
Ken
4 boxes
H ? 30 K ? 40
3 boxes
Hank
H ? 32 K ? 32
4 boxes
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Hank should always choose 4 boxes
Now what about Ken?
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Hank chooses 3
Ken
3 boxes
4 boxes
H ? 36 K ? 36
H ? 30 K ? 40
3 boxes
Hank
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Hank chooses 4
Ken
3 boxes
4 boxes
Hank
4 boxes
H ? 40 K ? 30
H ? 32 K ? 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 agrees to
coordinate its activities is called a cartel
A cartel that includes all the firms in
an industry 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
of collusion 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
explicit agreement, they are engaging in tacit
collusion
Typically, players adopt strategies along the
following lines
In general, I will set a high price
If my rival sets a low price this time, I
will punish 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
both players will set a high price over time
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Another type of tacit collusion occurs when
there is 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