An overview on best-response mechanisms (Nisan, Schapira, Zohar, Valiant 2011) and their applications to BGP and other protocols. Also a probabilistic extension in which players can make mistakes (Ferraioli-Penna 2013).

1. Paolo Penna
LIAFA Univ. Paris Diderot
joint work with
Diodato Ferratioli
Univ. Rome “La Sapienza”
Imperfect Best-Response
Mechanisms
DISPLEXITY 2nd Workshop on Distributed Computing: Computability and Complexity

2. Game Theory
In practice…
Ideally

3. The Internet
No central authority, open, self organized, anarchic
Different “entities” which
• have their own goal
• may not follow the “protocol”
Rational (selfish)

4. This talk
DISTRIBUTED
COMPUTING
GAME THEORY
Self Stabilization Incentives
Best-Response Mechanisms
(Nisan, Schapira, Valiant, Zohar, ICS11)
Imperfect Best-Response Mechanisms
(Ferraioli&P., SAGT13)

5. TCP
Probe-increase educated-decrease
(increase rate if no packet lost, decrease othw)
share
Ignore
Protocol
Decrease

6. AT&T
Border Gateway Protocol
Routes to destination
destination
Swisscom
Telecom
Comcast
Local choice (“next hop”)
Autonomous (“best for me”)
Prefer
Swisscom
Min
delay
Prefer
Telecom

7. Best-Response Mechanisms
(Nisan et al 2011)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)

8. Example
1 2
d
● Want to reach d
● Prefer not directly
Prefer route
through 1
Prefer route
through 2

9. Example
1 2
d
● Want to reach d
● Prefer not directly
Prefer route
through 1
Prefer route
through 2

10. 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

11. 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?)

12. Example
1 2
d
Unstable
● Want to reach d
● Prefer not directly
Prefer route
through 1
Prefer route
through 2
BR BR

13. Example
1 2
d
Unstable
● Want to reach d
● Prefer not directly
Prefer route
through 1
Prefer route
through 2
BR BR

14. Example
1 2
d
No convergence!!
● Want to reach d
● Prefer not directly
Prefer route
through 1
Prefer route
through 2
BR BR

15. BGP does not work...
..yes it does!! (Gao-Rexford'01)

16. When do best-respond converge?
(self-stabilization)

17. Converge?
1
2
6
10
5
d
Min
latency
NBR = Never Best Response
NBR
NBR

18. Converge?
1
2
6
10
5
d
Min
latency
NBR = Never Best Response
Min
latency
Min
latency

19. Converge?
1
2
6
10
5
d
Min
latency
NBR = Never Best Response
Min
latency
Min
latency

20. Converge?
1 0
0 2
2
1
2 0
3 1
NBR
In 2 rounds:
equilibrium

21. Converge?
1 0
0 2
2 0
3 1
In 2 rounds:
equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once

22. Converge?
1 0
0 2
2 0
3 1
In 2 rounds:
equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once

23. Converge?
1 0
0 2
2 0
3 1
In 2 rounds:
equilibrium
Adversary: Initial state and activation sequence
Round: All players activated at least once

24. Convergence
For NBR-solvable games
repeated best-response
converge to (Nash)
equilibrium
NENumber of rounds =
number of NBR eliminations
...
...

25. 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)

26. Incentive Compatibility
1 0
0 2
2
1
2 0
3 1
NBR

27. Incentive Compatibility
1 0
0 2
2
1
2 0
3 1
Higher payoff
Nash equilibrium
BR
BR
Clear outcome: payoff Nash at least payoff “discarded”

28. Best-Response Mechanisms
(Nisan et al 2011)
Protocol: “repeatedly best respond” (greedy)
Asynchronous setting (adversarial schedule)
Convergence and Incentive Compatibility
together (NBR-solvable games with clear
outcome).
● Many applications (BGP, TCP-games, Intern-Hospital
Matching, Auctions)

29. Part 2
Mistakes and faults...what happens?

30. 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

31. 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

32. 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

33. Proof of Lower Bound
Game with a fragile equilibrium
1 1 1 0 11
Adversary: R = exponential
... ...

34. ...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

35. Proof of Lower Bound
Adversary:
1
12
1213
12131214
1213121412131215
12131214121312151213121412131216
R= 2n-1
p ≤ 1/R
1 1 1 0 11

36. Lower Bound
For some games, there is an adversary, such that
convergence requires p exponentially small
p ≤ 1/R = 1/2n-1
Upper Bound
For convergence always enough p small in the inverse of
“total time”
p ≤ 1/(mRL)

37. 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

38. Incentive Compatibility
and mistakes
● Need stronger condition
● Some games are not robust (TCP games)

39. Open Questions
Tighter bounds (specific games, adversaries)
BGP
Gao-Rexford
routing
Commercial
relationships
Two-layer games
More general games (restricted dynamics)
Nash
equilibria“Equilibria selection”

40. Thank You!