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![Page 1: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/1.jpg)
Selfish Flows over Time
Umang Bhaskar, Lisa FleischerDartmouth College
Elliot AnshelevichRensselaer Polytechnic Institute
![Page 2: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/2.jpg)
Selfish Flows over Time
Umang Bhaskar, Lisa FleischerDartmouth College
Elliot AnshelevichRensselaer Polytechnic Institute
(I have animations)
![Page 3: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/3.jpg)
Uncoordinated Traffic
• on roads • in communication
• and in other networks
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Uncoordinated TrafficA
B
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Uncoordinated TrafficA
B
Players choose their route selfishly
(i.e., to minimize some objective)
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System Performance
For a given objective,how well does the system perform,for uncoordinated traffic routing?
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System Performance
For a given objective,how well does the system perform,for uncoordinated traffic routing?
![Page 8: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/8.jpg)
Price of Anarchy
Objective for uncoordinated traffic routing
Objective for coordinated routing which minimizes objective
PriceOf
Anarchy
=
For a given objective,how well does the system perform,for uncoordinated traffic routing?
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Price of Anarchy
Time taken for uncoordinated traffic routing
Minimum time taken
Objective: Time taken by all players to reach destination
=
For a given objective,how well does the system perform,for uncoordinated traffic routing?
PriceOf
Anarchy
we will refine this later
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Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
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Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
Routing games with static flows
- allow rigorous analysis- capture player “selfishness”- network flows, game theory
Tight bounds on PoA in this model
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Static Flows
s t
fe
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Flows over Time
s t
- Edges have delays
- Flow on an edge varies with time
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Flows over Time
1000 bits
Total time: 11 seconds
2 seconds
100 bps
bitsper
second
1 2 3 4 5 6 7 8 9 10 11 12
100
Arrival graph:
time
What’s the “quickest flow”?
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Flows over Time
Edge delay de
Edge capacity ces t
Flow value v
Total time: ?
What’s the “quickest flow”?
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Flows over Time
c1 , d1s t
Flow value v
c2 , d2
c3 , d3
c4 , d4
c5 , d5
c6 , d6
c7 , d7
c8 , d8
c9 , d9
c10 , d10
c11 , d11
Total time: ?
What’s the “quickest flow”?
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• Flows over time have been studied since [Ford, Fulkerson ’62]• Used for traffic engineering, freight, evacuation planning, etc.
Flows over Time
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• Quickest flow: flow over time which gets flow value v from s to t in shortest time• [FF ‘62] showed how to compute quickest flow in
polynomial time
Total time: ?
c1 , d1s t
Flow value v
c2 , d2
c3 , d3
c4 , d4
c5 , d5
c6 , d6
c7 , d7
c8 , d8
c9 , d9
c10 , d10
c11 , d11
Flows over Time
What’s the “quickest flow”?
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• Traffic in networks is uncoordinated• Players pick routes selfishly to minimize travel time
Selfish Flows over Time
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Motivation I & II: Networks
• Data networks• Road traffic
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Motivation III : Evacuation
Safe zone
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A Queuing Model
st
c = 2, d = 2c = 1, d = 1
But if players are selfish …
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A Queuing Model
st
c = 2, d = 2c = 1, d = 1
?
Queue forms here
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A Closer Look at Queues
st
c = 2, d = 2c = 1, d = 1
Queue
• Queues are formed when inflow exceeds capacity on an edge• Queues are first in, first out (FIFO)• Player’s delay depends on queue as well
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A Game-Theoretic Model
s t
Assumptions:
• Players are infinitesimal
time
flow at t
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A Game-Theoretic Model
s t
Assumptions:
• Players are infinitesimal
Model:
• Players are ordered at s• Each player picks a path from s to t• Minimizes the time it arrives at t
time
flowrateat t
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Equilibrium
s t?
Delay along a path depends on Queues depend on Other players
?
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Equilibrium
s t?
Delay along a path depends on Queues depend on Other players
?
?
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Equilibrium
• At equilibrium, every player minimizes its delay w. r. t. others; thus no player wants to change
s t
• Equilibria are stable outcomes
! !
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Features of the Model
s t
• Various nice properties, including existence of equilibrium in single-source, single-sink case[Koch, Skutella ‘09]
our case
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•We’ve seen a game-theoretic model of selfish flows over time, based on queues
So Far…
s t
• Equilibrium exists in this model
But how bad is equilibrium?
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(Quickest flow minimizes time for flow to reach t)
The Price of Anarchy
• Price of Anarchy (PoA) =
Time taken at equilibrium for all flow to reach tTime taken by quickest flow
So, what is the Price of Anarchy for selfish flows over time?[KS ‘09]
s t
In static flow games, PoA is essentially unbounded
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The Price of Anarchy• Lower bound of e/(e-1) ~ 1.6 [KS ‘09]
s t
Flow rate at t
Time
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The Price of Anarchy• Lower bound of e/(e-1) ~ 1.6 [KS ‘09]
i.e., flow rate at t increases to maximum in one step
• Upper bounds?
Flow rate at t
Time
s t
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Enforcing a bound on the PoA
We show (to appear in SODA ‘11): The network administrator can enforce a bound of e/(e-1) on the Price of Anarchy
In a network with reduced capacity, equilibrium takes time≤ e/(e-1) ~ 1.6 times the minimum in original graph
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Enforcing a bound on the PoA
1. Modify network so that quickest flow is unchanged
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA = e/(e-1)
In a network with reduced capacity, equilibrium takes time≤ e/(e-1) ~ 1.6 times the minimum in original graph
Corollary: Equilibrium in modified network takes time ≤ e/(e-1) times the quickest flow
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Enforcing a bound on the PoA
1. Modify network so that quickest flow is unchanged
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA = e/(e-1)
In a network with reduced capacity, equilibrium takes time≤ e/(e-1) ~ 1.6 times the minimum in original graph
Corollary: Equilibrium in modified network takes time ≤ e/(e-1) times the quickest flow
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1. Modify network so that quickest flow is unchanged
s ta. Compute quickest flow in the original network
b. On every edge, remove capacity in excess of quickest flow
s t
c, d
c', d
Enforcing a bound on the PoA
([FF ‘62] gave a polynomial-time algorithm for computing quickest flow)
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Enforcing a bound on the PoA
i.e., PoA is largest when flow rate at t increases in one step
(PoA of [KS ‘09] example is e/(e-1) )
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA
s t
Flow rate at t
Time
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Open Questions1. If we don’t remove excess capacity, can PoA exceed
e/(e-1) ?
3. What if players have imperfect information?
4. …
2. PoA for multiple sources
Thanks for listening!
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Thanks for listening!
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Enforcing a bound on the PoA
1. Modify network so that quickest flow is unchanged
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA = e/(e-1)
In a network with reduced capacity, equilibrium takes time≤ e/(e-1) ~ 1.6 times the minimum in original graph
Corollary: Equilibrium in modified network takes time ≤ e/(e-1) times the quickest flow
![Page 43: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/43.jpg)
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Enforcing a bound on the PoA
We show: the network administrator can enforce a bound of e/(e-1) on the Price of Anarchy
1. Modify network so that quickest flow is unchanged
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA = e/(e-1)
- In modified network, equilibrium takes at most e/(e-1) of the time taken by quickest flow
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A Closer Look at Queues - II
• Queues are time-varying• Players should anticipate queue at an edge in the future, i.e.,
at time when player reaches the edge
s t
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• Capacity ce bounds rate of outflow; rate of inflow is unbounded
• Excess flow forms a queue on the edge
A Simple Example
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A Closer Look at Queues - II
s t
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A Closer Look at Queues - II
• We assume that path chosen by each player is known
s t
• So each player can calculate queue on an edge at any time
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A Closer Look at Queues
st
c = 2, d = 2c = 1, d = 1
Queue
• Queues are time-varying• Assume: players know time taken along a path
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Price of Anarchy
vs
• Distributed usage of resources leads to inefficiency, e.g.,
Central coordination Distributed usage
slowing down of traffic overuse of some resources, underuse of others
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Price of Anarchy
vs
Central coordination Distributed usage
For a given objective (e.g., average speed, resource usage)Price of Anarchy measures worst-case inefficiencydue to distributed usage
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Price of Anarchy
(i) (ii) (iii)
For a given objective (e.g., traffic slowdown, resource usage),Price of Anarchy measures worst-case inefficiencydue to distributed usage
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Price of Anarchy
• Guide design of systems
Uses of Price of Anarchy:
(Murphy’s Law!)
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• Traffic in networks varies with time• Edges have delays
• Common models assume static traffic, no delays
Flows over Time
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ThePrice of
Anarchy(and how to control it)
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Enforcing a bound on the PoA
s Time
Flow rate at t
t
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA
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Enforcing a bound on the PoA
s Time
Flow rate at t
t
2. Main Lemma: In modified network, the example shown in [KS ’09] has largest PoA
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Equilibrium
s t?
Delay along a path depends on Queues depend on Other players
?
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Equilibrium
s t? ?
Delay along a path depends on Queues depend on Other players
?
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Properties at Equilibrium
s
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
tc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10Flow rate
at t
Time
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Properties at Equilibrium
sc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
Flow rate at t
Timet
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Properties at Equilibrium
sc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
Flow rate at t
Timet
![Page 62: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/62.jpg)
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Properties at Equilibrium
sc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
Flow rate at t
Timet
![Page 63: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/63.jpg)
63
Properties at Equilibrium
sc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
Flow rate at t
Timet
![Page 64: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/64.jpg)
64
Properties at Equilibrium
sc = 3, d = 0 c = 2, d = 0 c = 1, d = 0
c = 1, d = 1
c = 1, d = 10
• At any time there is a quickest-path network (least delay s-t paths)• At equilibrium, players use path in quickest-path network
[Koch, Skutella ‘09]
Flow rate at t
Timet
![Page 65: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/65.jpg)
Static Flows
s t
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Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
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Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
- Direct simulation
- flexible- only for small instances- no rigorous analysis
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Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
- Mathematical models
- allow rigorous analysis- assume probabilistic traffic- difficult to analyse
- Direct simulation
![Page 69: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/69.jpg)
Modeling Uncoordinated Traffic
How do we model uncoordinated traffic?
- Mathematical models
- Routing games with static flows
- allow rigorous analysis- capture player “selfishness”- network flows, game theory
- Direct simulation
Tight bounds on PoA in this model
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Static Flows
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Flows over Time
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Flows over Time
- Edges have delays
- Flow on an edge varies with time
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Motivation IV : Machine Scheduling
Each machine i has a capacity ci and delay di
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Features of the Model
• Continuous time
• Preserves FIFO
• Queuing model used since ‘70s for studying road traffic
s t
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Price of Anarchy
• Guide design of systems
Uses of Price of Anarchy:
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Price of Anarchy
• Guide design of systems
Uses of Price of Anarchy:
• Guide design of policies, e.g., tollbooths to influence traffic routing
![Page 77: Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute.](https://reader035.fdocuments.us/reader035/viewer/2022062619/55163753550346b2068b4f38/html5/thumbnails/77.jpg)
Price of AnarchyObjective: Time taken by all players to reach destination
A
B
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Price of Anarchy
A
B
Objective: Time taken by all players to reach destination
Uncoordinatedrouting
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System PerformanceObjective: Time taken by all players to reach destination
A
B
Coordinated,optimal routing
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System Performance
For a given objective,how well does the system perform,for uncoordinated traffic routing?
Time taken by uncoordinated traffic routing
Time taken by optimal routing
Objective: Time taken by all players to reach destination
Priceof
Anarchy=
we will refine this later