Risk-averseSelfishRou1ng

ThanasisLianeasUTAus-n/NTUA

EvdokiaNikolovaUTAus-n

NicolasS-er-MosesFacebook

Risk-averseSelfishRou-ng

Trafficcondi-onsareuncertain

EvdokiaNikolova

Commuterspadtravel-mesWorstcase>twicefreeflow-me

Source:TexasTransporta-onIns-tute;ABCNewsSurvey.EvdokiaNikolova Risk-averseSelfishRou-ng

Goal

– Uncertaintravel-mesinfluenceusers’decisions

– Equilibriumexistence,encoding,efficiency*

– PriceofRiskAversion**

Risk-averseSelfishRou-ng

* E. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research, 2014

** T. Lianeas, E. Nikolova, N. Stier-Moses. Math of OR, forthcoming

Understandeffectofrisk-aversiononconges-on,bystudyingresul-ngtrafficassignment:

EvdokiaNikolova

Risk-averseSelfishRou-ng

Understandingtrafficconges-on•  PriceofAnarchy[Koutsoupias,Papadimitriou’99]

measuresthedegrada-onofsystemperformanceduetofreewill(selfishbehavior)

•  4/3ingeneralgraphs,lineartravel-mesasfunc-onof

traffic;2forquar-ctravel-mes(Roughgarden,Tardos’02;Correa,Schulz,S-er-Moses‘04,‘08)

EvdokiaNikolova

Cost Optimum SocialCost mEquilibriusup

instancesproblem

Priceofanarchy=4/3

•  Example:Oneunitoftraffic(flow)fromStoT

•  Equilibrium:Routeallflowontop;cost1hour•  Socialop-mum:Routeflow½oneachlink;cost¾hour

•  Priceofanarchy:(Equil.Cost/Op-mumCost)=4/3

S T

1hour

xhours

flowx

flow1-x

Risk-averseSelfishRou-ngEvdokiaNikolova

Risk-averseSelfishRou-ng

Risksensi-vityofpriceofanarchy•  Rou-nggameswithuncertaindelaysresul-ngfrom“uniform

schedulers”•  Priceofanarchyoflinearconges-ongamesunderriskaltudes:

–  Wald’sminimaxcost 2–  Savage’sminimaxregret [4/3,1]–  MinimizingExpectedcost 5/3–  Averagecaseanalysis 5/3–  Win-or-Go-Home unbounded –  Secondmomentmethod unbounded

•  Conclusion:Riskcri1callyaffectspredic1onsofsystemperformance

EvdokiaNikolova * G. Piliouras, E. Nikolova, J. Shamma. EC 2013 / ACM Transactions on Economics and Computation 2016

RelatedWork

•  Rou-ngGames:Wardrop’52,Beckmannetal.’56,…SurveysinNisanetal.’07,Correa&S-er-Moses’11

•  Stochas-cEquilibriummodels:Dial’71,Gupta-Stahl-Whinston’97

•  Risk-aversioninrou-nggames:afewreferencesintransporta-on(butnottoomany),Ordóñez&S-er-Moses’10,Nie’11,Angelidakis-Fotakis-Lianeas’13,Cominel-Torico’13,Meir-Parkes’15,Kleer-Schäfer’16-‘17.

EvdokiaNikolova Risk-averseSelfishRou-ng

Rou-nggameswithstochas-cdelays

•  DirectedgraphG=(V,E)Unitdemandbetweensource-dest.pair(s,t)

•  Nonatomicplayers(flowmodel)choosefeasibles-tpathsPlayers’decisions:flowvector

•  Edgedelayfunc-ons:

x ∈ R|Ρaths|

)()( eeee xxl ξ+

Risk-averseSelfishRou-ngEvdokiaNikolova E. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research, 2014

Rou-nggameswithstochas-cdelays

•  DirectedgraphG=(V,E)Unitdemandbetweensource-dest.pair(s,t)

•  Nonatomicplayers(flowmodel)choosefeasibles-tpathsPlayers’decisions:flowvector

•  Edgedelayfunc-ons:

•  Playersminimizerisk-aversepathcost:–  Mean-stdev–  Mean-var

x ∈ R|Ρaths|

)()( eeee xxl ξ+

Risk-averseSelfishRou-ngEvdokiaNikolovaE. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research, 2014

Qpath (x) = le(xe )e∈path∑ + r σ e(xe )

e∈path∑

2

Qpath (x) = le(xe )e∈path∑ + r σ e(xe )

e∈path∑

2= le(xe )+ rσ e(xe )

2( )e∈path∑

Risk-aversevsRisk-neutralEquilibrium

•  Usersselectminimum-riskpathwithrisk

•  Defini-on:Aflowxisatequilibriumifforeverysource-des-na-onpairkandforeverypathwithposi-veflow

•  WecallitaRisk-AverseWardropEquilibrium(RAWE)ifQisthemean-varianceormean-stdevcostofapath

•  WecallitaRisk-NeutralWardropEquilibrium(RNWE)ifQisthemeancostofapath

'every for ),()( ' pathxQxQ pathpath ≤

)(xQpath

Risk-averseSelfishRou-ngEvdokiaNikolova

Equilibriumcharacteriza-onformean-stdevrisk

Equilibriumcharacteriza1on

Uncertaintyindependentofflow

(σconstant)

Uncertaintydependingonflow(σdependsonflow)

Non-atomicmodel

Eq.existsItsolvesaconvexprogram(exponen-allylarge)

Eq.existsItsolvesvaria1onalineq.(alsoexponent.large)

Atomicmodel

Eq.existsGameispoten1al

Noequilibrium!(inpurestrategies)

EvdokiaNikolova Risk-averseSelfishRou-ng E. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research, 2014

AreRisk-AverseEquilibriaEfficient?

•  POA:Impactofselfishbehaviorbycomparingequilibriumtosocialop-mumflow(flowminimizingtotalusercost)Theorem*:POAwithriskaversion=POAinclassicrou-nggameswhenuncertaintydoesnotdependonflow.

•  Problem:selfishbehaviorandriskaversioncoupledtogether.Notclearwhichcausestheinefficiency

•  Decoupleeffectsofselfishnessandriskbycomparingtotherisk-neutralequilibrium

EvdokiaNikolova Risk-averseSelfishRou-ng * E. Nikolova, N. Stier-Moses. SAGT 2011 / Operations Research, 2014

PriceofRiskAversion CostofFlowC(x):althoughusersarerisk-averse,

centralplannerisrisk-neutral.•  ConsiderthesumofexpectedtravelCmes

PriceofRiskAversion(PRA):capturesinefficiencyintroducedbyuserrisk-aversionbycomparingwiththerisk-neutralcase

Risk-averseequilibriumRisk-neutralequilibrium

)C(x )C(xsup 0

instancesproblem

r

EvdokiaNikolova Risk-averseSelfishRou-ng T. Lianeas, E. Nikolova, N. Stier-Moses. Math of OR, forthcoming

Risk-aversevsRisk-neutralequilibria

•  Example:SendoneunitofflowfromStoT

•  Risk-averseeq.:Routeallflowontop;cost(1+rk)•  Risk-neutraleq.:Routeflowonbothlinks;cost1•  Priceofriskaversion:(1+rk)

S T

mean1,vark

mean(1+rk)x,var0

x

1-x

Risk-averseSelfishRou-ngEvdokiaNikolova

•  PriceofRiskAversion(PRA)isunboundedingeneral,butuncertaintyisnotarbitraryinrealworld

•  Consideraboundedvariance-to-meanra-o:

σ2e (xe)/le (xe)≤k

•  GOAL:ComputePRAforfixedk

•  Asfunc-onoftopology,forgeneraledgedelays

•  Asfunc-onofedgedelays,forgeneraltopologies

PriceofRiskAversion(PRA)

EvdokiaNikolova Risk-averseSelfishRou-ng

PriceofRiskAversion:UpperBoundforArbitraryLatencyFunc-ons

Theorem:Inageneralgraph,PRA≤1+ηrk•  Here,ηisagraphtopologyparameter:#forwardsubpathsinanalterna-ngpath[η≤½|V|]

Intui-on:

•  For2-linknetworks: PRA≤1+1rk

•  Forseries-parallelnetworks: PRA≤1+1rk

•  ForBraessnetworks: PRA≤1+2rk

EvdokiaNikolova Risk-averseSelfishRou-ng T. Lianeas, E. Nikolova, N. Stier-Moses. Math of OR, forthcoming

PriceofRiskAversion:UpperBoundforArbitraryLatencyFunc-ons

Theorem:Inageneralgraph,PRA≤1+ηrk•  Here,ηisagraphtopologyparameter:#forwardsubpathsinanalterna-ngpath[η≤½|V|]

Proofidea:Compareequilibriaonanalterna-ngpath:forwardedgeshavehigherrisk-neutralequilibriumflow,andbackwardedgeshavehigherrisk-averseequilibriumflow.

EvdokiaNikolova Risk-averseSelfishRou-ng

Theorem:Inageneralgraph,PRA≥1+ηrk

PriceofRiskAversion:LowerBoundforArbitraryLatencyFunc-ons

•  Ingraphswithgeneralmean,variancefunc-onswhereusersminimize(mean+r*variance):

Cost(Risk-averseeq.)≤(1+ηrk)Cost(Risk-neutraleq.)•  η=1forseries-parallelgraphs,η=2forBraessgraph,

η≤|V|/2forageneralgraph

PriceofRiskAversion

Risk-averseSelfishRou-ngEvdokiaNikolova*Lianeas,Nikolova,S-erMoses.“Risk-averseselfishrou1ng.”ForthcominginMathema-csofOpera-onsResearch.

T. Lianeas, E. Nikolova, N. Stier-Moses. Mathematics of Operations Research, forthcoming

PriceofRiskAversion

Risk-averseSelfishRou-ngEvdokiaNikolova*Lianeas,Nikolova,S-erMoses.“Risk-averseselfishrou1ng.”ForthcominginMathema-csofOpera-onsResearch.

T. Lianeas, E. Nikolova, N. Stier-Moses. Mathematics of Operations Research, forthcoming

•  Ingraphswithgeneralmean,variancefunc-onswhereusersminimize(mean+r*variance):

Cost(Risk-averseeq.)≤(1+ηrk)Cost(Risk-neutraleq.)•  η=1forseries-parallelgraphs,η=2forBraessgraph,

η≤|V|/2forageneralgraph

•  Alterna-veboundwithrespecttolatencyfunc-ons:Cost(Risk-averseeq.)≤(1+rk)POACost(Risk-neutraleq.)

•  Open:extendtootherriskaltudes.

Heterogeneousplayers

•  Doesheterogeneity(diversity)ofusersreducethecostofequilibrium?Usersmin(delay+αicost)

•  Diversityhelpsifandonlyifthenetworkisseries-parallelforsingleorigin-des-na-on.

•  Diversityhelpsifandonlyifthenetworkis“block-matched”formul-pleorigin-des-na-onpairs.

Risk-averseSelfishRou-ngEvdokiaNikolova R. Cole, T. Lianeas, E. Nikolova, 2017. https://arxiv.org/abs/1702.07806

EvdokiaNikolova Risk-averseSelfishRou-ng

Summary•  Goal:Developtoolkitofalgorithmsandgametheory

techniquesforriskmi-ga-oninnetworks

•  Lotsofopenproblemsin–  Algorithms(sta-c,dynamic,online,etc)–  AlgorithmicGameTheory(sta-c,dynamicgames,learning)–  AlgorithmicMechanismDesign(whatareop-mal/simplemechanismswithrisk-averseorrisk-lovingagents?)

•  Opportuni-esforimpactintransporta-on,communica-ons,smart-grid,evacua-onfromnaturaldisasters,etc.