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Introduction to Game Theory and its Applications in Computer Networks John C.S. Lui Dept. of Computer Science & Engineering The Chinese University of Hong Kong Daniel R. Figueiredo School of Computer and Communication Sciences Swiss Federal Institute of Technology – Lausanne (EPFL) ACM SIGMETRICS / IFIP Performance June 2006

Tutorial Organization ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

What is Game Theory About? ,[object Object],[object Object],[object Object],[object Object],[object Object],2 2

Applications of Game Theory ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Limitations of Game Theory ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

What is a Game? ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Classification of Games ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Matrix Game (Normal form) ,[object Object],[object Object],[object Object],Player 1 Player 2 Strategy set for Player 1 Strategy set for Player 2 Payoff to Player 1 Payoff to Player 2 ,[object Object],(3, 4) (0, 0) B (3, -1) (-5, 1) B (-2, -1) (2, 2) A C A

More Formal Game Definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Two-person Zero-sum Games ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Analyzing the Game ,[object Object],Player 1 Player 2 Strictly dominated strategy (dominated by C) Strictly dominated strategy (dominated by B) -1 2 1 -16 D 3 4 2 5 C -18 3 1 3 B 0 1 -1 12 A D C B A

Dominance ,[object Object],[object Object],[object Object],[object Object],[object Object]

Solving the Game Player 1 Player 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],3 2 3 B 4 -1 -2 T R M L

Solving the Game Player 1 Player 2 ,[object Object],[object Object],[object Object],[object Object],-1 0 -16 D 3 2 5 C 0 -1 12 A D B A

Analyzing the Game Player 1 Player 2 ,[object Object],[object Object],-1 0 -16 D 3 2 5 C 0 -1 12 A D B A

Saddle Points ,[object Object],[object Object],[object Object],[object Object],[object Object]

Why Play Saddle Points? ,[object Object],[object Object],[object Object],[object Object],Player 1 Player 2 -1 0 -16 D 3 2 5 C 0 -1 12 A D B A

Why Play Saddle Points? ,[object Object],[object Object],[object Object],[object Object],Player 1 Player 2 -1 0 -16 D 3 2 5 C 0 -1 12 A D B A Powerful arguments to play saddle point!

Solving the Game (min-max algorithm) ,[object Object],[object Object],[object Object],Player 1 Player 2 ,[object Object],[object Object],[object Object],[object Object],-5 -4 8 0 D 3 1 5 7 C -1 0 2 -10 B 5 2 3 4 A D C B A -5 1 -10 2 5 2 8 7

Multiple Saddle Points Player 1 Player 2 ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],-5 -4 0 8 D 3 2 2 5 C -1 0 -10 2 B 5 2 2 3 A D C B A -5 2 -10 2 5 2 2 8

Games With no Saddle Points ,[object Object],[object Object],Player 1 Player 2 3 0 B 1 -5 B -1 2 A C A

Mixed Strategies ,[object Object],[object Object],[object Object],Player 1 Payoff to P1 when playing A = 1/3(4) + 2/3(0) = 4/3 Payoff to P1 when playing B = 1/3(-5) + 2/3(3) = 1/3 ,[object Object],3 0 D -5 B 4 A C 2/3 1/3

Mixed Strategies ,[object Object],[object Object],[object Object],Player 1 Payoff to P1 when playing A = x(4) + (1-x)(0) = 4x Payoff to P1 when playing B = x(-5) + (1-x)(3) = 3 – 8x 4x = 3 – 8x, thus x = 1/4 ,[object Object],3 0 D -5 B 4 A C (1-x) x

Mixed Strategies ,[object Object],[object Object],[object Object],[object Object],Player 1 Payoff to P2 when playing C = x(-4) + (1-x)(5) = 5 - 9x Payoff to P2 when playing D = x(0) + (1-x)(-3) = -3 + 3x 5 – 9x = -3 + 3x, thus x = 2/3 Player 2 Payoff to P2 = -1 3 0 D -5 B 4 A C (1-x) x

Minimax Theorem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Two-person Non-zero Sum Games ,[object Object],[object Object],Player 1 Player 2 ,[object Object],[object Object],-1, 2 2, 0 B 5, 1 B 3, 4 A A

What is the Solution? ,[object Object],[object Object],[object Object],[object Object],Player 1 Player 2 Player 1 Player 2 2, 1 -1, 4 B 3, 2 B 5, 0 A A -1, 2 2, 0 B 3, 1 B 5, 4 A A

Multiple Solution Problem ,[object Object],[object Object],[object Object],Player 1 Player 2 equilibria 2, 2 1, 1 B 0, 1 B 1, 4 A A

The Good News: Nash’s Theorem ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

The Prisoner’s Dilemma ,[object Object],[object Object],[object Object],[object Object],Suspect 1 Suspect 2 payoff is years in jail ( smaller is better ) single NEP better outcome 5, 5 10, 1 C 1, 10 C 2, 2 S S

Pareto Optimal ,[object Object],Suspect 1 Suspect 2 ,[object Object],[object Object],[object Object],Pareto Optimal 5, 5 10, 1 C 1, 10 C 2, 2 S S

Game of Chicken Revisited ,[object Object],[object Object],Driver 1 Driver 2 Drivers want to do opposite of one another Will prior communication help? 2 2 -10, -10 -1, 5 stay 5, -1 stay 0, 0 swerve swerve

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],How much should firm i produce?

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],value chosen by firm 2 value chosen by firm 1

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],and and

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],less quantity produced higher price Competition can be good!

Example: Cournot Model of Duopoly ,[object Object],[object Object],[object Object],NEP : strategies are mutual best responses ,[object Object]

Game Trees (Extensive form) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Game Trees Example ,[object Object],Player 1 Player 2 Player 2 L L R R R L 3, 1 1, 2 -2, 1 0, -1 ,[object Object],what to do when P1 plays L what to do when P1 plays R ,[object Object],Payoff to Player 2 Payoff to Player 1

More Formal Extensive Game Definition ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Game Tree Example ,[object Object],[object Object],[object Object],[object Object],Microsoft Mozilla Mozilla .net .net java java java .net 3, 1 1, 0 0, 0 2, 2

Converting to Matrix Game ,[object Object],[object Object],Microsoft Mozilla 0, 0 1, 0 java, .net 2, 2 3, 1 .net, java 2, 2 0, 0 java 1, 0 3, 1 .net java, java .net, .net .net .net java java java .net 3, 1 1, 0 0, 0 2, 2

NEP and Incredible Threats Microsoft Mozilla NEP incredible threat ,[object Object],[object Object],.net .net java java java .net 3, 1 1, 0 0, 0 2, 2 0, 0 1, 0 java, .net 2, 2 3, 1 .net, java 2, 2 0, 0 java 1, 0 3, 1 .net java, java .net, .net

Solving the Game (backward induction) ,[object Object],[object Object],Best strategy for Mozilla: .net, java (follow Microsoft) Best strategy for Microsoft: .net ,[object Object],[object Object],Equilibrium outcome .net java 3, 1 2, 2 .net .net java java java .net 3, 1 1, 0 0, 0 2, 2

Backward Induction on Game Trees ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Leaders and Followers ,[object Object],[object Object],Mozilla: java Microsoft: .net, java ,[object Object],[object Object],[object Object],[object Object],Mozilla Microsoft Microsoft .net .net java java java .net 1, 3 0, 1 0, 0 2, 2

The Subgame Concept ,[object Object],[object Object],[object Object],[object Object],Microsoft Mozilla Mozilla .net .net java java java .net 3, 1 1, 0 0, 0 2, 2

Subgame Perfect Nash Equilibrium ,[object Object],[object Object],[object Object],[object Object],[object Object]

Subgame Perfect Nash Equilibrium ,[object Object],[object Object],[object Object],MS Mozilla J N Subgame Perfect NEP Not subgame Perfect NEP Microsoft Mozilla Mozilla .net .net java java java .net 3, 1 1, 0 0, 0 2, 2 0,0 1,0 JN 2,2 3,1 NJ 2,2 0,0 J 1,0 3,1 N JJ NN

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