Mechanism Design in Large Congestion Games

Transcription

1 Mechanism Design in Large Congestion Games Ryan Rogers, Aaron Roth, Jonathan Ullman, and Steven Wu July 22, 2015

2 Routing Game l e (y)

3 Routing Game

4 Routing Game A routing game G is defined by

5 Routing Game A routing game G is defined by A set of n players

6 Routing Game A routing game G is defined by A set of n players A graph with vertices V and edges E where E = m and each edge has a latency function l e : R + [0, 1].

7 Routing Game A routing game G is defined by A set of n players A graph with vertices V and edges E where E = m and each edge has a latency function l e : R + [0, 1]. A set of types T = source destination pair t i = (s i, d i ) T.

8 Routing Game A routing game G is defined by A set of n players A graph with vertices V and edges E where E = m and each edge has a latency function l e : R + [0, 1]. A set of types T = source destination pair t i = (s i, d i ) T. A set of actions A = routes for each source destination pair.

9 Routing Game A routing game G is defined by A set of n players A graph with vertices V and edges E where E = m and each edge has a latency function l e : R + [0, 1]. A set of types T = source destination pair t i = (s i, d i ) T. A set of actions A = routes for each source destination pair. A cost function c : T A n R depends on congestion y e on each edge c(t i, a) = l e (y e (a)). e a i

10 Incomplete Information Setting Players may not know each other s type - incomplete information.

11 Incomplete Information Setting Players may not know each other s type - incomplete information. Sources and destinations may be sensitive information, n may be HUGE!!

12 First Goal - Equilibrium Selection Definition An action profile a = (a 1,..., a n ) is an η-nash equilibrium if for every player i of type t i T and every deviation a i we have c(t i, a) c(t i, (a i, a i )) + η

13 First Goal - Equilibrium Selection Definition An action profile a = (a 1,..., a n ) is an η-nash equilibrium if for every player i of type t i T and every deviation a i we have c(t i, a) c(t i, (a i, a i )) + η Finding a NE requires knowing the types of every player

14 First Goal - Equilibrium Selection Definition An action profile a = (a 1,..., a n ) is an η-nash equilibrium if for every player i of type t i T and every deviation a i we have c(t i, a) c(t i, (a i, a i )) + η Finding a NE requires knowing the types of every player Goal 1: Coordinate players to play an approximate Nash equilibrium as if we knew the types, even in settings of incomplete information.

15 Second Goal - Social Welfare Definition An action profile a is an η-socially Optimal Routing if where C(a) = e E y el e (y e ) C(a) min a A n C(a ) + η

16 Second Goal - Social Welfare Definition An action profile a is an η-socially Optimal Routing if where C(a) = e E y el e (y e ) C(a) min a A n C(a ) + η Finding the social optimal requires knowing the players types.

17 Second Goal - Social Welfare Definition An action profile a is an η-socially Optimal Routing if where C(a) = e E y el e (y e ) C(a) min a A n C(a ) + η Finding the social optimal requires knowing the players types. Goal 2: Coordinate selfish players to play an approximate social optimal routing as if we knew the types, in settings of incomplete information.

18 Braess Paradox Is it true that the Social Optimal is a NE?! x! 10! 10!!0! 10! 10! x!

19 Braess Paradox Is it true that the Social Optimal is a NE?! x! 10! 10!!0! 10! 10! x! Total Cost

20 Braess Paradox Is it true that the Social Optimal is a NE?! x! 10! 10!!0! 10! 10! x! Total Cost = = 200.

21 Braess Paradox Is it true that the Social Optimal is a NE?! x! 10! 10!!0! 10! 10! x! Total Cost = = 200. Equilibrium?

22 Braess Paradox Is it true that the Social Optimal is a NE?! x! 5" 10! 10!!0! 10! 10! 5" x! Total Cost

23 Braess Paradox Is it true that the Social Optimal is a NE?! x! 5" 10! 10!!0! 10! 10! 5" x! Total Cost = 5 (5 + 10) + 5 (5 + 10) = 150.

24 Braess Paradox Is it true that the Social Optimal is a NE?! x! 5" 10! 10!!0! 10! 10! 5" x! Total Cost = 5 (5 + 10) + 5 (5 + 10) = 150. Equilibrium?

25 Goal One: Equilibrium Selection Create a mechanism (or a mediator) that takes reported source destinations as input and then suggests an action (route) for each player to take. Mediator for a Routing Game:

26 Goal One: Equilibrium Selection Create a mechanism (or a mediator) that takes reported source destinations as input and then suggests an action (route) for each player to take. Mediator for a Routing Game:

27 Mediated Game G "#$%&'()*)! "#+%,&-*.%#!!!" /)0.-*%,! 1%&'()*)! "#+%,&-*.%#! G! A mediator is an algorithm M : (T ) n (A ) n.

28 Mediated Game G "#$%&'()*)! "#+%,&-*.%#!!!" /)0.-*%,! 1%&'()*)! "#+%,&-*.%#! G! A mediator is an algorithm M : (T ) n (A ) n. The game has changed - Mediated Game G M.

29 Mediated Game G "#$%&'()*)! "#+%,&-*.%#!!!" /)0.-*%,! 1%&'()*)! "#+%,&-*.%#! G! A mediator is an algorithm M : (T ) n (A ) n. The game has changed - Mediated Game G M. Players actions include how they will interact with M.

30 Weak Mediator Mediator cannot force people to use it.

31 Weak Mediator Mediator cannot force people to use it. Players need not follow its suggested action.

32 Weak Mediator Mediator cannot force people to use it. Players need not follow its suggested action. Players may lie to the mediator if they choose to use it.

33 Good Behavior Players should have little incentive from deviating from:

34 Good Behavior Players should have little incentive from deviating from: Using the Mediator M.

35 Good Behavior Players should have little incentive from deviating from: Using the Mediator M. Reporting their true type to M.

36 Good Behavior Players should have little incentive from deviating from: Using the Mediator M. Reporting their true type to M. Following the suggested actions of M.

37 Solution Attempt 1 Naive Solution: Have M output an approximate NE for the game induced by the reported types.

38 Solution Attempt 1 Naive Solution: Have M output an approximate NE for the game induced by the reported types. If all players report truthfully, then there is little incentive to not follow M s suggestion.

39 Solution Attempt 1 Naive Solution: Have M output an approximate NE for the game induced by the reported types. If all players report truthfully, then there is little incentive to not follow M s suggestion. If one person changes her type, the game has changed and the NE may be very different - costs to players may be very different between different NE.

40 Solution Attempt 1 Naive Solution: Have M output an approximate NE for the game induced by the reported types. If all players report truthfully, then there is little incentive to not follow M s suggestion. If one person changes her type, the game has changed and the NE may be very different - costs to players may be very different between different NE. How do we control the impact any one player has on the outcome of M?

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43 Differential Privacy [DMNS 06] Definition A randomized algorithm M : T n O is ɛ-dp if for all neighboring datasets t and t and all outcome sets B O we have P(M(t) B) e ɛ P(M(t ) B)

44 Differential Privacy [DMNS 06] Definition A randomized algorithm M : T n A n is ɛ-dp if for all neighboring datasets t and t and all outcome sets B A n we have P(M(t) B) e ɛ P(M(t ) B) where M(t) = (M 1 (t),, M n (t))

45 Differential Privacy [DMNS 06] Definition A randomized algorithm M : T n A n is ɛ-dp if for all neighboring datasets t and t and all outcome sets B A n we have P(M(t) B) e ɛ P(M(t ) B) where M(t) = (M 1 (t),, M n (t)) Too strong of a definition in our case. We want player s actions to depend on their reported type.

46 Relaxation of DP - Joint DP [KPRU 14] Definition A randomized algorithm M : T n A n is ɛ-jdp if for every player i, t i T n 1, t i, t i T, and all outcome sets B A n 1, P[ M(t i, t i ) }{{ i } B] e ɛ P[ M(t i, t i ) }{{ i } B] everyone except i everyone except i where M(t) i = (M 1 (t),, M i 1 (t), M i+1 (t),, M n (t))

47 Relaxation of DP - Joint DP [KPRU 14] Definition A randomized algorithm M : T n A n is ɛ-jdp if for every player i, t i T n 1, t i, t i T, and all outcome sets B A n 1, P[ M(t i, t i ) }{{ i } B] e ɛ P[ M(t i, t i ) }{{ i } B] everyone except i everyone except i where M(t) i = (M 1 (t),, M i 1 (t), M i+1 (t),, M n (t)) Allows outcome for player i to depend on i s report t i.

48 JDP Mediators Key Property: A JDP mediator that also computes an equilibrium of the underlying game is approximately truthful.

49 JDP Mediators Key Property: A JDP mediator that also computes an equilibrium of the underlying game is approximately truthful. Theorem Let G be any game with costs in [0, m], and let M be a mediator such that It is ɛ-jdp For any set of reported types t, it outputs an η-approximate pure strategy Nash Equilibrium. Then good behavior is an η -approximate ex-post Nash Equilibrium for the incomplete information game G M where η = 2mɛ + η

50 Main Result for Equilibrium Selection There exists such a mediator from the previous theorem for large congestion games.

51 Main Result for Equilibrium Selection There exists such a mediator from the previous theorem for large congestion games. Further, we show that good behavior is an η -approximate ex-post equilibrium for the incomplete information game G M where ( (m ) η 5 1/4 ) = Õ n

52 Main Result for Equilibrium Selection There exists such a mediator from the previous theorem for large congestion games. Further, we show that good behavior is an η -approximate ex-post equilibrium for the incomplete information game G M where ( (m ) η 5 1/4 ) = Õ n Resulting play of good behavior is an η approximate NE of the complete information game.

53 Large Games

54 Large Games We assume that each player cannot significantly change the cost of another player by changing her route. l e (y e ) l e (y e + 1) 1 n for y e [n] and e E. The costs then satisfy for j i and a j a j A c(t i, (a j, a j )) c(t i, (a j, a j )) m n.

55 How to Construct Such a Mechanism? Simulate Best Response Dynamics = obtains a NE in routing games [MS 96].

56 How to Construct Such a Mechanism? Simulate Best Response Dynamics = obtains a NE in routing games [MS 96]. Compute Best Responses privately = costs only depend on the number of people on each edge.

57 How to Construct Such a Mechanism? Simulate Best Response Dynamics = obtains a NE in routing games [MS 96]. Compute Best Responses privately = costs only depend on the number of people on each edge. Limit the number of times a single player can change routes = uses the largeness assumption.

58 Billboard Lemma! If a mechanism M : T n O is (ɛ, δ)-dp and consider any function φ : T O A. Define M : T n A n to be Then M is (ɛ, δ)- JDP. M (t) i = φ(t i, M(t)).

59 Goal Two - Social Welfare Recall that we want to minimize the cost to all players: Definition An action profile a is an η-socially Optimal Routing if where C(a) = e E y el e (y e ) C(a) min a A n C(a ) + η

60 Goal Two - Social Welfare Recall that we want to minimize the cost to all players: Definition An action profile a is an η-socially Optimal Routing if where C(a) = e E y el e (y e ) C(a) min a A n C(a ) + η How to get selfish agents to play the socially optimal routing without knowing their types?

61 Classical Approach - Tolls Add constant tolls τ = (τ e ) e E to the edges such that a NE of the game with tolls (tolled game) is the socially optimal in the game without tolls. However these tolls depend on players types.

62 Classical Approach - Tolls Add constant tolls τ = (τ e ) e E to the edges such that a NE of the game with tolls (tolled game) is the socially optimal in the game without tolls. However these tolls depend on players types.! x+4! 10! 10!!0! 10! 10! x+4!

63 Modified Mediator for Social Welfare Mediator still suggests routes to each player a = (a 1,, a n ) that they may or may not follow, but it also outputs tolls τ = (τ e ) e E on each edge, that every player must pay. Modified Mediator:

64 Modified Mediator for Social Welfare Mediator still suggests routes to each player a = (a 1,, a n ) that they may or may not follow, but it also outputs tolls τ = (τ e ) e E on each edge, that every player must pay. Modified Mediator:!

65 JDP Mediators + Tolls Recall: JDP + NE = truthfulness

66 JDP Mediators + Tolls Recall: JDP + NE = truthfulness Theorem Let M i : (T ) A [0, U] m where M i (t) = (M A i (t), M τ (t)) outputs a suggested route and tolls for each edge. If M = (M 1,, M n ) satisfies both ɛ-jdp and for any input types t, the action profile a = M A (t) is an η-approximate NE in the modified routing game with l M e (y) = l e (y) + M τ e (t) then good behavior is an η approximate ex-post NE in the mediated tolled game, where η = η + 2m(U + 1)ɛ

67 Main Result for Social Welfare There exists a mediator M from the previous theorem for large routing games with convex latencies.

68 Main Result for Social Welfare There exists a mediator M from the previous theorem for large routing games with convex latencies. Further, with this mediator M, good behavior forms an η approximate ex-post NE in the mediated tolled game, where ( ) η = Õ m 3/2 n 1/5.

69 Main Result for Social Welfare There exists a mediator M from the previous theorem for large routing games with convex latencies. Further, with this mediator M, good behavior forms an η approximate ex-post NE in the mediated tolled game, where ( ) η = Õ m 3/2 n 1/5. Resulting play of good behavior is an Õ(mn 4/5 ) socially optimal routing.

70 Main Result for Social Welfare There exists a mediator M from the previous theorem for large routing games with convex latencies. Further, with this mediator M, good behavior forms an η approximate ex-post NE in the mediated tolled game, where ( ) η = Õ m 3/2 n 1/5. Resulting play of good behavior is an Õ(mn 4/5 ) socially optimal routing. As long as the optimal solution grows ω(n 4/5 ), then we get a (1 + o(1)) multiplicative approximation to the true optimal.

71 How to Construct such a Mediator? Compute an approximately optimal flow a subject to JDP via a privacy preserving projected gradient descent algorithm.

72 How to Construct such a Mediator? Compute an approximately optimal flow a subject to JDP via a privacy preserving projected gradient descent algorithm. Given target flow a, we find the necessary tolls τ so that most players are nearly best responding in this tolled game when playing a.

73 How to Construct such a Mediator Allow the few players that are not approximately best responding to then best respond in the tolled game. This will modify a only slightly (by largeness assumption) and so will remain nearly optimal in the original game.

74 How to Construct such a Mediator Allow the few players that are not approximately best responding to then best respond in the tolled game. This will modify a only slightly (by largeness assumption) and so will remain nearly optimal in the original game. Resulting flow â is then nearly optimal in the original game and an approximate NE in the tolled game with tolls ˆτ.

75 Recap Key Property: NE + JDP = Approx Truthful Mediator

76 Recap Key Property: NE + JDP = Approx Truthful Mediator Equilibrium Selection Mediator Main Result: Output routing that is a õ(1)-ne

77 Recap Key Property: NE + JDP = Approx Truthful Mediator Equilibrium Selection Mediator Main Result: Output routing that is a õ(1)-ne Social Welfare Mediator Main Result: Output routing within (1 + õ(1)) OPT!

78 QUESTIONS?