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8/12/2019 Game Theory 91 2 Slide01
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In the Name of God
Sharif University of Technology Graduate School of Management and Economics
Game Theory44812 (1391-92 2nd term)
Dr. S. Farshad Fatemi
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Game Theory Dr. F. Fatemi Page 2
Graduate School of Management and Economics – Sharif University of Technology
Introduction
Game: A situation in which intelligent decisions are necessarily
interdependent. A situation where utility (payoff) of an individual
(player) depends upon her own action, but also upon the actions of other
agents.
How does a rational individual choose?
My optimal action depends upon what my opponent does but his
optimal action may in turn depend upon what I do
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Game Theory Dr. F. Fatemi Page 3
Graduate School of Management and Economics – Sharif University of Technology
Examples:
Matching Pennies
Rock, Scissors, Paper
Dots & Crosses (Tick-Tack-Toe)
Meeting in New York
Prisoners’ Dilemma
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Game Theory Dr. F. Fatemi Page 4
Graduate School of Management and Economics – Sharif University of Technology
Game Components:
Players: Rational agents who participate in a game and try tomaximise their payoff.
Strategy (action): An action which a player can choose from a set of
possible actions in every conceivable situation.
Strategy profile: A list of strategies including one strategy for each
player.
Order of play: Shows who should play when? At each history, which
player should to choose his action. Player may move simultaneously or
sequentially.
Game might have one round which players act simultaneously.
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Game Theory Dr. F. Fatemi Page 5
Graduate School of Management and Economics – Sharif University of Technology
Information set: What players know about previous actions when it istheir turn?
Outcome: For each set of actions (at each terminal history) by the
players what is outcome of the game.
Payoff: The utility (payoff) that a player receives depending on the
strategy profile chosen (outcome). Players need not be concerned only
with money and could be altruistic, or could be concerned not to violate
a norm.
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Game Theory Dr. F. Fatemi Page 6
Graduate School of Management and Economics – Sharif University of Technology
Each player seeks to maximize his expected payoff (in short, is
rational).
Furthermore he knows that every other player is also rational, and
knows that every other player knows that every player is rational and so
on (rationality is common knowledge).
The theory of rational choice:The action chosen by a decision maker is as least as good as any
other available action (according to his preferences).
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Game Theory Dr. F. Fatemi Page 8
Graduate School of Management and Economics – Sharif University of Technology
An action can be a contingent plan that is why it is sometimes called
a strategy. But since in this setting the time is absent then the two
are equivalent.
Strategic form of a simultaneous move game:
Column
Player
C D
RowPlayer
A
B
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Game Theory Dr. F. Fatemi Page 9
Graduate School of Management and Economics – Sharif University of Technology
Example: Prisoners’ Dilemma
Prisoner 2
Confess Don’t Confess
Prisoner 1Confess -4 , -4 -1 , -10
Don’t Confess -10 , -1 -2 , -2
Since the preferences are ordinal it is equivalent to:
Prisoner 2
Confess Don’t Confess
Prisoner 1Confess 1 , 1 3 , 0
Don’t Confess 0 , 3 2 , 2
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Game Theory Dr. F. Fatemi Page 10
Graduate School of Management and Economics – Sharif University of Technology
Example: Working on a Project
Player 2
Work hard Don’t Bother
Player 1Work hard 2 , 2 0 , 3
Don’t Bother 3 , 0 1 , 1
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Game Theory Dr. F. Fatemi Page 11
Graduate School of Management and Economics – Sharif University of Technology
Example: The money sharing game:
B
Share Grab
A
Share M/2 , M/2 0 , M
Grab M , 0 0 , 0
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Game Theory Dr. F. Fatemi Page 12
Graduate School of Management and Economics – Sharif University of Technology
Example: Battle of Sexes:
Two players are to choose simultaneously whether to go to the cinema
or theatre. They have different preferences, but they both would prefer
to be together rather than go on their own.
Wife
Cinema Theatre
HusbandCinema 2 , 1 0 , 0
Theatre 0 , 0 1 , 2
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Game Theory Dr. F. Fatemi Page 13
Graduate School of Management and Economics – Sharif University of Technology
Example: Duopoly
Two firms competing in a market should decide simultaneouslywhether to price high or low.
Firm 2
High Low
Firm 1High 1000 , 1000 -200 , 1200
Low 1200 , -200 600 , 600
This game is equivalent to Prisoners’ Dilemma.
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Game Theory Dr. F. Fatemi Page 14
Graduate School of Management and Economics – Sharif University of Technology
Nash Equilibrium
Definition: A Nash Equilibrium is an action profile with the
property that no player can do better by choosing an action different
from , given that all other players stick to .
(we show an action profile for player ’s opponents by )
Definition (Os 23.1): The action profile in a strategic game is a
Nash Equilibrium if for every player :
() ( ) for every action of player .
Where ( ) is a payoff function representing player ’s preferences.
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Game Theory Dr. F. Fatemi Page 15
Graduate School of Management and Economics – Sharif University of Technology
Example: Prisoners’ Dilemma
Prisoner 2
C NC
Prisoner 1C -4 , -4 -1 , -10
NC -10 , -1 -2 , -2
(NC , NC) is not a Nash eq. Prisoner 1 would prefer to play C instead of
NC.
(NC , C) is not a Nash eq. Prisoner 1 would prefer to play C instead of
NC. (a similar argument for (C , NC)).
(C , C) is the only Nash eq. of the game. No one has incentive to deviate
form this strategy profile.
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Game Theory Dr. F. Fatemi Page 16
Graduate School of Management and Economics – Sharif University of Technology
Example: Battle of Sexes:
Wife
Cinema Theatre
Husband
Cinema 2 , 1 0 , 0
Theatre 0 , 0 1 , 2
(Cinema , Cinema) and (Theatre, Theatre) are the two NEs of this game.
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Game Theory Dr. F. Fatemi Page 17
Graduate School of Management and Economics – Sharif University of Technology
Example: Matching Pennies:
Player 2
Head Tail
Player 1
Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1
This game has no NE.
We will return to this game to study the likely outcome of the game.
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Game Theory Dr. F. Fatemi Page 18
Graduate School of Management and Economics – Sharif University of Technology
Example: Stag Hunt:
Two hunters should help each other to be able to catch a stag.Alternatively each can hunt a hare on their own (the initial idea of this
game is from Jean-Jacques Rousseau). The payoffs are:
Player 2
Stag Hare
Player 1
Stag 2 , 2 0 , 1
Hare 1 , 0 1 , 1
(Stag , Stag) and (Hare, Hare) are the two NEs of this game.
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Game Theory Dr. F. Fatemi Page 19
Graduate School of Management and Economics – Sharif University of Technology
Example: Stag Hunt with Hunters (Os Exercise 30.1)
There are hunters. Only hunters are enough to catch a stag where
. Assume there is only a single stag.
What is the NE of the game if :
a) Each hunter prefers the fraction ⁄ of the stag to a hare.
b) Each hunter prefers the fraction ⁄
of the stag to a hare
( ), but prefers a hare to any smaller fraction of the stag.
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Game Theory Dr. F. Fatemi Page 20
Graduate School of Management and Economics – Sharif University of Technology
Example: a different version of battle of sexes (a coordination game)
Wife
Cinema Theatre
Husband
Cinema 2 , 2 0 , 0
Theatre 0 , 0 1 , 1
(Cinema , Cinema) and (Theatre, Theatre) are the two NEs of this game.
Which eq. is the most likely outcome of this game? Why?
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Game Theory Dr. F. Fatemi Page 21
Graduate School of Management and Economics – Sharif University of Technology
Example: Chicken Game (Hawk-Dove game)
Driver 2
Swerve Straight
Driver 1
Swerve 0 , 0 -1 , 2
Straight 2 , -1 -5 , -5
NE: (Sw , St) and (St , Sw).
Game is symmetric but the pure NE is asymmetric.
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Game Theory Dr. F. Fatemi Page 22
Graduate School of Management and Economics – Sharif University of Technology
Why NE?
Why NE makes sense as an equilibrium concept for a given game?
Why should we believe that players will play a Nash equilibrium?
a) NE as a consequence of rational inference.
b) Self enforcing agreement: If the players reach an agreement about how
to play the game, then a necessary condition for the agreement to hold
up is that the strategies constitute a Nash equilibrium. Otherwise, at least
one player will have an incentive to deviate.
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Game Theory Dr. F. Fatemi Page 23
Graduate School of Management and Economics – Sharif University of Technology
c) Any prediction about the outcome of a non-cooperative game is self-
defeating if it specifies an outcome that is not a Nash equilibrium. (NE as
a necessary condition if there is a unique predicted outcome to a game).
d) Result of process of adaptive learning: Suppose players play a game
repeatedly. The idea is that a reasonable learning process, if it converges,
should converge to a Nash equilibrium (NE as a stable socialconvention).
e) If a game has a focal point, then it is necessary a NE.
(Schelling (The strategy of conflict ; 1960) introduced the concept of focal
points).
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Game Theory Dr. F. Fatemi Page 24
Graduate School of Management and Economics – Sharif University of Technology
Example: Provision of a Public Good (Os Exercise 33.1):
Players:
}
Strategies: } : Contribute for the public good
: Don’t contribute for the public good
Outcome: public good is provided if at least people contribute.
{
Payoff:
( ) ( ) ( ) ( )
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Game Theory Dr. F. Fatemi Page 25
Graduate School of Management and Economics – Sharif University of Technology
What are the NE of this game?
- Is there a NE where more than players contribute?
- Is there a NE where exactly players contribute?
- Is there a NE where less than
players contribute?
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Game Theory Dr. F. Fatemi Page 26 Graduate School of Management and Economics – Sharif University of Technology
Best Response Functions (Correspondence)
Best response function of player denotes a player’s best reaction (utility
maximising reaction) to any strategy profile chosen by other players.
Definition: Player ’s best response function (correspondence) in a
strategic game is the function that assigns to each the set:() ( ) ( ) } .
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Game Theory Dr. F. Fatemi Page 27 Graduate School of Management and Economics – Sharif University of Technology
Proposition (Os 36.1): The action profile is a Nash equilibrium of a
strategic game if and only if every player’s action in this profile is a best
response to the other players’ actions ( ) :
( )
All individual strategies in are best responses to each other.
Recall: A Nash Equilibrium is an action profile with the property
that no player can do better by choosing an action different from ,given that all other players stick to .
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Game Theory Dr. F. Fatemi Page 28 Graduate School of Management and Economics – Sharif University of Technology
Using BR function to find NE
a) Find the BR function of each player
b) Find the action profiles that satisfy:
(
)
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Game Theory Dr. F. Fatemi Page 29 Graduate School of Management and Economics – Sharif University of Technology
Example: Matching Pennies:
Player 2
Head Tail
Player 1
Head -1 , 1 1 , -1
Tail 1 , -1 -1 , 1
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Game Theory Dr. F. Fatemi Page 30 Graduate School of Management and Economics – Sharif University of Technology
Example: Chicken Game
Driver 2
Swerve Straight
Driver 1
Swerve 0 , 0 -1 , 2
Straight 2 , -1 -5 , -5
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Game Theory Dr. F. Fatemi Page 31 Graduate School of Management and Economics – Sharif University of Technology
Example: Cournot Duopoly
Two firms, 1 and 2, producing a homogeneous good
The inverse demand function for the good is .
They choose quantity simultaneously.
For simplicity suppose marginal costs are zero.
Total quantity is placed on the market and determines the
price.
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Game Theory Dr. F. Fatemi Page 33 Graduate School of Management and Economics – Sharif University of Technology
And similarly for firm 2:
To find the NE we have to set ()
()
And the only NE is:
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Game Theory Dr. F. Fatemi Page 34 Graduate School of Management and Economics – Sharif University of Technology
Remember the monopoly quantity is: .
Easy to calculate that:
,
And
,
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Game Theory Dr. F. Fatemi Page 35 Graduate School of Management and Economics – Sharif University of Technology
0
20
40
60
80
100
120
0 50 100 150
q 2
q1
BR1
BR2
Example: Bertrand Duopoly
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Game Theory Dr. F. Fatemi Page 36 Graduate School of Management and Economics – Sharif University of Technology
Example: Bertrand Duopoly
Same context, but firms choose prices. Prices can be continuously varied,i.e. is any real number. Firms have the same marginal cost of .
If prices are unequal, all consumers go to lower price firm.
If equal, market is shared.
1) There is a Nash equilibrium where : None of the firms has
incentive to deviate from this strategy.
2) There is no other Nash equilibrium in pure strategies.
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