algorithms and economics of networks: coordination mechanisms
DESCRIPTION
Algorithms and Economics of Networks: Coordination Mechanisms. Abraham 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. - PowerPoint PPT PresentationTRANSCRIPT
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Algorithms and Economics of Networks:Coordination MechanismsAbraham Flaxman and Vahab Mirrokni,
Microsoft Research
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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
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Price of Anarchy Concept
Not algorithmic
Only analysis
What to do if PoA is large
How to influence the system
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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)
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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
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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)
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Unrelated Machines Scheduling
Machine A
A
B
A
B? ?
Machine B
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Coordination Mechanism for Scheduling
Policy for each machine (algorithm) which
decides how to schedule jobs assigned to it
Each Policy induces NE on jobs
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Local Scheduling Policies
A A
Shortest-First Policy Longest-First Policy
B B
A A
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Local Scheduling Policies
Shortest-first Policy Longest-first Policy Random Order Policy MakeSpan Policy
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Challenge
Design policies that results in
good NE (i.e. low PoA)
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Unrelated Machines Scheduling
Machine A
A
B
A
B? ?
Machine B
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Equilibrium for Longest First
A
B
A
B
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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
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Machine Scheduling Models
Identical Machines: P||Cmax
Related Machines: Q||Cmax
(Different Speeds)
Restricted Assignment: B||Cmax
Unrelated Machines: R||Cmax
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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)
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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
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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)
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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
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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
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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
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Example for Shortest-First
?
B
C
A
? ?
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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
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Optimal Assignment
A
B
C
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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
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Equilibrium for Shortest-First
?
B
C
A
? ?
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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
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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)
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Equilibrium Improves
?
B
C
A
? ?
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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)
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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)
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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)
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Discussion and Open ProblemsNon ordering strategies – get below log m
Extend to network routing
Show more effective usage of coordination mechanism