Asymptotically Truthful Equilibrium Selection in Large Congestion Games
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1 Asymptotically Truthful Equilibrium Selection in Large Congestion Games Ryan Rogers - Penn Aaron Roth - Penn June 12, 2014
2 Related Work Large Games Roberts and Postlewaite 1976 Immorlica and Mahdian 2005, Kojima and Pathak 2009, Kojima et al 2010 Bodoh-Creed 2013 Azevedo and Budish 2011 Incorporating a Mediator Monderer and Tennenholtz 2003, 2009 Ashlagi et al 2009 Work most closely related to ours Kearns et al 2014.
3 Routing Game l e (y)
4 Routing Game A game G is defined by A set of n players A set of types U = source destination pair s i U. A set of actions A = routes for each source destination pair. A cost function c : U A n R c(s i, a) = e a i l e (y e (a))
5 Routing Game A game G is defined by A set of n players A set of types U = source destination pair s i U. A set of actions A = routes for each source destination pair. A cost function c : U A n R c(s i, a) = e a i l e (y e (a)) Players may not know each others type.
6 Routing Game A game G is defined by A set of n players A set of types U = source destination pair s i U. A set of actions A = routes for each source destination pair. A cost function c : U A n R c(s i, a) = e a i l e (y e (a)) Players may not know each others type. n may be HUGE!!
7 Routing Game A game G is defined by A set of n players A set of types U = source destination pair s i U. A set of actions A = routes for each source destination pair. A cost function c : U A n R c(s i, a) = e a i l e (y e (a)) Players may not know each others type. n may be HUGE!! Types may be sensitive information
8 Routing Game A game G is defined by A set of n players A set of types U = source destination pair s i U. A set of actions A = routes for each source destination pair. A cost function c : U A n R c(s i, a) = e a i l e (y e (a)) Players may not know each others type. n may be HUGE!! Types may be sensitive information Main Goal : Have players play a pure strategy Nash equilibrium of the complete information game in settings of partial information.
9 Mediator for a Routing Game
10 Mediator for a Routing Game
11 Introduce a Mediator G "#$%&'()*)! "#+%,&-*.%#!!!" /)0.-*%,! 1%&'()*)! "#+%,&-*.%#! G! A mediator is an algorithm M : (U ) n (A ) n.
12 Weak Mediator
13 Weak Mediator Mediator cannot force people to use it
14 Weak Mediator Mediator cannot force people to use it Players need not follow its suggested action
15 Weak Mediator Mediator cannot force people to use it Players need not follow its suggested action Players may lie to the mechanism if they choose to use it.
16 Augmented Game Define the augmented game G M (Kearns et al 2014): Action Space: A = {(s, f ) : s U, f : (A ) A} Costs for g = ((s i, f i)) n i=1 : g i = (s i, f i ) A c M (s i, g ) = E a M(s ) [c(s i, f(a))]
17 Player s should: Good Behavior
18 Player s should: Use the Mediator M Good Behavior
19 Good Behavior Player s should: Use the Mediator M Report her true type to M
20 Good Behavior Player s should: Use the Mediator M Report her true type to M Follow the suggested action of M = f i = identity map.
21 Joint Differential Privacy (Kearns et al 2014) Let M : D n O n. Then M satisfies ɛ-joint differential privacy if for every s D n, for every i [n], s i D and for every B O n 1 P[M(s) i B] e ɛ P[M(s i, s i ) i B]
22 Billboard Lemma!
23 Billboard Lemma! If a mechanism M : U n O is (ɛ, δ)-differentially private and consider any function φ : U O A n. Define M : U n A n to be M (s) i = φ(s i, M(s)). Then M is (ɛ, δ)- joint differentially private.
24 Motivating Theorem Let G be any game with costs in [0, m], and let M be a mediator such that
25 Motivating Theorem Let G be any game with costs in [0, m], and let M be a mediator such that It is ɛ-joint differentially private
26 Motivating Theorem Let G be any game with costs in [0, m], and let M be a mediator such that It is ɛ-joint differentially private For any set of reported types s, it outputs an η-approximate pure strategy Nash Equilibrium.
27 Motivating Theorem Let G be any game with costs in [0, m], and let M be a mediator such that It is ɛ-joint differentially private For any set of reported types s, it outputs an η-approximate pure strategy Nash Equilibrium. Then good behavior g is an η -approximate ex-post equilibrium for the incomplete information game G M where η = 2mɛ + η
28 Main Theorem There exists such a mechanism from the motivating theorem for large congestion games. Further, we show that good behavior g is an η -approximate ex-post equilibrium for the incomplete information game G M where ( (m ) η 5 1/4 ) = Õ 0 as n n
29 Large Games
30 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(s i, (a j, a j )) c(s i, (a j, a j )) m n.
31 How to Construct Such a Mechanism?
32 How to Construct Such a Mechanism? Simulate Best Response Dynamics
33 How to Construct Such a Mechanism? Simulate Best Response Dynamics Compute Best Responses privately
34 How to Construct Such a Mechanism? Simulate Best Response Dynamics Compute Best Responses privately Limit the number of times a single player can change routes.
35 Best Responses In congestion games, allowing each player to best respond given the other players routes will converge to an approximate Nash Equilibrium.
36 Best Responses In congestion games, allowing each player to best respond given the other players routes will converge to an approximate Nash Equilibrium. We will have an algorithm that will have each player move if she can improve her cost by more than α: α-best Response
37 Best Responses In congestion games, allowing each player to best respond given the other players routes will converge to an approximate Nash Equilibrium. We will have an algorithm that will have each player move if she can improve her cost by more than α: α-best Response There can be no more than T = mn α best responses.
38 Best Responses In congestion games, allowing each player to best respond given the other players routes will converge to an approximate Nash Equilibrium. We will have an algorithm that will have each player move if she can improve her cost by more than α: α-best Response There can be no more than T = mn α best responses. We need to only maintain a count of the number of people on every edge to compute α-best Responses for each player
39 Binary Mechanism Chan et al 2011 and Dwork et al 2010 give a way to obtain an online count of a sensitivity 1 stream ω {0, 1} T such that the output ŷ t for any t = 1, 2,, T is ɛ differentially private Has high accuracy to the exact count y t for every t = 1,, T ( ) 1 ŷ t y t Õ ɛ 0! 1! 1! 0! 1! 1! 0! 0!! 47!! 1! 0! 1!! 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1!
40 ! 0! 1! 1! 0! 1! 1! 0! 0!! 47!! 1! 0! 1! Generalized Binary Mechanism 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1! 1! 0! 0! 1! 0! &1! &1! 1!!!!!!!16!!!1! 1! 0!!! 0! 1! 1! 1! 0! 0! 0! &1!!!!!!!!8!!!1! 1! 1! 0! 0! 0! 0! 1! &1! 1! 1!!!!!!!37!!!1! &1! &1!
41 ! 0! 1! 1! 0! 1! 1! 0! 0!! 47!! 1! 0! 1! Generalized Binary Mechanism 47! 0! 1! 1! 0! 1! 1! 0! 0!!! 1! 0! 1!! 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1! 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1! 1! 0! 0! 1! 0! &1! &1! 1!!!!!!!16!!!1! 1! 0 1!! 0!! 0!! 1! 0! &1! &1! 1!!!!!!!16!!!1! 1! 0!!! 0! 1! 1! 1! 0! 0! 0! &1!!!!!!!!8!!!1! 1! 1! 0! 1! 1! 1! 0! 0! 0! &1!!!!!!!!8!!!1! 1! 1! 0! 0! 0! 0! 1! &1! 1! 1!!!!!!!37!!!1! &1! &1! 0! 0! 0! 0! 1! &1! 1! 1!!!!!!!37!!!1! &1! &1!
42 ! 0! 1! 1! 0! 1! 1! 0! 0!! 47!! 1! 0! 1! Generalized Binary Mechanism 47! 0! 1! 1! 0! 1! 1! 0! 0!!! 1! 0! 1!! 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1! 0! &1! 1! &1! 0! 0! 1! 1!!!!!!!!21!!!1! 0! &1! 1! 0! 0! 1! 0! &1! &1! 1!!!!!!!16!!!1! 1! 0 1!! 0!! 0!! 1! 0! &1! &1! 1!!!!!!!16!!!1! 1! 0!!! 0! 1! 1! 1! 0! 0! 0! &1!!!!!!!!8!!!1! 1! 1! 0! 1! 1! 1! 0! 0! 0! &1!!!!!!!!8!!!1! 1! 1! 0! 0! 0! 0! 1! &1! 1! 1!!!!!!!37!!!1! &1! &1! 0! 0! 0! 0! 1! &1! 1! 1!!!!!!!37!!!1! &1! &1! Each of the m streams are k-sensitive, so we get ɛ differentially private counters With high probability ( ) km ŷe t ye t Õ e E, t = 1,, T ɛ
43 The Gap After a player i has made an α-private best response, how many times must other players move before i can move again? We will call this the gap γ.
44 The Gap After a player i has made an α-private best response, how many times must other players move before i can move again? We will call this the gap γ. Due to the largeness condition, each time a player does not move, her cost can increase by at most m n and can only move once her cost has increased by α ( αn ) γ = Ω m
45 The Gap After a player i has made an α-private best response, how many times must other players move before i can move again? We will call this the gap γ. Due to the largeness condition, each time a player does not move, her cost can increase by at most m n and can only move once her cost has increased by α ( αn ) γ = Ω m All the players can only make T = Õ ( ) mn α (with high probability).
46 The Gap After a player i has made an α-private best response, how many times must other players move before i can move again? We will call this the gap γ. Due to the largeness condition, each time a player does not move, her cost can increase by at most m n and can only move once her cost has increased by α ( αn ) γ = Ω m All the players can only make T = Õ ( ) mn α (with high probability). A player only changes routes k times ( ) m 2 k = O α 2
47 Equilibrium Analysis of our Algorithm With high probability, after T = Õ ( ) mn α moves by all players, no player will be able to improve her private cost by more than α. If we set ( (m ) 4 1/3 ) α = Θ nɛ then we know no player will be able to improve her actual cost by more than ( (m ) 4 1/3 ) η α + Error from BM = Õ nɛ
48 Equilibrium Analysis of our Algorithm Is it Joint Differentially Private?
49 Equilibrium Analysis of our Algorithm Is it Joint Differentially Private? Recall our motivating theorem that says good behavior is an η -approximate ex-post equilibrium for G M and we can set ɛ (which is a parameter we control) to satisfy the following ( (m ) η 5 1/4 ) = Õ 0 as n n
50 Open Questions Can Nash Equilibria of the complete information game be implemented as exact ex-post or Bayes Nash Equilibria of the incomplete information game? Does there exist a jointly differentially private algorithm for computing approximate Nash Equilibria for general large games?
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