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