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

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

Uncoordinated Traffic on roads in communication and in other networks

Uncoordinated Traffic A B

A B Players choose their route selfishly (i.e., to minimize some objective)

System Performance For a given objective, how well does the system perform, for uncoordinated traffic routing?

System Performance For a given objective, how well does the system perform, for uncoordinated traffic routing?

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?

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

Modeling Uncoordinated Traffic How do we model uncoordinated traffic?

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

Static Flows st f e

Flows over Time st - Edges have delays - Flow on an edge varies with time

Flows over Time 1000 bits Total time: 11 seconds 2 seconds 100 bps 14 bits per second Arrival graph: time Whats the quickest flow?

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

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?

Flows over time have been studied since [Ford, Fulkerson 62] Used for traffic engineering, freight, evacuation planning, etc. Flows over Time 17

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?

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

Motivation I & II: Networks Data networks Road traffic 20

Motivation III : Evacuation 21 Safe zone

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

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

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

A Game-Theoretic Model s t 25 Assumptions: Players are infinitesimal time flow at t

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

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

Equilibrium s t ? 28 Delay along a path depends on Queues depend on Other players ? ?

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

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

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

(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

The Price of Anarchy Lower bound of e/(e-1) ~ 1.6 [KS 09] s t 33 Flow rate at t Time

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

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

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

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

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

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

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!

41

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

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

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

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

A Closer Look at Queues - II s t 46

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

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

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

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

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

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

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

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

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

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

Equilibrium s t ? 57 Delay along a path depends on Queues depend on Other players ?

Equilibrium s t ? ? 58 Delay along a path depends on Queues depend on Other players ?

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

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

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

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

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

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

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?

Static Flows st

Modeling Uncoordinated Traffic How do we model uncoordinated traffic?

Modeling Uncoordinated Traffic How do we model uncoordinated traffic? - Direct simulation - flexible - only for small instances - no rigorous analysis

Modeling Uncoordinated Traffic How do we model uncoordinated traffic? - Mathematical models - allow rigorous analysis - assume probabilistic traffic - difficult to analyse - Direct simulation

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

Static Flows

Flows over Time

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

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

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

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

Price of Anarchy Guide design of systems Uses of Price of Anarchy: Guide design of policies, e.g., tollbooths to influence traffic routing 77

Price of Anarchy Objective: Time taken by all players to reach destination A B

Price of Anarchy A B Objective: Time taken by all players to reach destination Uncoordinated routing

System Performance Objective: Time taken by all players to reach destination A B Coordinated, optimal routing

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