GAME THEORY
STRATEGIC DECISION
MAKING
What is Game Theory?
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GT is a tool used to analyze strategic
interactions or conflict situations.
Strategic interaction: When actions of a player
influence other player(s).
Where can we use GT?
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Any situation that requires us to anticipate our
rival’s response to our action is a potential
context for GT.
§ Games: Checkers, poker, chess, tennis, soccer etc.
§ Economics: Industrial Organization, Micro/Macro/
§
§
§
§
International/Labor/Natural resource Economics, and
Public Finance
Political science: war/peace
Law: Designing laws that work
Biology: animal behavior, evolution
Information systems: System competition/evolution
Where can we use GT? (cont.)
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Business:
§ Games against rival firms:
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Pricing, advertising, marketing, auctions, R&D,
joint ventures, investment, location, quality,
M&A, etc.
§ Games versus other players
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Employee/employer, managers/stockholders
Supplier/buyer, producer/distributor,
firm/government, etc.
Strategic (Normal)
Form Games
Static Game of Complete
Information
What is a Normal Form Game?
A normal (strategic) form game consists of:
l Players: list of players
l Strategies: all actions available to players
l Payoffs: a payoff assigned to every contingency
(every possible strategy profile as the outcome
of the game)
Prisoners’ Dilemma
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Two suspects are caught and put in different
rooms (no communication). They are offered the
following deal:
§ If both of you confess, you will both get 5 years in
§
§
prison (-5 payoff)
If one of you confesses whereas the other does not
confess, you will get 0 (0 payoff) and 10 (-10 payoff)
years in prison respectively.
If neither of you confess, you both will get 2 years in
prison (-2 payoff)
Easy to Read Format of
Prisoner’s Dilemma
Prisoner 2
Prisoner 1
Confess
Confess
Don’t
Confess
Don’t
Confess
-5, -5 0, -10
-10, 0 -2, -2
Assumptions in Static (Normal
Form) Games
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All players are rational (self-interest and very
smart).
Rationality is common knowledge.
Players move simultaneously. (They do not
know what the other player has chosen).
Solution of a Static (Normal
Form) Game
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Equilibrium in strictly dominant strategies
§ A strictly dominant strategy is the one that
§
§
always yields the highest payoff.
Dominant Strategy: I’m doing the best I can,
no matter what you do.
Rational players will always play their strictly
dominant strategies.
Solution of Prisoners’ Dilemma
Dominant Strategy Equilibrium
Prisoner 2
Prisoner 1
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
Solution of a Static Normal
Form Game
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Iterated elimination of strictly dominated
strategies
§ Rational players will never play their
§
dominated strategies.
Eliminating dominated strategies may solve
the game.
Solution of Prisoners’ Dilemma
Iterated Elimination Procedure
Prisoner 2
Prisoner 1
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
Solution of a Static Normal
Form Game (cont.)
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Nash Equilibrium (NE):
§ Nash Strategy: I’m doing the best I can given
§
§
what you are doing.
In equilibrium neither player has an incentive
to deviate from his/her strategy, given the
equilibrium strategies of rival players.
The strategies are mutually best response in
equilibrium.
1994 Economics Noble Prize Winner
John Nash
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Prize motivation: "for their
pioneering analysis of
equilibria in the theory of
non-cooperative games"
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Contribution: Nash
Equilibrium; Nash
Bargaining
Celebrities of Game Theory
Nash Equilibrium
A Beautiful Mind
NE of
positioning,
coalition, and
strategic
interactions.
Introduced the
coalition and
positioning
strategies to get
girls.
John Nash
Russell Crowe
Noble Prize Winner
Oscar Winner
Solution of Prisoners’ Dilemma
Cell-by-cell Inspection
Prisoner 2
Prisoner 1
Confess
Don’t
Confess
Confess
-5, -5 0, -10
Don’t
Confess
-10, 0 -2, -2
NE of Prisoners’ Dilemma
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The strategy profile {confess, confess} is
the unique pure strategy NE of the game.
In equilibrium both players get a payoff of
–5.
Inefficient equilibrium; (don’t confess, don’t
confess) yields higher payoffs for both.
A Pricing Example
Firm 2
Firm 1
High
Price
Low
Price
High Price 100, 100 -10, 140
Low Price 140, -10
0, 0
A Coordination Game
Battle of the Sexes
Husband
Movie
Opera
2, 1
0, 0
Movie
0, 0
1, 2
Wife
Opera
A Strictly Competitive Game
Matching Pennies
Player 2
Player 1
Heads
Heads
Tails
Tails
1, -1
-1, 1
No NE in pure
strategies
-1, 1
1, -1
International Investment Game
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3 Chinese firms invest in US.
A new law is being debated and they all want the
law to be favorable for the firms.
The president is very powerful. He promises to
match the total donation made to a state
university in terms of favorable tax cuts for the
firms.
The 3 firms have to decide whether to contribute
or not.The more they contribute the more
favorable the law.
International Investment Game
Firm 1 is the row player. Firm 2 is the column player.
Firm 3 is the page player.
Firm 3 Donates
Firm 3 does not Donate
Donate Don’t
Donate Don’t
Donate 5, 5, 5 3, 6, 3
Donate 3, 3, 6 1, 4, 4
Don’t
Don’t
6, 3, 3 4, 4, 1
4, 1, 4 2, 2, 2
NE is: (Don’t, Don’t, Don’t)
Extensive Form
Games
Dynamic Games of
Complete Information
Dynamic (Extensive Form)
Games
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Players move sequentially. (Therefore, also
called sequential games)
Players have complete and perfect information
§ Players can see the full game tree including the
§
payoffs
Players can observe and recall all previous moves
What is a Game Tree?
Player 1
Right
Left
Player 2
Player 2
A
B
C
D
P11
P12
P13
P14
P21
P22
P23
P24
An Entry Game
Target
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Stay out
Enter
Stay out
680
730
700
800
-50
0
400
0
Solution of an Extensive Form
Game
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Subgame Perfect Equilibrium (SPE): For an equilibrium
to be subgame perfect, it has to be a NE for all the
subgames as well as for the entire game.
§ A subgame is a decision node from the original game
along with the decision nodes and end nodes.
§ Backward induction is used to find SPE
§ Looking forward, reasoning backward
Entry Example:
3 proper subgames
Target
Wal-Mart
Wal-Mart
680
730
700
800
-50
0
400
0
Solution of the Entry Game
Subgame 1
Subgame 2
Wal-Mart
Wal-Mart
Enter
Stay out
Enter
Stay out
680
730
700
800
-50
0
400
0
Solution of the Entry Game
(cont.)
Target
Aggressive
Normal
730
700
0
400
SPE of the game is the strategy profile:
{aggressive, (stay out, enter)}
Properties of SPE
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The outcome that is selected by the backward
induction procedure is always a NE of the game
with perfect information.
SPE is a stronger equilibrium concept than NE
SPE eliminates NE that involve incredible
threats.
Suppose WM threatens to enter no matter
what Target does. Is this a credible threat?
Target
Normal
Aggressive
Wal-Mart
Wal-Mart
Enter
Stay out
Enter
Stay out
680
730
700
800
-50
0
400
0
A 3 Player Sequential Game
P1
Left
Right
Middle
2
P3
P2
1
X
1
Y
P3
P3
A
B
C D
E
3
-2
4
-3
10
2
1
4
3
2
2
5
6
3
5
F
G
3
-9
3
2
3
1
Backwards Induction
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Obviously, Player 3’s choices are B, C, and F in
the three last period subgames.
Eliminating the non-equilibrium strategies will
make the game tree simpler.
The game tree reduces to:
Reduced Game Tree
P1
Left
Middle
2
1
Right
3
P2
X
Y
1
-2
4
1
4
5
6
3
3
SPE is when player 1 plays middle, 2
plays Y, and 3 plays C.
A 3 Player Sequential Game
(cont.)
P1
Left
Right
Middle
2
P3
P2
1
X
1
Y
P3
P3
A
B
C D
E
3
-2
4
-3
10
2
1
4
3
2
2
5
6
3
5
F
G
3
-9
3
2
3
1
1994 Economics Noble Prize Winner
Reinhard Selten
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Prize motivation: "for their
pioneering analysis of
equilibria in the theory of
non-cooperative games"
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Contribution: Subgame
Perfection and Dynamic
Games.
An Entrant Game
Entrant
Enter
Stay out
Incumbent
Incumbent
Maximize
Maximize
Limit P.
Limit P.
540
-250
0
0
540
265
1275
865
Commitment Game
Incumbent
Inflexible
Technology
Flexible
Technology
Entrant
Entrant
Enter
Enter
Stay out
Stay out
1000
2000
500
3000
-100
0
500
0
Entry Deterrence
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Find equilibrium for the following game.
Not fight
enter
(40,40)
Incumbent
Entrant
Do not enter
market
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Fight entry
(-20,30)
(0,100)
Suppose the incumbent builds a large plant which is loss
making unless it has a large share of the market. This reduces
the incumbent's payoff to sharing the market to 20. Then what
does the incumbent want to do?
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Establishing Credibility
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Establish and use a reputation. Example: Never
negotiate with terrorists. Always fight facing
entrant
Write contracts. Example: Supplier agrees to a
punishment if he fails to deliver on time.
(Strategic) commitments. Examples: Burn
bridges behind you; Firm investing in inflexible
technology or building up extra capacities;
Automatic nuclear response to nuclear attacks.
Bargaining game
Example: Wage negotiations
•
•
•
Management and labor are bargaining over a
fixed sum of $200 million
Two possible strategies are available to each
player: bargain hard or accommodate.
•
•
•
If both bargain hard, no deal is reached. Neither side
gains.
If both accommodate, they split the gains from trade.
If one player bargains hard and the other
accommodates, then the player who bargains hard
takes 75% of the pie
First-mover advantage
•
Management wins by moving first
Management
low offer
generous offer
Union
strike
accept
0,0
150 , 50
strike
0,0
accept
50 , 150
Changing the game
•
Union can change the outcome by credibly
committing to strike if a low offer is made
Management
low offer
generous offer
Union
strike
0,0
strike
0,0
accept
50 , 150
2005 Economics Noble Prize Winner
Thomas C. Schelling
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Prize motivation: "for
having enhanced our
understanding of conflict
and cooperation through
game-theory analysis"
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Contribution: Strategic
Commitment; War and
Peace; Focal Point
JUST PLAYING!
Individual home work 1
1.
2.
Find the NE for the following game. Is there a
“prisoner’s dilemma”? If there is, how can you
provide solutions to solve it?
Let one player 1 move first, draw the game
three, and find the SPE.
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Game 1
Firm 1
Firm 2
Low Price
High Price
Low Price
40, 40
60, 30
High Price
30, 60
50, 50
Individual home work 3
1.
2.
Find the NE for the following game.
Let Boeing move first, draw the game three,
and find the SPE. Is there the first-mover
advantage? If there is, how can the second
mover (Airbus) to change the game outcome?
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R&D rivalry
Airbus
Large
R&D
Boeing
Small
R&D
Large
R&D 1
1
Small
R&D 2
4
2
4
3
3
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REPEATED
GAMES
Repeated Normal Form
Games
Prisoners’ Dilemma Revisited
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Suppose that the two suspects play the same
game every time they get caught.
Can they coordinate their choices in order to get
the best outcome for both of them?
Finitely repeated game
Infinitely repeated game
Prisoner’s Dilemma
Prisoner 2
Prisoner 1
Confess
Confess
Don’t
Confess
Don’t
Confess
-5, -5 0, -10
-10, 0 -2, -2
N-times repeated PD
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In a finitely repeated (n times repeated game
where n ³ 2) PD game, the cooperative
outcome (don’t confess, don’t confess) cannot
be enforced.
Since in the last stage (nth stage) the NE is
(confess, confess) and all players know this, in
all previous stages the same NE will prevail.
Infinitely Repeated PD
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When the game is played infinitely or players do
not know when the game is going to end, the
backward induction breaks down. Following
trigger strategies can enforce the cooperative
outcome.
§ Trigger strategy: A player cooperates as long as the
other players cooperate, but any defection from
cooperation on the part of the rivals triggers the player
to behave noncooperatively for a specified period of
time (period of punishment).
Trigger Strategies
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Grim strategy: A trigger strategy in which the
punishment period lasts till the end of the game.
§ Grim strategy for PD game: Play “don’t
confess” in the first period. In period t, play
“don’t confess” if the outcome was (don’t
confess, don’t confess) in all preceding t-1
periods, and play “confess” otherwise.
Trigger Strategies (cont.)
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A trigger strategy in which the punishment period lasts
as long as the rival keeps on cheating (returning back to
cooperative periods of game play is possible).
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Tit For Tat strategy for PD game: Play “don’t confess”
in the first period. In period t, play “don’t confess” if the
rival’s most recent play was (don’t confess, don’t
confess), and play “confess” otherwise.
A Pricing Example
Firm 2
Firm 1
High
Price
Low
Price
High Price 100, 100 -10, 140
Low Price 140, -10
0, 0
2005 Economics Noble Prize Winner
Robert J. Aumann
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Prize motivation: "for
having enhanced our
understanding of conflict
and cooperation through
game-theory analysis"
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Contribution: Repeated
games.
Game Theory: Summary
Static Game: Strategic Form (matrix)
Dominant Strategy equilibrium; Nash equilibrium;
Prisoner’s dilemma; Coordination game, etc.
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Dynamic Game: Extensive Form (game tree)
Subgame perfect equilibrium; Backward induction;
Strategic commitment and establish credibility; etc.
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Repeated Game
Long-term relationship; cooperation; trigger strategy, etc.
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Managers need to understand
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what kind of game to play
with whom to play
which strategies are available to each
player
what payoff each outcome will yield
whether we can design/change the game
(rule) and how?
Intelligence is a style of life
The true test of intelligence is not how
much we know how to do, but how we
behave when we don’t know what to do.
----- John Holt
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