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

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 Atomic Congestion Games 1 / 29

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¨ oder Dynamic Atomic Congestion Games 2 / 29

Model Parallel networks Open problems Ongoing research Model A directed network N = ( V , E , ( τ e ) e ∈ E , ( γ e ) e ∈ E ) with a single source and sink, where - τ e ∈ N is the travel time, - γ e ∈ N is the capacity. Time is discrete and players are atomic. Inflow is deterministic, but need not be constant. Marc Schr¨ oder Dynamic Atomic Congestion Games 3 / 29

Model Parallel networks Open problems Ongoing research Model At each stage t , a generation G t of δ t 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 + τ e at the exit of e . Marc Schr¨ oder Dynamic Atomic Congestion Games 4 / 29

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 γ e 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 ). Marc Schr¨ oder Dynamic Atomic Congestion Games 5 / 29

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¨ oder Dynamic Atomic Congestion Games 6 / 29

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¨ oder Dynamic Atomic Congestion Games 7 / 29

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 γ e . e 1 e 2 s t e 3 We assume that δ t = γ for all t ∈ N . Marc Schr¨ oder Dynamic Atomic Congestion Games 8 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Example Inflow (2,2,2,. . . ). What happens in equilibrium? τ 1 = 1 s t τ 2 = 2 Marc Schr¨ oder Dynamic Atomic Congestion Games 9 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Equilibrium t=1 Marc Schr¨ oder Dynamic Atomic Congestion Games 10 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Equilibrium 2 1 t=1 Marc Schr¨ oder Dynamic Atomic Congestion Games 10 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Equilibrium 2 1 2 1 * t=1 t=2 Marc Schr¨ oder Dynamic Atomic Congestion Games 10 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Equilibrium 2 1 2 1 2 1 * * * t=1 t=2 t=3 Marc Schr¨ oder Dynamic Atomic Congestion Games 10 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Equilibrium 2 1 2 1 2 1 * * * . . . in the long-run total costs are 4 t=1 t=2 t=3 Marc Schr¨ oder Dynamic Atomic Congestion Games 10 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Optimum What happens in the (social) optimum? τ 1 = 1 s t τ 2 = 2 Marc Schr¨ oder Dynamic Atomic Congestion Games 11 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Optimum t=1 Marc Schr¨ oder Dynamic Atomic Congestion Games 12 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Optimum 2 1 t=1 Marc Schr¨ oder Dynamic Atomic Congestion Games 12 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Optimum 2 2 1 1 * t=1 t=2 Marc Schr¨ oder Dynamic Atomic Congestion Games 12 / 29

Model Parallel networks Open problems Ongoing research Uniform departures Optimum 2 2 1 1 * . . . total costs are 3 t=1 t=2 Marc Schr¨ oder Dynamic Atomic Congestion Games 12 / 29

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 τ e , � Opt ( N , γ ) = γ e · τ e . e ∈ E PoA ( N , γ ) = WEq ( N , γ ) Opt ( N , γ ) ≤ max e τ e . min e τ e Marc Schr¨ oder Dynamic Atomic Congestion Games 13 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Periodic departures Inflow is a K -periodic vector: δ = ( δ 1 , . . . , δ K ) ∈ N K such that � K k =1 δ k = K · γ . We denote N K ( γ ) the set of such sequences. When δ is not-uniform, queues have to be created when there is a surge of players. Marc Schr¨ oder Dynamic Atomic Congestion Games 14 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Example Inflow (3,1,2). What happens to the equilibrium? τ 1 = 1 s t τ 2 = 2 Marc Schr¨ oder Dynamic Atomic Congestion Games 15 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 2 3 1 t=1 Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 2 3 1 1 * * t=1 t=2 Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 2 3 1 1 2 1 * * * t=1 t=2 t=3 Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 3 2 3 1 1 2 1 2 * 1 1 * * * * * * * t=1 t=2 t=3 t=4 t=5 Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 3 2 3 1 1 2 1 2 * 1 1 2 1 * * * * * * * * * t=1 t=2 t=3 t=4 t=5 t=3 Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Equilibrium 3 2 3 1 1 2 1 2 * 1 1 2 . . . 1 * * * * * * * * * t=1 t=2 t=3 t=4 t=5 t=3 In the long-run total costs are 3 · 4 + 1 = 13. Marc Schr¨ oder Dynamic Atomic Congestion Games 16 / 29

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). Marc Schr¨ oder Dynamic Atomic Congestion Games 17 / 29

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¨ oder Dynamic Atomic Congestion Games 17 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Measure of periodicity Definition For any two elements δ, δ ′ ∈ N K ( γ ), we say that δ ′ is obtained from δ by an elementary operation if there exist a time i , with such that δ i > γ , δ ′ i = δ i − 1, δ ′ i +1 = δ i + 1. Let D ( δ ) be the minimal number of elementary operations one has to perform to transform δ into γ K . Marc Schr¨ oder Dynamic Atomic Congestion Games 18 / 29

Model Parallel networks Open problems Ongoing research Periodic departures Theorem Let N be a parallel network and δ ∈ N K ( γ ) . Then WEq ( N , K , δ ) = K · γ · max e ∈ E τ e + D ( δ ) , � Opt ( N , K , δ ) = K · γ e · τ e + D ( δ ) . e ∈ E Marc Schr¨ oder Dynamic Atomic Congestion Games 19 / 29

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¨ oder Dynamic Atomic Congestion Games 20 / 29

Model Parallel networks Open problems Ongoing research General networks Parallel network τ 1 = 1 τ 2 = 2 s t τ 3 = 3 τ 4 = 3 Equilibrium. If δ = 3, then WEq ( N , 1 , δ ) = 9. If δ = (6 , 0), then WEq ( N , 2 , δ ) = 16. Marc Schr¨ oder Dynamic Atomic Congestion Games 21 / 29