Approximate Equilibria Lower Bound Asymmetric. Michael Erjemenko

Transcription

1 Approximate Equilibria Lower Bound Asymmetric

2 Topic of the Paper First hardness result about approximibility of PNE Computing an equilibrium from an initial state is PSPACE-complete PLS-hard to compute an -approximate equilibrium of a given congestion game for any polynomial time computable α> 1 α 2

3 Topic of the Paper Reference to the work of Chien & Sinclair: Need only polynomial number of steps Symmetric congestion games (1+ ϵ )-approximate equilibrium (1+ ϵ )-greedy players Bounded-jump condition Disprove for asymmetric congestion games with an appropriate family construction 3

4 Def: -approximate equilibrium A state of the game in which none of the players can unilaterally improve by at least a factor of α. α-approximate equilibrium s s i * other strategy of player i u i (s i-1, s i *) < α u i (s) 4

5 Def: -greedy step Is an improvement step of an individual player that decrease the player's delay by at least a factor of α. d i (s) = 2 α x d i (s') = α x d i (s'') = x 5

6 Def: -bounded jump condtion A congestion game satisfies a -bounded jump condition, if there exists a polynomially bounded parameter with: 6

7 Strategies of Player i Resources Delays (1) a i b i c i (2) b i /5 3/7 (3) b i-1 10/12 c 1/55 i-1 d 1 ((1,1)) = d 1 ((1,2)) = d 2 ((1,3)) =

8 Theorem α For every > 2, there is a > 1 such that, for every n, there is a congestion game G(n) and a state s with the following properties. The description length of G(n) is polynomial in n. The length of every sequence of α-greedy improvement steps leading from s to an α-approximate equilibrium is exponential in n. All delay functions of G(n) satisfy the -bounded jump condition. 8

9 Idea Exponential many steps Exponential many improvement steps Player i cause player i-1 to change strategy Recursive invokation Each run cause 2 new runs => 2 i many improvement steps for player i 9

10 Gadgets G i = (Main i, Block i1,..., Block i8 ) = (M i, B i1,..., B i8 ) If it runs, then M i chooses each strategy after another G(n) consists of G 1,..., G n 10

11 Side Remark 11

12 Main i : 12

13 Main i : 13

14 Blocking Player Start in (1) 14

15 Main i : 15

16 Blocking Player Start in (1) If Block j choose (1) then Main cannot choose i i (j+2) Block j change to (2) if Main choose (j+1) i i 16

17 Main i : 17

18 Main i : 18

19 i-th player (i-1)-th player 19

20 1 Player M i+1 takes resource e i (i.e. in strategy 3 or 7) Delay of M i becomes = Delay of M i with strategy 2 20

21 i-th player (i-1)-th player 21

22 i-th player occupies t i j Block i j chooses (2) 22

23 i-th player (i-1)-th player 23

24 i-th player (i-1)-th player 24

25 i-th player (i-1)-th player 25

26 i-th player (i-1)-th player 26

27 M i-1 starts run of G i-2 8 M i-2 reaches (9), takes with that resource t i-2 8 => b i-2 is free => M i-1 can continue his run Finally M i-1 reaches (9) 27

28 i-th player (i-1)-th player 28

29 i-th player (i-1)-th player 29

30 i-th player (i-1)-th player 30

31 i-th player (i-1)-th player 31

32 i-th player (i-1)-th player j = 8 Before switch to (5) free After switch to (5) 32

33 i-th player (i-1)-th player j = 8 Before switch to (5) free After switch to (5) free 33

34 i-th player (i-1)-th player 34

35 i-th player (i-1)-th player 35

36 i-th player (i-1)-th player 36

37 i-th player (i-1)-th player 37

38 i-th player (i-1)-th player 38

39 i-th player (i-1)-th player 39

40 Check Theorem Properties G(n)'s description length is polynomial in n α-approximate equilibrium α-greedy steps -bounded jump condition holds with 40

41 G(n)'s description length is polynomial in n α-approximate equilibrium α-greedy steps -bounded jump condition holds with 41

42 Summary Disproved polynomial upper bound of improvement steps in asymmetric congestion games Sequence for our family of asymmetric congestion games is exponentially in n 42

43 Thanks for your attention!

44 Approximate Equilibria Lower Bound Asymmetric

45 Topic of the Paper First hardness result about approximibility of PNE Computing an equilibrium from an initial state is PSPACE-complete PLS-hard to compute an α -approximate equilibrium of a given congestion game for any polynomial time computable α> 1 2

46 Topic of the Paper Reference to the work of Chien & Sinclair: Need only polynomial number of steps Symmetric congestion games (1+ ϵ )-approximate equilibrium (1+ ϵ )-greedy players Bounded-jump condition Disprove for asymmetric congestion games with an appropriate family construction 3 We will show that the assumption that this result holds also for asymmetric congestion games does not hold, by constructing a family of asymmertric congestion games. Chien & Sinclair showed upper bound

47 Def: -approximate equilibrium A state of the game in which none of the players can unilaterally improve by at least a factor of α. α-approximate equilibrium s s i * other strategy of player i u i (s i-1, s i *) < α u i (s) 4 No unilateral strategy change increases the utility of the corresponding player by more than some constant factor alpha.

48 Def: -greedy step Is an improvement step of an individual player that decrease the player's delay by at least a factor of α. d i (s) = 2 α x d i (s') = α x d i (s'') = x 5

49 Def: -bounded jump condtion A congestion game satisfies a -bounded jump condition, if there exists a polynomially bounded parameter with: 6

50 Strategies of Player i Resources Delays (1) a i b i c i (2) b i /5 3/7 (3) b i-1 10/12 c 1/55 i-1 d 1 ((1,1)) = d 1 ((1,2)) = d 2 ((1,3)) = Two player game in the example Player 1; i = 1 Player 2; i = 2

51 Theorem α For every > 2, there is a > 1 such that, for every n, there is a congestion game G(n) and a state s with the following properties. The description length of G(n) is polynomial in n. The length of every sequence of α-greedy improvement steps leading from s to an α-approximate equilibrium is exponential in n. All delay functions of G(n) satisfy the -bounded jump condition. 8

52 Idea Exponential many steps Exponential many improvement steps Player i cause player i-1 to change strategy Recursive invokation Each run cause 2 new runs => 2 i many improvement steps for player i 9 Because the i-th player invokes the (i-1)-th player the with the highest number (n) starts.

53 Gadgets G i = (Main i, Block i1,..., Block i8 ) = (M i, B i1,..., B i8 ) If it runs, then M i chooses each strategy after another G(n) consists of G 1,..., G n 10

54 Side Remark 11

55 Main i : 12 When M i chooses the strategy 3 or 7, he starts a run of G i-1 Something must be added so that M i-1 doesn't choose the 9 th strategy

56 Main i : 13 We want to go through all steps/strategies because this causes at the end two runs of G i-1

57 Blocking Player Start in (1) 14

58 Main i : 15 The resources of B j 's first strategy are distributed. i Tj i is at M i 's (j+1)-th strategy and b j at the (j+2)-th i strategy.

59 Blocking Player Start in (1) If Block j choose (1) then Main cannot choose i i (j+2) Block j change to (2) if Main choose (j+1) i i 16

60 Main i : 17 M i should start G i-1 not before M i-1 is finish with his run.

61 Main i : 18 B i-1 8 and b i-1 8 provided a way to make sure that M i does not continue before G i-1 is finish. Now we need something that makes sure that we do not trigger a run befor everything is set back to idle (every player of a Gadget chooses his first strategy). Therefore we use c i-11,..., c i-18. So we cannot enter this steps if a c i-1 j is used by a B i-1 j

62 i-th player (i-1)-th player 19 Gadgeds G i and G i-1 are idle

63 1 Player M i+1 takes resource e i (i.e. in strategy 3 or 7) Delay of M i becomes = Delay of M i with strategy 2 20

64 i-th player (i-1)-th player 21

65 i-th player occupies t i j Block i j chooses (2) 22

66 i-th player (i-1)-th player 23

67 i-th player (i-1)-th player 24

68 i-th player (i-1)-th player 25

69 i-th player (i-1)-th player 26

70 M i-1 starts run of G i-2 8 M i-2 reaches (9), takes with that resource t i-2 8 => b i-2 is free => M i-1 can continue his run Finally M i-1 reaches (9) 27

71 i-th player (i-1)-th player 28

72 i-th player (i-1)-th player 29

73 i-th player (i-1)-th player 30

74 i-th player (i-1)-th player 31

75 i-th player (i-1)-th player j = 8 Before switch to (5) free After switch to (5) 32

76 i-th player (i-1)-th player j = 8 Before switch to (5) free After switch to (5) free 33

77 i-th player (i-1)-th player 34

78 i-th player (i-1)-th player 35

79 i-th player (i-1)-th player 36

80 i-th player (i-1)-th player 37

81 i-th player (i-1)-th player 38

82 i-th player (i-1)-th player 39

83 Check Theorem Properties G(n)'s description length is polynomial in n α-approximate equilibrium α-greedy steps -bounded jump condition holds with 40

84 G(n)'s description length is polynomial in n α-approximate equilibrium α-greedy steps -bounded jump condition holds with 41

85 Summary Disproved polynomial upper bound of improvement steps in asymmetric congestion games Sequence for our family of asymmetric congestion games is exponentially in n 42

86 Thanks for your attention!