1. ecs289m Spring, 2008 Non-cooperative Games S. Felix Wu Computer Science Department University of California, Davis wu@cs.ucdavis.edu http://www.cs.ucdavis.edu/~wu/

2. 05/06/2008Non-cooperative Game Theory2 The Structure of the Game –Four elements –Extensive versus Strategic/Normalized forms –The Structure of the Game Strategies –Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game

3. 05/06/2008Non-cooperative Game Theory3 The Structure of the Game –Four elements –Extensive versus Strategic/Normalized forms –The Structure of the Game Strategies –Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game

4. 05/06/2008Non-cooperative Game Theory4 Four Elements A game: Formal representation of a situation of strategic interdependence –Set of agents, I (|I|=n) AKA players –Each agent, j, has a set of actions, Aj AKA moves –Actions define outcomes For each possible set of actions there is an outcome. –Outcomes define payoffs Agents’ derive utility from different outcomes

5. 05/06/2008Non-cooperative Game Theory5 Matching Pennies Agents: {Alice, Bob} Actions: {Head, Tail} Outcomes: {Matched, Not} Payoffs: {(-1, 1), (1, -1)} Alice gives 1 dollar to Bob!

6. 05/06/2008Non-cooperative Game Theory6 Matching Pennies Agents: {Alice, Bob} Actions: {Head, Tail} Outcomes: {Matched, Not} Payoffs: {(-1, 1), (1, -1)} Bob gives 1 dollar to Alice!

7. 05/06/2008Non-cooperative Game Theory7 Matching Pennies Simultaneous moves Sequential moves

8. 05/06/2008Non-cooperative Game Theory8 Extensive Form (or Game Tree) Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs (-1,+1) (+1,-1)

9. 05/06/2008Non-cooperative Game Theory9 Tick-Tack-Toe

10. 05/06/2008Non-cooperative Game Theory10 Matching Pennies Sequential moves Simultaneous moves –“Bob doesn’t know Alice’s move” –Not Perfect Information

11. 05/06/2008Non-cooperative Game Theory11 Bob can not distinguish… Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs (-1,+1) (+1,-1)

12. 05/06/2008Non-cooperative Game Theory12 Agent Alice Agent Bob H H H T T T Action Terminal node (outcome) Payoffs Another representation … (-1,+1) (+1,-1)

13. 05/06/2008Non-cooperative Game Theory13 Information Sets Agent Alice Agent Bob H H H T T T (-1,+1) (+1,-1) Action Terminal node (outcome) Payoffs

14. 05/06/2008Non-cooperative Game Theory14 Matching Pennies Sequential moves Simultaneous moves –“Bob doesn’t know Alice’s move” –Not Perfect Information Assumption: “Perfect Recall” –“remember the history”

15. 05/06/2008Non-cooperative Game Theory15 Example #1: Perfect Recall? p q x x y y a b ba

16. 05/06/2008Non-cooperative Game Theory16 Example #2: Perfect Recall? p q x x yy a bbababa

17. 05/06/2008Non-cooperative Game Theory17 Example #2: Perfect Recall? p q x x yy a bbababa Felix likes that for sure!

18. 05/06/2008Non-cooperative Game Theory18 Matching Pennies Sequential moves Simultaneous moves –“Bob doesn’t know Alice’s move” –Not Perfect Information Assumption: “Perfect Recall” Perfect Information: each information set contains a single decision node.

19. 05/06/2008Non-cooperative Game Theory19 Matching Pennies Sequential moves Simultaneous moves –“Bob doesn’t know Alice’s move” Assumption: “Perfect Recall” Perfect Information: each information set contains a single decision node. Random moves –Flip a coin to decide Head or Tail

20. 05/06/2008Non-cooperative Game Theory20 Common Knowledge Structure of the Game All the players know about the structure of the game, all players know that their rivals know it, and,…

21. 05/06/2008Non-cooperative Game Theory21 Strategy Let H X denote the collection of agent X ’s information sets, A the set of possible actions in the game, and C(h) A the set of actions possible at information set h. A strategy for agent X is a function: – Per player (Info Set => Action)

22. 05/06/2008Non-cooperative Game Theory22 Strategies for Matching Pennies Bob’s four possible strategies: –S 1 : Play H if Alice plays H; play H if Alice plays T –S 2 : Play H if Alice plays H; play T if Alice plays T –S 3 : Play T if Alice plays H; play H if Alice plays T –S 4 : Play T if Alice plays H; play T if Alice plays T Information Sets

23. 05/06/2008Non-cooperative Game Theory23 Strategies imply… A sequence of moves actually taken A probability distribution over the terminal nodes of the game Strategies ~ Outcomes ~ Payoffs

24. 05/06/2008Non-cooperative Game Theory24 Strategies imply… A sequence of moves actually taken A probability distribution over the terminal nodes of the game Strategies ~ Outcomes ~ Payoffs –“Strategic/Normal Forms”

25. 05/06/2008Non-cooperative Game Theory25 Matching Pennies Alice Bob H H T T -1, +1 +1, -1 Action Outcome Payoffs

26. 05/06/2008Non-cooperative Game Theory26 Bob doesn’t know Alice’s move… Alice Bob H H T T -1, +1 +1, -1 Action Outcome Payoffs Strategies

27. 05/06/2008Non-cooperative Game Theory27 Strategies for Matching Pennies Bob’s four possible strategies: –S 1 : Play H if Alice plays H; play H if Alice plays T –S 2 : Play H if Alice plays H; play T if Alice plays T –S 3 : Play T if Alice plays H; play H if Alice plays T –S 4 : Play T if Alice plays H; play T if Alice plays T Information Sets

28. 05/06/2008Non-cooperative Game Theory28 Bob does know Alice’s move… Alice Bob H S1S1 T -1, +1 +1, -1 S2S2 S3S3 S4S4 -1, +1 +1, -1 -1, +1

29. 05/06/2008Non-cooperative Game Theory29 Alice Bob H S1S1 T -1, +1 +1, -1 S2S2 S3S3 S4S4 -1, +1 +1, -1 -1, +1 S 2 : Play H if Alice plays H; play T if Alice plays T

30. 05/06/2008Non-cooperative Game Theory30 Non-cooperative Game Theory The Structure of the Game –Four elements –Extensive versus Strategic/Normalized forms –The Structure of the Game Strategies –Mixed, Rationalizable, Dominance Nash Equilibrium Dynamic Game

31. 05/06/2008Non-cooperative Game Theory31 Strategies Strategy: –A strategy, s j, is a complete contingency plan; defines actions agent j should take for all possible information sets of the world Strategy profile: s =(s 1,…,s n ) –s -i = (s 1,…,s i-1,s i+1,…,s n ) Utility function: u i (s) –Note that the utility of an agent depends on the strategy profile, not just its own strategy –We assume agents are expected utility maximizers

32. 05/06/2008Non-cooperative Game Theory32 Mixed Strategy Randomization over a set of pure and deterministic strategies for each agent

33. 05/06/2008Non-cooperative Game Theory33 Mixed Strategy Randomization over a set of pure and deterministic strategies for each agent

34. 05/06/2008Non-cooperative Game Theory34 Dominant/Dominated Strategy

35. 05/06/2008Non-cooperative Game Theory35 Dominant/Dominated Strategy Strictly dominant Strictly dominated Weakly dominated

36. 05/06/2008Non-cooperative Game Theory36 Dominant Strategies Agents’ will play best-response strategies –Rationalizable A dominant strategy is –a best response for all s -i –They do not always exist –Inferior strategies are called dominated

37. 05/06/2008Non-cooperative Game Theory37 Alice Bob H S1S1 T -1, +1 +1, -1 S2S2 S3S3 S4S4 -1, +1 +1, -1 -1, +1 S 2 : Play H if Alice plays H; play T if Alice plays T Assuming Alice and Bob simultaneously choose a strategy..

38. 05/06/2008Non-cooperative Game Theory38 Iterated Elimination of “Dominated” Let Ri  Si be the set of removed strategies for agent i Initially Ri=Ø Choose agent i, and strategy si such that si  Si\Ri and there exists si’  Si\Ri such that Add si to Ri, continue ui(si’,s -i )>ui(si,s -i ) for all s -i  S -i \R -i

39. 05/06/2008Non-cooperative Game Theory39 Prisoner’s Dilemma Art and Bob been caught stealing a car: sentence is 2 years in jail. DA wants to convict them of a big bank robbery: sentence is 10 years in jail. DA has no evidence and to get the conviction, he makes the prisoners play a “game”.

40. 05/06/2008Non-cooperative Game Theory40 Rules of the Game Players cannot communicate with one another.  If both confess to the larger crime, each will receive a sentence of 3 years for both crimes.  If one confesses and the accomplice does not, the one who confesses will receive a sentence of 1 year, while the accomplice receives a 10-year sentence.  If neither confesses, both receive a 2-year sentence.

41. 05/06/2008Non-cooperative Game Theory41 Another one

42. 05/06/2008Non-cooperative Game Theory42 Strategies The strategies of a game are all the possible outcomes of each player. The strategies in the prisoners’ dilemma are: Confess to the bank robbery Deny the bank robbery Four outcomes:  Both confess.  Both deny.  Art confesses and Bob denies.  Bob confesses and Art denies.

43. 05/06/2008Non-cooperative Game Theory43 How would Bob play?

44. 05/06/2008Non-cooperative Game Theory44 Iterated Elimination of “Dominated” Let Ri  Si be the set of removed strategies for agent i Initially Ri=Ø Choose agent i, and strategy si such that si  Si\Ri and there exists si’  Si\Ri such that Add si to Ri, continue ui(si’,s -i )>ui(si,s -i ) for all s -i  S -i \R -i

45. 05/06/2008Non-cooperative Game Theory45 Iterated Elimination of Dominated Strategies Let Ri  Si be the set of removed strategies for agent i Initially Ri=Ø Choose agent i, and strategy si such that si  Si\Ri and there exists si’  Si\Ri such that Add si to Ri, continue We might not be able to eliminate many! ui(si’,s -i )>ui(si,s -i ) for all s -i  S -i \R -i

46. 05/06/2008Non-cooperative Game Theory46 Art versus Bob Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

47. 05/06/2008Non-cooperative Game Theory47 Can Bob eliminate ONE? Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

48. 05/06/2008Non-cooperative Game Theory48 Dominant Strategy for Bob Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

49. 05/06/2008Non-cooperative Game Theory49 Dominant Strategy for Bob Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

50. 05/06/2008Non-cooperative Game Theory50 Best Response for Bob

51. 05/06/2008Non-cooperative Game Theory51 Dominant Strategy for Bob Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

52. 05/06/2008Non-cooperative Game Theory52 Dominant Strategy for Art Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

53. 05/06/2008Non-cooperative Game Theory53 Dominant Strategy for both Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

54. 05/06/2008Non-cooperative Game Theory54 Dominant Strategy for both Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

55. 05/06/2008Non-cooperative Game Theory55 Stable, Equilibrium Can we converge into a “stable” strategy such that we satisfy all the players to some degree? Can we converge into a “stable” strategy such that none of the players wanted to change or deviate “unilaterally”?

56. 05/06/2008Non-cooperative Game Theory56 Unilateral Change/Deviate Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

57. 05/06/2008Non-cooperative Game Theory57 Nash Equilibrium

58. 05/06/2008Non-cooperative Game Theory58 Nash Equilibrium Given The strategy s is a Nash Equilibrium if

59. 05/06/2008Non-cooperative Game Theory59 Dominant Strategy for both Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

60. 05/06/2008Non-cooperative Game Theory60 3-player Game (1-B, 2-R, 3-G) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

61. 05/06/2008Non-cooperative Game Theory61 Nash Equilibrium? 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

62. 05/06/2008Non-cooperative Game Theory62 (u 1,s 2,s 3 )? 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

63. 05/06/2008Non-cooperative Game Theory63 (u1,s2,s3)(u1,s2,s3) 0,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,3 3,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 3,3,03,3,0 3,0,33,0,3 4,1,14,1,1

64. 05/06/2008Non-cooperative Game Theory64 (u1,s2,s3)(u1,s2,s3) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

65. 05/06/2008Non-cooperative Game Theory65 (u1,s2,s3)(u1,s2,s3) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 3,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 0,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 1,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 4,1,14,1,13,3,03,3,0 4,1,14,1,1

66. 05/06/2008Non-cooperative Game Theory66 (u 1,s 2,s 3 ) (s 1,u 2,t 3 )? 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

67. 05/06/2008Non-cooperative Game Theory67 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

68. 05/06/2008Non-cooperative Game Theory68 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

69. 05/06/2008Non-cooperative Game Theory69 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Which one will the players converge into?

70. 05/06/2008Non-cooperative Game Theory70 The Prediction Problem Given a game, how can we describe or specify the mechanism(s) behind it such that we will have a better prediction? –Social network formation –Economical outcomes –Many other applications –The basic Nash Equilibrium may still be too loose…

71. 05/06/2008Non-cooperative Game Theory71 Nash Equilibrium Extensions Undominated (uNE) Strong (SNE) Coalition-Proof (CPNE)

72. 05/06/2008Non-cooperative Game Theory72 Undominated Nash Equilibrium Nash Equilibrium –Strategy s given (and so was s -i ) Undominated Nash Equilibrium (uNE) –Consider all possible s -i

73. 05/06/2008Non-cooperative Game Theory73 Undominated Nash Equilibrium If all agents/players choose the “best response”… A dominant strategy equilibrium is a strategy profile where the strategy for each player is dominant Agents do not need to counter-speculate!

74. 05/06/2008Non-cooperative Game Theory74 Iterated Elimination of Dominated Strategies Let Ri  Si be the set of removed strategies for agent i Initially Ri=Ø Choose agent i, and strategy si such that si  Si\Ri and there exists si’  Si\Ri such that Add si to Ri, continue Thm: If a unique strategy profile, s*, survives then it is a Nash Eq. ui(si’,s -i )>ui(si,s -i ) for all s -i  S -i \R -i

75. 05/06/2008Non-cooperative Game Theory75 Dominant Strategy for both Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1

76. 05/06/2008Non-cooperative Game Theory76 Microsoft versus Yahoo Y 6-4 N (7,3)(0,0) 5-5 split 7-3 YNYN (6,4)(0,0)(5,5)(0,0)

77. 05/06/2008Non-cooperative Game Theory77 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Which ONES are Nash Equilibrium?

78. 05/06/2008Non-cooperative Game Theory78 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Seven! Which ONES are Undominated Nash Equilibrium?

79. 05/06/2008Non-cooperative Game Theory79 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Seven! Which ONES are Undominated Nash Equilibrium?

80. 05/06/2008Non-cooperative Game Theory80 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Seven! Which ONES are Undominated Nash Equilibrium? Only YYY for player 2!

81. 05/06/2008Non-cooperative Game Theory81 Strong Nash Equilibrium Nash Equilibrium (NE) –No Coalition allowed Strong Nash Equilibrium (SNE) –Works for ALL possible coalitions

82. 05/06/2008Non-cooperative Game Theory82 Coalition Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1 Not Strong NE (SNE) Not stable/NE at all!!

83. 05/06/2008Non-cooperative Game Theory83 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Seven! Which ONES are Strong Nash Equilibrium?

84. 05/06/2008Non-cooperative Game Theory84 The Splitting Game 7,37,37,37,37,37,3 6,46,46,46,40,00,0 5,55,50,00,05,55,5 YYYYYNYNY 7-3 6-4 5-5 7,37,30,00,00,00,0 0,00,06,46,46,46,4 0,00,05,55,50,00,0 YNNNYYNYN 0,00,00,00,0 0,00,00,00,0 5,55,50,00,0 NNYNNN Seven! Which ONES are Strong Nash Equilibrium? ALL of them!

85. 05/06/2008Non-cooperative Game Theory85 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

86. 05/06/2008Non-cooperative Game Theory86 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Coalition: 2&3

87. 05/06/2008Non-cooperative Game Theory87 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Coalition: 1&3

88. 05/06/2008Non-cooperative Game Theory88 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3

89. 05/06/2008Non-cooperative Game Theory89 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Is any of them “Strong”? No!!

90. 05/06/2008Non-cooperative Game Theory90 Coalition-Proof Nash Equilibrium NE is too loose & SNE is too restrictive, and CPNE is somewhere in between… Under SNE, a coalition can move from a NE to any other cell, but that cell might not be stable… Under CPNE, a coalition can be only allowed to move a “self-enforcing” cell (I.e., no further deviation from that cell).

91. 05/06/2008Non-cooperative Game Theory91 (-2,-2) is not Self-Enforcing Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1 Not Strong NE (SNE) Not stable/NE at all!! In Prisoner’s Dilemma, therefore, (-3,-3) is NE, uNE, also CPNE(?), but only not SNE!!

92. 05/06/2008Non-cooperative Game Theory92 (u 1,s 2,s 3 ) is not SNE, how about CPNE? 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Deviation to [0,3,3] disallowed under CPNE!

93. 05/06/2008Non-cooperative Game Theory93 (u 1,s 2,s 3 ) is not SNE, how about CPNE? 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 Deviation to [2,2,2] is, however, allowed, &, therefore [4,1,1] (u 1,s 2,s 3 ) is not CPNE!

94. 05/06/2008Non-cooperative Game Theory94 The game of 1,11,13,03,01,11,1 0,30,33,33,31,41,4 1,11,14,14,12,22,2 s3s3 t3t3 u3u3 s2s2 t2t2 u2u2 u1u1

95. 05/06/2008Non-cooperative Game Theory95 The game of 1,11,13,03,01,11,1 0,30,33,33,31,41,4 1,11,14,14,12,22,2 s3s3 t3t3 u3u3 s2s2 t2t2 u2u2 u1u1

96. 05/06/2008Non-cooperative Game Theory96 SNE versus CPNE SNE: regardless of any possible coalitions, the NE will survive. –Assuming the agents within the coalition are selfless CPNE: considering any possible coalitions, but also consider the relation between the bigger game and the inner-circle game. –Agents in a coalition might move away from a NE if the new state will be better for ALL members of the coalition, but then, we need to consider the sub-game. –Assuming the agents within the coalition are non- cooperative

97. 05/06/2008Non-cooperative Game Theory97 SNE versus CPNE SNE: regardless of any possible coalitions, the NE will survive. –Assuming the agents within the coalition are selfless CPNE: considering any possible coalitions, but also consider the relation between the bigger game and the inner-circle game. –Agents in a coalition might move away from a NE if the new state will be better for ALL members of the coalition, but then, we need to consider the sub-game. –Assuming the agents within the coalition are non- cooperative

98. 05/06/2008Non-cooperative Game Theory98 (u 1,s 2,s 3 ) (s 1,u 2,t 3 ) (t 1,t 2,u 3 ) (u 1,u 2,u 3 ) 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 4 NEs 0 SNE

99. 05/06/2008Non-cooperative Game Theory99 (u 1,u 2,u 3 ) is the unique CPNE! 3,3,03,3,00,0,00,0,03,3,03,3,0 3,0,33,0,33,0,33,0,33,0,33,0,3 4,1,14,1,13,0,33,0,34,1,14,1,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 0,0,00,0,00,3,30,3,30,3,30,3,3 3,3,03,3,00,3,30,3,31,4,11,4,1 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 3,3,03,3,00,3,30,3,31,4,11,4,1 3,0,33,0,31,1,41,1,41,1,41,1,4 4,1,14,1,11,1,41,1,42,2,22,2,2 s2s2 t2t2 u2u2 s1s1 t1t1 u1u1 s3s3 t3t3 u3u3 4 NEs 0 SNE 1 CPNE

100. 05/06/2008Non-cooperative Game Theory100 Coalition in Prisoner’s Dilemma Art Bob Confess Deny -3, -3 -2, -2 -1, -10 -10, -1 Not Strong NE (SNE) Not stable/NE at all!!

101. 05/06/2008Non-cooperative Game Theory101 Updated Prisoner’s Dilemma Art Bob Confess Deny -3, -3 -2, -2 -2, -10 -10, -2 NE, ~CPNE, ~SNE NE, CPNE, SNE

102. 05/06/2008Non-cooperative Game Theory102 Definition of CPNE “Coalition-Proof Nash Equilibria: I. Concept” Berheim/Peleg/Whiston, J. of Economic Theory, 42, 1-12 (1987). In a single player game is a Coalition-Proof Nash Equilibrium if and only if maximizes. Let n > 1 and assume that Coalition-Proof Nash Equilibrium has been defined for games with fewer than n players. Then, a)For any game with n players, is self-enforcing if,., is a Coalition-Proof Nash Equilibrium in the game of. b)For any game with n players, is a Coalition-Proof Nash Equilibrium if it is self-enforcing and if there does not exist another self-enforcing strategy vector such that.

103. 05/06/2008Non-cooperative Game Theory103 What’s next? Many mathematical tools, definitions, and theories… Now, let’s try to apply them to online social networks…

104. 05/06/2008Non-cooperative Game Theory104 Materials Mainly Covered… Chapter 5 Chapters 7~9