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
“Alright people, listen up. The harder you push,the faster we will all get out of here.”
Tax deadline
Deadlines:
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
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
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
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
Deadline Cost – Few slots
Theorem: In a symmetric equilibrium, whenever
there are more agents than time slots until
deadline,
agents transmit (transmission probability 1)
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
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
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
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]
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
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
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”
• 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
Deadline Cost Function
Deadline utility (scaled):
• Success before deadline – cost 0
• Success after deadline – cost 1
Cost
TimeD (Deadline)
(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
=
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
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
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
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
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
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
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
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
> 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 >
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
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
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
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
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
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
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