Nash Flows Over Time Leon Sering and Martin Skutella COGA - - PowerPoint PPT Presentation

Nash Flows Over Time

Leon Sering and Martin Skutella

COGA

Multi-Source Multi-Sink

Motivation

• traffic of selfish driver

Study:

• user equilibrium?
• dynamic traffic assignment

Motivation

• traffic of selfish driver

Study:

chooses fastest routes

• user equilibrium?
• dynamic traffic assignment

continuous time

Motivation

• traffic of selfish driver

Study:

chooses fastest routes

• user equilibrium?

Simulation (e.g. MATSim)

• atomic players (cars)
• deterministic queuing model
• multi origin-destination
• iterative approach (approximation)
• large scale
• additional features (spill back, ...)
• dynamic traffic assignment

continuous time

Multi-Agent Transport Simulation by Nagel [2004 - today]

Motivation

• traffic of selfish driver

Study:

chooses fastest routes

• user equilibrium?

Simulation (e.g. MATSim)

• atomic players (cars)
• deterministic queuing model
• multi origin-destination
• iterative approach (approximation)
• large scale
• additional features (spill back, ...)
• non-atomic players (flow)
• deterministic queuing model
• one source, one sink
• Nash equilibrium (exact)
• no efficient algorithm known
• no additional features (yet)

Theory (Nash flows over time) vs.

• dynamic traffic assignment

continuous time

Multi-Agent Transport Simulation by Nagel [2004 - today]

Motivation

• traffic of selfish driver

Study:

chooses fastest routes

• user equilibrium?

Simulation (e.g. MATSim)

• atomic players (cars)
• deterministic queuing model
• multi origin-destination
• iterative approach (approximation)
• large scale
• additional features (spill back, ...)
• non-atomic players (flow)
• deterministic queuing model
• one source, one sink
• Nash equilibrium (exact)
• no efficient algorithm known
• no additional features (yet)

Theory (Nash flows over time) vs.

• dynamic traffic assignment

continuous time

multi multi Multi-Agent Transport Simulation by Nagel [2004 - today]

State of the Art

Koch & Skutella [2009]: Cominetti & Correa et. all. [2011, 2015, 2017]:

• construction of Nash flows via thin flows with resetting

(single source, single sink)

• existance of Nash flows over time.

constructive for single source, single sink non-constructive for multiple origin-destination-pairs non-constructive for arbitrary inflow rates

• long time behavior of Nash flows over time

State of the Art

Koch & Skutella [2009]: Cominetti & Correa et. all. [2011, 2015, 2017]:

• construction of Nash flows via thin flows with resetting

(single source, single sink)

• existance of Nash flows over time.

constructive for single source, single sink non-constructive for multiple origin-destination-pairs non-constructive for arbitrary inflow rates

• long time behavior of Nash flows over time

variational inequality

State of the Art

Koch & Skutella [2009]: Cominetti & Correa et. all. [2011, 2015, 2017]:

• construction of Nash flows via thin flows with resetting

(single source, single sink)

• existance of Nash flows over time.

constructive for single source, single sink non-constructive for multiple origin-destination-pairs non-constructive for arbitrary inflow rates

• long time behavior of Nash flows over time

Our Contribution

• construction for multiple sources, multiple sinks
• demands on sinks

variational inequality

Routing Game

t3 t2 t1 s1 s2

Routing Game

t3 t2 t1 s1 s2 2 1

Routing Game

. . . 1 2 3 4 1 flow R≥0 5

t3 t2 t1 s1 s2 2 1

Routing Game

. . . 1 2 3 4 flow R≥0 5

t3 t2 t1 s1 s2 2 1

Routing Game

. . . 1 2 3 4 flow R≥0 5

t3 t2 t1 s1 s2 2 1

1 2 1 3 1 6

Routing Game

. . . 1 2 3 4 flow R≥0 5

t3 t2 t1 s1 s2 2 1

1 2 1 3 1 6

Routing Game

. . . 1 2 3 4 flow R≥0 5

t3 t2 t1 s1 s2 2 1

1 2 1 3 1 6

Routing Game

. . . 1 2 3 4 flow R≥0 5

t3 t2 t1 s1 s2 2 1

1 2 1 3 1 6

traffic jam

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.

Flows Over Time

• Flow needs time to travel through an edge.
• constant speed
• edge-length
• capacity rate νe
• source inflow rate ri

transit time τe flow time

}

Flows Over Time

• Flow needs time to travel through an edge.
• constant speed
• edge-length
• capacity rate νe
• source inflow rate ri

transit time τe flow time Flow over time (f +

e , f − e )e∈E: • inflow rate f + e : R≥0 → R≥0

• outflow rate f −

e

: R≥0 → R≥0

• conservation at all times

time rate

}

(for v ∈ S ∪ T)

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze

Deterministic Queuing Model

νe = 0.5 edge e queue ze f −

e (θ) =

• νe

min{f +

e (θ − τe), νe}

if queue is positive if queue is empty

Inflow Distribution

. . . 1 2 3 4 flow R≥0 5

s1 s2 s3

Inflow Distribution

. . . 1 2 3 4 flow R≥0 5

s1 s2 s3 f1 f2 f3 fi : R≥0 → [0, 1]

Inflow Distribution

. . . 1 2 3 4 flow R≥0 5

s1 s2 s3 f1 f2 f3 fi : R≥0 → [0, 1]

Inflow Distribution

. . . 1 2 3 4 flow R≥0 5

s1 s2 s3 f1 f2 f3 fi : R≥0 → [0, 1] +

k

• i=1

fi ≡ 1

Nash Flows over Time (with one sink)

Nash Flows over Time (with one sink)

no particle can be faster by changing its route

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

no particle can be faster by changing its route

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

no particle can be faster by changing its route

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

no particle can be faster by changing its route

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

no particle can be faster by changing its route

waiting time in front of si earliest (possible) arrival time at si

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

• Flow only travels along current shortest paths.

no particle can be faster by changing its route

waiting time in front of si earliest (possible) arrival time at si

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

• Flow only travels along current shortest paths.

f +

uv(ℓu(φ)) > 0

⇒ uv active for φ

no particle can be faster by changing its route

waiting time in front of si earliest (possible) arrival time at si

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

• Flow only travels along current shortest paths.

f +

uv(ℓu(φ)) > 0

⇒ uv active for φ

no particle can be faster by changing its route

waiting time in front of si earliest (possible) arrival time at si earliest arrival time at u part of a current shortest path

Nash Flows over Time (with one sink)

flow over time (f +, f −) & inflow distribution (fi):

• Entering a source directly is always a fastest option.

Fi(φ)/ri = ℓsi(φ)

• Flow only travels along current shortest paths.

⇒ flow always takes fastest routes to the sink f +

uv(ℓu(φ)) > 0

⇒ uv active for φ

no particle can be faster by changing its route

waiting time in front of si earliest (possible) arrival time at si earliest arrival time at u part of a current shortest path

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)
• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: Given: [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)
• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: Given: [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

l′

si = x′ i /ri

l′

si ≤ min e=usi ρe(l′ u, x′ e)

l′

v = min e=uv ρe(l′ u, x′ e)

for v ∈ S l′

v = ρe(l′ u, x′ e)

if x′

e > 0

ρe(l′

u, x′ e) :=

• {x′

e/νe}

if e ∈ E ∗ max{l′

u, x′ e/νe}

if e ∈ E ′\E ∗. where

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

2 2 1 3 2

s2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 1 Given: φ [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

2 2 1 3 2

s2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 Given: φ [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

2 2 1 3 2 1

3 4

s2

1 8 3 4 3 4 1 4 1 4 3 4 1 4 3 4 1 2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 Given: φ [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

Thm: [Cominetti et. all. ’11] There always exists a thin flow with resetting. 2 2 1 3 2 1

3 4

s2

1 8 3 4 3 4 1 4 1 4 3 4 1 4 3 4 1 2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 Given: φ [Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

Thm: [Cominetti et. all. ’11] There always exists a thin flow with resetting. 2 2 1 3 2 1

3 4

s2

1 8 3 4 3 4 1 4 1 4 3 4 1 4 3 4 1 2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 Given: φ

MIP

[Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Thin Flows with Resetting

• current shortest path network G ′ = (V , E ′)

s1 t1

Thm: [Cominetti et. all. ’11] There always exists a thin flow with resetting. Theorem: [Koch & Skutella ’09] The derivatives of a Nash flow are almost everywhere a thin flow with resetting. 2 2 1 3 2 1

3 4

s2

1 8 3 4 3 4 1 4 1 4 3 4 1 4 3 4 1 2

• edges with queue (resetting edges) E ∗ ⊆ E ′.

Find: 2 1 Given: φ

MIP

[Koch & Skutella ’09]

• Nash flow, and a fixed particle φ.
• earliest arrival time derivatives ℓ′

v

• static flow x′

e and x′ i

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3 1 3 1 2 1 6

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ

1 3 1 2 1 6

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ

1 3 1 2 1 6

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

1 1 1 1 2 3 6

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ

1 3 1 2 1 6

1

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

1 1 1 1 2 3 6

φ ℓv (φ) 1 2 3 4 5 10 3 9 5 1 1 15 θ f + e (θ) θ 3 2 3

1 3 1 2 1 6

1

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

φ ℓv (φ) 1 2 3 4 5 10 3 9 5 1 1 15 θ f + e (θ) θ

1

1 3

1 2 1 2

3 2 3

1 3 2 3 1 3 1 2 1 6

1

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ

1

1 3

1 2 1 2

2 5 6 11 6 4

1 3 2 3 1 3 1 2 1 6

1

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ 2 5 6 11

3 4

1 1.5 1 1.5

6 4

3 4 1 3 1 2 1 6 1 4 1 4

1

Constructing Nash Flows from Thin Flows

s1 t1

1 1

1 2

4 1 1 6

1 3

1 2 3 4 5

Thin Flow:

φ ℓv (φ) 1 2 3 4 5 10 1 15 θ f + e (θ) θ

3 4

1 1.5 1 1.5

3 4 1 3 1 2 1 6 1 4 1 4

1

Demands

s1 s2

1 2 1 3

t1 t2 t3

1 6

Demands

s1 s2

1 2 1 3

t1 t2 t3

super sink t 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3 4 5 3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3 4 5 3 1 2

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

1 2 1 3

super sink t 1 2

1 6

1 6

thin flow

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

super sink t 1 2 1 6

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

1 2 1 3

super sink t 1 2

1 6

1 6

thin flow

Demands

s1 s2

1 2 1 3 1 6

ε· 1

3

ε· ε· t1 t2 t3

1 2 1 3

super sink t 1 2

1 6

1 6

Thank you!

thin flow