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

2. 05/15/2008Network Formation2 Coalition Game Game Characteristic Function Super-Additive* Payoff Allocation *Ankush Garg Pointed out my mistake earlier about the definition. Thanks!.

3. 05/15/2008Network Formation3 Payoff Allocation Efficient Individually Rational Imputation Set

4. 05/15/2008Network Formation4 Core Imputation Set Core

5. 05/15/2008Network Formation5 Interpretation of “Core” No incentive for splitting from the “Grand Coalition” No subgroup coalition adds more value than the “Grand Coalition”

6. 05/15/2008Network Formation6 Does “Core” always exist? Imputation Set Core?

7. 05/15/2008Network Formation7 3-player coalition game Super-additive? Efficient allocation? 1 23 1 23 1 23 1 23 1 23

8. 05/15/2008Network Formation8 3-player game Super-additive? Efficient allocation?

9. 05/15/2008Network Formation9 3-player game Super-additive? Efficient allocation?

10. 05/15/2008Network Formation10 Imputation and Core

11. 05/15/2008Network Formation11 Examples of

12. 05/15/2008Network Formation12 Examples of Efficient but Irrational Rational, Imputation, not Core

13. 05/15/2008Network Formation13 3-player game Super-additive? Efficient allocation? Imputation and Core? 1 23 1 23 1 23 1 23 1 23

14. 05/15/2008Network Formation14 Examples of Rational, Imputation, Core? 1 23 1 23

15. 05/15/2008Network Formation15 Examples of

16. 05/15/2008Network Formation16 The Class of Balanced Game is called a balanced map, if A vector!

17. 05/15/2008Network Formation17 The Class of Balanced Game is called a balanced map, if is called a balanced game, if “balanced maps”

18. 05/15/2008Network Formation18 Bondareva (1963) Shapley (1967) is balanced

19. 05/15/2008Network Formation19 Bondareva (1963) Shapley (1967) is balanced The first paper is in Russian, while the second one was written in French!

20. 05/15/2008Network Formation20 3-player game 1 23 1 23 1 23 1 23 1 23

21. 05/15/2008Network Formation21 3-player game 1 23 1 23 1 23 1 23 1 23

22. 05/15/2008Network Formation22 Balanced Map

23. 05/15/2008Network Formation23 Balanced Map

24. 05/15/2008Network Formation24 Solving the Equations?

25. 05/15/2008Network Formation25 Solving the Equations?

26. 05/15/2008Network Formation26 Check Balance

27. 05/15/2008Network Formation27 Jack Chan’s Question Can we configure the Kappa differently?

28. 05/15/2008Network Formation28 3-player game 1 23 1 23 1 23 1 23 1 23

29. 05/15/2008Network Formation29 3-player game

30. 05/15/2008Network Formation30 3-player game

31. 05/15/2008Network Formation31 3-player game

32. 05/15/2008Network Formation32 Two Issues Non-existence of Core Too many possible cores ~ select ONE

33. 05/15/2008Network Formation33 Shapley Value If you are in the coalition, you equally share the value with others. If you are not, you get nothing.

34. 05/15/2008Network Formation34 Unanimity Coefficients

35. 05/15/2008Network Formation35 Unanimity Coefficients

36. 05/15/2008Network Formation36 Networks Not all network topologies are possible:

37. 05/15/2008Network Formation37 Network Components

38. 05/15/2008Network Formation38 Internal versus Connected 12 56 34 7

39. 05/15/2008Network Formation39 Internally Connected 12 56 34 7

40. 05/15/2008Network Formation40 Network Empty: all players are isolated Complete: mesh-directly connected Connected: exactly one component Cycle-Free (e.g., star, line) Cycle-Complete

41. 05/15/2008Network Formation41 Cycle Complete

42. 05/15/2008Network Formation42 Wheel A cycle with no further links

43. 05/15/2008Network Formation43 Example 12 56 34 78

44. 05/15/2008Network Formation44 Connected Hull The intersections of all internally connected sets that contains S

45. 05/15/2008Network Formation45 Shortest Communication Path If (N,L) is cycle-complete, then between any two nodes, there exist a unique shortest communication path. How to prove it?

46. 05/15/2008Network Formation46 Example 12 56 34 78

47. 05/15/2008Network Formation47 Example 12 56 34 78

48. 05/15/2008Network Formation48 Connected Hull Players, necessary/sufficient, for a coalition to cooperate/communicate!

49. 05/15/2008Network Formation49 Example 12 45 3 6

50. 05/15/2008Network Formation50 Example 12 45 3 6

51. 05/15/2008Network Formation51 Cycle-Complete and Hull “Cycle Complete” if and only if “Connected Hull”

52. 05/15/2008Network Formation52 Network Restricted Games

53. 05/15/2008Network Formation53 Network Formation Game

54. 05/15/2008Network Formation54 Value of ( x,y ) The value for between any two nodes x and y in G is the best trust path value between x and y:

55. 05/15/2008Network Formation55 Value of x (and G ) The value for a node x is the summation of the path values from x to all other nodes in the graph: Will these definitions handle sybil attacks? How about mobility?

56. 05/15/2008Network Formation56 Trust Network Formation Incentive model to maximize v ( G ) and yet the allocation of v ( x ) is core! How do we define the update of t ( m i,m i+1 ) as a game?

57. 05/15/2008Network Formation57 Action and Game Sending a message M from x to y With certain probability distribution P, y might or might not like the message M. If y likes it, v ( RP ( x,y,n )) will be upgraded. Otherwise, the trust value t along the path will be downgraded. At the beginning of the game, the trust value t is 0.5 for all direct links

58. 05/15/2008Network Formation58 Each Player Decide whether they want to sent a message Upon receiving a message, flip a coin to decide whether he likes it or not The game ends when no player wants to send any more message

59. 05/15/2008Network Formation59 Issues Let’s assume simple topology, e.g. line. What are the strategies for each player? What are the possible Nash Equilibrium states?

60. 05/15/2008Network Formation60 3-Persons Game 12 3

61. 05/15/2008Network Formation61 A simplified game Aumann/Myerson 1988 –“Endogenous Formation of Links between Players and Coalitions: an Application of the Shapley value”, pp. 175-191, The Shapley Value, Cambridge University Press. Each link will get a chance, in some particular round-robin sequential order, to form a link, which can never be broken. The game ends when no pairs want to form any new link.

62. 05/15/2008Network Formation62 The Given Order

63. 05/15/2008Network Formation63 The Game 1 23 1 23 1 23 1 23 1 23

64. 05/15/2008Network Formation64 Unanimity Coefficients

65. 05/15/2008Network Formation65 The Network Restricted Game 1 23 1 23 1 23 1 23 1 23 1 23

66. 05/15/2008Network Formation66 Interpretation {1,2,3} are in the same coalition, but… “1” can not directly talk to “3” In order to form a friend link between “1” and “3”, they must go through “2” “2” is more valuable, but how much? 1 23 1 23

67. 05/15/2008Network Formation67 Unanimity Coefficients

68. 05/15/2008Network Formation68 Unanimity Coefficients

69. 05/15/2008Network Formation69 Unanimity Coefficients Shapley Myerson

70. 05/15/2008Network Formation70 The Network Restricted Game

71. 05/15/2008Network Formation71 Nash Equilibrium?

72. 05/15/2008Network Formation72 Link Probe Order ~ {12,13,23} Should {12} agree or not? –Yes, should {13} agree? Yes, should {23} agree? –Yes, the game ends –No, should {12} agree?… No, should {23} agree? –Yes, should {12} agree?… –No, should {12} agree?… –No, should {13} agree? Yes, should {23} agree? –Yes, should {12} agree? –No, should {12} agree?

73. 05/15/2008Network Formation73 Link Probe Order ~ {12,13,23} Should {12} agree or not? –Yes, should {13} agree? Yes, should {23} agree? –Yes, the game ends –No, the game ends No, should {23} agree? –Yes, should {13} agree?… –No, should {13} agree?… –No, should {13} agree? Yes, should {23} agree? –Yes, should {12} agree? –No, should {12} agree?

74. 05/15/2008Network Formation74 The Extensive Form {12} ? y{13} ?n{13} ? yy{23} ?yn{23} ?ny{23} ?nn{23} ? yyy yny{13} ? yynynn ynyyynyn nnn

75. 05/15/2008Network Formation75 Nash Equilibrium Only one link will be formed! –{12}, {13}, or {23}

76. 05/15/2008Network Formation76 5-person game

77. 05/15/2008Network Formation77 Wheel? 1 2 5 43

78. 05/15/2008Network Formation78 Wheel? 1 2 5 43

79. 05/15/2008Network Formation79 Strategic Forms

80. 05/15/2008Network Formation80 Nash Equilibrium Alice Bob No Link No link Link 0, 0 1, 1 0, 0 Strategic Form of Coalition

81. 05/15/2008Network Formation81 Three Properties Component Efficiency Weak Link Symmetry Improvement Property

82. 05/15/2008Network Formation82 Network Formation Theorem: For any super-additive coalitional game with an allocation rule on communication situations satisfying component efficiency, weak link symmetry, and the improvement property, then any network can be supported by a Nash Equilibrium of the network formation game. NE has no power in predicting the network formation in strategic form

83. 05/15/2008Network Formation83 How about SNE, UnNE, CPNE?

84. 05/15/2008Network Formation84 Strong Nash Equilibrium Nash Equilibrium (NE) –No Coalition allowed Strong Nash Equilibrium (SNE) –Works for ALL possible coalitions

85. 05/15/2008Network Formation85 (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!!

86. 05/15/2008Network Formation86 NE versus SNE Alice Bob No Link No link Link 0, 0 1, 1 0, 0 Strategic Form of Coalition

87. 05/15/2008Network Formation87 NE versus SNE Alice Bob No Link No link Link 0, 0 1, 1 0, 0 Strategic Form of Coalition

88. 05/15/2008Network Formation88 Three Person Coalition Game 1 23 1 23 1 23 1 23

89. 05/15/2008Network Formation89 Three Person Coalition Game 1 23 1 23 1 23 1 23 Each one of them will get 24!

90. 05/15/2008Network Formation90 Three Person Coalition Game 1 23 1 23 1 23 1 23 Each one of them will get 24! But, say {2,3} deviate by breaking 12 and 13, they both can get 30!

91. 05/15/2008Network Formation91 Three Person Coalition Game 1 23 1 23 1 23 1 23 Player 2 get “44”!

92. 05/15/2008Network Formation92 Three Person Coalition Game 1 23 1 23 1 23 1 23 Player 2 get “44”! Again, players 1 and 3 can collaborate and break their links with 2 to get “30” each from merely “14”!

93. 05/15/2008Network Formation93 Three Person Coalition Game 1 23 1 23 1 23 1 23 Players 2 and 3 get “30”!

94. 05/15/2008Network Formation94 Three Person Coalition Game 1 23 1 23 1 23 1 23 Players 2 and 3 get “30”! Again, players 1 and 2 can collaborate such that 2 will get “44”, while 1 will get “14”!

95. 05/15/2008Network Formation95 Three Person Coalition Game 1 23 1 23 1 23 1 23 No Strong NE! How about CPNE?

96. 05/15/2008Network Formation96 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).

97. 05/15/2008Network Formation97 Three Person Coalition Game 1 23 1 23 1 23 1 23

98. 05/15/2008Network Formation98 Three Person Coalition Game 1 23 1 23 1 23 1 23 Is it NE itself?

99. 05/15/2008Network Formation99 Three Person Coalition Game 1 23 1 23 1 23 1 23 Is it NE itself? Each individual player can NOT add a link. But they can break the link! The answer is YES!

100. 05/15/2008Network Formation100 Three Person Coalition Game 1 23 1 23 1 23 1 23 Is it NE itself? No!

101. 05/15/2008Network Formation101 Three Person Coalition Game 1 23 1 23 1 23 1 23 Two NE’s!

102. 05/15/2008Network Formation102 Three Person Coalition Game 1 23 1 23 1 23 1 23 Which one is CPNE?

103. 05/15/2008Network Formation103 Three Person Coalition Game 1 23 1 23 1 23 1 23 Only ONE CPNE!

104. 05/15/2008Network Formation104 Without Perfect Information Each player doesn’t know whether the other players have formed coalition or not!

105. 05/15/2008Network Formation105 Without Perfect Information Players 1 & 2 already formed a coalition. Player 3 now negotiate with both 1 & 2 by asking for a new friendship.

106. 05/15/2008Network Formation106 Game to link with player 3 Player 1 Player 2 No Link No link Link 30, 30 24,24 14, 44 44, 14

107. 05/15/2008Network Formation107 Prisoner’s Dilemma! Player 1 Player 2 No Link No link Link 30, 30 24,24 14, 44 44, 14