gametheory.ppt
TRANSCRIPT
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GAME THEORY
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Game Theory
• Basic Concepts
- The payoff matrix
- The Nash Equilibrium
- Dominant Strategies
- Dominated Strategies- Maximin Strategies
- Mixed Strategies
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Game Theory and Oligopoly
• Non co-operative games :Prisoner’s
Dilemma
• Cooperative Games: Enforcing a Cartel
• Repeated games : dealing with cheaters
• Sequential games :The advantage of being
first
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Game Theory and Economics
Game theory is the study of how people
behave in strategic situations.Strategic decisions are those in which
each person, in deciding what actions to
take, must consider how others mightrespond to that action.
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Game Theory and Economics
If the market is composed by a small number of firms, each firm must act strategically.
Each firm affects the market price changing the
quantity produced.
Suppose 2 firms are producing 100 units.
If one of the firms decides to increase the
production by 10 units.
The market supply will increase from 200 to 210
and the price has to drop to reach an equilibrium.
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Game Theory and Economics
Therefore, it also affects the profits of other
firms.
Each firm knows that its profit depends notonly on how much it produced but also on
how much the other firms produce.
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What is a Game?
A game is a situation where the
participants’ payoffs depend not only on
their decisions, but also on their rivals’decisions.
This is called Strategic Interactions:
My optimal decisions will depend on whatothers do in the game.
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A Game
Four elements to describe a game:
players;
rules: when each player moves, what are the
possible moves, what is known to each player
before moving;
outcomes of the moves;
payoffs of each possible outcome: how muchmoney each player receive for any specific
outcome.
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A situation of competitive rivalry must involve two or more players whose choice of actions affect eachother.
Players and their actions
•A “player” can be a firm, an interest group or coalition,
a military leader, government official.
•Games generally consider only one kind of action —
e.g., number of daily departures, fares, in-flight services,schedules, advertising, choice of hubs, ordering planes,
expanding terminals, use of computerized reservations
systems, mergers and acquisitions, and human resource
decisions.
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Outcomes and Payoffs
The firm’s action, together with the actions ofits rivals, determine its payoff
•In the standard “business” game, the payoff can
be in the form of profit, market share, ratings
points,
•In war games, the payoff might be measured in
enemy killed or territory seized.
•In political games, payoffs may be measured in
votes or campaign contributions.
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Underlying “rules”
The rules of the game define the range ofpossible outcomes and payoffs
•For example, collusion to fix prices or a merger
among direct rivals in a concentrated market
structure may be against the rules.
•Another set of rules specifies whether players move
sequentially or simultaneously, who moves first, and
what does each player know about the other players’
preference and prior to actions?
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Game theory in business
In applying game theory to the behaviour of firms we can suggest that
firms face a number of strategic choices which govern their ability to
achieve a desired pay-off, including:
Decisions on price and output, such as whether to:
• Raise• Lower
• Hold
Decisions on products, such as whether to:
• Keep existing products
• Develop new ones
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.
Decisions on promoting products, such as whether to:
• Spend more on advertising
• Spend less
• Keep spending constant
Firms could derive a range of possible pay-offs from theirstrategy choices, including:
• More profits for shareholders
• Greater market share
• Improved chances of survival• Getting rid of a rival
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Matching pennies
Each player selects one side of a coin;
if the coins match player 1 wins and gets 1dollar from player 2;
if the coins don’t match player 2 wins and
gets 1 dollar from player 1.
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Matrix Representation of
Matching Pennies
Player 2
Head Tail
Player 1
Head 1,-1 -1,1
Tail -1,1 1,-1
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Boeing-Airbus game
Boeing and Airbus have to decide whether toinvest in the development of a Super Jumbo for long distance travel;
if they both develop successfully the new plane,
their profits will drop by 50 millions a year;if only one develop the Super Jumbo, it will
make 80 millions a year in additional profits,
whereas the profits of the other firm will drop by30 millions a year;
if no firm develops the plane, nothing changes.
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Matrix Representation of
Boeing-Airbus game
Airbus
Develop Do notdevelop
Boeing
Develop -50,-50 80,-30
Do not
develop-30,80 0,0
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Solutions of the Games
To predict what will be the
solution/outcome of the game we needsome tools:
dominated and dominant strategies;
Nash equilibrium.
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Prisoners Dilemma
Two individuals have been arrested for possession of guns. The police suspects thatthey have committed 10 bank robberies;
if nobody confesses the police, they will be jailed for 2 years.
if only one confesses, he’ll go free and his
partner will be jailed for 40 years.if they both confess, they get 16 years
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Matrix Representation of
Prisoners Dilemma
B
Confess Do notConfess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
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We want to predict the outcome of the game
B
ConfessDo not
Confess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
Suppose that C decides to confess. What is the best decision for B?
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We want to predict the outcome of the game
B
ConfessDo not
Confess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
Suppose that C decides to remain silent. What is the best decision
for B?
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We want to predict the outcome of the game
B
ConfessDo not
Confess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
Suppose that B decides to confess. What is the best decision for C?
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We want to predict the outcome of the game
B
ConfessDo not
Confess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
Suppose that B decides to remain silent. What is the best decision
for C?
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Outcome of the Game
B
ConfessDo not
Confess
C
Confess 16,16 0,40
Do not
Confess40,0 2,2
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Dominated and Dominant Strategy
Dominant Strategy:
a strategy that gives best possible payoffs no
matter what the opponent does;Dominated Strategy:
a strategy is dominated if there exists another strategy that is dominant.
So far we have only assumed that each player is rational to determine the outcomeof the game.
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Nash Equilibrium
A list of strategies, one for each player, is a Nashequilibrium if each player’s strategy maximizes his
(or her) payoff given the strategies selected by the
other players and if this condition holds for all
players simultaneously.
Any outcome that is best response for both
players is a Nash equilibrium
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Nash Equilibrium
The decisions of the players are a Nash
Equilibrium if no individual prefers a
different choice.In other words, each player is choosing the
best strategy, given the strategies chosen by
the other players.
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Dominant strategy
Confess is the dominant strategy in this case,
since it gives the shortest sentence
irrespective of whether the other prisoner selects the “confess” or “do not confess”
strategy
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Price wars in a duopoly
The preceding is what we call a “non-
cooperative” game. Cooperation among
duopolists is a strategy that maximizes joint profits. So why do price wars break out?
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The payoff matrix for a running
shoe duopoly
HighPrice LowPrice
High
Price
$10 million,
$10 million
$5 million,
$12 million
LowPrice
$12 million,$5 million
$7 million,$7 million
REEBOK
NIKE
Notice that “low price” is the
dominant strategy
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Pizza Planet and Luigi’s are rivals in the
market for home-delivered pizza. Each rivalseeks to gain an advantage through advertising(product differentiation).
Advertising is presumed NOT to affect marketdemand--only market share.
Market share depends on the intensity of advertising relative to one’s rival.
Advertising rivalry
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Let
P = $15
Q = 100 pizzas (market quantity-demanded)
AC (w/o advertising expense) = $5.
Hence:
/Pizza = (TR - TC)/Q = ($1500 - $500)/100 = $10
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If neither seller advertises, each will
sell 50 pizzas and earn a profit of $500. However, advertising could
potentially increase sales to 75
pizzas.
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The payoff matrix for
a pizza duopoly
Low
Advertising
High
Advertising
Low
Advertising
$400,
$400
$150,
$550
High
Advertising
$550,
$150
$300,
$300
LUIGI’S
PIZZA PLANET
Notice that “high advertising” is the
dominant strategy
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Nash Equilibrium and the Shoe,
Pizza Duopoly Games•A Nash equilibrium is given by the “low-
low” strategy in the running shoe duopoly
game •A Nash equilibrium is given by the “high-
high strategy” in the pizza duopoly game.
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Dominated Strategy
• Example
• Example of an iterated deletion of
dominated strategy equilibrium.doc
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Maximin Strategy
• This strategy is based on the fact that
sometimes it is more prudent to maximize
the minimum gains achievable in a gamingsituation. Let's take the Meat and Potatoes
example to appreciate this strategy.
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Dominant Strategy -? ;
Nash Equilibrium - ?
Nash Equilibriumis based on the fact
that both M and P
behave rationally.
However, can Palways count on M
being rational?
Can P take the risk of selling potatoes if he is conservative and afraid
that M may not behave rationally ? No.
So, what does P do? P will sell meat, for he is assured of making at least
$80,000. In this scenario, P is pursuing a Maximin Strategy of
maximizing the minimum gains that can be earned.
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Mixed Strategies
• A mixed strategy is an assignment of a
probability to each pure strategy. This
allows for a player to randomly select a pure strategy.
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• ..\Reference\mixed strategies.pdf
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Game theory and Oligopoly
• Non Cooperative Games
• Cooperative Games
• Repeated Games
• Sequential games
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Non Cooperative games• The Prisoner’s Dilemma model
• This model explains a number of interesting
phenomena in business
Example:
30, 30 10,40
40,10 20,20
Firm 2
Low- level Advg High level Advg
Low level advg
High level advgFirm 1
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Cooperative Games: Enforcing a Cartel
The experience of the pan-masala industry during the 1980s is an
interesting example of how a non cooperative game was turned, bygovernment policy, into a cooperative game. For many years, the
major pan-masala manufacturers had been spending large sums on TV
advertising to promote their products. Much of this advertising was
mutually offsetting, so the firms probably were in the Prisoner’s
Dilemma. They would have been better off if they had reduced their advertising expenditures, but no one firm could afford to do so unless
there was some assurance that their lead would be followed by the
other firms.
In mid-1980s, the government banned advertising of pan-masala on
Doordarshan. Initially, the companies fought the ban, but it soon became apparent that any lost sales were more than made up by the
savings in advertising. With this ban, the government did for pan-
masala manufacturers what they were unable to do for themselves-
enforce a reduction in advertising expenditure.
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Repeated Games- Dealing with
cheaters• What is the optimal strategy for firms playing non
cooperaive repeated games?
AN approach described as “tit-for-tat”
TIT FOR TAT is a very simple strategy, which
cooperates on the first round, and thereafter
simply copies what its opponent did on the
previous round — thereby rewarding cooperationwith more cooperation and punishing defection
with defection.
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Sequential Games
• The advantage of being first
No new
product
Introduce new
product
No New product 2,2 (5), 10
Introduce new
product 10, (5) (7), (7)Firm 1
Firm2
Assumption: Maximin Strategy