Game Theory

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).