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