Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston
Bst rout dpnds on othrs Ntwork Congstion Gams
Travl tim incrass with congstion Highway congstion costs wr $115 billion in 2009. Avg. commutr travls 100 minuts a day. Ntwork Congstion Gams
Gam modl Dirctd graph G = (V,E Multipl sourc-dst. pairs (s k,t k, dmand d k Playrs (usrs: nonatomic (infinitsimally small Stratgy st: paths P k btwn (s k,t k for all k Ρ Playrs dcisions: flow vctor R or Ρ Somtims will us f R for path flow. R E ( Edg dlay (latncy functions: typically assumd continuous and nondcrasing. Ntwork Congstion Gams
Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
Wardrop s First Principl Travl tims on usd routs ar qual and no gratr than travl tims on unusd routs. Dfinition: A flow is a Wardrop Equilibrium (WE if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, whr path path ( path' (, ( = ( path Also calld Usr Equilibrium or Nash Equilibrium. Equilibrium flow is calld Nash flow.. for all path ' P k Ntwork Congstion Gams
Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
Wardrop s Scond Principl Th avrag [total] journy tim is minimum. Th cost of flow is dfind as th total journy tim : Dnot C( = ( = all paths E c path ( path path path ( : = ( = all paths = E path path (. (, assumd conv. Ntwork Congstion Gams
Wardrop s Scond Principl Th avrag [total] journy tim is minimum. Dfinition: A flow is a Social Optimum if it minimizs total dlay: Ntwork Congstion Gams. for all 0, for all, for all s.t. ( ( min : P p f K k f d E f c p P p p k p P p p E E k = = = flow constraints
Social Optimum Th avrag [total] journy tim is minimum. Dfinition: A flow is a Social Optimum if it minimizs total dlay: min s.t. E c ( = E flow constraints ( Ntwork Congstion Gams
Social Optimum Dfinition: A flow is a Social Optimum if it solvs Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, whr c' ( min = E c ( = ( E c' ( c' ' (, p c' ( Proof sktch: p marginal bnfit of marginal cost of rducing traffic on p incrasing traffic on p p. p Ntwork Congstion Gams
Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min E c ( = ( E Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, Corollary 1: If costs ar conv, local opt is a global opt, and lmma givs quivalnt dfn of SO. c' ( c' ' (. p c ( p Ntwork Congstion Gams
Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min E c ( = ( E Lmma: A flow vctor is locally optimal if for ach path p with positiv flow and ach path p, Corollary 2: If costs ar conv, SO is an quilibrium with rspct to modifid latncis * c' ( c' ' (. p c ( ( = c' ( = ( + p '(. Ntwork Congstion Gams
Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min c ( = ( Mchanism Dsign Intrprtation: If usrs valu tim and mony qually, E imposing E tolls '( pr unit flow on ach dg will caus slfish Lmma: A flow vctor is locally optimal if for playrs ach path to rach p with th positiv Social flow Optimum! and ach path p, c' ( c' ' (. p c ( Corollary 2: If costs ar conv, SO is an quilibrium with rspct to modifid latncis * ( = c' ( = ( + p '(. Ntwork Congstion Gams
Computing Social Optimum (SO Dfinition: A flow is a Social Optimum if it solvs min s.t. E c ( flow constraints c ( ( Corollary 3: If costs ar conv, SO ists and can b found fficintly by solving conv program abov. = E Ntwork Congstion Gams
Outlin Rvisit Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, path compar with: ( '(, for all path' path SO Dfinition: A flow vctor is a Social Optimum for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, c' ( c' ' (, for all path' path path Ntwork Congstion Gams
Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, h whr path h '( h ''(, for all path' path SO Dfinition: A flow vctor is a Social Optimum for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, c' ( c' ' (, for all path' path '( : = ( path. Ntwork Congstion Gams
Equilibrium istnc WE Dfinition: A flow is a Wardrop Equilibrium if for vry sourc-dst. pair k and for vry path with positiv flow btwn this pair, h path '( h ''(, for all path' path whr h '( : = (. Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min E c ( Ntwork Congstion Gams
Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: whr h '( min E : = ( Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min E h (,. c ( E.g., h ( = 0 ( z dz Ntwork Congstion Gams
Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: whr h '( min : = E ( Altrnativ SO Dfinition: A flow vctor is a Social Optimum if it solvs: min 0 E ( z dz,. c ( Ntwork Congstion Gams
Equilibrium istnc Altrnativ WE Dfinition: A flow vctor is a Wardrop Equilibrium if it solvs: min s.t. E 0 ( z dz flow constraints Thorm: A Wardrop Equilibrium ists and can b computd in polynomial tim. Also, if program abov is strictly conv, quilibrium is uniqu, up to sam flow cost. Ntwork Congstion Gams
Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams
Eampl: Infficincy of quilibria hours 1/2 1 hour Dlay is 1.5 hours for vrybody at th uniqu Nash quilibrium Town A Town B 1 hour hours Suppos drivrs (total 1 unit of flow lav from town A towards town B. Evry drivr wants to minimiz hr own travl tim. What is th traffic on th ntwork? 1/2 In any unbalancd traffic pattrn, all drivrs on th most loadd path hav incntiv to switch thir path. Ntwork Congstion Gams
Eampl: Infficincy of quilibria hours 1 1 hour Dlay is 2 hours for vrybody at th uniqu Nash quilibrium Town A 0 hours Town B 1 hour hours A bnvolnt mayor builds a suprhighway conncting th fast highways of th ntwork. What is now th traffic on th ntwork? No mattr what th othr drivrs ar doing it is always bttr for m to follow th zig-zag path. Ntwork Congstion Gams
Eampl: Infficincy of quilibria 1/2 1 hours 1 hour hours 1 hour A B vs A B 1 hour hours 1 hour hours 1/2 Adding a fast road on a road-ntwork is not always a good ida! Brass s parado In th RHS ntwork thr ists a traffic pattrn whr all playrs hav dlay 1.5 hours. Pric of Anarchy: Ntwork Congstion Gams masurs th loss in systm prformanc du to fr-will
Pric of Anarchy Cost of Flow: total usr cost Social optimum: flow minimizing total usr cost Pric of anarchy: (Koutsoupias, Papadimitriou 99 sup problm instancs Equilibrium Cost Social Optimum Cost Ntwork Congstion Gams
Variational Inquality rprsntation of quilibria Thorm: Equilibria in nonatomic gams ar solutions to th Variational Inquality (VI ( ( ' 0 for all fasibl flows ' Proof: whr Flow f is an quilibrium = (,..., if and only if [flow l(. vctor] <= l(.. 1 P ( = ( 1(,..., P ( [latncy vctor] Proof: (=> Equilibrium flow routs along minimum-cost paths l(. Fiing path costs at l(, any othr flow that assigns flow to highr-cost paths will rsult in highr ovrall cost l(.. VI Solution ists ovr compact conv st with l( continuous [Hartman, Stampacchia 66]. (<= Suppos is not an q. Thn thr is a flow-carrying path p VI Solution uniqu if l( is monoton: (l(-l ( (- 0. with lp( > lp (. Shifting flow from p to p will obtain [Ercis: vrify] l(. < l(., contradiction. Ntwork Congstion Gams
Thorm**: Th pric of anarchy (PoA is 4/3 in gnral graphs and latncis i.. whr is WE and * is SO flow. Pf: 0,,, ( + = b a b a l Pric of Anarchy with linar latncis Ntwork Congstion Gams * ( ( ( E E C =, ( 3 4 ( * C C ] 4 1 ( [ ( * 2 * * * 2 E E b a b a + + + = + + = E E a b a 2 * * 4 1 ( **Rfrncs: Roughgardn, Tardos 02; Corra, Schulz, Stir-Moss 04, 08 QED., ( ( 4 3 * C C 2 * 2 * 4 1 + 2 * 2 1 ( 0 ( 4 1 ( * C C + * ( E b a + =
Tak-away points Equilibrium and Social Optimum in nonatomic routing gams ist and can b found fficintly via conv programs. Social optimum is an quilibrium with rspct to modifid latncis = original latncis plus toll. Pric of anarchy: 4/3 for linar latncis, can b found similarly for mor gnral classs of latncy functions. Ntwork Congstion Gams
Rfrncs Wardrop 52, Bckmann t al. 56, A lot of work in AGT community and othrs. Survys of rcnt work: AGT Book Nisan t al. 07 Corra, Stir-Moss 11 Ntwork Congstion Gams
Som opn qustions What is th pric of anarchy with rspct to othr Social Cost functions? Dynamic (tim-changing latncy functions? Uncrtain dlays? Ntwork Congstion Gams
Ntwork Congstion Gams
Announcmnt I am looking for Ph.D. studnts and a postdoc with a strong thortical background. Email application to: nikolova@tamu.du Ntwork Congstion Gams