Wardrop vs Nesterovtraffic equilibrium concept.

Georg StillUniversity of Twente

joint work with Walter Kern

(9th International Conference on Operations Research,Havana, February, 2010)

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Dutch Highway System

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Traffic Network: V: nodes; E: edges (roads)• (sw , tw) : origin-destination nodes, w ∈ W• dw : traffic demands (cars/hour)• xe, fp : edge-, path-flow (cars/hour)• ce(x) ∈ C : travel time (“costs”) on edge e ∈ E

e xe1

1f1

f2s

t

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• Pw : set of (sw , tw)-paths p , P = ∪w Pwcp(x) =

∑e∈p ce(x), p ∈ P: path costs

feasible flow: (x, f ) ∈ RE × RP satisfying

Λf = d | Λ path-demand incidence-∆f − x = 0 | ∆ path-edge incidence matrixf ≥ 0

Notation: given demand dI (x, f ) ∈ Fd : feasible setI x ∈ Xd : projection of Fd onto x-space.

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Wardrop Equilibrium (52): (x, f ) ∈ Fd is WE if

∀w ∈ W , p, q ∈ Pw

fp > 0⇒ cp(x) = cq(x) if fq > 0cp(x) ≤ cq(x) if fq = 0

Meaning: For each used path p ∈ Pw betweenO-D pairs (sw , tw ) the path-costs must be the same.

“traffic user equilibrium”, (Nash-equilibrium)

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Relation: Wardrop-equilibrium↔ optimization

Assume ce(x) = ce(xe), increasing. Consider the program

P : minx,f

N(x) :=∑

e∈E

∫ xe

0ce(t)dt s.t. (x, f ) ∈ Fd

KKT conditions: (x, f ) ∈ Fd is sol. of P iff

c(x) = λ

0 = ΛT γ −∆T λ + µ

f T µ = 0 f , µ ≥ 0

or equivalentely: for any path p ∈ Pw

γw = [∆T c(x)]p − µp

{= cp(x) if fp > 0≤ cp(x) if fp = 0

These are the W-equilibrium conditions.

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Th.1 The following are equivalent for x ∈ Xd

(i) x is an W-equilibrium flow.(ii) c(x)T (x − x) ≥ 0 ∀x ∈ Xd .(iii) x solves min{c(x)T x | x ∈ Xd}.(iv) [in case ce(x) = ce(xe)] x minimizes N(x) on Xd

I More generally, (iv) holds if there exists N(x) such that

∇N(x) = c(x)

By Poincare’s Lemma this holds (on convex sets) if:

c(x) ∈ C1 and∂ci

∂xj=

∂cj

∂xi

Existence of a Wardrop equilibrium x ?

I case c(x) = ∇N(x): By the Weierstrass TheoremI general case: By a Fixed Point Theorem

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Existence Theorem: (Stampacchia 1966) Let c : X → Rm

be continuous on the convex, compact set X ⊂ Rm. Thenthere exists a vector x ∈ X such that

c(x)T (x − x) ≥ 0 ∀x ∈ X

Stampacchia’s Lemma←→ Brouwer’s Fixed Point Theorem

Brouwer’s Fixed Point Theorem:Let f : X → X be continuous , X ⊂ Rm convex, compact.Then f has a fixed point x ∈ X : f (x) = x

Pf. “→”: Choose c(y) := −[f (y)− y].

Then, there exists x ∈ X such that

c(x)T (x − x) = −[f (x)− x]T (x − x) ≥ 0 ∀x ∈ X

Choose x = f (x) ∈ X −→ −‖f (x)− x‖2 ≥ 0

−→ f (x) = x

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Objectives: user (Nash-eq.) ↔ government

minimize: N(x) ↔ S(x) :=∑

e ce(xe)xe

Braes example:

edge costs: 1, 1, c,xflow : x

s-t demand: 1

1

x 1

x

c

s

t

u v

c ≥ 1: Nash flow x , S(x) = 3/2

p1 = s−u−t , f1 = 1/2, c1 = 3/2p2 = s−v−t , f2 = 1/2, c2 = 3/2

c = 0: Nash flow x , S(x) = 2

p = s−u−v−t , f = 1, c1 = 2

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Nesterov’s new model (2000):

Based on the “queering model”

ce(xe) =

{te for 0 ≤ xe < ueM for xe = ue

I ue: max capacity of e ∈ EI te: costs (travel times) without

congestion (e ∈ E).modified, generalized concept (with te(x) ∈ C)

ce(x) =

{te(x) for 0 ≤ xe ≤ ueM for xe > ue

This function is lower semicontinuous (lsc).

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Def. Nesterov E. (NE): Given costs t : RE+ → RE

+ in C, acapacity vector u, then (x, t) ∈ RE+E

+ , x ∈ Xd is a NE if1. x ≤ u , t ≥ t(x) and2. x is a WE relative to the costs t .

Related capacity constr. program: Find x solving

Pt(x) : minx,f

t(x)T x s.t.

Λf = d∆f − x = 0

x ≤ u |νf ≥ 0

Changes in KKT-condition compared with WE:

t(x) = λ − ν and (u − x)T ν = 0

or eqivalentely for any path p ∈ Pw

γw = [∆T (t(x) + ν)]p − µp

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So: consider costs t = t(x) + ν.

Th.2 (x, t) is a NE if and only if x (with L-multiplier ν) is asolution of Pt(x) and t = t(x) + ν.

I The existence of a NE follows also by Stampacchia’sLemma.

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Wardrop’s model for non-continuous costs

Def. lower-, upper limit, c−e , c+

e :c−

e (x) := lim infxn→x

ce(xn)

c+e (x) := lim sup

xn→xce(xn)

Similarly: c−p (x), c+

p (x) for pathcosts.

Model conditions: If fp > 0, p ∈ Pw , then:

• cp(x) ≤ c+q (x) ∀q ∈ Pw

should be a necessary condition and

• cp(x) ≤ lim infε↓0

cq(x + ε1q − ε1p)

∀q ∈ Pw a sufficient condition for “stability”

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This leads to the assumpions:ce(x) are lower semicontinuous (lsc), i.e.

ce(x) ≤ c−e (x), ∀x

and satisfy the regularity condition: ∀q, p ∈ Pw ,e ∈ q, e /∈ p

(?) c+e (x) ≤ lim inf

ε↓0ce(x + ε1q − ε1p)

Def. (Wardrop equilibrium:) Suppose the ce’s are lscand satisfy the link regularity (?). We then callx =

∑p fp1p ∈ Xd a Wardrop equilibrium if:

fp > 0⇒ cp(x) ≤ c+q (x) ∀q ∈ Pw

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Th.3 Let the link costs ce be lsc and satisfy the linkregularity condition. Assume x ∈ Xd and c ∈ [c(x), c+(x)].Then (iii)⇔ (ii)⇒ (i) holds for

(i) x is a Wardrop equilibrium.(ii) cT (x − x) ≥ 0 ∀x ∈ Xd .(iii) x solves min{cT x | x ∈ Xd}

Def. A WE satisfying the sufficient condition (ii) (or (iii)) ofthe theorem is called a strong WE.

Th.4 For lsc regular link costs strong Wardrop equilibriaexist.

Pf. Use cke (x) ↑ ce(x) with continuous ck

e (x) and theexistence of WE wrt. ck

e (x).

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NE as special case of WE: NE is based on costs,

(?) ce(x) =

{te(x) for 0 ≤ xe ≤ ueM for xe > ue

This function is lsc and regular. By comparing theKKT-conditions for a strong WE wrt. the costs (?):

c = λ and 0 = ΛT γ −∆T λ + µ

c ∈ [c(x), c+(x)] , ce

{= te(x) for xe < ue∈ [te(x), M] if xe = ue

with the KKT-conditions for the NE-program Pt(x) wedirectly find:

Cor.1 (x, t) is a Nesterov equilibrium (wrt. te(x) and u)if and only if x is a strong WE (wrt. ce(x) in (?)).

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Parametric Aspects:

How do the equilibrated travel times γw(·) depend onchanges in the demand d and/or costs ce(x)?

Dependence on c(x): No monotonicity

I c(x)↗ ; γw ↗ (see Braes)

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Dependence on d: Let N(x) be convex with∇N(x) = c(x),i.e., c(x) satisfies the “monotonicity” condition

(?) [c(x ′)− c(x)]T (x ′ − x) ≥ 0 ∀x ′, x

Consider the W-equilibrium problem: d parameter

P(d) : minx,f N(x) s.t. (x, f ) ∈ Fd

Λf = d | γ

Parametric Opt.: For solutions x(d) with L-Mult. γ(d):I the value function v(d) of P(d) is convex (in d).I ∂v(d) = {γ(d)} (maximal) monotone:

[γ(d)− γ(d)]T (d − d) ≥ 0 ∀d, d

Even if the W-equilibrium cannot be modelled as anoptimization problem: Monotonicity of γ(d) still holdsunder (?).

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interpretation: of (Hall’s result)

[γ(d)− γ(d)]T (d − d) ≥ 0 ∀d, d

Let d d then

I γw(d) ≥ γw(d) for at least one w ∈ W

I even if d > d: possibly

γw ′(d) > γw ′(d) for one w ′ ∈ W

γw(d) < γw(d) for the other w 6= w ′

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Pf. of Hall’s result: solutions x ′, x corresp. to d ′, d

[c(x ′)− c(x)]T (x ′ − x) ≥ 0 •[c(x ′)− c(x)]T ∆(f ′ − f ) ≥ 0

[µ′ − µ]T (f ′ − f )︸ ︷︷ ︸≤0 by 3.

+ [ΛT γ′ − ΛT γ]T (f ′ − f ) ≥ 0

[γ′ − γ]T Λ(f ′ − f ) ≥ 0[γ′ − γ]T (d ′ − d) ≥ 0 •

Use:

1. x = ∆f , Λf = d 2. ∆T c(x) = µ + ΛT γ

3. E.g.: f ′p > 0, fp = 0⇒ µ′

p = 0, µp ≥ 0⇒ ( µ′

p︸︷︷︸=0

−µp)(f ′p − fp︸︷︷︸

=0

) ≤ 0

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Monotonicity of γ(d) in the general W-concept?

For strong W-equilibria: Under the “monotonicitycondition”,

[c(x ′)− c(x)]T (x ′ − x) ≥ 0 ∀x ′, x

monotonicity of the equilibrated travel-times γ(d) stillholds:

[γ(d)− γ(d)]T (d − d) ≥ 0 ∀d, d

However: For (weak) W-equilibria this monotonicity mayfail.

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Example network with 4 O-D pairs; identify edge e:

44

b1 b21 1 2 2 3 e 3

e2e1 e3

e e

The link cost for e, ei bj are zero except for

ce(t) :=

{0 0 ≤ t ≤ 2M else cei (t) = t, cbj (t) ≡ 1.

demands: d1 = d2 = d3 = 1, d4 = ε ≥ 0.

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A weak W-E x and the unique strong equilibrium x:

x : 2 1 1 1− ε ε ε

x : 2 23 + 1

3ε 23 + 1

3ε 13 − 1

3ε 13 + 2

3ε 13 + 2

3ε

Corresponding γ, γ

γ : 1 1 1− ε 3− ε ←− •γ : 1 1 1

3 − 13ε 5

3 + 13ε

with objective

N(x) =32

+ ε +12ε2 , N(x) =

76

+53ε +

16ε2 .

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Many other interesting aspects:

I Generalization to elastic demand is easy.I Tolling policy to ’improve’ the traffic flow.I Computation of traffic equilibria in large networksI Dynamic traffic equilibrium models

(demand changes with time)

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