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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
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“Alright people, listen up. The harder you push,the faster we will all get out of here.”
Tax deadline
Deadlines:
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Deadline Analysis: 2 Symmetric Agents / 2 Time slots / Service takes 1 time
slotBoth agents are aggressive with prob. q, and polite
with prob. 1-q
Deadline
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
Slot #16 Slot #17
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2 agents 1 Slot before deadline
And Samson said, "Let me die with the Philistines!" Judges 16:30
Deadline
Let Lisa be polite with prob. qIf Bart is:• polite - cost is 1• aggressive - expected cost is q
Aggression is dominant strategy
Slot #17
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Solving with MATHEMATICAq20(t): Prob. of aggression when 20 agents are pending as a function of the time t , in equilibrium
20 40 60 80 100
0.2
0.4
0.6
0.8
1
Time
“Aggression”Probability
deadline
19
0.05
Blocking no one getsserved
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Solving with MATHEMATICAqk(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)
#agents20 40 60 80
10
20
30
40
50
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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)
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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 > 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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Switch Subject: Broadcast Channel / Latency
Slot #1 Slot #2 Slot #3 Slot #4 Slot #5 Slot #6 time
• n agents (with a packet each) at time 0 • No arrivals• Known number of agents
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Broadcast Channel
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, agent is better off transmitting all the time and has constant latency
time
Transmission probability 1/n is not in equilibrium
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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
Protocol in equilibrium: No incentive not to follow protocol
Agent strategy: Transmission probability is a function of the number of pending agents k and current waiting time t
Symmetry: All agents are symmetric
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Summary of (Latency) Results
1. All protocols where transmission probabilities do not depend on the time have exponential latency
2. 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:
• 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
Very high “Price of Anarchy”
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• 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)
Translate Latency Minimization to Deadline
Cost
TimeDeadline
Effectively, no message
gets through here
T
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Deadline Cost Function
Deadline utility (scaled):
• Success before deadline – cost 0
• Success after deadline – cost 1
Cost
TimeD (Deadline)
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(t+1) +(1- ) Ck,t+1 Ck-1,t+1 + (1 - ) Ck,t+1
Equilibrium Equations (Deadline, Latency, etc.)
* Ck,t = expected cost of k agents at time t
(t) = cost of leaving at time t
=
QuiescenceTransmit
Probability one of the other k-1
agents leaves
Probability the other k-1 agents
are silent
=
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1 –Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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Equilibrium Equations
k,t((t+1)-Ck,t+1) = k,t(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1-Ck,t+1)
(1-qk,t)k-1((t+1)-Ck,t+1) = (k-1)qk,t(1-qk,t)k-2(Ck-1,t+1- (t+1)+(t+1)-Ck,t+1)
(1-qk,t)k-1(Fk,t+1) = (k-1)qk,t(1-qk,t)k-2(Fk,t+1-Fk-1,t+1)
(1-qk,t) Fk,t+1 = (k-1)qk,t (Fk,t+1-Fk-1,t+1)
k,t((t+1))+(1- k,t )Ck,t+1 = k,t Ck-1,t+1 + (1- k,t ) Ck,t+1
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> 1/2
Transmission Probability in Equilibrium
Lemma (Manipulating equilibrium equations):
>01/k <
Benefit from losing one
agent
* Fk,t = Ck,t - (t) ; expected future cost
Ck,t = expected cost of k agents at time t
Transmission probability when k players at time tObservation:
– Either transmission probability in [1/k,2/k]
– Or, limited benefit from loosing one agent
<1/2
2/k >
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Analysis of Deadline utility
Fk,t = Fk-1,t+1 + (1- ) Fk,t+1
We seek an upper bound for Cn,0 = Fn,0
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
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Fn,t Fn,t+1
Fn-1,t+1
Upper Bound on Cost
Two descendants One descendant
(Fn,t+1 > 2 Fn-1,t+1 )
Fn,t+1 < 2 Fn-1,t+1
1-
Fn,t = Fn-1,t+1 + (1-) Fn,t+1 Fn,t < Fn,t+1 < 2 Fn-1,t+1
<2
Good edges Doubling edges
Fn,t+1
Fn-1,t+1
Fn,t
Fn,t / F
n-1,t+1
<0.8
< 0.3
Transmission probability
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Upper Bound on Cost
# Agents
TimeDeadline
Fn,0 Fn,1
F17,D = 1
Fn-3,4
Fn-1,1
Fn-2,2
Fn-3,3
Fn-4,4
F1,D-9 = 0
cost=0
L1
cost=1
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Upper Bound on Cost
• The weight of such a path:
– At least D-n good edges
– Weight at most (1-β)D-n2n
• Number of paths at most:
cost=0
1
Set D > 20n to get an upper bound of e-c n on
cost
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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
• Sub-game perfect equilibrium
• Social optimum achieved with high probability
Equilibriu
m
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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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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: don’t leave home until 9:00 AM
and get to work earlier