algorithms and economics of networks: coordination mechanisms abraham flaxman and vahab mirrokni,...
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Algorithms and Economics of Networks:Coordination MechanismsAbraham Flaxman and Vahab Mirrokni,
Microsoft Research
Price on Anarchy [KP, RT]
Selfish users
User goal: minimize its cost
Nash Equilibrium (NE)
System goal (e.g. Social welfare)
The worst ratio of NE cost to OPT cost
Price of Anarchy Concept
Not algorithmic
Only analysis
What to do if PoA is large
How to influence the system
Possible Solutions
Change the system (add tolls, payments)
Stackelberg strategy = control some users
Disadvantages: changing the settings, global knowledge
Challenge: influence within the same setting and locally (distributed)
Coordination Mechanism [CKN]Mechanism: local policy (algorithm) that assigns a cost for each strategy of the user
Advantages: local, same type of cost
Goal: achieving good NE
Example: scheduling jobs on machines
Unrelated Machine Scheduling m unrelated machines
n jobs – each owned by different user
p(i,j) - processing time of job i on machine j
Social Objective: minimize completion time
User goal: minimize its own completion time (Makespan: Cmax)
Unrelated Machines Scheduling
Machine A
A
B
A
B? ?
Machine B
Coordination Mechanism for Scheduling
Policy for each machine (algorithm) which
decides how to schedule jobs assigned to it
Each Policy induces NE on jobs
Local Scheduling Policies
A A
Shortest-First Policy Longest-First Policy
B B
A A
Local Scheduling Policies
Shortest-first Policy Longest-first Policy Random Order Policy MakeSpan Policy
Challenge
Design policies that results in
good NE (i.e. low PoA)
Unrelated Machines Scheduling
Machine A
A
B
A
B? ?
Machine B
Equilibrium for Longest First
A
B
A
B
PoA of Longest First
Results in poor NE
The PoA is unbounded even for 2 machines
The optimum completion time is low
The completion time of NE is large
Machine Scheduling Models
Identical Machines: P||Cmax
Related Machines: Q||Cmax
(Different Speeds)
Restricted Assignment: B||Cmax
Unrelated Machines: R||Cmax
PoA Results
MakeSpanPolicy
Shortest-first Policy
Longest-first Policy
Randomized Policy
P||Cmax 2-1/m2-1/m 4/3-1/3m 2-2/m+1
Q||Cmax O(log m) O(log m) 2-1/m O(log m)
B||Cmax O(log m) O(log m) O(log m) O(log m)
R||Cmax Unbounded
O(m) Unbounded O(m)
Pure NE Results
MakeSpanPolicy
Shortest-first Policy
Longest-first Policy
Randomized Policy
P||Cmax Exists Exists Exists Exists
Q||Cmax Exists Exists Exists Exists
B||Cmax Exists Exists Exists Exists
R||Cmax Exists Exists ??? OPEN
Type of Policies Local policy – depends on jobs
assigned to machine Strongly local policy - depends only
on processing time of jobs on that machine
Ordering Policy = IIA (independence of irrelevant alternative)
Lower Bound for Strongly Local Policy
We start with Shortest-First
Extend it to arbitrary strongly local
IIA policy
Shortest-First is interesting by its
own
Shortest-First
Approx factor known to be at most m
NE can be computed by shortest-first
greedy algorithm
(Alg D by Ibarra and Kim)
An open question from 1977
We show it is at least m/2
Idea of the Proof
m types of jobs
Type j can be scheduled on machines j &
j+1
Processing time of type j on machine j is
low and on machine j+1 is high (ratio is j)
All jobs on machine j have almost the
same processing time
Example for Shortest-First
?
B
C
A
? ?
Idea of the Proof
OPT assign all jobs of type j to machine j
Number of jobs is chosen such that OPT
has the same completion time for all
machines
Optimal Assignment
A
B
C
Idea of the Proof
In NE about half jobs of type j are on
machine j and half on machine j+1
Completion time of NE grows linearly in
m
Equilibrium for Shortest-First
?
B
C
A
? ?
Extend to Arbitrary Strongly Local Structure is similar to lower bound for
Shortest-First
Arbitrary ordering function is given
for each machine
Indices of jobs are chosen to behave
similar to the above example
Efficiency Based Algorithm
Order jobs on each machine by their
efficiency
Efficiency of job on machine is:
The ratio between job’s best
processing time to its processing time
on this machine
PoA of algorithm is O(log m)
Equilibrium Improves
?
B
C
A
? ?
Efficiency Based Algorithm
Unfortunately – pure NE may not
exist
Iterative improvement may cycle
Modified algorithm guarantees
convergence and pure NE with PoA
of O(log^2 m)
Modified Algorithm
Each machine simulate log m submachines
(by round robin)
Submachine k of machine j handles jobs on
efficiency between 2^{-k} and 2^{-k+1}
Jobs are ordered on submachine by
Shortest-First
PoA of algorithm is O(log^2 m)
Summary
Coordination Mechanism: Influence on the quality of the equilibrium
Unrelated Machines: m – lower bound Shortest-First is at least m Local order by efficiency O(log m) –
optimal Pure + Convergence O(log^2 m)
Discussion and Open ProblemsNon ordering strategies – get below log m
Extend to network routing
Show more effective usage of coordination mechanism