Optima and Equilibria for Traffic Flow on a Network

Transcription

1 Opima and Equilibria for Traffic Flow on a Nework Albero Bressan Deparmen of Mahemaics, Penn Sae Universiy bressan@mah.psu.edu Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 1 / 1

2 A Traffic Flow Problem Car drivers saring from a locaion A (a residenial neighborhood) need o reach a desinaion B (a working place) a a given ime T. There is a cos ϕ(τ d ) for deparing early and a cos ψ(τ a ) for arriving lae. ϕ() ψ() T A B Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 2 / 1

3 Elemenary soluion L = lengh of he road, v = speed of cars τ a = τ d + L v Opimal deparure ime: τ op d = argmin { ( ϕ() + ψ + L ) }. v If everyone depars exacly a he same opimal ime, a raffic jam is creaed and his sraegy is no opimal anymore. Speed of cars depends on raffic densiy! Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 3 / 1

4 An opimizaion problem for raffic flow Given deparure cos ϕ() and arrival cos ψ(), choose a deparure rae ū() so ha he soluion of he conservaion law ρ + [ρ v(ρ)] x = 0 x [0, L] ρ(, 0)v(ρ(, 0)) = ū() minimizes he sum of he coss o all drivers arrival rae u(,l) deparure rae _ u(,0) = u() 0 ρ L flux u = ρv( ) Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 4 / 1

5 Mahemaical formulaion u(, x). = ρ(, x) v(ρ(, x)) = flux of cars Minimize oal cos: J(ū). = ϕ() u(, 0) d + ψ() u(, L) d for a soluion o ρ + [ρ v(ρ)] x = 0 x [0, L] ρ(, 0)v(ρ(, 0)) = ū() Choose he opimal deparure rae ū(), subjec o he consrain ū() 0, ū() d = κ = [oal number of drivers] Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 5 / 1

6 Exisence of a globally opimal soluion (A1) The flux funcion ρ f (ρ) = ρ v(ρ) is sricly concave down. f (0) = f (ρ max) = 0, f < 0. (A2) The cos funcions ϕ, ψ saisfy ϕ < 0, ψ, ψ 0, ( ) lim ϕ() = +, lim ϕ() + ψ() + = + Theorem (A.B. and K. Han, SIAM J. Mah. Anal., 2012). Le (A1)-(A2) hold. Then, for any κ > 0, here exiss a unique admissible iniial daa ū minimizing he oal cos J( ). Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 6 / 1

7 Characerizaion of he globally opimal soluion If ρ = ρ(, x) is he densiy in a globally opimal soluion, hen here exiss a consan C such ha, for any characerisic line = (x) ϕ((0)) + ψ((l)) = C if (0) Supp(ū) ϕ((0)) + ψ((l)) C if (0) / Supp(ū) =(x) arrival rae u(,l) deparure rae _ u(,0) = u() 0 flux u = ρv( ρ) L x Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 7 / 1

8 An Example Cos funcions: ϕ() =, ψ() = { 0, if 0 2, if > 0 L = 1, u = ρ(2 ρ), M = 1, κ = Bang-bang soluion Pareo opimal soluion L=1 x L=1 x τ 0 0 τ 1 τ 0 0 τ 1 τ 0 = , τ 1 = oal cos = τ 0 = , τ 1 = oal cos = Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 8 / 1

9 Does everyone pay he same cos? Cos Deparure ime Cos vs. deparure ime in a globally opimal soluion Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 9 / 1

10 The Nash equilibrium soluion A soluion u = u(, x) is a Nash equilibrium if no driver can reduce his/her own cos by choosing a differen deparure ime. This implies ha all drivers pay he same cos. To find a Nash equilibrium, inroduce he inegraed variable U(, x). = ρ(s, x) v(ρ(s, x)) ds = [number of drivers ha have crossed he poin x along he road wihin ime ] This solves a Hamilon-Jacobi equaion U x + F (U ) = 0 U(, 0) = Q() Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 10 / 1

11 Noe: a queue can form a he enrance of he highway κ Q() β U(,L) q τ ( β) a τ ( ) β Q() = number of drivers who have sared heir journey before ime (possibly joining he queue) L = lengh of he road U(, L) = number of drivers who have reached desinaion before ime Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 11 / 1

12 Characerizaion of a Nash equilibrium κ Q() β U(,L) q τ ( β) a τ ( ) β β [0, κ] = Lagrangian variable labeling one paricular driver τ q (β) = ime when driver β joins he queue τ a (β) = ime when driver β arrives a desinaion Nash equilibrium = ϕ(τ q (β)) + ψ(τ a (β)) = c Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 12 / 1

13 Exisence and Uniqueness of Nash equilibrium Theorem (A.B. - K. Han). Le he flux f and cos funcions ϕ, ψ saisfy he assumpions (A1)-(A2). Then, for every κ > 0, he Hamilon-Jacobi equaion U x + F (U ) = 0 admis a unique Nash equilibrium soluion wih oal mass κ Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 13 / 1

14 Skech of he proof 1. For a given cos c, le Q c be he se of all deparure disribuions Q( ) for which every driver has a cos c: ϕ(τ q (β)) + ψ(τ a (β)) c for a.e. β [0, Q(+ )]. { } 2. Claim: Q. () = sup Q() ; Q Q c is he iniial daa for a Nash equilibrium wih common cos c. Q () Q() Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 14 / 1

15 3. There exiss a minimum cos c 0 such ha κ(c) = 0 for c c 0. The map c κ(c) is sricly increasing and coninuous from [c 0, + [ o [0, + [. κ κ(c) c 0 c Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 15 / 1

16 An example of Nash equilibrium x S τ 0 S τ τ q () 0 τ τ Q() τ 1 A queue of size δ 0 forms insanly a ime τ 0 The las driver of his queue depars a τ 2, and arrives a exacly 0. The queue is depleed a ime τ 3. A shock is formed. The las driver depars a τ 1. δ 0 τ 0 τ 4 τ 1 Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 16 / 1

17 Characerizaion of he Nash equilibrium soluion If ρ = ρ(, x) is he densiy in a Nash equilibrium soluion, hen here exiss a consan C such ha, for any paricle line = τ(x) (describing a car rajecory) ϕ(τ(0)) + ψ(τ(l)) = C if τ(0) Supp(ū) ϕ(τ(0)) + ψ(τ(l)) C if τ(0) / Supp(ū) arrival rae u(,l) deparure _ rae u(,0) = u() (joining he queue) = τ(x) τ(0) 0 flux u = ρv( ρ) L x Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 17 / 1

18 A comparison Toal cos of he Pareo opimal soluion: J op = Toal cos of he Nash equilibrium soluion: J Nash = Price of anarchy: J Nash J op Can one eliminae his inefficiency, ye allowing freedom of choice o each driver? (goal of non-cooperaive game heory: devise incenives) Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 18 / 1

19 Opimal pricing Suppose a fee b() is colleced a a oll booh a he enrance of he highway, depending on he deparure ime. New deparure cos: ϕ() = ϕ() + b() Is here an opimal choice of b()? Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 19 / 1

20 Cos cos = p (τ d ) Deparure ime p() = cos o a driver saring a ime, in a globally opimal soluion Choose addiional fee: b() = p max p()+ consan = Nash equilibrium coincides wih he globally opimal soluion ϕ ϕ ~ = ϕ + b ψ 0 Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 20 / 1

21 Traffic Flow on a Nework Nodes: A 1,..., A m arcs: γ ij n groups of drivers wih differen origins and desinaions, and differen coss A i γ ji γ ij A j Γ Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 21 / 1

22 Traffic Flow on a Nework Drivers in he k-h group depar from A d(k) and arrive o A a(k) Deparure cos: ϕ k () arrival cos: ψ k () ( ) (A2) ϕ < 0, ψ k > 0, lim ϕ() + ψ() = A d(k) Aa(k) Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 22 / 1

23 Traffic Flow on a Nework A a(2) A d(2) drivers can use differen pahs Γ 1, Γ 2,... o reach desinaion Does here exis a globally opimal soluion, and a Nash equilibrium soluion for raffic flow on a nework? Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 23 / 1

24 Admissible deparure raes G k = oal number of drivers in he k-h group, k = 1,..., n Γ p = viable pah (concaenaion of viable arcs γ ij ), p = 1,..., N ū k,p () = deparure rae of k-drivers raveling along he pah Γ p The se of deparure raes {ū k,p } is admissible if ū k,p () 0, p ū k,p () d = G k k = 1,..., n τ p (). = arrival ime for a driver saring a ime, raveling along Γ p (depends on he overall raffic condiions) Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 24 / 1

25 Global opima and Nash equilibria on neworks An admissible family {ū k,p } of deparure raes is globally opimal if i minimizes he sum of he oal coss of all drivers. J(ū) = ( ϕ k () + ψ k (τ p ())) ū k,p () d k,p An admissible family {ū k,p } of deparure raes is a Nash equilibrium soluion if no driver of any group can lower his own oal cos by changing deparure ime or swiching o a differen pah o reach desinaion. Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 25 / 1

26 Exisence resuls Theorem (A.B. - Ke Han, Neworks & Heerogeneous Media, 2013). On a general nework of roads, here exiss a leas one globally opimal soluion, and a leas one Nash equilibrium soluion. Proof: By finie dimensional approximaions + opological mehods No uniqueness, in general Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 26 / 1

27 Finie dimensional approximaions Fix a ime sep > 0 Consider piecewise consan deparure raes u = (u k,p ), wih bounded suppor Solving a variaional inequaliy on a compac finie dimensional se K, we obain a Galerkin approximaion o a Nash equilibrium u k,p Φ () = ϕ () + ψ (τ ) k,p k k p () u k,p Φ () = ϕ k,p () + k ψ (τ p() ) k m l Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 27 / 1

28 Exisence of a Nash equilibrium on a nework Leing he discreizaion sep 0, aking subsequences: deparure raes: ū ν k,p ( ) ū k,p( ) weakly, in L (R) arrival imes: τ ν p ( ) τ p ( ) uniformly The deparure raes ū k,p ( ) provide a Nash equilibrium Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 28 / 1

29 Sabiliy of Nash equilibrium? To jusify he pracical relevance of a Nash equilibrium, we need o analyze a suiable dynamic model check wheher he rae of deparures asympoically converges o he Nash equilibrium Assume: drivers can change heir deparure ime on a day-o-day basis, in order o decrease heir own cos (one group of drivers, one single road) Inroduce an addiional variable θ couning he number of days on he calendar.. ū(, θ) = rae of deparures a ime, on day θ]. Φ(, θ) = [cos o a driver saring a ime, on day θ] Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 29 / 1

30 A conservaion law wih non-local flux Model 1: drivers gradually change heir deparure ime, drifing oward imes where he cos is smaller. If he rae of change is proporional o he gradien of he cos, his leads o he conservaion law ū θ + [Φ ū] = 0 Φ() u _ Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 30 / 1

31 An inegral evoluion equaion Model 2: drivers jump o differen deparure imes having a lower cos. If he rae of change is proporional o he difference beween he coss, his yields d [ ] [ ] dθ ū() = ū(s) Φ(s) Φ() ds ū() Φ() Φ(τ) dτ + + Φ _ u τ s Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 31 / 1

32 Numerical experimens (Wen Shen, 2011) Quesion: as θ, does he deparure rae u(, θ) approach he unique Nash equilibrium? Flux funcion: f (ρ) = ρ (2 ρ) v(ρ) = 2 ρ Road lengh: L = 2 Deparure and arrival coss: ϕ() =, ψ() = e Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 32 / 1

33 x L y() 0 z() main difficuly: non-local dependence linearized equaion: d [ ( )] dθ Y () = α() β()y () Y (z()) Albero Bressan (Penn Sae) Opima and equilibria for raffic flow 33 / 1