Game Theory

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Game Theory

Quick Intro to Game Theory

Analysis of Games

Design of Games (Mechanism Design)

Some References

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John von Neumann The Genius who created two intellectual currents in the 1930s, 1940s

Founded Game Theory with Oskar Morgenstern (1928-44)

Pioneered the Concept of a Digital Computer and Algorithms (1930s)

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Robert AumannNobel 2005

Recent Excitement : Nobel Prizes for Game Theory and Mechanism DesignThe Nobel Prize was awarded to two Game Theorists in 2005

The prize was awarded to three mechanism designers in 2007 Thomas Schelling

Nobel 2005

Leonid HurwiczNobel 2007

Eric MaskinNobel 2007

Roger MyersonNobel 2007

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Game TheoryMathematical framework for rigorous study of conflict

and cooperation among rational, intelligent agents

Market

Buying Agents (rational and intelligent)

Selling Agents (rational and intelligent)

Social Planner

In the Internet Era, Game Theory has become a valuable tool for analysis and design

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Microeconomics, Sociology, Evolutionary Biology

Auctions and Market Design: Spectrum Auctions, Procurement Markets, Double Auctions

Industrial Engineering, Supply Chain Management, E-Commerce, Procurement, Logistics

Computer Science: Algorithmic Game Theory, Internet and Network Economics,

Protocol Design, Resource Allocation, etc.

Applications of Game Theory

A Familiar Game

Sachin Tendulkar IPL Franchisees

1

2

3

4

Mumbai Indians

Kolkata Knight Riders

Bangalore RoyalChallengers

Punjab Lions

IPL CRICKET AUCTION

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Sponsored Search Auction Advertisers

CPC

1

2

n

Major money spinner for all search engines and web portals

DARPA Red Balloon Contest

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Mechanism Design Meets Computer Science, Communications of the ACM, August 2010

Procurement Auctions

Buyer

SUPPLIER 1

SUPPLIER 2

SUPPLIER n

Budget Constraints, Lead Time Constraints, Learning by Suppliers,Learning by Buyer, Logistics constraints, Combinatorial Auctions,

Cost Minimization, Multiple Attributes

Supply (cost) Curves

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KEY OBSERVATIONS

Players are rational,Intelligent, strategic

Both conflict and cooperation are “issues”

Some information is“common knowledge”

Other information is “private”, “incomplete”,

“distributed”

Our Goal: To implement a system wide solution (social choice function) with desirable properties

Game theory is a natural choice for modelingsuch problems

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Strategic Form Games (Normal Form Games)

S1

Sn

U1 : S R

Un : S R

N = {1,…,n}

PlayersS1, … , Sn

Strategy Sets

S = S1 X … X Sn

Payoff functions

(Utility functions)

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Example 1: Coordination Game

B A

RVCE MG Road

RVCE 100,100 0,0

MG Road 0,0 10,10

Models the strategic conflict when two players have to choose their priorities

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Example 2: Prisoner’s Dilemma

No Confess

NCConfess

CNo Confess

NC - 2, - 2 - 10, - 1Confess

C -1, - 10 - 5, - 5

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Pure Strategy Nash Equilibrium

A profile of strategies is said to bea pure strategy Nash Equilibrium if is a best response strategy against *

is ni ,...,2,1

**2

*1 ,...,, nsss

*is

A Nash equilibrium profile is robust to unilateral deviations and captures a stable, self-enforcing

agreement among the players

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Nash Equilibria in Coordination Game

B A

College Movie

College 100,100 0,0

Movie 0,0 10,10

Two pure strategy Nash equilibria: (College,College) and (Movie, Movie);

one mixed strategy Nash equilibrium

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Nash Equilibrium in Prisoner’s Dilemma

No Confess

NCConfess

CNo Confess

NC - 2, - 2 - 10, - 1Confess

C -1, - 10 - 5, - 5(C,C) is a Nash equilibrium

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Relevance/Implications of Nash Equilibrium

Players are happy the way they are;Do not want to

deviate unilaterally

Stable, self-enforcing,self-sustaining

agreement

Provides a principled way of predicting a

steady-state outcome of a dynamic

Adjustment process

Need not correspondto a socially optimal or

Pareto optimalsolution

45C

2

45

x/100

x/100

B

D

ASource

Destination

Example 3: Traffic Routing Game

N = {1,…,n}; S1 = S2 = … = Sn = {C,D}

45C

2

45

x/100

x/100

B

D

ASource

Destination

Traffic Routing Game: Nash Equilibrium

Assume n = 4000

U1 (C,C, …, C) = - (40 + 45) = - 85

U1 (D,D, …, D) = - (45 + 40) = - 85

U1 (D,C, …, C) = - (45 + 0.01) = - 45.01

U1 (C, …,C;D, …,D) = - (20 + 45) = - 65

Any Strategy Profilewith 2000 C’s and 2000 D’s is a Nash Equilibrium

45C

2

45

x/100

x/100

B

D

ASource

Destination

Traffic Routing Game: Braess’ Paradox

Assume n = 4000

S1 = S2 = … = Sn = {C,CD, D}

U1 (CD,CD, …, CD) = - (40+0+40) = - 80

U1 (C,CD, …, CD) = - (40+45) = - 85

U1 (D,CD, …, CD) = - (45+40) = - 85

Strategy Profile with 4000 CD’s is the uniqueNash Equilibrium

0

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Nash’s Beautiful Theorem

Every finite strategic form game has at least one mixed strategy Nash equilibrium;

Computing NE is one of thegrand challenge problems in CS

Game theory is all about analyzing games through such solution concepts and

predicting the behaviour of the playersNon-cooperative game theory and cooperative

game theory are the major categories

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MECHANISM DESIGN

Game Theory involves analysis of games – computing NE, DSE, MSNE, etc and

analyzing equilibrium behaviour

Mechanism Design is the design of games orreverse engineering of games; could be called

Game Engineering

Involves inducing a game among the players such that in some equilibrium of the game,

a desired social choice function is implemented

Example 1: Mechanism Design Fair Division of a Cake

MotherSocial PlannerMechanism Designer

Kid 1Rational and Intelligent

Kid 2Rational and Intelligent

Example 2: Mechanism Design Truth Elicitation through an Indirect Mechanism

Tenali Rama(Birbal)Mechanism Designer

Mother 1Rational and Intelligent Player

Mother 2Rational and Intelligent Player

Baby

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William Vickrey(1914 – 1996 )

Nobel Prize: 1996

Winner = Winner = 4 Price = 4 Price =

6060

1122

3344

4400445566008800

BuyersBuyers

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Mechanism Design: Example 3 Vickrey Auction

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Four Basic Types of Auctions

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nnSellerSeller

BuyersBuyers

Winner = 4 Winner = 4 Price = 60Price = 60

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3344

Dutch AuctionDutch Auction

Vickrey AuctionVickrey Auction

Winner = 4 Winner = 4 Price = 60Price = 60

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22

33

44

5050

First Price AuctionFirst Price Auction

5555

6060

4040 4040

4545

6060

8080

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nnAuctioneerAuctioneer oror sellerseller

English AuctionEnglish Auction

BuyersBuyers

BuyersBuyers BuyersBuyers

0, 10, 20, 30,0, 10, 20, 30,40, 45, 50, 55,40, 45, 50, 55,58, 60, stop.58, 60, stop.

100, 90, 85, 100, 90, 85, 75, 70, 65, 60, 75, 70, 65, 60, stop.stop.

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Vickrey-Clarke-Groves (VCG) Mechanisms

Only mechanisms under a quasi-linear setting satisfyingAllocative Efficiency

Dominant Strategy Incentive Compatibility

Vickrey Clarke Groves

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Concluding RemarksGame Theory and Mechanism Design havenumerous, high impact applications in the

Internet era

Game Theory, Machine Learning, Optimization,and Statistics have emerged as the most

important mathematical tools for engineers

Algorithmic Game Theory is now one of the mostactive areas of research in CS, ECE, Telecom, etc.

Mechanism Design is extensively being used in IEM

It is a wonderful idea to introducegame theory and mechanism design at the

BE level for CS, IS, EC, IEM; to be done with care

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REFERENCES

Martin Osborne. Introduction to Game Theory.Oxford University Press, 2003

Roger Myerson. Game Theory and Analysis of Conflict. Harvard University Press, 1997

A, Mas-Colell, M.D. Whinston, and J.R. Green.Microeconomic Theory, Oxford University Press, 1995

N. Nisan, T. Roughgarden, E. Tardos, V. VaziraniAlgorithmic Game Theory, Cambridge Univ. Press, 2007

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REFERENCES (contd.)

Y. Narahari, Essentials of Game Theory and Mechanism Design

IISc Press, 2012 (forthcoming)

http://www.gametheory.netA rich source of material on game theory and game

theory courses

http://lcm.csa.iisc.ernet.in/hariCourse material and

several survey articles can be downloaded

Y. Narahari, Dinesh Garg, Ramasuri, and HastagiriGame Theoretic Problems in Network Economicsand Mechanism Design Solutions, Springer, 2009

Cooperative Game with Transferable Utilities

coalitions possible 12 are There

. a called is 0)( 2:

},...,2,1{),(

||

N

N

NCvv

nNvNT

coalition ; functionsticcharacteri

players of set

Divide the Dollar GameThere are three players who have to share 300 dollars. Each one proposes a particular allocation of dollars to

players.

}300 ;0;0;0:),,{(

}3,2,1{

321

3213

321321

xxxxxxxxxSSS

N

Divide the Dollar : Version 1 The allocation is decided by what is proposed by player 0

Characteristic Function

300})3,2,1({})3,1({})2,1({0})3,2({})3({})2({

300})1({

vvvvvv

v

otherwise 0 ),,( if ),,( 3211321

xxxsxsssu ii

Divide the Dollar : Version 2

It is enough 1 and 2 propose the same allocation

Players 1 and 2 are equally powerful; Characteristic Function is:

300})3,2,1({0})3,2({})3,1({

300})2,1({0})3({})2({})1({

vvv

vvvv

otherwise 0 ),,( if ),,( 32121321

xxxssxsssu ii

Divide the Dollar : Version 3 Either 1 and 2 should propose the same allocation or 1 and 3

should propose the same allocation

Characteristic Function

300})3,2,1({})3,1({})2,1({0})3,2({})3({})2({})1({

vvvvvvv

otherwise 0 ),,(or ),,( if ),,( 3213132121321

xxxssxxxssxsssu ii

Divide the Dollar : Version 4 It is enough any pair of players has the same proposal

Also called the Majority Voting Game

Characteristic Function

300})3,2,1({})3,2({})3,1({})2,1({0})3({})2({})1({

vvvvvvv

otherwise 0 ),,(or ),,(or ),,( if ),,(

32132

32131

32121321

xxxssxxxssxxxssxsssu ii

Shapley Value of a Cooperative Game

Captures how competitive forces influence the outcomes of a game

Describes a reasonable and fair way of dividing the gains from cooperation given the strategic realities

Shapley value of a player finds its average marginal contribution across all permutation orderings

Unique solution concept that satisfies symmetry, preservation of carrier, additivity, and Pareto optimality

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Lloyd Shapley

Shapley Value : A Fair Allocation Scheme

Provides a unique payoff allocation that describes a fair way

of dividing the gains of cooperation in a game (N, v)

iNCi

n

CviCvNCNCv

vvv

)}(}){({|!|

)!1|||(||!|)(

))(),...,(()( 0

where

Shapley Value: Examples

Version of Divide-the-Dollar Shapley Value

Version 1

Version 2

Version 3

Version 4

(150, 150, 0)

(300, 0, 0)

(200, 50, 50)

(100, 100, 100)