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  • 1. 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
  • 2. Tutorial Organization
    • Two parts of 90 minutes
      • 15 minutes coffee break in between
    • First part: introduction to game theory
      • definitions, important results, (simple) examples
      • divided in two 45 minutes sessions (Daniel + John)
    • Second part: game theory and networking
      • game-theoretic formulation of networking problems
      • 1 st 45 minute session (Daniel)
        • routing games and congestion control games
      • 2 nd 45 minute session (John)
        • overlay games and wireless games
  • 3. What is Game Theory About?
    • Analysis of situations where conflict of interests are present
    • Goal is to prescribe how conflicts can be resolved
    • Game of Chicken
      • driver who steers away looses
    • What should drivers do?
    2 2
  • 4. Applications of Game Theory
    • Theory developed mainly by mathematicians and economists
      • contributions from biologists
    • Widely applied in many disciplines
      • from economics to philosophy, including computer science (Systems, Theory and AI)
      • goal is often to understand some phenomena
    • “ Recently” applied to computer networks
      • Nagle, RFC 970, 1985
        • “ datagram networks as a multi-player game”
      • paper in first volume of IEEE/ACM ToN (1993)
      • wider interest starting around 2000
  • 5. Limitations of Game Theory
    • No unified solution to general conflict resolution
    • Real-world conflicts are complex
      • models can at best capture important aspects
    • Players are (usually) considered rational
      • determine what is best for them given that others are doing the same
    • No unique prescription
      • not clear what players should do
    • But it can provide intuitions, suggestions and partial prescriptions
      • best mathematical tool we currently have
  • 6. What is a Game?
    • A Game consists of
      • at least two players
      • a set of strategies for each player
      • a preference relation over possible outcomes
    • Player is general entity
      • individual, company, nation, protocol, animal, etc
    • Strategies
      • actions which a player chooses to follow
    • Outcome
      • determined by mutual choice of strategies
    • Preference relation
      • modeled as utility (payoff) over set of outcomes
  • 7. Classification of Games
    • Many, many types of games
      • three major categories
    • Non-Cooperative (Competitive) Games
      • individualized play, no bindings among players
    • Repeated and Evolutionary Games
      • dynamic scenario
    • Cooperative Games
      • play as a group, possible bindings
  • 8. Matrix Game (Normal form)
    • Simultaneous play
      • players analyze the game and write their strategy on a paper
    • Combination of strategies determines payoff
    Player 1 Player 2 Strategy set for Player 1 Strategy set for Player 2 Payoff to Player 1 Payoff to Player 2
    • Representation of a game
    (3, 4) (0, 0) B (3, -1) (-5, 1) B (-2, -1) (2, 2) A C A
  • 9. More Formal Game Definition
    • Normal form (strategic) game
      • a finite set N of players
      • a set strategies for each player
      • payoff function for each player
        • where is the set of strategies chosen by all players
    • A is the set of all possible outcomes
    • is a set of strategies chosen by players
      • defines an outcome
  • 10. Two-person Zero-sum Games
    • One of the first games studied
      • most well understood type of game
    • Players interest are strictly opposed
      • what one player gains the other loses
      • game matrix has single entry (gain to player 1)
    • Intuitive solution concept
      • players maximize gains
      • unique solution
  • 11. Analyzing the Game
    • Player 1 maximizes matrix entry, while player 2 minimizes
    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
  • 12. Dominance
    • Strategy S strictly dominates a strategy T if every possible outcome when S is chosen is better than the corresponding outcome when T is chosen
    • Dominance Principle
      • rational players never choose strictly dominated strategies
    • Idea: Solve the game by eliminating strictly dominated strategies!
      • iterated removal
  • 13. Solving the Game Player 1 Player 2
    • Iterated removal of strictly dominated strategies
      • Player 1 cannot remove any strategy (neither T or B dominates the other)
      • Player 2 can remove strategy R (dominated by M)
      • Player 1 can remove strategy T (dominated by B)
      • Player 2 can remove strategy L (dominated by M)
      • Solution: P 1 -> B, P 2 -> M
        • payoff of 2
    3 2 3 B 4 -1 -2 T R M L
  • 14. Solving the Game Player 1 Player 2
    • Removal of strictly dominates strategies does not always work
    • Consider the game
    • Neither player has dominated strategies
    • Requires another solution concept
    -1 0 -16 D 3 2 5 C 0 -1 12 A D B A
  • 15. Analyzing the Game Player 1 Player 2
    • Outcome (C, B) seems “stable”
      • saddle point of game
    -1 0 -16 D 3 2 5 C 0 -1 12 A D B A
  • 16. Saddle Points
    • An outcome is a saddle point if it is both less than or equal to any value in its row and greater than or equal to any value in its column
    • Saddle Point Principle
      • Players should choose outcomes that are saddle points of the game
    • Value of the game
      • value of saddle point outcome if it exists
  • 17. Why Play Saddle Points?
    • If player 1 believes player 2 will play B
      • player 1 should play best response to B (which is C)
    • If player 2 believes player 1 will play C
      • player 2 should play best response to C (which is B)
    Player 1 Player 2 -1 0 -16 D 3 2 5 C 0 -1 12 A D B A
  • 18. Why Play Saddle Points?
    • Why should player 1 believe player 2 will play B?
      • playing B guarantees player 2 loses at most v (which is 2)
    • Why should player 2 believe player 1 will play C?
      • playing C guarantees player 1 wins at least v (which is 2)
    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!
  • 19. Solving the Game (min-max algorithm)
    • choose minimum entry in each row
    • choose the maximum among these
    • this is maximin value
    Player 1 Player 2
    • choose maximum entry in each column
    • choose the minimum among these
    • this is the minimax value
    • if minimax == maximin, then this is the saddle point of game
    -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
  • 20. Multiple Saddle Points Player 1 Player 2
    • In general, game can have multiple saddle points
    • Same payoff in every saddle point
      • unique value of the game
    • Strategies are interchangeable
      • Example: strategies (A, B) and (C, C) are saddle points
      • then (A, C) and (C, B) are also saddle points
    -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
  • 21. Games With no Saddle Points
    • What should players do?
      • resort to randomness to select strategies
    Player 1 Player 2 3 0 B 1 -5 B -1 2 A C A
  • 22. Mixed Strategies
    • Each player associates a probability distribution over its set of strategies
      • players decide on which prob. distribution to use
    • Payoffs are computed as expectations
    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
    • How should players choose prob. distribution?
    3 0 D -5 B 4 A C 2/3 1/3
  • 23. Mixed Strategies
    • Idea: use a prob. distribution that cannot be exploited by other player
      • payoff should be equal independent of the choice of strategy of other player
      • guarantees minimum gain (maximum loss)
    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
    • How should Player 2 play?
    3 0 D -5 B 4 A C (1-x) x
  • 24. Mixed Strategies
    • Player 2 mixed strategy
      • 1/4 C , 3/4 D
      • maximizes its loss independent of P1 choices
    • Player 1 has same reasoning
    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
  • 25. Minimax Theorem
    • Every two-person zero-sum game has a solution in mixed (and sometimes pure) strategies
      • solution payoff is the value of the game
      • maximin = v = minimax
      • v is unique
      • multiple equilibrium in pure strategies possible
        • but fully interchangeable
    • Proved by John von Neumann in 1928!
      • birth of game theory…
  • 26. Two-person Non-zero Sum Games
    • Players are not strictly opposed
      • payoff sum is non-zero
    Player 1 Player 2
    • Situations where interest is not directly opposed
      • players could cooperate
    -1, 2 2, 0 B 5, 1 B 3, 4 A A
  • 27. What is the Solution?
    • Ideas of zero-sum game: saddle points
    • pure strategy equilibrium
    • mixed strategies equilibrium
      • no pure strategy eq.
    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
  • 28. Multiple Solution Problem
    • Games can have multiple equilibria
      • not equivalent: payoff is different
      • not interchangeable: playing an equilibrium strategy does not lead to equilibrium
    Player 1 Player 2 equilibria 2, 2 1, 1 B 0, 1 B 1, 4 A A
  • 29. The Good News: Nash’s Theorem
    • Every two person game has at least one equilibrium in either pure or mixed strategies
    • Proved by Nash in 1950 using fixed point theorem
      • generalized to N person game
      • did not “invent” this equilibrium concept
    • Def: An outcome o* of a game is a NEP (Nash equilibrium point) if no player can unilaterally change its strategy and increase its payoff
    • Cor: any saddle point is also a NEP
  • 30. The Prisoner’s Dilemma
    • One of the most studied and used games
      • proposed in 1950s
    • Two suspects arrested for joint crime
      • each suspect when interrogated separately, has option to confess or remain silent
    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
  • 31. Pareto Optimal
    • Prisoner’s dilemma: individual rationality
    Suspect 1 Suspect 2
    • Another type of solution: group rationality
      • Pareto optimal
    • Def: outcome o* is Pareto Optimal if no other outcome is better for all players
    Pareto Optimal 5, 5 10, 1 C 1, 10 C 2, 2 S S
  • 32. Game of Chicken Revisited
    • Game of Chicken (aka. Hawk-Dove Game)
      • driver who swerves looses
    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
  • 33. Example: Cournot Model of Duopoly
    • Several firms produce exactly same product
      • : quantity produced by firm
    • Cost to firm i to produce quantity
    • Market clearing price (price paid by consumers)
      • where
    • Revenue of firm i
    How much should firm i produce?
  • 34. Example: Cournot Model of Duopoly
    • Consider two firms:
    • Simple production cost
      • no fixed cost, only marginal cost with constant c
    • Simple market (fixed demand a )
      • where
    • Revenue of firm
    • Firms choose quantities simultaneously
    • Assume c < a
  • 35. Example: Cournot Model of Duopoly
    • Two player game: Firm 1 and Firm 2
    • Strategy space
      • production quantity
      • since if ,
    • What is the NEP?
    • To find NEP, firm 1 solves
    • To find NEP, firm 2 solves
    value chosen by firm 2 value chosen by firm 1
  • 36. Example: Cournot Model of Duopoly
    • Solution to maximization problem
      • first order condition is necessary and sufficient
    • Best response functions
      • best strategy for player 1, given choice for player 2
    • At NEP, strategies are best response to one another
      • need to solve pair of equations
      • using substitution…
    and and
  • 37. Example: Cournot Model of Duopoly
    • NEP is given by
    • Total amount produced at NEP:
    • Price paid by consumers at NEP:
    • Consider a monopoly (no firm 2, )
    • Equilibrium is given by
    • Total amount produced:
    • Price paid by consumers:
    less quantity produced higher price Competition can be good!
  • 38. Example: Cournot Model of Duopoly
    • Graphical approach: best response functions
    • Plot best response for firm 1
    • Plot best response for firm 2
    NEP : strategies are mutual best responses
      • all intersections are NEPs
  • 39. Game Trees (Extensive form)
    • Sequential play
      • players take turns in making choices
      • previous choices can be available to players
    • Game represented as a tree
      • each non-leaf node represents a decision point for some player
      • edges represent available choices
    • Can be converted to matrix game (Normal form)
      • “plan of action” must be chosen before hand
  • 40. Game Trees Example
    • Strategy set for Player 1: {L, R}
    Player 1 Player 2 Player 2 L L R R R L 3, 1 1, 2 -2, 1 0, -1
    • Strategy for Player 2: __, __
    what to do when P1 plays L what to do when P1 plays R
    • Strategy set for Player 2: {LL, LR, RL, RR}
    Payoff to Player 2 Payoff to Player 1
  • 41. More Formal Extensive Game Definition
    • An extensive form game
      • a finite set N of players
      • a finite height game tree
      • payoff function for each player
        • where s is a leaf node of game tree
    • Game tree: set of nodes and edges
      • each non-leaf node represents a decision point for some player
      • edges represent available choices (possibly infinite)
    • Perfect information
      • all players have full knowledge of game history
  • 42. Game Tree Example
    • Microsoft and Mozilla are deciding on adopting new browser technology (.net or java)
      • Microsoft moves first, then Mozilla makes its move
    • Non-zero sum game
      • what are the NEP?
    Microsoft Mozilla Mozilla .net .net java java java .net 3, 1 1, 0 0, 0 2, 2
  • 43. Converting to Matrix Game
    • Every game in extensive form can be converted into normal form
      • exponential growth in number of strategies
    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
  • 44. NEP and Incredible Threats Microsoft Mozilla NEP incredible threat
    • Play “java no matter what” is not credible for Mozilla
      • if Microsoft plays .net then .net is better for Mozilla than java
    .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
  • 45. Solving the Game (backward induction)
    • Starting from terminal nodes
      • move up game tree making best choice
    Best strategy for Mozilla: .net, java (follow Microsoft) Best strategy for Microsoft: .net
    • Single NEP
    • Microsoft -> .net, Mozilla -> .net, java
    Equilibrium outcome .net java 3, 1 2, 2 .net .net java java java .net 3, 1 1, 0 0, 0 2, 2
  • 46. Backward Induction on Game Trees
    • Kuhn’s Thr: Backward induction always leads to saddle point (on games with perfect information)
      • game value at equilibrium is unique (for zero-sum games)
    • In general, multiple NEPs are possible after backward induction
      • cases with no strict preference over payoffs
    • Effective mechanism to remove “bad” NEP
      • incredible threats
  • 47. Leaders and Followers
    • What happens if Mozilla is moves first?
    • NEP after backward induction:
    Mozilla: java Microsoft: .net, java
    • Outcome is better for Mozilla, worst for Microsoft
      • incredible threat becomes credible!
    • 1 st mover advantage
      • but can also be a disadvantage…
    Mozilla Microsoft Microsoft .net .net java java java .net 1, 3 0, 1 0, 0 2, 2
  • 48. The Subgame Concept
    • Def: a subgame is any subtree of the original game that also defines a proper game
      • includes all descendents of non-leaf root node
    • 3 subtrees
      • full tree, left tree, right tree
    Microsoft Mozilla Mozilla .net .net java java java .net 3, 1 1, 0 0, 0 2, 2
  • 49. Subgame Perfect Nash Equilibrium
    • Def: a NEP is subgame perfect if its restriction to every subgame is also a NEP of the subgame
    • Thr: every extensive form game has at least one subgame perferct Nash equilibrium
      • Kuhn’s theorem, based on backward induction
    • Set of NEP that survive backward induction
      • in games with perfect information
  • 50. Subgame Perfect Nash Equilibrium
    • (N, NN) is not a NEP when restricted to the subgame starting at J
    • (J, JJ) is not a NEP when restricted to the subgame starting at N
    • (N, NJ) is a subgame perfect Nash equilibrium
    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
  • 51. Title