Optima and Equilibria for a Model of Traffic Flow Alberto Bressan - PowerPoint PPT Presentation

Optima and Equilibria for a Model of Traffic Flow Alberto Bressan Mathematics Department, Penn State University Alberto Bressan (Penn State) Optima and equilibria for traffic flow 1 / 34

NSF grant: “A Theory of Complex Transportation Network Design” with Terry Friesz, Tao Yao (Industrial Engineering, PSU) Collaborators: Ke Han, Wen Shen (PSU), Chen Jie Liu, Fang Yu (Shanghai Jiao Tong University) Alberto Bressan (Penn State) Optima and equilibria for traffic flow 2 / 34

A Traffic Flow Problem Car drivers starting from a location A (a residential neighborhood) need to reach a destination B (a working place) at a given time T . There is a cost ϕ ( τ d ) for departing early and a cost ψ ( τ a ) for arriving late. ϕ (t) (t) ψ t T B A Alberto Bressan (Penn State) Optima and equilibria for traffic flow 3 / 34

Elementary solution L = length of the road, v = speed of cars τ a = τ d + L v Optimal departure time: � t + L �� � τ opt = argmin ϕ ( t ) + ψ . d v t If everyone departs exactly at the same optimal time, a traffic jam is created and this strategy is not optimal anymore. Alberto Bressan (Penn State) Optima and equilibria for traffic flow 4 / 34

An optimization problem for traffic flow Problem: choose the departure rate ¯ u ( t ) in order to minimize the total cost to all drivers. . u ( t , x ) = ρ ( t , x ) · v ( ρ ( t , x )) = flux of cars � � minimize: ϕ ( t ) · u ( t , 0) dt + ψ ( t ) u ( t , L ) dt for a solution of  ρ t + [ ρ v ( ρ )] x = 0 x ∈ [0 , L ]  ρ ( t , 0) v ( ρ ( t , 0)) = ¯ u ( t )  Choose the optimal departure rate ¯ u ( t ), subject to the constraint � ¯ u ( t ) dt = κ = [total number of drivers] Alberto Bressan (Penn State) Optima and equilibria for traffic flow 5 / 34

Equivalent formulations Boundary value problem for the density ρ : conservation law: ρ t + [ ρ v ( ρ )] x = 0 , ( t , x ) ∈ R × [0 , L ] control (on the boundary data): ρ ( t , 0) v ( ρ ( t , 0)) = ¯ u ( t ) Cauchy problem for the flux u : conservation law: u x + f ( u ) t = 0 , u = ρ v ( ρ ) , f ( u ) = ρ control (on the initial data): u ( t , 0) = ¯ u ( t ) � + ∞ � + ∞ Cost: J ( u ) = ϕ ( t ) u ( t , 0) dt + ψ ( t ) u ( t , L ) dt −∞ −∞ � + ∞ Constraint: ¯ u ( t ) dt = κ −∞ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 6 / 34

The flux function and its Legendre transform u ρ * ρ M * f (p) f(u) ρ v( ) ρ p 0 ρ * 0 u 0 ρ M ’ f (0) u = ρ v ( ρ ) , ρ = f ( u ) . � � Legendre transform: f ∗ ( p ) = max pu − f ( u ) (1) u Solution to the conservation law is provided by the Lax formula Alberto Bressan (Penn State) Optima and equilibria for traffic flow 7 / 34

The globally optimal (Pareto) solution � � minimize: J ( u ) = ϕ ( x ) · u (0 , x ) dx + ψ ( x ) u ( T , x ) dx u t + f ( u ) x = 0    subject to: � u (0 , x ) = ¯ u ( x ) , u ( x ) dx = κ ¯   (A1) The flux function f : [0 , M ] �→ R is continuous, increasing, and strictly convex. It is twice continuously differentiable on the open interval ]0 , M [ and satisfies u → M − f ′ ( u ) = + ∞ , f ′′ ( u ) ≥ b > 0 f (0) = 0 , lim for 0 < u < M . (2) ϕ ′ < 0, ψ, ψ ′ ≥ 0, (A2) The cost functions ϕ, ψ satisfy � � x →−∞ ϕ ( x ) = lim + ∞ , lim ϕ ( x ) + ψ ( x ) = + ∞ x → + ∞ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 8 / 34

Existence and characterization of the optimal solution Theorem (A.B. and K. Han, 2011). Let (A1)-(A2) hold. Then, for any given T , κ , there exists a unique admissible initial data ¯ u minimizing the cost J ( · ). In addition, No shocks are present, hence u = u ( t , x ) is continuous for t > 0. Moreover sup u ( t , x ) < M t ∈ [0 , T ] , x ∈ R For some constant c = c ( κ ), this optimal solution admits the following characterization: For every x ∈ R , let y c ( x ) be the unique point such that ϕ ( y c ( x )) + ψ ( x ) = c Then, the solution u = u ( t , x ) is constant along the segment with endpoints (0 , y c ( x )), ( T , x ). x − y c ( x ) Indeed, either f ′ ( u ) ≡ , or u ≡ 0 T Alberto Bressan (Penn State) Optima and equilibria for traffic flow 9 / 34

Necessary conditions ϕ (x) ψ (x) x 0 t x T f(u) γ x 0 y (x) u c ϕ ( y c ( x )) + ψ ( x ) = c f ′ ( u ) = x − y c ( x ) on the characteristic segment γ x T Alberto Bressan (Penn State) Optima and equilibria for traffic flow 10 / 34

An Example � 0 , if t ≤ 0 Cost functions: ϕ ( t ) = − t , ψ ( t ) = t 2 , if t > 0 L = 1, u = ρ (2 − ρ ), M = 1, κ = 3 . 80758 Bang-bang solution Pareto optimal solution x x L=1 L=1 τ 0 τ 1 τ 0 τ 1 0 t 0 t τ 0 = − 2 . 78836 , τ 1 = 1 . 01924 τ 0 = − 2 . 8023 , τ 1 = 1 . 5976 total cost = 5 . 86767 total cost = 5 . 5714 Alberto Bressan (Penn State) Optima and equilibria for traffic flow 11 / 34

Does everyone pay the same cost? 3 2.5 2 Cost 1.5 1 0.5 0 −2.8022 −2.4273 −2.0523 −1.6773 −1.3023 −0.9273 −0.5523 −0.1773 0.1977 0.5727 0.9477 1.3227 Departure time Departure time vs. cost in the Pareto optimal solution Alberto Bressan (Penn State) Optima and equilibria for traffic flow 12 / 34

The Nash equilibrium solution A solution u = u ( t , x ) is a Nash equilibrium if no driver can reduce his/her own cost by choosing a different departure time. This implies that all drivers pay the same cost. To find a Nash equilibrium, write the conservation law u t + f ( u ) x = 0 in terms of a Hamilton-Jacobi equation U t + f ( U x ) = 0 U (0 , x ) = Q ( x ) (3) � x . U ( t , x ) = u ( t , y ) dy −∞ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 13 / 34

No constraint can be imposed on the departing rate, so a queue can form at the entrance of the highway. x �→ Q ( x ) = number of drivers who have started their journey before time x (joining the queue, if there is any). Q ( −∞ ) = 0 , Q (+ ∞ ) = κ x �→ U ( T , x ) = number of drivers who have reached destination within time x T f ∗ � x − y � � � U ( T , x ) = min + Q ( y ) T y ∈ R Alberto Bressan (Penn State) Optima and equilibria for traffic flow 14 / 34

Characterization of a Nash equilibrium κ β Q(x) U(T,x) q x a x ( ) β x ( ) β β ∈ [0 , κ ] = Lagrangian variable labeling one particular driver x q ( β ) = time when driver β joins the queue x a ( β ) = time when driver β arrives at destination Alberto Bressan (Penn State) Optima and equilibria for traffic flow 15 / 34

Existence and Uniqueness of Nash equilibrium Departure and arrival times are implicitly defined by Q ( x q ( β ) − ) ≤ β ≤ Q ( x q ( β )+) , U ( T , x a ( β )) = β ϕ ( x q ( β )) + ψ ( x a ( β )) ≡ c Nash equilibrium = ⇒ Theorem (A.B. - K. Han, SIAM J. Applied Math., to appear). Let the flux f and cost functions ϕ, ψ satisfy the assumptions (A1)-(A2). Then, for every κ > 0, the Hamilton-Jacobi equation U t + f ( U x ) = 0 admits a unique Nash equilibrium solution with total mass κ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 16 / 34

Sketch of the proof 1. For a given cost c , let Q c be the set of all initial data Q ( · ) for which every driver has a cost ≤ c : ϕ ( x q ( β )) + ψ ( x a ( β )) ≤ c for a.e. β ∈ [0 , Q (+ ∞ )] . . � � 2. Claim: Q ∗ ( x ) = sup Q ( x ) ; Q ∈ Q c is the initial data for a Nash equilibrium with common cost c . 3. For a given cost c , the Nash equilibrium is unique. Alberto Bressan (Penn State) Optima and equilibria for traffic flow 17 / 34

4. There exists a minimum cost c 0 such that κ ( c ) = 0 for c ≤ c 0 . The map c �→ κ ( c ) is strictly increasing and continuous from [ c 0 , + ∞ [ to [0 , + ∞ [ . κ (c) κ c c 0 Alberto Bressan (Penn State) Optima and equilibria for traffic flow 18 / 34

An example of Nash equilibrium x t t S A queue of size δ 0 S forms instantly at time τ 0 The last driver of this queue departs at τ 2 , τ q τ t τ τ 0 τ τ 1 (t) 4 and arrives at exactly 0. 0 2 3 The queue is depleted at time τ 3 . A shock is formed. Q(t) The last driver departs at τ 1 . δ 0 t τ τ 4 τ 1 0 Alberto Bressan (Penn State) Optima and equilibria for traffic flow 19 / 34

Numerical results L = 1 , u ( ρ ) = ρ (2 − ρ ) , M = 1 , κ = 3 . 80758 , c = 2 . 7 x t t S S τ 0 = − 2 . 7 τ 2 = − 0 . 9074 τ q t 0 τ (t) τ 0 τ 2 τ 3 τ 1 4 flux τ 3 = 0 . 9698 τ 4 = 1 . 52303 Q’(t) τ 1 = 1 . 56525 t S = 2 . 0550 δ 0 = 1 . 79259 M total cost = 10 . 286 t 0 τ τ 2 τ 3 τ 0 τ 1 4 √ √ Q ( t ) = 1 . 7 + t + 2 . 7 + 1 / (4( t + 2 . 7 + 2 . 7)) √ √ � t + 2 . 7 + 2 . 7) 2 ) � Q ′ ( t ) = 1 − 1 / (4( / (2 t + 2 . 7) Alberto Bressan (Penn State) Optima and equilibria for traffic flow 20 / 34

A comparison J opt = 5 . 5714 Total cost of the Pareto optimal solution: Total cost of the Nash equilibrium solution: J Nash = 10 . 286 J Nash − J opt ≈ 4 . 715 Price of anarchy: Can one eliminate this inefficiency, yet allowing freedom of choice to each driver ? (goal of non-cooperative game theory: devise incentives) Alberto Bressan (Penn State) Optima and equilibria for traffic flow 21 / 34