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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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Example: Prisoners ’ Dilemma



Two suspects are charged with a joint crime,

and are held separately by the police.



Each prisoner is told the following (assume

that a plea bargain is allowed):



If both confess, each receives 3 years imprisonment.



If neither confesses, both receive 1 year.



If one confesses and the other one does not, the

former will be set free immediately (0 payoff) and

the latter receives 5 years.

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Prisoners’ Dilemma: Payoff Matrix

 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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How to Use (Read) Bi-Matrices



Any two players game (with finite number of

strategies) can be expressed as a bi-matrix, called

payoff bi-matrix or payoff matrix.

 The payoffs to the two players when a particular pair of strategies is chosen are given in the appropriate cell.

 The payoff to the row player (player 1) is given first, followed by the payoff to the column player (player 2).

How can we solve this game?

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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 s is called a dominant strategy if playing s is

optimal for any combination of other players’

strategies.

Pareto efficiency:



An outcome of games is Pareto efficient 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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Another Formulation of PD

 The larger the payoff, the better the corresponding result.

 Desirability of outcomes for each player:

Player 2 Player 1

Cooperation Defection

Cooperation 2

2

3 0

Defection 0

3

1 1

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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. 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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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 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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Further Exercises

 Find a social problem which can be described as a prisoner’s dilemma game.

 Find a social problem which can be described as a prisoner’s dilemma 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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