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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

(Lecture 2-4)

(Lecture 11-12)

### Equilibrium

(Lecture 5, 7-10)

(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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## 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



### 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).

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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



### 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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



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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

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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## 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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## 参照

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