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ECO290E Game Theory

Lecture 2: Static Games and Nash Equilibrium

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Review of Lecture 1

Game Theory

 

studies strategically inter-dependent situations.

 

provides us tools for analyzing most of problems in social

science.

 

employs Nash equilibrium as a solution concept.

 

made a revolution in Economics.

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Two Frameworks of Game Theory 

  Non-cooperative Game Theory

  examine individual decision making in strategic settings.

  assume a person decides her action on her own.

  does NOT rule out cooperative behaviors.

  Cooperative Game Theory

  examine group decision making in strategic settings.

  assume players can agree on their joint action, or can make binding contracts.

  simplifies strategic analysis by NOT modeling the negotiation process explicitly.

The two tools are complements to one another, but this lecture focuses mainly on Non-cooperative games.

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Timing and Information 

Complete

Information

Incomplete

Information

Static Nash Equilibrium

(Lecture 2-4)

Bayesian NE

(Lecture 11-12)

Dynamic Subgame Perfect

Equilibrium

(Lecture 5, 7-10)

Perfect Bayesian

Equilibrium

(not covered)

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Look into Games

 

We consider static situations in which each player

simultaneously chooses a strategy, and the combination of

strategies determines a payoff for each player.

 

The players need not literally act simultaneously.

  Each chooses her own action without knowing others’ choices.

Formally, representation of a game specifies:

1.

Players in the game.

2.

Strategies available to each player.

3.

Payoff received by each player (for every possible

combination of strategies chosen by the players).

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Review: Prisoners’ Dilemma 

 

Here we set payoffs as (negative of) years.

  Other numbers such that larger number shows better outcomes can express the same situations.

Player 2 Player 1

Silent Confess

Silent -1 -1

0 -5

Confess -5 0

-3 -3

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Prisoners’ Dilemma: Analysis 

 

(Silent, Silent) looks mutually beneficial outcomes, though

  Playing Confess is optimal regardless of other player’s choice!

 

Acting optimally (Confess, Confess) rends up realizing!!

  This is why the game is called Prisoners’ “Dilemma”. Player 2

Player 1

Silent Confess

Silent -1 -1

0 -5

Confess -5 0

-3 -3

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Prisoners’ Dilemma: Remarks

 

Playing “Confess” is optimal no matter how the

opponent takes “Confess” or “Silent.”

  “Confess” is an optimal (dominant) strategy.

  Combination of dominant strategies is Nash equilibrium.

  There are many games where no dominant strategy exists.

 

Individually best decision Socially efficient outcome

  Optimality for individuals does not necessary imply optimality (Pareto efficiency) for a group or society.

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Terminology

Dominant strategy:

 

A strategy X is called a dominant strategy if playing X

is optimal for any combination of other players’

strategies.

Pareto efficiency:

 

An outcome of games is Pareto efficient (or Pareto

optimal) if it is not possible to make one person

better off (through moving to another outcome)

without making someone else worse off.

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Prisoners’ Non-Dilemma 

  If plea bargain is not allowed: (Silent, Silent) realizes

  Playing Silent becomes optimal regardless of other player’s choice!

  Police cannot achieve (Confess, Confess) any more… Player 2

Player 1

Silent Confess

Silent -1 -1

-3 -3

Confess -3 -3

-3 -3

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Application: Price Competition 

  (Keep, Keep) are joint-profit maximizing outcomes, though

  Cprice-cut is optimal regardless of other player’s choice!

  Acting optimally (Price-cut, Price-cut) realize!!

  Competing firms may be difficult to escape from price wars.

Player 2 Player 1

Keep Price-cut

Keep 2 2

3 -1

Price-cut -1 3

0 0

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General Formulation of PD 

 

The larger the payoff, the better the corresponding result.

 

Desirability of outcomes for each player:

  g > c > d > l, that is, (D, C) > (C, C) > (D, D) > (C, D) Player 2

Player 1

Cooperation Defection

Cooperation c c

g l

Defection l g

d d

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Applications of PD 

Examples Players “Cooperation” “Defection”

Arms races Countries Disarm Arm

International trade policy

Countries Lower trade barriers

No change

Marital cooperation

Couple Obedient Demanding

Provision of public goods

Citizen Contribute Free-ride Deforestation Woodmen Restrain cutting Cut down

maximum

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Example: Coordination Game 

 

Two students need a new computer each for joint-work.

  Having different OS generates no value.

  They (are assumed to) prefer (Mac, Mac) to (Win, Win) Player 2

Player 1

Windows Mac

Windows 1 1

0 0

Mac 0

0

2 2

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Coordination Game: Optimal Play (1) 

 

If the opponent chooses Win,

  Selecting Win is optimal since 1 > 0. Player 2

Player 1

Windows Mac

Windows 1 1

0 0

Mac 0

0

2 2

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Coordination Game: Optimal Play (2) 

 

If the opponent chooses Mac,

  Selecting Mac is optimal since 2 > 0. Player 2

Player 1

Windows Mac

Windows 1 1

0 0

Mac 0

0

2 2

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Coordination Game: Analysis 

 

There is NO dominant (optimal) strategy.

  Choosing Mac (Win) is optimal if the others is Mac (Win).

  Best strategy varies depending on other’s choice.

 

Game cannot be solved merely from individual rationality.

  Coordination game is NOT like Prisoner’s dilemma.

  We need to look at Nash equilibrium!

 

(Mac , Mac) looks a unique reasonable outcome…

  Let’s review the definition of Nash equilibrium!

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The Solution: Nash Equilibrium 

 

John Nash discovered a path breaking solution

concept, called Nash equilibrium!

 

No one can benefit if she unilaterally changes her

action from the original Nash equilibrium.

NE describes a stable situation.

 

Everyone correctly predicts other players’ actions

and takes best-response against them.

NE serves as a rational prediction.

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Solving Coordination Game

 

There are two equilibria: ( W , W ) and ( M , M ).

Games, in general, can have more than one Nash

equilibrium.

 

Everybody prefers one equilibrium ( M , M ) to the

other ( W , W ).

Several equilibria can be Pareto-ranked.

 

However, bad equilibrium can be chosen.

This is called “coordination failure.”

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Finding NE: Underline Method 

 

Draw underlines for optimal strategy for each player.

  If both numbers are underlined => Nash equilibrium

  Otherwise => outcome is NOT Nash equilibrium Player 2

Player 1

Windows Mac

Windows 1 1

0 0

Mac 0

0

2 2

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Example: Battle of the Sexes 

 

A wife (player 1) and a husband (player 2) are supposed

to choose between going to a musical and a soccer game.

 

Presumably, the wife prefers the musical and the husband

the soccer game.

Player 2 Player 1

Musical Soccer

Musical 1 3

0 0

Soccer 0 0

3 1

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Battle of the Sexes: Underline Method 

 

There are two Nash equilibria:

  (Musical, Musical) and (Soccer, Soccer)

 

Unlike Coordination Game, two equilibria cannot be

(Pareto) comparable: conflict of interests

Player 2 Player 1

Musical Soccer

Musical 1 3

0 0

Soccer 0 3

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Example: Chicken Game 

  Two teenagers take their cars to opposite ends of main street and start to drive toward each other.

  The one who swerves to prevent a collision is the chicken and the one who keeps going straight (tough) is the winner.

Player 2 Player 1

Chicken Tough

Chicken 2 2

3 0

Tough 0 3

-1 -1

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Chicken Game: Optimal Play (1) 

  If the opponent chooses Chicken,

  Playing Tough is optimal since 3 > 2.

  This is the best outcome!

Player 2 Player 1

Chicken Tough

Chicken 2 2

3 0

Tough 0 3

-1 -1

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Chicken Game: Optimal Play (2) 

  If the opponent chooses Tough,

  Playing Chicken is optimal since 0 > -1.

  Low payoff, but still better than the worst outcome.

Player 2 Player 1

Chicken Tough

Chicken 2 2

3 0

Tough 0 3

-1 -1

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Chicken Game: Underline Method 

  There are two Nash equilibria:

  (Chicken, Tough) and (Tough, Chicken).

  They are asymmetric and look unfair, but still equilibrium!

Player 2 Player 1

Chicken Tough

Chicken 2 2

3 0

Tough 0 3

-1 -1

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Further Exercises 

 

Find a social problem which can be described as a

  Coordination game,

  Battle of sexes, and

  Chicken game.

 

Explain why rational decision made by each individual

does not necessarily result in socially optimal outcome (in

the presence of strategic interdependence).

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