Efficient Contention Resolution Protocols for Selfish Agents

1
Efficient Contention Resolution Protocols for
Selfish Agents
Uri Nadav, Joint work with Amos Fiat and Yishay
Mansour Tel-Aviv University, Israel
Second Israeli Seminar on Computational Game
Theory, 22/2/2007
2
Broadcast Channel
Transmission probability 1/n is not in equilibrium
time
Slot 1
Slot 2
Slot 3
Slot 4
Slot 5
Slot 6

• Symmetric solution every agent transmits with
  probability 1/n, the expected waiting time is
  O(n) slots. (Social optimum)

• If all others transmit with probability 1/n, I am
  better off transmitting all the time

3
Classical Results

• Maximizing the throughput
• Aloha (fixed probability) 0.37
• More advanced algorithms 0.48 MoH85
• Impossibility result 0.56 TsL88

4

• Well established research.
• Mostly in the 80s
• To learn more

5
Classical View versus Our view

• The classical view
• Find a good protocol
• Assumes agents follow any protocol.
• Our view
• What would happen if agents are selfish
• Agents can adjust their transmission
  probabilities
• Rather than optimization consider equilibrium.

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Equilibrium

• Utility Waiting time until success

Strategy Transmission probability is a function
of the number of pending agents k and current
waiting time t
Equilibrium Following the protocol is best
response
Protocol Symmetric equilibrium
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Simplified model

• n agents (with a packet each) at time 0
• No arrivals
• Known number of agents

8
Two users Equilibrium

• Best response is to Quiescence

• Always transmit

2-agents Eq. q ½, minimizes time to first
success
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Broadcast Channel
Strategy Always transmit!
Slot 1
Slot 2
Slot 3
Slot 4
Slot 5
Slot 6

• Equilibrium
• The channel is blocked anyway
• Also in subgame perfect equilibrium
• Remark For at least 3 players
• Not quite what we look for
• Is this the only equilibrium?

10
Summary of Results

• All protocols where transmission probabilities do
  not depend on the time have exponential latency
• We give a time-dependent protocol where all
  agents are successful in linear time

11
Related Work Strategic MAC

• Altman et al 04
• Incomplete information number of agents
• Stochastic arrival flow to each source
• Restricted to a single retransmission probability
• Shows the existence of an equilibrium
• Numerical results
• MacKenzie Wicker 03
• Multi-packet reception
• Transmission cost due to power loss
• Characterize the equilibrium and its stability
• Also Gang, Marbach Yuen

12
Time-Independent Equilibrium

• Theorem There is a unique time-independent,
  symmetric, non-blocking protocol in equilibrium
  for latency cost with transmission probabilities

Very high Price of Anarchy

• Expected Delay of the first transmitted packet
• Probability even one agent successful within
  polynomial time bound is negligible
• Compare to social optimum
• All agents successful in linear time bound, with
  high probability

13
Latency Equilibrium

• Proof idea (assuming q qk qk-1)
• For the other k-1 agents
• ak-1 Prall silent (1-q)k-1
• ßk-1 Prsuccess q(k-1)(1-q)k-2
• Consider always Transmit
• Expected Cost 1/ak-1
• Consider Quiescence and then Transmit
• Expected cost 1/ßk-11/ak-2

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Latency Equilibrium

• Proof idea (assuming q qk qk-1)
• Equilibrium Equation
• 1/ak-1 1/ßk-11/ak-2

• Simplifying 1-q-(k-1)q20
• Solution q 1/vk
• A major simplification qk qk-1

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Main Intuition
Effectively, no message gets through here
Cost
Time

• Fight for every slot
• Cooperation is more important when trying to
  prevent a large payment
• How to create a large leap in cost function?
• Using external payments
• Agents go crazy everyone continuously
  transmits
• Time dependent
• Analyze step cost function

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Deadline Cost Function
Cost
Time
D (Deadline)

• Deadline utility (scaled)
• Success before deadline cost 0
• Success after deadline cost 1

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Deadlines
Alright people, listen up. The harder you
push, the faster we will all get out of here.
crowd in post office at tax filing deadline
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2 agents 1 Slot before deadline

• Suppose a non-blocking equilibrium exist
• Transmission probability q lt 1

Non-blocking equilibrium does not exists

• Let Lisa play according to protocol
• If Bart plays
• Quiescent cost is 1
• Transmit expected cost is q

Transmit is dominant strategy
Deadline
Slot 17
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Deadline Cost Few slots

• Theorem In a symmetric equilibrium, whenever
  there are more agents than time slots until
  deadline,agents transmit (transmission
  probability 1)

• Proof By backward induction (on the time t)
• At any time more agents than time slots
• At times tgtt no successful transmission
• Fight for the chance to succeed

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Finite horizon Prisoners Dilemma

• Deadline reminds us of finite horizon prisoners
  dilemma
• Defect the last game played
• Inductively, no cooperation on any game

Not our case successful agents leave
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Deadline Analysis 2 Agents

• 2 time slots left

Deadline
?
?
Slot 16
Slot 17

• Bart plays quiescent
• With probability q Lisa will transmit and leave

• Bart plays transmit
• With probability 1-q Lisa will play quiescent

q 1-q ) q ½
Generally 2 agents any number of time slots q
½ n agents, n time slots left Transmission
probability lt 1
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Deadline non-blocking Equilibrium

• Theorem There exists a symmetric equilibrium,
  such that whenever there are at least as many
  time slots as agents, transmission probability is
  less than 1

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Solving with MATHEMATICA

• q20(t) Transmission probability when 20 agents
  are pending as a function of the time t , in
  equilibrium

20
Transmission Probability
0.05
Time
deadline
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Solving with MATHEMATICA

• qk(4k) Transmission probability when k agents
  are pending at time 4k, before deadline, in
  equilibrium

agents
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Efficiency of a linear deadline

• Theorem
• There exists a symmetric equilibrium for
• D-deadline cost function such that
• if the deadline D gt 20n
• then, the probability that not all agents succeed
  prior to the deadline is negligible (e-cD)

If there is enough time for everyone, a nice
equilibrium
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Equilibrium Equations

Probability one of the other k-1 agents leaves
Probability the other k-1 agents are silent
Quiescent
Transmit
In a strictly mixed Equilibrium, individual is
indifferent between Transmit and Quiescent
? Ck-1,t1
? ?(t1)



(1 - ?) Ck,t1
(1- ? ) Ck,t1
Ck,t expected cost of k agents at time
t ?(t) cost of leaving at time t
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Equilibrium Equations
)
?k,t(?(t1))(1- ?k,t )Ck,t1 ?k,t Ck-1,t1
(1- ?k,t ) Ck,t1
)

• ?k,t(?(t1)-Ck,t1) ?k,t(Ck,t1-Ck-1,t1)

)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck,t1-Ck-1,t1)
)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck,t1- ?(t1)?(t1)-Ck-1,t1)
)
(1-qk,t)k-1(Fk,t1) (k-1)qk,t(1-qk,t)k-2(Fk,t1-
Fk-1,t1)
)
(1-qk,t) Fk,t1 (k-1)qk,t (Fk,t1-Fk-1,t1)
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Transmission Probability in Equilibrium

• Lemma (Manipulating equilibrium equations)

Benefit from losing one agent
2/k gt
lt1/2
1/k lt
gt 1/2
lt1

• Observation
• Either transmission probability in 1/k,2/k
• Or, limited benefit from loosing one agent

Fk,t Ck,t - ?(t) expected future
cost Ck,t expected cost of k agents at
time t
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Upper Bound on Cost

• Two descendants

One descendant
Fn,t1 gt 2 Fn-1,t1
Fn,t1 lt 2 Fn-1,t1
) Transmission probability in 1/k, 2/k
lt 2
1-? lt 0.8
? lt 0.3
Fn,t ? Fn-1,t1 (1-?) Fn,t1
Fn,t lt Fn,t1 lt 2 Fn-1,t1
Good edges
Doubling edges
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Upper Bound on Cost
Agents
F17,D 1
Time
Deadline
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Upper Bound on Cost

• Consider all runs that fail
• Some agent does not meet the deadline
• The weight of such a path
• At least D-n good edges
• Weight at most (1-ß)D-n2n
• Number of paths at most

Set D gt 20n to get an upper bound of e-c n on cost

• Let D an, then total weight at most
• 2(1-ß)D-n2n anen 22e(1-ß)a-1an

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Protocol Design from Deadline to Latency

• Embed artificial deadline into deadline protocol

• Deadline Protocol
• Before time 20n transmission probability as in
  equilibrium
• If not transmitted until 20n
• Set transmission probability 1 (blocking)
• For exponential number of time slots

• Equilibrium

• Sub-game perfect equilibrium
• Social optimum achieved with high probability

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Summary

• Unique non-blocking equilibrium for Aloha like
  Protocols
• Exponential latency
• Deadlines
• If enough (linear) time, equilibrium is
  efficient
• Protocol Design
• Make ill behaved latency cost act more polite
• Using virtual deadlines
• No monetary bribes or penalties

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Two users Equilibrium

• Best response is to Quiescence

• Always transmit

2-agents Eq. q ½, minimizes time to first
success
Notation Ck,t expected latency with k agents
at time t Fk,t Ck,t - t