Imperfect Best-Response Mechanisms
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Transcript of Imperfect Best-Response Mechanisms
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Paolo PennaLIAFA Univ. Paris Diderot
joint work with
Diodato FerratioliUniv. Rome “La Sapienza”
Imperfect Best-Response Mechanisms
DISPLEXITY 2nd Workshop on Distributed Computing: Computability and Complexity
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Game Theory
In practice…
Ideally
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The Internet
No central authority, open, self organized, anarchic
Different “entities” which• have their own goal• may not follow the “protocol”
Rational (selfish)
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This talk
DISTRIBUTED COMPUTING GAME THEORY
Self Stabilization Incentives
Best-Response Mechanisms (Nisan, Schapira, Valiant, Zohar, ICS11)
Imperfect Best-Response Mechanisms (Ferraioli&P., SAGT13)
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TCP
Probe-increase educated-decrease
(increase rate if no packet lost, decrease othw)
share
Ignore Protocol
Decrease
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AT&T
Border Gateway Protocol
Routes to destination
destination
Swisscom
Telecom
Comcast
Local choice (“next hop”)Autonomous (“best for me”)
Prefer Swisscom
Mindelay
PreferTelecom
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Best-Response Mechanisms(Nisan et al 2011)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)
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Example
1 2
d
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
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Example
1 2
d
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
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Example
1 2
d
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
(Nash) Equilibrium: no reason to change
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Example
1 2
d
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
(Nash) Equilibrium: no reason to change
Convergence (how to reach it?)
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Example
1 2
d
Unstable
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
BR BR
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Example
1 2
d
Unstable
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
BR BR
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Example
1 2
d
No convergence!!
● Want to reach d● Prefer not directly
Prefer route through 1
Prefer route through 2
BR BR
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BGP does not work...
..yes it does!! (Gao-Rexford'01)
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When do best-respond converge?(self-stabilization)
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Converge?
1
2
6
10
5d
Min latency
NBR = Never Best Response
NBR
NBR
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Converge?
1
2
6
10
5d
Min latency
NBR = Never Best Response
Min latency
Min latency
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Converge?
1
2
6
10
5d
Min latency
NBR = Never Best Response
Min latency
Min latency
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Converge?
1 0
0 2
2
12 0
3 1
NBRIn 2 rounds:equilibrium
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Converge?
1 0
0 22 0
3 1
In 2 rounds:equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once
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Converge?
1 0
0 22 0
3 1
In 2 rounds:equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once
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Converge?
1 0
0 22 0
3 1
In 2 rounds:equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once
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Convergence
For NBR-solvable games repeated best-response converge to (Nash) equilibrium
NENumber of rounds = number of NBR eliminations
...
...
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Best-Response Mechanisms(Nisan et al 2011)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)
Convergence for NBR-solvable games.● Many applications (BGP, TCP-games, Intern-Hospital
Matching, Auctions)
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Incentive Compatibility
1 0
0 2
2
12 0
3 1
NBR
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Incentive Compatibility
1 0
0 2
2
12 0
3 1
Higher payoff
Nash equilibrium
BR
BR
Clear outcome: payoff Nash at least payoff “discarded”
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Best-Response Mechanisms(Nisan et al 2011)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)
Convergence and Incentive Compatibilitytogether (NBR-solvable games with clear outcome).● Many applications (BGP, TCP-games, Intern-Hospital
Matching, Auctions)
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Part 2Mistakes and faults...what happens?
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Each time they respond, with small probability p
do something else
Imperfect Best-Response Mechanisms
(Ferraioli & Penna, 2013)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)
What if players sometimes take wrong decision?
Are these protocol robust to faults?
Length of round is R with “good” probability
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Convergence with mistakes
Convergence: Reach the (Nash) equilibrium with “good” probability
Obs: Probability p must be small enough...
Adversary: Initial state and activation sequence
...R R R R
L rounds
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Convergence with mistakes
Convergence: Reach the (Nash) equilibrium with “good” probability
Obs: Probability p must be small enough...
Thm (Lower Bound). Even for deterministic non-adaptive adversary, convergence may require p exponentially small in the number of players
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Proof of Lower Bound
Game with a fragile equilibrium
1 1 1 0 11
Adversary: R = exponential
... ...
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...first the game
payoffi(S1,...,Si,...,Sn) =
1 if Si = AND(1,S1,...,Si-1)
0 otherwise
Play 1 if all before you play 1 0 otherwise
1 1 1 1 1
1 1 1 0 11
1
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Proof of Lower Bound
Adversary:
1
12
1213
12131214
1213121412131215
12131214121312151213121412131216
R= 2n-1
p ≤ 1/R
1 1 1 0 11
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Lower Bound
For some games, there is an adversary, such that convergence requires p exponentially small
p ≤ 1/R = 1/2n-1
Upper BoundFor convergence always enough p small in the inverse of “total time”
p ≤ 1/(mRL)
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An application to BGP
Lower bound applies to “real” BGP instances (Gao-Rexford model)
1 i0 2 n
a
dddd
● No faults (p=0): BGP converges and Incentive Compatible (Levin,Schapira, Zohar'11)(Nisan et al'11)
● Faults (p>0): BGP does not converge unless p exponentially small
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Incentive Compatibilityand mistakes
● Need stronger condition● Some games are not robust (TCP games)
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Open Questions
Tighter bounds (specific games, adversaries)
BGP
Gao-Rexford
routing
Commercialrelationships
Two-layer games
More general games (restricted dynamics)
Nashequilibria“Equilibria selection”
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Thank You!