Efficient Contention Resolution Protocols for Selfish Agents

1
Efficient Contention Resolution Protocols for
Selfish Agents
Amos Fiat, Joint work with Yishay Mansour and
Uri Nadav Tel-Aviv University, Israel
Workshop on Algorithmic Game Theory, University
of Warwick, UK
2
Deadlines
Alright people, listen up. The harder you
push, the faster we will all get out of here.
Tax deadline
3
Deadline Analysis 2 Symmetric Agents / 2 Time
slots / Service takes 1 time slot

• Both agents are aggressive with prob. q, and
  polite with prob. 1-q

?
?
Deadline
Slot 16
Slot 17
Bart is polite With probability q Lisa will
get service and depart
Bart is aggressive With probability 1-q Lisa
will be polite and Bart will be successful
4
2 agents 1 Slot before deadline

• And Samson said, "Let me die with the
  Philistines!"
• Judges 1630

• Let Lisa be polite with prob. q
• If Bart is
• polite - cost is 1
• aggressive - expected cost is q

Aggression is dominant strategy
Deadline
Slot 17
5
Solving with MATHEMATICA

• q20(t) Prob. of aggression when 20 agents are
  pending as a function of the time t , in
  equilibrium

Aggression Probability
19
Blocking no one gets served
0.05
Time
deadline
6
Solving with MATHEMATICA

• qk(4k) Aggression prob. when k agents are
  pending before deadline in 4k time slots
• (Deadline when lunch trays are removed at U.
  Warwick, CS department)

agents
7
Deadline Cost Few slots

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

8
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

9
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
10
Switch Subject Broadcast Channel / Latency
time
Slot 1
Slot 2
Slot 5
Slot 6
Slot 3
Slot 4

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

11
Broadcast Channel
time
Slot 1
Slot 2
Slot 3
Slot 5
Slot 6
Slot 4
Transmission probability 1/n is not in equilibrium

• 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,
  agent is better off transmitting all the time and
  has constant latency

12
Classical Results

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

13

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

14
Related Work Strategic MAC (Multiple Access
Channel)

• 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

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Protocol in Equilibrium

• Agent utility Minimize latency

Agent strategy Transmission probability is a
function of the number of pending agents k and
current waiting time t
Protocol in equilibrium No incentive not to
follow protocol
Symmetry All agents are symmetric
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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?

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Summary of (Latency) 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

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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

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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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Translate Latency Minimization to Deadline
Effectively, no message gets through here
Cost
Time

• Fight for every slot
• Cooperation is more important when trying to
  avoid a large payment (deadline)
• How can one create a sudden jump in cost?
• Using external payments
• Agents go crazy everyone continuously
  transmits
• Time dependence
• Analyze step cost function (Deadline)

22
Deadline Cost Function
Cost
Time
D (Deadline)

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

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Deadline vs. Repeated Prisoners Dilemma

• For finite horizon prisoners dilemma Defect on
  last game.
• Inductively, no cooperation on any game

Not our case successful agents leave
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Equilibrium Equations (Deadline, Latency, etc.)
Probability one of the other k-1 agents leaves

Quiescence
Transmit
? ?(t1) (1- ? ) Ck,t1
? Ck-1,t1 (1 - ?) Ck,t1

Probability the other k-1 agents are silent
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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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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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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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)
32
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-1,t1-Ck,t1)

(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1-Ck,t1)
(1-qk,t)k-1(?(t1)-Ck,t1) (k-1)qk,t(1-qk,t)k-2(
Ck-1,t1- ?(t1)?(t1)-Ck,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
gt0

• 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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Analysis of Deadline utility

• We seek an upper bound for Cn,0 Fn,0

Fk,t ?Fk-1,t1 (1- ?) Fk,t1
Recall

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

Consider a tree of recursive computation for Fn,0
35
Upper Bound on Cost

• Two descendants

One descendant
Fn,t1 lt 2 Fn-1,t1
Transmission probability
(Fn,t1 gt 2 Fn-1,t1 )
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
37
Upper Bound on Cost

• 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
38
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

40
Future Research

• General cost functions
• Does the time-independent equilibrium induces an
  optimal expected latency?
• Protocol in equilibrium for an arrival process
• Arrival times / duration in general congestion
  games
• Atomic traffic flow dont leave home until 900
  AM and get to work earlier

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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
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Two users Equilibrium

• Best response is to be quiescent

• Always transmit

2-agents Eq. q ½, minimizes time to first
success
43
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
44
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, until
  success

45
Upper Bound on Cost

• 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