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