Economics of Strategy
1. What is Game Theory?
2. Simultaneous Game (one shot game)
• Dominant Strategy
• Nash Equilibrium
Game Theory: Study of strategic interactions.
Whenever two players compete, we have a game (strategic
interaction) between the two players.
In a game, my actions affect my opponent’s behavior and
payoff. At the same time, my opponent’s actions affect my
behavior and payoff.
Examples:
• GM and Ford
• Bears and Colts
• US and Mexico on trading
• US and North Korea on nuclear weapon
• Cops and Criminals
• Republicans and Democrats
• Chess players
• Arms race
What drives the game? Self-interests, wanting to win.
Basics of Game Theory: What we need.
1. Players: we will assume two competing players
2. Actions (or Strategies): we will assume that each player has two
actions that he/she can choose from.
3. Payoffs: Given player’s choice of actions, his/her payoff is
determined.
I. Simultaneous (one shot) Game:
Players choose their actions (strategies) simultaneously and
their payoffs are determined simultaneously.
Example)
GM and Ford choose their production capacity levels
simultaneously.
U.S. and Russia choose their military capacity levels
simultaneously.
I. Simultaneous (one shot) Game:
Strategic form (normal form): A way to represent a game.
Consider a game with two player: GM and Ford
Each player can choose either high or low.
Given their choice of actions (either high or low), both players payoffs
are determined.
There is an easy way of summarizing the game. => Normal form game.
GM
High Low
Ford High Ford payoff1, GM payoff1 Ford payoff2, GM payoff2
Low Ford payoff3, GM payoff3 Ford payoff4, GM payoff4
I. Simultaneous (one shot) Game:
Strategic form (normal form)
Let’s be more specific in terms of payoffs.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
I. Simultaneous (one shot) Game:
Strategic form (normal form)
This normal form game shows the players, actions and payoffs.
Miller
High Low
Busch High $400, $300 $500, $100
Low $150, $400 $600, $600
I. Simultaneous (one shot) Game:
Strategic form (normal form)
This normal form game shows the players, actions and payoffs.
(Another example)
Nash Equilibrium:
A pair of actions (a*,b*) in a two-player game is called Nash
equilibrium if a* is an optimal action for player A against player
B’s action, b*, and b* is an optimal action for player B against
player A’s action, a*.
Given player B chooses b*, a* is player A’s best choice.
On the other hand,
Given player A chooses a*, b* is player B’s best choice.
=>
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For Ford, if GM chooses High => Ford gets $100 if it chooses High.
Ford gets $50 if it chooses Low.
Finding Nash Equilibrium using the normal form game
Hence, Ford should choose High when GM chooses High.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For Ford, if GM chooses Low => Ford gets $300 if it chooses High.
Ford gets $200 if it chooses Low.
Hence, Ford should choose High when GM chooses Low.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For GM, if Ford chooses High => GM gets $100 if it chooses High.
Ford gets $50 if it chooses Low.
Hence, GM should choose High when Ford chooses High.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
Hence, GM should choose High when Ford chooses Low.
For GM, if Ford chooses Low => GM gets $300 if it chooses High.
Ford gets $200 if it chooses Low.
Ford should choose High when GM chooses High.
Ford should choose High when GM chooses Low.
GM should choose High when Ford chooses High.
GM should choose High when Ford chooses Low.
=> Ford should always choose High!
=> GM should always choose High!
(Ford,GM) = (High,High) is a Nash Equilibrium
(it’s a unique Nash Equilibrium in this case).
GM
High Low
Ford High $150, $120 $230, $100
Low $100, $200 $250, $250
Exercise: Find Nash Equilibrium (or equilibria)
GM
High Low
Ford High $150, $120 $230, $100
Low $100, $200 $250, $250
Two Nash Equilibria in this case:
(High,High) and (Low,Low)
Dominant Strategy
Strategy a* is called dominant strategy (action) when you choose
a* over all the other strategies regardless of what your opponent
does.
Example: Rcall GM and Ford game
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
Example: Rcall GM and Ford game
For Ford, High is a dominant strategy.
For GM, High is a dominant strategy.
GM
High Low
Ford High $50, $100 $200, $50
Low $70, $300 $220, $200
Example2: GM and Ford game
For Ford, Low is a dominant strategy.
For GM, High is a dominant strategy.
Dominant Strategy
Can you see that (Ford,GM)=(Low,High) is a Nash?
Any Dominant Strategy is a Nash Equilibrium.
However, not all Nash Equilibrium is a Dominant Strategy
Is Nash Equilibrium outcome the best outcome for the players?
(Is Nash Equilibrium outcome necessarily the efficient outcome?)
David
confess not conf
Mike confess 4,4 0,8
not conf 8,0 1,1
(Not conf, Not conf) is the best outcome (efficient outcome) but
not be chosen.
=> reason: coordination problem!
Prisoner’s Dilemma
Dilbert on Simultaneous Games
Dilbert on Simultaneous GamesDilbert on Simultaneous Games
Dilbert on Simultaneous Games
Nash and Coordination Problem
Nash equilibrium is not necessarily the first best outcome.
See the following example below.
GM
High Low
Ford High $100, $100 $180, $0
Low $0, $180 $170, $170
Clearly (Low, Low) is the best outcome but is not a Nash, hence is
not chosen. (High, High) is a Nash.Nash and Coordination Problem
Nash equilibrium is not necessarily the first best outcome.
See the following example below.
GM
High Low
Ford High $100, $100 $180, $0
Low $0, $180 $170, $170
Clearly (Low, Low) is the best outcome but is not a Nash, hence is
not chosen. (High, High) is a Nash.
Technology and Coordination
HDTV market
Two Player: Network and Manufacturers
Two Actions: Invest on HDTV or Do not invest on HDTV
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Find Nash?Technology and Coordination
HDTV market
Two Player: Network and Manufacturers
Two Actions: Invest on HDTV or Do not invest on HDTV
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Find Nash?
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Two Nash Equilibria (I,I) and (NI,NI).
Best solution: (I,I) can be reached if two players can coordinate
the outcome of the game.
If coordination fails, players may choose (NI,NI) instead of (I,I).TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Two Nash Equilibria (I,I) and (NI,NI).
Best solution: (I,I) can be reached if two players can coordinate
the outcome of the game.
If coordination fails, players may choose (NI,NI) instead of (I,I).
Economics of Strategy
1. What is Game Theory?
2. Simultaneous Game (one shot game)
• Dominant Strategy
• Nash Equilibrium
Game Theory: Study of strategic interactions.
Whenever two players compete, we have a game (strategic
interaction) between the two players.
In a game, my actions affect my opponent’s behavior and
payoff. At the same time, my opponent’s actions affect my
behavior and payoff.
Examples:
• GM and Ford
• Bears and Colts
• US and Mexico on trading
• US and North Korea on nuclear weapon
• Cops and Criminals
• Republicans and Democrats
• Chess players
• Arms race
What drives the game? Self-interests, wanting to win.
Basics of Game Theory: What we need.
1. Players: we will assume two competing players
2. Actions (or Strategies): we will assume that each player has two
actions that he/she can choose from.
3. Payoffs: Given player’s choice of actions, his/her payoff is
determined.
I. Simultaneous (one shot) Game:
Players choose their actions (strategies) simultaneously and
their payoffs are determined simultaneously.
Example)
GM and Ford choose their production capacity levels
simultaneously.
U.S. and Russia choose their military capacity levels
simultaneously.
I. Simultaneous (one shot) Game:
Strategic form (normal form): A way to represent a game.
Consider a game with two player: GM and Ford
Each player can choose either high or low.
Given their choice of actions (either high or low), both players payoffs
are determined.
There is an easy way of summarizing the game. => Normal form game.
GM
High Low
Ford High Ford payoff1, GM payoff1 Ford payoff2, GM payoff2
Low Ford payoff3, GM payoff3 Ford payoff4, GM payoff4
I. Simultaneous (one shot) Game:
Strategic form (normal form)
Let’s be more specific in terms of payoffs.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
I. Simultaneous (one shot) Game:
Strategic form (normal form)
This normal form game shows the players, actions and payoffs.
Miller
High Low
Busch High $400, $300 $500, $100
Low $150, $400 $600, $600
I. Simultaneous (one shot) Game:
Strategic form (normal form)
This normal form game shows the players, actions and payoffs.
(Another example)
Nash Equilibrium:
A pair of actions (a*,b*) in a two-player game is called Nash
equilibrium if a* is an optimal action for player A against player
B’s action, b*, and b* is an optimal action for player B against
player A’s action, a*.
Given player B chooses b*, a* is player A’s best choice.
On the other hand,
Given player A chooses a*, b* is player B’s best choice.
=>
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For Ford, if GM chooses High => Ford gets $100 if it chooses High.
Ford gets $50 if it chooses Low.
Finding Nash Equilibrium using the normal form game
Hence, Ford should choose High when GM chooses High.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For Ford, if GM chooses Low => Ford gets $300 if it chooses High.
Ford gets $200 if it chooses Low.
Hence, Ford should choose High when GM chooses Low.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
For GM, if Ford chooses High => GM gets $100 if it chooses High.
Ford gets $50 if it chooses Low.
Hence, GM should choose High when Ford chooses High.
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
Hence, GM should choose High when Ford chooses Low.
For GM, if Ford chooses Low => GM gets $300 if it chooses High.
Ford gets $200 if it chooses Low.
Ford should choose High when GM chooses High.
Ford should choose High when GM chooses Low.
GM should choose High when Ford chooses High.
GM should choose High when Ford chooses Low.
=> Ford should always choose High!
=> GM should always choose High!
(Ford,GM) = (High,High) is a Nash Equilibrium
(it’s a unique Nash Equilibrium in this case).
GM
High Low
Ford High $150, $120 $230, $100
Low $100, $200 $250, $250
Exercise: Find Nash Equilibrium (or equilibria)
GM
High Low
Ford High $150, $120 $230, $100
Low $100, $200 $250, $250
Two Nash Equilibria in this case:
(High,High) and (Low,Low)
Dominant Strategy
Strategy a* is called dominant strategy (action) when you choose
a* over all the other strategies regardless of what your opponent
does.
Example: Rcall GM and Ford game
GM
High Low
Ford High $100, $100 $300, $50
Low $50, $300 $200, $200
Example: Rcall GM and Ford game
For Ford, High is a dominant strategy.
For GM, High is a dominant strategy.
GM
High Low
Ford High $50, $100 $200, $50
Low $70, $300 $220, $200
Example2: GM and Ford game
For Ford, Low is a dominant strategy.
For GM, High is a dominant strategy.
Dominant Strategy
Can you see that (Ford,GM)=(Low,High) is a Nash?
Any Dominant Strategy is a Nash Equilibrium.
However, not all Nash Equilibrium is a Dominant Strategy
Is Nash Equilibrium outcome the best outcome for the players?
(Is Nash Equilibrium outcome necessarily the efficient outcome?)
David
confess not conf
Mike confess 4,4 0,8
not conf 8,0 1,1
(Not conf, Not conf) is the best outcome (efficient outcome) but
not be chosen.
=> reason: coordination problem!
Prisoner’s Dilemma
Dilbert on Simultaneous Games
Dilbert on Simultaneous GamesDilbert on Simultaneous Games
Dilbert on Simultaneous Games
Nash and Coordination Problem
Nash equilibrium is not necessarily the first best outcome.
See the following example below.
GM
High Low
Ford High $100, $100 $180, $0
Low $0, $180 $170, $170
Clearly (Low, Low) is the best outcome but is not a Nash, hence is
not chosen. (High, High) is a Nash.Nash and Coordination Problem
Nash equilibrium is not necessarily the first best outcome.
See the following example below.
GM
High Low
Ford High $100, $100 $180, $0
Low $0, $180 $170, $170
Clearly (Low, Low) is the best outcome but is not a Nash, hence is
not chosen. (High, High) is a Nash.
Technology and Coordination
HDTV market
Two Player: Network and Manufacturers
Two Actions: Invest on HDTV or Do not invest on HDTV
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Find Nash?Technology and Coordination
HDTV market
Two Player: Network and Manufacturers
Two Actions: Invest on HDTV or Do not invest on HDTV
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Find Nash?
TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Two Nash Equilibria (I,I) and (NI,NI).
Best solution: (I,I) can be reached if two players can coordinate
the outcome of the game.
If coordination fails, players may choose (NI,NI) instead of (I,I).TVManufacturers
I NI
Network I $100, $100 -$100, $0
NI $0, -$100 $0, $0
Two Nash Equilibria (I,I) and (NI,NI).
Best solution: (I,I) can be reached if two players can coordinate
the outcome of the game.
If coordination fails, players may choose (NI,NI) instead of (I,I).