Dynamic Atomic Congestion Games with Seasonal Flows Marc Schr oder - - PowerPoint PPT Presentation

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Dynamic Atomic Congestion Games with Seasonal Flows Marc Schr oder - - PowerPoint PPT Presentation

Jan 17, 2023 253 likes •704 views

Model Parallel networks Open problems Ongoing research Dynamic Atomic Congestion Games with Seasonal Flows Marc Schr oder M. Scarsini, T. Tomala Maastricht University Department of Quantitative Economics Marc Schr oder Dynamic

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Model Parallel networks Open problems Ongoing research

Dynamic Atomic Congestion Games with Seasonal Flows

Marc Schr¨

  • der
  • M. Scarsini, T. Tomala

Maastricht University Department of Quantitative Economics

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 1 / 29

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Model Parallel networks Open problems Ongoing research

Dynamic congestion games

Main questions: How do the dynamics evolve over time? Inflow is rarely constant, although often (nearly) periodic. How does this affect the steady state?

Marc Schr¨

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Dynamic Atomic Congestion Games 2 / 29

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Model Parallel networks Open problems Ongoing research

Model

A directed network N = (V , E, (τe)e∈E, (γe)e∈E) with a single source and sink, where

  • τe ∈ N is the travel time,
  • γe ∈ N is the capacity.

Time is discrete and players are atomic. Inflow is deterministic, but need not be constant.

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Dynamic Atomic Congestion Games 3 / 29

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Model Parallel networks Open problems Ongoing research

Model

At each stage t, a generation Gt of δt players departs from the source. Players are ordered by priority ⊳. At time t, player [it] observes the choices of players [js] ⊳ [it] and chooses an edge e = (s, v) ∈ E. Player [it] arrives at time t + τe at the exit of e.

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Dynamic Atomic Congestion Games 4 / 29

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Model Parallel networks Open problems Ongoing research

Model

At this exit a queue might have formed by

1 players who entered e before [it], 2 players who entered e at the same time as [it], but have higher

priority.

Recall at most γe players can exit e simultaneously. When exiting edge e = (s, v), player [it] chooses an outgoing edge e′ = (v, v′). This is repeated until player [it] arrives at the destination. This defines a game with perfect information Γ(N , K, δ).

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Dynamic Atomic Congestion Games 5 / 29

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Model Parallel networks Open problems Ongoing research

Solution concepts

  • Equilibrium. Each player minimizes her own total cost

(travel+waiting), given the queues in the system.

  • Optimum. A social planner controls all players and seeks to

minimize the long-run total latency, averaged over a period.

Marc Schr¨

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Dynamic Atomic Congestion Games 6 / 29

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SLIDE 7

Model Parallel networks Open problems Ongoing research

Overview

1 Model 2 Parallel networks

Uniform departures Periodic departures

3 Open problems

General networks Presence of initial queues

4 Ongoing research

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  • der

Dynamic Atomic Congestion Games 7 / 29

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SLIDE 8

Model Parallel networks Open problems Ongoing research

Uniform inflow

In a parallel network each route is made of a single edge. The capacity of the network is γ =

e γe.

s t

e1 e2 e3 We assume that δt = γ for all t ∈ N.

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Dynamic Atomic Congestion Games 8 / 29

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Model Parallel networks Open problems Ongoing research Uniform departures

Example

Inflow (2,2,2,. . . ). What happens in equilibrium?

s t

τ1 = 1 τ2 = 2

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Dynamic Atomic Congestion Games 9 / 29

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Model Parallel networks Open problems Ongoing research Uniform departures

Equilibrium

t=1

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Dynamic Atomic Congestion Games 10 / 29

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SLIDE 11

Model Parallel networks Open problems Ongoing research Uniform departures

Equilibrium

2 1 t=1

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Dynamic Atomic Congestion Games 10 / 29

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SLIDE 12

Model Parallel networks Open problems Ongoing research Uniform departures

Equilibrium

2 1 t=1 1 2 * t=2

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Dynamic Atomic Congestion Games 10 / 29

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SLIDE 13

Model Parallel networks Open problems Ongoing research Uniform departures

Equilibrium

2 1 t=1 1 2 * t=2 1 2 * * t=3

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Dynamic Atomic Congestion Games 10 / 29

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SLIDE 14

Model Parallel networks Open problems Ongoing research Uniform departures

Equilibrium

2 1 t=1 1 2 * t=2 1 2 * * t=3 . . . in the long-run total costs are 4

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Dynamic Atomic Congestion Games 10 / 29

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SLIDE 15

Model Parallel networks Open problems Ongoing research Uniform departures

Optimum

What happens in the (social) optimum?

s t

τ1 = 1 τ2 = 2

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Dynamic Atomic Congestion Games 11 / 29

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Model Parallel networks Open problems Ongoing research Uniform departures

Optimum

t=1

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Dynamic Atomic Congestion Games 12 / 29

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SLIDE 17

Model Parallel networks Open problems Ongoing research Uniform departures

Optimum

2 1 t=1

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Dynamic Atomic Congestion Games 12 / 29

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SLIDE 18

Model Parallel networks Open problems Ongoing research Uniform departures

Optimum

2 1 t=1 2 1 * t=2

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Dynamic Atomic Congestion Games 12 / 29

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SLIDE 19

Model Parallel networks Open problems Ongoing research Uniform departures

Optimum

2 1 t=1 2 1 * t=2 . . . total costs are 3

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Dynamic Atomic Congestion Games 12 / 29

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SLIDE 20

Model Parallel networks Open problems Ongoing research Uniform departures

Price of anarchy

Lemma Let N be a parallel network. Then WEq(N , γ) = γ · max

e∈E τe,

Opt(N , γ) =

  • e∈E

γe · τe. PoA(N , γ) = WEq(N , γ) Opt(N , γ) ≤ maxe τe mine τe .

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  • der

Dynamic Atomic Congestion Games 13 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Periodic departures

Inflow is a K-periodic vector: δ = (δ1, . . . , δK) ∈ NK such that K

k=1 δk = K · γ. We denote NK(γ) the set of such

sequences. When δ is not-uniform, queues have to be created when there is a surge of players.

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Dynamic Atomic Congestion Games 14 / 29

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SLIDE 22

Model Parallel networks Open problems Ongoing research Periodic departures

Example

Inflow (3,1,2). What happens to the equilibrium?

s t

τ1 = 1 τ2 = 2

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  • der

Dynamic Atomic Congestion Games 15 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1

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Dynamic Atomic Congestion Games 16 / 29

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SLIDE 24

Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1 1 * * t=2

Marc Schr¨

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Dynamic Atomic Congestion Games 16 / 29

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SLIDE 25

Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1 1 * * t=2 1 2 * t=3

Marc Schr¨

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Dynamic Atomic Congestion Games 16 / 29

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SLIDE 26

Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1 1 * * t=2 1 2 * t=3 3 1 2 * * t=4 * 1 * * t=5

Marc Schr¨

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Dynamic Atomic Congestion Games 16 / 29

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SLIDE 27

Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1 1 * * t=2 1 2 * t=3 3 1 2 * * t=4 * 1 * * t=5 1 2 * * t=3

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Dynamic Atomic Congestion Games 16 / 29

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SLIDE 28

Model Parallel networks Open problems Ongoing research Periodic departures

Equilibrium

2 3 1 t=1 1 * * t=2 1 2 * t=3 3 1 2 * * t=4 * 1 * * t=5 1 2 * * t=3 . . . In the long-run total costs are 3 · 4 + 1 = 13.

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Dynamic Atomic Congestion Games 16 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Measure of periodicity

1 2 3 1 2 3 1 2 3

Figure: 1 operation needed to transform (3, 1, 2) into (2, 2, 2).

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Dynamic Atomic Congestion Games 17 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Measure of periodicity

1 2 3 1 2 3 1 2 3

Figure: 1 operation needed to transform (3, 1, 2) into (2, 2, 2).

1 2 3 1 2 3 1 2 3 1 2 3

Figure: 2 operations needed to transform (3, 2, 1) into (2, 2, 2).

Marc Schr¨

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Dynamic Atomic Congestion Games 17 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Measure of periodicity

Definition For any two elements δ, δ′ ∈ NK(γ), we say that δ′ is obtained from δ by an elementary operation if there exist a time i, with such that δi > γ, δ′

i = δi − 1, δ′ i+1 = δi + 1.

Let D(δ) be the minimal number of elementary operations one has to perform to transform δ into γK.

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Dynamic Atomic Congestion Games 18 / 29

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Model Parallel networks Open problems Ongoing research Periodic departures

Theorem Let N be a parallel network and δ ∈ NK(γ). Then WEq(N , K, δ) = K · γ · max

e∈E τe + D(δ),

Opt(N , K, δ) = K ·

  • e∈E

γe · τe + D(δ).

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 19 / 29

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SLIDE 33

Model Parallel networks Open problems Ongoing research

Overview

1 Model 2 Parallel networks

Uniform departures Periodic departures

3 Open problems

General networks Presence of initial queues

4 Ongoing research

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 20 / 29

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Model Parallel networks Open problems Ongoing research General networks

Parallel network

s t

τ1 = 1 τ2 = 2 τ3 = 3 τ4 = 3 Equilibrium. If δ = 3, then WEq(N , 1, δ) = 9. If δ = (6, 0), then WEq(N , 2, δ) = 16.

Marc Schr¨

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Dynamic Atomic Congestion Games 21 / 29

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Model Parallel networks Open problems Ongoing research General networks

Chain-of-parallel network

s v t

τ1 = 1 τ2 = 2 τ3 = 3 τ4 = 3 τ5 = 1 γ5 = 3

  • Equilibrium. If δ = (6, 0), then

1 WEq∗(N , 2, δ) = 22 (earliest-arrival property). 2 WEq∗∗(N , 2, δ) = 25 (no overtaking). 3 WEq(N , 2, δ) = 2 · 12 + 3 = 27 (allows overtaking if arrival

in same period).

Marc Schr¨

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Dynamic Atomic Congestion Games 22 / 29

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Model Parallel networks Open problems Ongoing research General networks

Braess’s network s v t w

τ2 = 1 τ1 = 0 τ4 = 1 τ3 = 0 τ5 = 0

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Dynamic Atomic Congestion Games 23 / 29

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Model Parallel networks Open problems Ongoing research General networks

Braess’s network

Proposition For every even integer n, there exists a network N in which |V | = n and such that PoS(N , γ) = 1, PoA(N , γ) = n − 1.

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Dynamic Atomic Congestion Games 24 / 29

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Model Parallel networks Open problems Ongoing research Presence of initial queues

Series-parallel network s v t

τ1 = 1 τ2 = 0 γ2 = 2 τ3 = 0 τ4 = 1

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 25 / 29

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Model Parallel networks Open problems Ongoing research Presence of initial queues

Series-parallel network s v t

τ1 = 1 τ2 = 0 γ2 = 2 τ3 = 0 τ4 = 1 Equilibrium. Player [11] chooses e2e3, [21] chooses e2e4, [31] chooses e2e3. For t ≥ 2, [1t] chooses e1, [2t] chooses e2e3, [3t] chooses e2e4. Total costs are: 1+1+2=4

Marc Schr¨

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Dynamic Atomic Congestion Games 25 / 29

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Model Parallel networks Open problems Ongoing research Presence of initial queues

Series-parallel network s v t

τ1 = 1 τ2 = 0 γ2 = 2 τ3 = 0 τ4 = 1 Suppose e3 contains a queue, then total costs decrease to 3 Another view on Braess’s paradox: initial queues can improve total costs.

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 26 / 29

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Model Parallel networks Open problems Ongoing research

Overview

1 Model 2 Parallel networks

Uniform departures Periodic departures

3 Open problems

General networks Presence of initial queues

4 Ongoing research

Marc Schr¨

  • der

Dynamic Atomic Congestion Games 27 / 29

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Model Parallel networks Open problems Ongoing research

Ongoing research

General networks Effect of initial queues Discrete vs continuous time

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  • der

Dynamic Atomic Congestion Games 28 / 29

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Model Parallel networks Open problems Ongoing research

Apologies for congesting your brain.

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  • der

Dynamic Atomic Congestion Games 29 / 29