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Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute

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Selfish Flows over Time Umang Bhaskar, Lisa Fleischer Dartmouth College Elliot Anshelevich Rensselaer Polytechnic Institute (I have animations)

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Uncoordinated Traffic on roads in communication and in other networks

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Uncoordinated Traffic A B

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A 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?

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Price of Anarchy Objective for uncoordinated traffic routing Objective for coordinated routing which minimizes objective Price Of 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? Price Of 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 st f e

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Flows over Time st - 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 14 bits per second Arrival graph: time Whats the quickest flow?

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Flows over Time Edge delay d e Edge capacity c e st Flow value v Total time: ? 15 Whats the quickest flow?

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Flows over Time c 1, d 1 s t Flow value v c 2, d 2 c 3, d 3 c 4, d 4 c 5, d 5 c 6, d 6 c 7, d 7 c 8, d 8 c 9, d 9 c 10, d 10 c 11, d 11 Total time: ? 16 Whats 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 17

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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: ? c 1, d 1 s t Flow value v c 2, d 2 c 3, d 3 c 4, d 4 c 5, d 5 c 6, d 6 c 7, d 7 c 8, d 8 c 9, d 9 c 10, d 10 c 11, d Flows over Time Whats the quickest flow?

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Traffic in networks is uncoordinated Players pick routes selfishly to minimize travel time Selfish Flows over Time 19

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Motivation I & II: Networks Data networks Road traffic 20

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Motivation III : Evacuation 21 Safe zone

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A Queuing Model s t c = 2, d = 2 c = 1, d = 1 But if players are selfish … 22

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A Queuing Model s t c = 2, d = 2 c = 1, d = 1 ? Queue forms here 23

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A Closer Look at Queues s t c = 2, d = 2 c = 1, d = 1 Queue Queues are formed when inflow exceeds capacity on an edge Queues are first in, first out (FIFO) Players delay depends on queue as well 24

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A Game-Theoretic Model s t 25 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 26 time flow rate at t

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Equilibrium s t ? 27 Delay along a path depends on Queues depend on Other players ?

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Equilibrium s t ? 28 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 ! ! 29

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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] 30 our case

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Weve 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? 31

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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 t Time taken by quickest flow So, what is the Price of Anarchy for selfish flows over time? [KS 09] s t 32 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 33 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? 34 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 35

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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 36 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 37 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 t a. 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) 38

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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) ) 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 Questions 1. If we dont remove excess capacity, can PoA exceed e/(e-1) ? 3. What if players have imperfect information? 4. … 2. PoA for multiple sources 40 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 42 Corollary: Equilibrium in modified network takes time e/(e-1) times the quickest flow

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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 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 44

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Capacity c e bounds rate of outflow; rate of inflow is unbounded Excess flow forms a queue on the edge A Simple Example 45

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A Closer Look at Queues - II s t 46

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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 47

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A Closer Look at Queues s t c = 2, d = 2 c = 1, d = 1 Queue Queues are time-varying Assume: players know time taken along a path 48

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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 inefficiency due to distributed usage

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Price of Anarchy 51 (i) (ii)(iii) For a given objective (e.g., traffic slowdown, resource usage), Price of Anarchy measures worst-case inefficiency due to distributed usage

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Price of Anarchy Guide design of systems Uses of Price of Anarchy: 52 (Murphys Law!)

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Traffic in networks varies with time Edges have delays Common models assume static traffic, no delays Flows over Time 53

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The Price of Anarchy (and how to control it) 54

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Enforcing a bound on the PoA s 55 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 56 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 ? 57 Delay along a path depends on Queues depend on Other players ?

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Equilibrium s t ? ? 58 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] t c = 3, d = 0c = 2, d = 0c = 1, d = 0 c = 1, d = 1 c = 1, d = Flow rate at t Time

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Properties at Equilibrium s c = 3, d = 0c = 2, d = 0c = 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) 60 At equilibrium, players use path in quickest-path network [Koch, Skutella 09] Flow rate at t Time t

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Properties at Equilibrium s c = 3, d = 0c = 2, d = 0c = 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) 61 At equilibrium, players use path in quickest-path network [Koch, Skutella 09] Flow rate at t Time t

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Properties at Equilibrium s c = 3, d = 0c = 2, d = 0c = 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) 62 At equilibrium, players use path in quickest-path network [Koch, Skutella 09] Flow rate at t Time t

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Properties at Equilibrium s c = 3, d = 0c = 2, d = 0c = 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) 63 At equilibrium, players use path in quickest-path network [Koch, Skutella 09] Flow rate at t Time t

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Properties at Equilibrium s c = 3, d = 0c = 2, d = 0c = 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) 64 At equilibrium, players use path in quickest-path network [Koch, Skutella 09] Flow rate at t Time t

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So Far… s t At equilibrium, how quickly does all the flow reach t ? 65 For a given objective, how well does the system perform, for uncoordinated traffic routing?

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Static Flows st

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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

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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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- Edges have delays - Flow on an edge varies with time

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Motivation IV : Machine Scheduling Each machine i has a capacity c i and delay d i 74

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Features of the Model Continuous time Preserves FIFO Queuing model used since 70s for studying road traffic s t 75

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Price of Anarchy Guide design of systems Uses of Price of Anarchy: 76

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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 77

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Price of Anarchy Objective: 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 Uncoordinated routing

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System Performance Objective: 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 Price of Anarchy = we will refine this later

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