Chapter 12
Game Theory
Presented by Nahakpam
1213504
PhD Student
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Brief Contents
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•
•
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Game Theory
Strong or weak pig
Prisoner’s Dilemma
System for studying strategic behavior
– Positive
– Normative
• Pareto Optima
• Sequential Games
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What is Game theory?
• Game theory is a study of strategic decision
making.
• 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 might respond 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.
• Therefore, it also affects the profits of other
firms.
• Each firm knows that its profit depends not only
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 what others
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 much
money each player receive for any specific
outcome.
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Boeing-Airbus game
• Boeing and Airbus have to decide whether to
invest 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 by
30 millions a year;
• If no firm develops the plane, nothing changes.
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Matrix Representation of BoeingAirbus game
Airbus
Boeing
Develop
Do not
develop
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 need
some tools:
– dominated and dominant strategies;
– Nash equilibrium.
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Prisoners Dilemma
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Prisoners Dilemma
• Tom and Mary have been arrested for
possession of guns.
• The police suspects that they have committed
10 bank robberies.
• If nobody confesses the police, they will be
jailed for 2 years.
• If only one confesses, she’ll go free and her
partner will be jailed for 40 years.
• If they both confess, they get 16 years.
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Matrix Representation of Prisoners
Dilemma
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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We want to predict the outcome of the game
Suppose that Tom decides to confess. What is the best decision for
Mary?
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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We want to predict the outcome of the game
Suppose that Tom decides to remain silent. What is the best
decision for Mary?
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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Dominated and Dominant Strategy
• Dominant Strategy:
– a strategy that gives higher 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 outcome
of the game.
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We want to predict the outcome of the game
Suppose that Mary decides to confess. What is the best decision
for Tom?
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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We want to predict the outcome of the game
Suppose that Mary decides to remain silent. What is the best
decision for Tom?
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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Outcome of the Game
Mary
Tom
Confess
Do not
Confess
Confess
16,16
0,40
Do not
Confess
40,0
2,2
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No Dominant Strategies
• In most games there are no dominant
strategies for all players.
• We cannot use this method to predict the
outcome of the game.
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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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Nash equilibrium
• Each player’s predicted strategy is the best
response to the predicted strategies of other
players
• No incentive to deviate unilaterally
• Strategically stable or self-enforcing
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Nash’s Theorem
• Existence
– Any finite game will have at least one Nash
equilibrium possibly involving mixed strategies
• Finding a Nash equilibrium is not easy
– Not efficient from an algorithmic point of view
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“Battle of Sexes” or “Bach or Stravinsky”
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•
•
•
•
•
A couple deciding how to spend the evening
Wife would like to go for a opera
Husband would like to go for a boxing match
Both however want to spend the time together
Scope for strategic interaction
No means of communication
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Battle of Sexes
• Normal Form representation – Payoff Matrix
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The Copycat Game
• Dot is happy as long as she is alone; Ditto is
happy as long as he is with Dot.
• There is no Nash equilibrium in this game.
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Choosing Strategies
• Player choosing columns or rows
• Adjusting choice
• Nash equilibrium
– No deviation from choice by either player forms
outcome
– Take other player’s behavior as given
• Any outcome that survives this process of
elimination is called a Nash equilibrium outcome.
• An outcome is a Nash equilibrium if neither
player would want to deviate from it, taking his
opponent’s behavior as given.
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Mixed Strategies in Sports
• In the international tournaments organized by
the World Rock Paper Scissors Society
• The best players are the least predictable
players.
• In Nash equilibrium, everyone plays a mixed
strategy—1/3 “Rock,” 1/3 “Paper,” and 1/3
“Scissors.”
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Pareto Optima
• An outcome is Pareto-optimal if nothing sits above it in
the tree.
• The tree shows that outcomes A and D are Pareto-preferred
to C and B, and C is Pareto-preferred to B.
• A and D are Pareto optima, because nothing sits above
them in the tree.
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Sequential Games
• Simultaneous games
• Second player advantage
• Oligopoly problem
– Stackelberg equilibrium
• Player commits to strategy
• Importance of commitment
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Dynamic games
• Sequential moves
– One player moves
– Second player observes and then moves
• Examples
– Industrial Organization – a new entering firm in the
market versus an incumbent firm; a leader-follower
game in quantity competition
– Sequential bargaining game - two players bargain over
the division of a pie of size 1 ; the players alternate in
making offers
– Game Tree
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Cournot equilibrium:
A Nash equilibrium in a
game where each
company chooses its
quantity.
•The only Nash equilibrium is in the center square, where Kodak and Fuji
each earn profits of $15.
•But if the game is played sequentially and Kodak moves first, then Kodak
announces a policy of producing 100 rolls of film.
•Fuji’s best response is to produce 50, leading to the upper right-hand
square.
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Economic applications of game theory
• The study of oligopolies (industries containing
only a few firms)
• The study of cartels, e.g., OPEC
• The study of externalities, e.g., using a
common resource such as a fishery
• The study of military strategies
• The study of international negotiations
• Bargaining
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Summary
• Strategic situations can be represented by game
matrices,
– showing the outcome that results from each
combination of strategies that the players can
choose.
• A Nash equilibrium is an outcome from which
neither player would deviate, taking the other’s
behavior as given.
• A game can have one Nash equilibrium, no Nash
equilibrium, or many Nash equilibria.
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Summary
• A dominant strategy is a strategy that a player
would want to adopt regardless of his beliefs
about the other player’s strategy choice.
• The Prisoner’s Dilemma is an example of a game
where both players have dominant strategies.
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Thank you
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