Optima and Equilibria for Traffic Flow on a Network Alberto Bressan Department of Mathematics, Penn State University bressan@math.psu.edu Alberto Bressan (Penn State) Optima and equilibria for traffic flow 1 / 29
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 2 / 29
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 3 / 29
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 3 / 29
An optimization problem for a conservation law model of traffic flow Problem: choose the departure rate ¯ u ( t ) so that the solution of the conservation law ρ t + [ ρ v ( ρ )] x = 0 x ∈ [0 , L ] ρ ( t , 0) v ( ρ ( t , 0)) = ¯ u ( t ) minimizes the sum of the costs to all drivers. . u ( t , x ) = ρ ( t , x ) v ( ρ ( t , x )) = flux of cars � � . minimize: J (¯ u ) = ϕ ( t ) · u ( t , 0) dt + ψ ( t ) u ( t , L ) dt Choose the optimal departure rate ¯ u ( t ), subject to the constraint � ¯ u ( t ) ≥ 0 , ¯ u ( t ) dt = κ = [total number of drivers] Alberto Bressan (Penn State) Optima and equilibria for traffic flow 4 / 29
Existence of a globally optimal solution (A1) The flux function ρ �→ f ( ρ ) = ρ v ( ρ ) is strictly concave down. f ′′ < 0 . f (0) = f ( ρ max ) = 0 , ϕ ′ < 0, ψ, ψ ′ ≥ 0, (A2) The cost functions ϕ, ψ satisfy � � t →−∞ ϕ ( t ) = lim + ∞ , lim ϕ ( t ) + ψ ( t ) = + ∞ t → + ∞ Theorem (A.B. and K. Han, SIAM J. Math. Anal. , 2012). Let (A1)-(A2) hold. Then, for any κ > 0, there exists a unique admissible initial data ¯ u minimizing the total cost J ( · ). Alberto Bressan (Penn State) Optima and equilibria for traffic flow 5 / 29
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 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 6 / 29
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 7 / 29
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, introduce the integrated variable � t . U ( t , x ) = ρ ( s , x ) v ( ρ ( s , x )) ds = [number of drivers −∞ that have crossed the point x along the road within time t ] This solves a Hamilton-Jacobi equation U x + F ( U t ) = 0 U ( t , 0) = Q ( t ) Alberto Bressan (Penn State) Optima and equilibria for traffic flow 8 / 29
Note: a queue can form at the entrance of the highway Q(t) κ β U(t,L) q a t τ ( ) τ ( ) β β Q ( t ) = number of drivers who have started their journey before time t (possibly joining the queue) L = length of the road U ( t , L ) = number of drivers who have reached destination before time t Alberto Bressan (Penn State) Optima and equilibria for traffic flow 9 / 29
Characterization of a Nash equilibrium Q(t) κ β U(t,L) q a t τ ( ) τ ( ) β β β ∈ [0 , κ ] = Lagrangian variable labeling one particular driver τ q ( β ) = time when driver β joins the queue τ a ( β ) = time when driver β arrives at destination ϕ ( τ q ( β )) + ψ ( τ a ( β )) = c Nash equilibrium = ⇒ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 10 / 29
Existence and Uniqueness of Nash equilibrium Theorem (A.B. - K. Han). Let the flux f and cost functions ϕ, ψ satisfy the assumptions (A1)-(A2). Then, for every κ > 0, the Hamilton-Jacobi equation U x + F ( U t ) = 0 admits a unique Nash equilibrium solution with total mass κ Alberto Bressan (Penn State) Optima and equilibria for traffic flow 11 / 29
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 ( τ q ( β )) + ψ ( τ a ( β )) ≤ c for a.e. β ∈ [0 , Q (+ ∞ )] . . � � Q ∗ ( t ) Q ∈ Q − 2. Claim: = sup Q ( t ) ; c is the initial data for a Nash equilibrium with common cost c . * Q (t) Q(t) t Alberto Bressan (Penn State) Optima and equilibria for traffic flow 12 / 29
3. 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 13 / 29
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 14 / 29
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 15 / 29
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 15 / 29
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 15 / 29
Optimal pricing Suppose a fee b ( t ) is collected at a toll booth at the entrance of the highway, depending on the departure time. New departure cost: ϕ ( t ) = ϕ ( t ) + b ( t ) ˜ Is there an optimal choice of b ( t ) ? Alberto Bressan (Penn State) Optima and equilibria for traffic flow 16 / 29
3 2.5 2 Cost 1.5 cost = p ( τ d ) 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 p ( t ) = cost to a driver starting at time t , in a globally optimal solution Choose additional fee: b ( t ) = p max − p ( t )+ constant = ⇒ Nash equilibrium coincides with the globally optimal solution ~ b ψ ϕ = ϕ + ϕ t 0 Alberto Bressan (Penn State) Optima and equilibria for traffic flow 17 / 29
Traffic Flow on a Network Nodes: A 1 , . . . , A m arcs: γ ij n groups of drivers with different origins and destinations, and different costs γ ij A i A j Alberto Bressan (Penn State) Optima and equilibria for traffic flow 18 / 29
Traffic Flow on a Network k -drivers depart from A d ( k ) and arrive to A a ( k ) Departure cost: ϕ k ( t ) arrival cost: ψ k ( t ) � � ϕ ′ < 0, ψ ′ (A2) k > 0 , lim | t |→∞ ϕ ( t ) + ψ ( t ) = ∞ A d(1) A a(1) Alberto Bressan (Penn State) Optima and equilibria for traffic flow 19 / 29
Traffic Flow on a Network A a(2) A d(2) drivers can use different paths Γ 1 , Γ 2 , . . . to reach destination Does there exist a globally optimal solution, and a Nash equilibrium solution for traffic flow on a network ? Alberto Bressan (Penn State) Optima and equilibria for traffic flow 20 / 29
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