game theory: an intoduction

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GAME THEORYAN INTRODUCTION

Njdeh Tahmasian Savarani

Winter 2014

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History

Interdisciplinary (Economic and Mathematic)

approach to the study of human behavior

Founded in the 1920s by John von Neumann

1994 Nobel prize in Economics awarded to

three researchers

“Games” are a metaphor for wide range of

human interactions

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Applications

Market:

pricing of a new product when other firms have similar new

products

deciding how to bid in an auction

Networking:

choosing a route on the Internet or through a transportation

networks

Politic:

Deciding whether to adopt an aggressive or a passive stance in

international relations

Sport:

choosing how to target a soccer penalty kick and choosing how

to defend against

Choosing whether to use performance-enhancing drugs in a

professional sport

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What is a (Mathematical) Game?

Rules

Outcomes and payoffs

Uncertainty of the Outcome

Decision making

No cheating

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Classification of Games

Number of players

Simultaneous or sequential game

Random moves

Perfect information

Complete information

Zero-sum games

Communication

Cooperative or non-cooperative game

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Two Players Games

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Two Players, Zero-sum

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Three or More Players

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The Prisoner’s Dilemma

Two burglars, Jack and Tom, are captured

and separated by the police

Each has to choose whether or not to confess

and implicate the other

If neither confesses, they both serve one

year for carrying a concealed weapon

If each confesses and implicates the other,

they both get 4 years

If one confesses and the other does not, the

confessor goes free, and the other gets 8

years

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Prisoners dilemma

Introduction

Tom

Not

Confess

Confess

Jack

Not

Confess-1, -1 -8, 0

Confess 0, -8 -4, -4

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Jack’s Decision Tree

If Tom Does Not ConfessIf Tom Confesses

Jack

4 Years in Prison

8 Years in Prison

Free1 Years in

Prison

Jack

Not ConfessConfess Confess Not Confess

Best

StrategyBest

Strategy

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Dominant strategy

A players has a dominant strategy if that

player's best strategy does not depend on

what other players do.

P1(S,T) >= P1 (S’, T)

Strict Dominant strategy

P1(S,T) > P1 (S’, T)

Games with dominant strategies are easy to

play

No need for “what if …” thinking

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Prisoner's Dilemma

Strategies must be undertaken without the

full knowledge of what other players will do.

Players adopt dominant strategies

BUT they don't necessarily lead to the best

outcome.

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Nash Equilibrium

A Nash equilibrium is a situation in which

none of them have dominant Strategy and

each player makes his or her best response

(S, T) is Nash equilibrium if S is the best strategy to

T and T is the best strategy to S

John Nash shared the 1994 Nobel prize in

Economic for developing this idea!

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Coordination Game

Your Partner

Power

Point

Keynote

You

Power

Point1, 1 0, 0

Keynot

e0, 0 1, 1

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Other samples of Coordination Game

Using Metric units of measurement of English

Units

Two people trying to find each other in a

crowded mall with two entrance

…

These games has more than one Nash

Equilibrium

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Unbalanced Coordination Game

Your Partner

Power

Point

Keynote

You

Power

Point1, 1 0, 0

Keynot

e0, 0 2, 2

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Battle of the Sexes

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Mixed Strategies- Matching Pennies

Zero-sum GamePlayer 2

Head Tail

Player 1

Head -1, +1 +1, -1

Tail +1, -1 -1, +1

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Three or More Players

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References