## ECO290E Game Theory

Lecture 2: Static Games and Nash Equilibrium

## 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.

## 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.

## 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)

## 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).

## 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.

## 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

## 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?

## 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

## 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.

## 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.

## 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

## 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

## 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

## 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!

## 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.

## 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.”

## 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

## 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

## 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).