Network Congestion Games

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1 Ntwork Congstion Gams Assistant Profssor Tas A&M Univrsity Collg Station, TX TX Dallas Collg Station Austin Houston

2 Bst rout dpnds on othrs Ntwork Congstion Gams

3 Travl tim incrass with congstion Highway congstion costs wr $115 billion in Avg. commutr travls 100 minuts a day. Ntwork Congstion Gams

4 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

5 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams

6 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams

7 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

8 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 Outlin Rvisit Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams

18 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

19 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

20 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

21 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

22 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

23 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

24 Outlin Wardrop Equilibrium Social Optimum Pric of Anarchy Ntwork Congstion Gams

25 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

26 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

27 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

28 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

29 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

30 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 * * 2 1 ( 0 ( 4 1 ( * C C + * ( E b a + =

31 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

32 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

33 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

34 Ntwork Congstion Gams

35 Announcmnt I am looking for Ph.D. studnts and a postdoc with a strong thortical background. application to: nikolova@tamu.du Ntwork Congstion Gams

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